07_interval_labels.ipynb

{
 "cells": [
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "---\n",
    "description: _min-post_ intervals labeling and its extensions, as used in GRAIL, FERRARI\n",
    "  (and PReaCH)\n",
    "output-file: interval_labels.html\n",
    "title: DFS Intervals Labelling\n",
    "\n",
    "---\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#| default_exp labelling.dfs_intervals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The autoreload extension is already loaded. To reload it, use:\n",
      "  %reload_ext autoreload\n"
     ]
    }
   ],
   "source": [
    "#| output: false\n",
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Imports for the module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#| export\n",
    "import networkx as nx\n",
    "from copy import deepcopy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Definitions: out-tree, spanning tree / tree cover, etc."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![A tree in graph theory (Wikipedia)](https://upload.wikimedia.org/wikipedia/commons/thumb/2/24/Tree_graph.svg/162px-Tree_graph.svg.png)\n",
    "\n",
    "In graph theory, a **tree** is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20171027092037172-0306:9781316105450:08931fig13_6.png?pub-status=live\" width=\"60%\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In graph theory, a **directed rooted tree** or an **arborescence**, or **out-tree** is a directed graph in which, for a vertex $u$ called the root and any other vertex $v$, there is exactly one directed path from $u$ to $v$.  Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root.\n",
    "\n",
    "Every directed rooted tree is a directed acyclic graph (DAG), but not every DAG is a tree.\n",
    "\n",
    ":::{.callout-note}\n",
    "\n",
    "here _root_ denotes a source node / verted, with in-degree of zero.\n",
    "\n",
    ":::"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A **directed rooted forest** is a disjoint union of directed rooted trees.  It is called **branching** or **out-forest** if all its edges point away from the root in each rooted tree."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![A 4x4 grid graph and one of its spanning trees (Wikipedia)](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/4x4_grid_spanning_tree.svg/240px-4x4_grid_spanning_tree.svg.png)\n",
    "\n",
    "In the mathematical field of graph theory, a **spanning tree** $T$ of an undirected graph $G$ is a subgraph that is a tree which includes all of the vertices of $G$, with a minimum possible number of edges.\n",
    "\n",
    "A **full spanning forest** in an undirected graph (possibly not connected) is a maximal acyclic subgraph of the given graph, or equivalently a graph consisting of a spanning tree in each connected component of the graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a vertex $v$ on a directed multigraph $G$, an **oriented spanning out-tree**, or **tree cover** $T(G)$ rooted at $v$ is an acyclic subgraph of $G$ in which every vertex other than $v$ has in-degree 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Small DAG with the spanning out-forest overlay and min-post intervals](images/colab/small_DAG_spanning_tree_min-post.png)\n",
    "\n",
    "Given directed acyclic graph $G$, a **spanning forest**, or **oriented spanning out-forest** is such its subgraph,\n",
    "for which the following is true:\n",
    "- it includes all original (full) graph nodes\n",
    "- there is at most one incoming edge per node (every vertex has in-degree 1 or 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## _min-post_ intervals in directed trees"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The reachability problem on **_trees_** (out-forests) can be solved effectively by **interval labeling**,\n",
    "which takes linear time and space for constructing the index,\n",
    "and provides constant time querying.\n",
    "\n",
    "It labels each node $u$ with a **_min-post_** range $I_u = [s_u, e_u]$,\n",
    "where \n",
    "- $s_u$ denotes the rank of the node $u$ in a <em><u>post</u>-order traversal</em> of the tree,\n",
    "  where the ranks are assumed to begin at 1,\n",
    "  and all the children of a node are assumed to be ordered and fixed for that traversal.\n",
    "  \n",
    "  Note that the tree needs to be traversed in _depth-first search_ (DFS) manner.\n",
    "  \n",
    "- $s_u$ denotes the lowest rank for any node $x$,\n",
    "  or in other words <em><u>min</u>imum</em> of DFS number of a node\n",
    "  in the subtree rooted at $u$ (i.e., including $u$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Min-post interval** for a vertex (node) $u$ in a directed acyclic graph $G$,\n",
    "denoted as $I(u) \\equiv I_u = [s_u, e_u]$, can be computed in the following way:\n",
    "- $e_u$ is defined as $e_u = \\pi(u) = \\text{post}(u)$, i.e. post-order value (back traversal) in DFS traversal\n",
    "- $s_u = \\pi(u) = e_u$ for leaf nodes, also known as sink nodes (no outcoming edges, out-degree 0),  \n",
    "  otherwise $s_u = \\min\\{s_x \\colon x \\in \\text{out-neighbours}(s)\\}$\n",
    "  (out-neighbours are sometimes called children)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, **max-pre interval** for a vertex (node) $u$ in a directed acyclic graph $G$,\n",
    "denoted as $\\text{range}(u) \\equiv \\hat{I}_u = [\\hat{\\phi}(u), \\phi(u)]$, can be computed in the following way:\n",
    "- $\\phi(u)$ is defined as $\\phi(u) = \\text{pre}(u)$, i.e. pre-order value (forward traversal) in DFS traversal\n",
    "- $\\hat{\\phi}(u) = \\phi(u)$ for leaf nodes, also known as sink nodes (no outcoming edges, out-degree 0),  \n",
    "  otherwise $\\hat{\\phi}(u) = \\max\\{\\hat{\\phi}(x) \\colon x \\in \\text{out-neighbours}(s)\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Min-post interval properties for trees"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For out-forests (directed forests) the containment between intervals\n",
    "is equivalent to the reachability relationship between the nodes,\n",
    "since the post-order DFS traversal enters a node before all of its descendants,\n",
    "and leaves after having visited all of its descendants.\n",
    "\n",
    "In other words, $u \\leadsto v \\iff I_v \\subseteq I_u$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ":::{.callout-note}\n",
    "\n",
    "for trees it is enough to check that $s_v \\in I_u$, because that is equivalent to checking $I_v \\subseteq I_u$.\n",
    "\n",
    ":::"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This means that the reachability query for trees can be answered in constant time.\n",
    "\n",
    "Min-post and max-pre intervals for trees functions both as positive-cut and as negative-cut filter:\n",
    "- if $I_v \\subseteq I_u$, then $u$ can reach $v$ _(positive-cut filter)_\n",
    "- if $u$ can reach $v$, then $I_v \\subseteq I_u$ _(negative-cut filter)_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## _min-post_ intervals in directed acyclic graphs (DAGs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The DFS **post-order** value (back traversal) induces topological sorting for a DAG.\n",
    "\n",
    "This means that if $u$ can reach $v$ (i.e. $r(u,v)$ is true, or $u \\leadsto v$), and $u \\neq v$, then $\\text{post}(u) > \\text{post}(v)$.  \n",
    "Conversely, if $\\text{post}(u) < \\text{post}(v)$, then $u$ cannot reach $v$ (_negative-cut_ filter)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Commit graph with the spanning forest](images/slides/git_commit_graph-spanning_forest_first_parent.png)\n",
    "\n",
    "For example, as shown in the above image, $n_{18} \\leadsto n_{9}$, but $n_{18} \\not\\leadsto n_{19}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### _min-post_ interval for a spanning tree _(positive-cut filter)_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a directed acyclic graph $G$,\n",
    "we define a **tree cover** (or **forest cover**) of $G$, denoted as $T(G) = (V_T, E_T)$,\n",
    "as a directed spanning tree of $G$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ":::{.callout-tip}\n",
    "\n",
    "if we want to have a tree cover rather than a forest,\n",
    "\n",
    ":::\n",
    "> we can augment graph $G$ by introducing an artificial root node $r$\n",
    "> that is connected to every node with no incoming edge:\n",
    "> $$ G' := (V \\cup \\{r\\}, E \\cup \\{(r,v) \\colon v \\in V, \\text{indegree}(v) = 0\\}) $$\n",
    "> Note that this modification has no effect on the reachability relation among the existing nodes of $G$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define **min-post tree interval** as following\n",
    "\n",
    "$$ I_v = \\{\\pi(w) \\colon w \\in V_{T_v}\\} \n",
    "   = \\left[ \\min_{w \\in V_{T_v}} \\pi(w), \\max_{w \\in V_{T_v}} \\pi(w) \\right] \n",
    "   = \\left[ \\min_{w \\in V_{T_v}} \\pi(w), \\pi(v) \\right]$$\n",
    "   \n",
    "Here $T_v = (V_{T_v}, E_{T_v})$ denote the subtree of $T(G)$ rooted at node $v$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The spanning forest, or tree cover, or out-forest $T(G)$ of directed acyclic graph $G$ has the following properties:\n",
    "- if there exists a path from $u$ to $v$ in the spanning tree $T(G)$, then $u$ can reach $v$ in the full graph $G$\n",
    "- but the path from $u$ to $v$ could require going through edges outside the spanning tree $T(G)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **_min-post tree interval labels_** have therefore the following properties:\n",
    "- path from $u$ to $v$ in the spanning tree $T(G)$ exists if and only if $I_v \\subseteq I_u$ (or $\\pi(v) \\in I_u$)\n",
    "- if $I_v \\subseteq I_u$, then $u \\leadsto v$ ($u$ can reach $v$, also in the full graph $G$)\n",
    "\n",
    "It can be therefore used as _positive-cut filter_."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example:\n",
    "\n",
    "![Small DAG with the spanning out-forest overlay and min-post intervals](images/colab/small_DAG_spanning_tree_min-post.png)\n",
    "\n",
    "In the graph and its spanning tree shown in the above figure we have:\n",
    "- $[3,3] \\subseteq [1,5]$ then $a \\leadsto h$,\n",
    "- $[3,3] \\not\\subseteq [7,9]$, but $b \\leadsto h$ (via out-of-tree edge)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Such interval is called **tree interval** or **exact interval** and denoted $I_T(u)$ in the FERRARI paper,  \n",
    "and the corresponding max-pre interval as a range is called **full interval** (because every $v$ such that $\\phi(v) \\in I_T(u)$ is reachable from $v$) in the PReaCH paper"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### _min-post_ interval for a DAG _(negative-cut filter)_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While above technique (min-post intervals) can be used to easily answer reachability queries on trees,\n",
    "the case of general DAGs is much more challenging.\n",
    "\n",
    "The reason is that, in general, the reachable set $\\mathcal{R}(v)$ of a vertex $v$ in the DAG\n",
    "(i.e. the set of vertices reachable from $v$: $\\mathcal{R}(v) = \\{u \\in V\\colon v \\leadsto u\\}$)\n",
    "is only partly represented by the interval $I_T(v)$,\n",
    "as the tree interval only accounts for reachability relationships that are preserved in $T$.\n",
    "Vertices that can only be reached from a node $v$ by traversing one or more non-tree edges\n",
    "have to be handled seperately: instead of merely storing the tree intervals $I_T(v)$,\n",
    "every node $v$ is now assigned a set of intervals, denoted by $\\mathcal{I}(v)$.\n",
    "The purpose of this so-called **reachable interval set** is to capture the complete reachability information of a node."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's define the **approximate graph interval** as a single interval $\\tilde{I}_G(v) = [\\tilde{s}_u, \\tilde{e}_u]$\n",
    "approximating the reachable set $\\mathcal{R}(u)$ of a vertex $u$ in the DAG from the above (i.e. $\\mathcal{R}(u) \\subseteq \\tilde{I}_G(u)$)\n",
    "in the following way:\n",
    "\n",
    "$$ \\tilde{I}_G(v) \n",
    "   = \\left[ \\min_{w \\in \\mathcal{R}(v)} \\pi(w), \\max_{w \\in \\mathcal{R}_v} \\pi(w) \\right] \n",
    "   = \\left[ \\min_{w \\in V} \\pi(w), \\pi(v) \\right]$$\n",
    "   \n",
    "The second part of the above equation is true for DFS traversal; the fact that the end of interval is simply post-traversal value $\\pi(v)$ was mentioned earlier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This **_min-post approximate graph interval_** has the following properties:\n",
    "- if $u$ can reach $v$ (denoted as $u \\leadsto v$), then $\\tilde{I}_G(v) \\subseteq \\tilde{I}_G(u)$\n",
    "- this means that if $\\tilde{I}_G(v) \\not\\subseteq \\tilde{I}_G(u)$, then $u$ cannot reach $v$ _(negative-cut filter)_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Interval Labeling: Tree (a) and DAG: Single (b)](images/grail/GRAIL_vldb2010-fig2ab-interval_labeling_tree_and_DAG.png)\n",
    "\n",
    "For example, part (a) of the figure above (taken from GRAIL paper) shows the _interval labeling on a tree_,\n",
    "assuming that the children are ordered from left to right.\n",
    "It is easy to see that reachability can be answered by interval containment.\n",
    "For example, $1 \\leadsto 9$, since $I_9 = [2, 2] \\subset [1, 6] = L_1$,\n",
    "but $2 \\not\\leadsto 7$, since $I_7 = [1, 3] \\not\\subseteq [7, 9] = L_2$.\n",
    "\n",
    "For example, part (b) of the figure above shows an _interval labeling on a DAG_,\n",
    "assuming a left to right ordering of the children.\n",
    "As one can see, interval containment of nodes in a DAG is not exactly equivalent to reachability.\n",
    "For example, $5 \\not\\leadsto 4$, but $\\tilde{I}_4 = [1, 5] \\subseteq [1, 8] = \\tilde{I}_5$.\n",
    "In other words, $\\tilde{U}_v \\subseteq \\tilde{I}_u$ does not imply that $u \\leadsto v$.\n",
    "On the other hand, one can show that $\\tilde{I}_v \\not\\subseteq \\tilde{I}_u \\implies u \\not\\leadsto v$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extending _min-post_ intervals for graphs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This section describes various extensions to the _min-post_ intervals (DFS numbering) going beyond those negative-cut and positive-cut filters described above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple intervals labelling: GRAIL (TODO)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Extending interval labelling: PReaCH"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The PReaCH paper extends min-post intervals labelling with approach slightly different from GRAIL, where instead of using additional DFS searches, additional extra labels are used.\n",
    "\n",
    "**NOTE:** the PReaCH paper uses max-pre interval labelling instead of min-post one, but one can be translated into the other."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Original extension of max-pre intervals (Section 3.3 of PReaCH)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Section 3.3 \"*Pruning Based on DFS Numbering*\" in the PReaCH paper.**\n",
    "\n",
    "![Full and empty intervals derived from a DFS ordering](images/preach/PReaCH_arXiv-fig_3-full_and_empty_intervals_derived_from_DFS_numbering.png)\n",
    "\n",
    "Let $\\phi(v)$ denote the pre-visit DFS number of node $v$ and $\\hat{\\phi}(v)$ the largest DFS number of a node\n",
    "in the subtree of the DFS tree rooted at $v$. The properties of DFS ensure that the nodes in\n",
    "$\\text{range}(v) := \\phi(v)..\\hat{\\phi}(v)$ are all reachable from $v$ (they form the subtree of the DFS tree which\n",
    "is rooted at $v$) and that no nodes with DFS number exceeding $\\hat{\\phi}(v)$ is reachable from $v$.\n",
    "\n",
    "Let $u \\to v$ denote that $r(u,v)$ is true, and $u \\nrightarrow v$ that $r(u,v)$ is false.  \n",
    "(We would use $u \\leadsto v$/$u \\rightsquigarrow v$, but $u \\not\\leadsto v$ is broken, and there is no $\\nrightsquigarrow$).\n",
    "\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ t \\in \\text{range}(v) \\implies v \\to t$  \n",
    "    _(positive-cut)_\n",
    "*   $\\forall v,t \\in V \\colon\\ \\phi(t) > \\hat{\\phi}(v) \\implies v \\nrightarrow t$  \n",
    "    _(negative-cut)_\n",
    "\n",
    "Let's define node $w = p_{\\text{tree}}(v)$ as the node that has $v \\to w$ and $w \\notin \\text{range}(v)$\n",
    "which maximizes $\\lvert\\text{range}(w)\\rvert$.  Then, if $p_{\\text{tree}}(v)$ exists\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ p_{\\text{tree}}(v) \\neq \\bot \\land t \\in \\text{range}(p_{\\text{tree}}(v)) \\implies v \\to t$  \n",
    "    _(positive-cut)_\n",
    "    \n",
    "For any $v \\in V$, let $\\phi_{\\text{min}}(v)$ denote the smallest DFS number of a node reachable from $v$ (including $v$). Then\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ \\phi(t) < \\phi_{\\text{min}}(v) \\implies v \\nrightarrow t$  \n",
    "    _(negative-cut)_\n",
    "    \n",
    "We can also define $\\phi_{\\text{gap}}(v)$ in such way that\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ \\phi_{\\text{gap}}(v) < \\phi(t) < \\phi(v) \\implies v \\nrightarrow t$  \n",
    "    _(negative-cut)_\n",
    "    \n",
    "All those values: $\\phi(v), \\hat{\\phi}(v), p_{\\text{tree}}(v), \\phi_{\\text{min}}(v), \\phi_{\\text{gap}}(v)$ can be done while computing the DFS numbering."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**On choosing the DFS spanning-tree from PReaCH paper**\n",
    "\n",
    "There are many ways to define a DFS ordering: We are free to choose the order in\n",
    "which we scan the nodes for starting recursive exploration and we can choose the order in\n",
    "which we inspect edges leaving a node being explored.\n",
    "\n",
    "One heuristics that seems to be useful: Make sure that most nodes are in a small number of trees because this leads to large positive intervals. It therefore makes sense to only uses sources of the graph as tree roots.\n",
    "\n",
    "In addition, one can order the sources by the number of nodes reached from them during the search for topological levels."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Translating PReaCH extension to min-post intervals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- https://www.overleaf.com/project/5fd15159e5a50909877b2ddf -->\n",
    "![Full and empty intervals derived from a post-visit DFS ordering](images/full_and_empty_intervals_derived_from_min-post_DFS_numbering.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\pi(v)$ denote the post-visit DFS number of node $v$\n",
    "and $\\breve{\\pi}(v)$ the smallest post-visit DFS number of a node\n",
    "in the subtree of the DFS tree rooted at $v$.\n",
    "The properties of DFS ensure that the nodes in $I_T(v) := \\breve{\\pi}(v)..\\pi(v) = [\\breve{\\pi}(v),\\pi(v)]$\n",
    "are all reachable from $v$ (they form the subtree of the DFS tree which is rooted at $v$)\n",
    "and that no nodes with DFS number exceeding $\\pi(v)$ is reachable from $v$.\n",
    "\n",
    "Let $u \\to v$ denote that $r(u,v)$ is true, and $u \\nrightarrow v$ that $r(u,v)$ is false.  \n",
    "\n",
    "-   $\\forall v,t \\in V \\colon\\ t \\in I_T(v) \\implies v \\to t$  \n",
    "    _(positive-cut)_\n",
    "-   $\\forall v,t \\in V \\colon\\ \\pi(t) > \\pi(v) \\implies v \\nrightarrow t$  \n",
    "    _(negative-cut)_\n",
    "\n",
    "\n",
    "Let's define node $w = p_{\\text{tree}}(v) = p_{\\text{t}}(v)$ as the node that has $v \\to w$ and $w \\notin I_T(v)$\n",
    "which maximizes $\\lvert I_T(w)\\rvert$.  \n",
    "Then, if $p_{\\text{tree}}(v)$ exists (which we denote as $p_{\\text{tree}}(v) \\neq \\bot$)\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ p_{\\text{tree}}(v) \\neq \\bot \\land t \\in I_T(p_{\\text{tree}}(v)) \\implies v \\to t$  \n",
    "    _(positive-cut)_\n",
    "    \n",
    "For any $v \\in V$, let $\\pi_{\\text{min}}(v)$ denote\n",
    "the smallest post-visit DFS number of a node reachable from $v$ (including $v$) in the graph.\n",
    "\n",
    ":::{.callout-note}\n",
    "\n",
    "with this notation we can say that $\\tilde{I}_G(v) = [\\pi_{\\text{min}}(v), \\pi(v)]$.\n",
    "\n",
    ":::\n",
    "\n",
    "Then\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ \\pi(t) < \\pi_{\\text{min}}(v) \\implies v \\nrightarrow t$  \n",
    "    _(negative-cut)_\n",
    "    \n",
    "We can also define $\\pi_{\\text{gap}}(v)$ in such way that\n",
    "\n",
    "*   $\\forall v,t \\in V \\colon\\ \\pi_{\\text{gap}}(v) < \\pi(t) < \\pi(v) \\implies v \\nrightarrow t$  \n",
    "    _(negative-cut)_\n",
    "    \n",
    "All those values: $\\pi(v), \\breve{\\pi}(v), p_{\\text{tree}}(v), \\pi_{\\text{min}}(v), \\pi_{\\text{gap}}(v)$ can be done while computing the DFS numbering."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Extending interval labelling: FERRARI (TODO)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualizing DFS intervals on example graphs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "TODO"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# this is needed for example (small) graphs\n",
    "import git_commit_graph_ext.example_graphs as graphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# this is needed only for demonstrations\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Examples of spanning trees and min-post tree intervals labelling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Draw a small DAG (from FELINE paper) and its spanning tree, like on Figure 8 of the FELINE paper, together with its _min-post_ tree intervals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#@title Draw a small DAG and its spanning-tree, as in Figure 8\n",
    "from copy import deepcopy\n",
    "\n",
    "plt.axis('off')\n",
    "plt.title('Small DAG with the spanning tree overlay and min-post intervals')\n",
    "\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "sd.tree={\n",
    "  'a':(2,3),'b':(4.5,3),\n",
    "  'c':(1,2),'d':(2,2),'e':(3,2),'f':(4,2),'g':(5,2),\n",
    "  'h':(3,1)\n",
    "}\n",
    "nx.draw_networkx(sd,\n",
    "                 pos=sd.tree,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa',node_color='y')\n",
    "nx.draw_networkx_edges(sd,\n",
    "                       pos=sd.tree,\n",
    "                       edgelist=[\n",
    "                            ('a','c'),('a','d'),('a','e'),\n",
    "                            ('e','h'),\n",
    "                            ('b','f'),('b','g')\n",
    "                        ],\n",
    "                        node_size=500,width=2.0,\n",
    "                        arrowstyle='->')\n",
    "\n",
    "sd.lpos=deepcopy(sd.tree)\n",
    "for v in list('abh'):\n",
    "  sd.lpos[v] = (sd.lpos[v][0]+0.40,sd.lpos[v][1])\n",
    "for v in list('cdfg'):\n",
    "  sd.lpos[v] = (sd.lpos[v][0]+0.05,sd.lpos[v][1]-0.2)\n",
    "for v in list('e'):\n",
    "  sd.lpos[v] = (sd.lpos[v][0]+0.35,sd.lpos[v][1]+0.1)\n",
    "sd.mpi={\n",
    "    'a':(1,5),'b':(7,9),\n",
    "    'c':(1,1),'d':(2,2),'e':(3,4),'f':(7,7),'g':(8,8),\n",
    "    'h':(3,3),\n",
    "}\n",
    "sd.mpi_labels={\n",
    "  v:('[%d,%d]'%(i[0],i[1]))\n",
    "  for (v,i) in sd.mpi.items()\n",
    "}\n",
    "\n",
    "nx.draw_networkx_labels(sd,\n",
    "                        pos=sd.lpos,\n",
    "                        labels=sd.mpi_labels)\n",
    "plt.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above figure shows *Min-post indexing for a spanning tree* extracted from the small DAG of Figure 2 in the FELINE paper.  Each vertex is labeled with its *min-post* interval. Original edges of the DAG are marked in light gray.\n",
    "\n",
    "For instance, consider the vertex $h$ of above figure.  We can conclude that the query $r(a,h)$ will return true ($a \\leadsto h$), without search, because $[3,3] \\subseteq  [1,5]$.  However, we can say nothing about the query $r(b,h)$, because there is no tree edge connecting the two vertices."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Post-traversal index vs backward topological level drawing of example graphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.lines import Line2D\n",
    "\n",
    "#plt.axis('off')\n",
    "plt.title('Small DAG with the spanning tree overlay and min-post intervals\\n'+\n",
    "          'drawn using spanning tree position and levels')\n",
    "\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "sd.lvl = {\n",
    "    'h': 0, 'g': 0, 'd': 0,\n",
    "    'c': 1, 'e': 1, 'f': 1,\n",
    "    'a': 2, 'b': 2,\n",
    "}\n",
    "sd.mpi={\n",
    "    'a':(1,5),'b':(7,9),\n",
    "    'c':(1,1),'d':(2,2),'e':(3,4),'f':(7,7),'g':(8,8),\n",
    "    'h':(3,3),\n",
    "}\n",
    "sd.mpi_labels={\n",
    "  v:('[%d,%d]'%(i[0],i[1]))\n",
    "  for (v,i) in sd.mpi.items()\n",
    "}\n",
    "sd.mpilvl={\n",
    "  v: (sd.mpi[v][1],sd.lvl[v]) for v in sd\n",
    "}\n",
    "\n",
    "plt.xlabel('spanning-tree post-order $\\pi(v)$')\n",
    "plt.ylabel('node level $L(v)$')\n",
    "\n",
    "ax=plt.gca()\n",
    "for v in sd:\n",
    "  if sd.mpi[v][0] == sd.mpi[v][1]:\n",
    "    continue\n",
    "  ax.add_line(Line2D([sd.mpi[v][0],sd.mpi[v][1]],\n",
    "                     [sd.lvl[v],sd.lvl[v]],\n",
    "                     lw=5., alpha=0.3))\n",
    "  \n",
    "nx.draw_networkx(sd,\n",
    "                 pos=sd.mpilvl,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa',node_color='y')\n",
    "edges=nx.draw_networkx_edges(sd,\n",
    "                             pos=sd.mpilvl,\n",
    "                             edgelist=[\n",
    "                               ('a','c'),('a','d'),('a','e'),\n",
    "                               ('e','h'),\n",
    "                               ('b','f'),('b','g')\n",
    "                             ],\n",
    "                             node_size=500,width=2.0,\n",
    "                             arrowstyle='->')\n",
    "\n",
    "labels=nx.draw_networkx_labels(sd,\n",
    "                               pos={u:(x+0.55,y-0.08) for (u,(x,y)) in sd.mpilvl.items()},\n",
    "                               labels=sd.mpi_labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Commit graph with the spanning forest and min-post intervals:\n",
    "- commit graph from the Stolee blog post  \n",
    "  In the drawing of this graph below arrows (directed edges) point from right to left.  \n",
    "  ![An example Git commit graph](images/stolee.dev/commit-graph-example.png)\n",
    "  \n",
    "- the same commit graph, but with the spanning forest (with the tree cover)  \n",
    "  the vertices / nodes in this image are post-visit order, denoted $\\text{post}(v) = \\pi(v)$  \n",
    "  numbered dashed lines denote the backward topological level (sink-based) of the vertex / node  \n",
    "  ![Commit graph with the spanning forest](images/slides/git_commit_graph-spanning_forest_first_parent.png)\n",
    "  \n",
    "- The same graph, just drawn differently (post-visit order vs topological level)  \n",
    "  with the spanning forrest, post-visit order as label, and min-post tree interval marked  \n",
    "  ![Commit graph drawn with post-visit order vs level](images/slides/git_commit_graph-min-post_intervals.png)\n",
    "  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing DFS intervals labelling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here there would be all code that automatically computes the spanning tree (or rather the spanning forest), also known as tree cover of a graph, and all the DFS numbering based labels."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing the spanning tree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#| export\n",
    "def find_dfs_spanning(G):\n",
    "  \"\"\"Find edges of the DFS spanning tree for graph G\n",
    "\n",
    "  This is used to visualize the spanning tree (spanning forest,\n",
    "  tree cover) in example small graphs.  This function is also used\n",
    "  to compute DFS intervals, among others in `find_dfs_intervals()`\n",
    "\n",
    "  Parameters\n",
    "  ----------\n",
    "  G : NetworkX graph\n",
    "      Graph to find spanning-tree in.\n",
    "\n",
    "  Returns\n",
    "  -------\n",
    "  edges : list\n",
    "      A list of edges in the depth-first-search.\n",
    "\n",
    "  Examples\n",
    "  --------\n",
    "  >>> G = nx.path_graph(5)\n",
    "  >>> find_dfs_spanning(G)\n",
    "  [(0, 1), (1, 2), (2, 3), (3, 4)]\n",
    "\n",
    "  Notes\n",
    "  -----\n",
    "  The source is chosen arbitrarily and repeatedly until all\n",
    "  components in the graph are searched.\n",
    "  \"\"\"\n",
    "  # this may not cover all nodes\n",
    "  return list(nx.dfs_edges(G))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simple test that it works correctly for a very simple linear graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw(nx.path_graph(5),with_labels=True,node_size=500,width=2.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(0, 1), (1, 2), (2, 3), (3, 4)]\n"
     ]
    }
   ],
   "source": [
    "print(find_dfs_spanning(nx.path_graph(5)))\n",
    "assert find_dfs_spanning(nx.path_graph(5)) == [(0, 1), (1, 2), (2, 3), (3, 4)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Visualize a spanning tree in undirected graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G5=nx.path_graph(5)\n",
    "G5.pos={v:(v,0) for v in range(5)}\n",
    "\n",
    "nx.draw_networkx(G5,\n",
    "                 pos=G5.pos,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa')\n",
    "nx.draw_networkx_edges(G5,\n",
    "                       pos=G5.pos,\n",
    "                       edgelist=find_dfs_spanning(G5),\n",
    "                       node_size=500,width=2.0,\n",
    "                       arrowstyle='->')\n",
    "plt.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Draw a small DAG from the FELINE paper, and visualize automatically generated spanning tree (or rather one of the spanning trees)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#@title Draw a small DAG and its find_dfs_spanning\n",
    "\n",
    "# get graph\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "# TODO: move setting original positions to `small_DAG_FELINE()` to avoid repetitions\n",
    "sd.tree={\n",
    "  'a':(2,3),'b':(4.5,3),\n",
    "  'c':(1,2),'d':(2,2),'e':(3,2),'f':(4,2),'g':(5,2),\n",
    "  'h':(3,1)\n",
    "}\n",
    "\n",
    "# configure plot\n",
    "plt.axis('off')\n",
    "plt.title('Small DAG with the spanning tree overlay '+\n",
    "          '(find_dfs_spanning)')\n",
    "\n",
    "# draw graph and its spanning tree\n",
    "nx.draw_networkx(sd,\n",
    "                 pos=sd.tree,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa',node_color='y')\n",
    "nx.draw_networkx_edges(sd,\n",
    "                       pos=sd.tree,\n",
    "                       edgelist=find_dfs_spanning(sd),\n",
    "                       node_size=500,width=2.0,\n",
    "                       arrowstyle='->')\n",
    "plt.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Draw a crown DAG, known as $S_0^3$ graph, used in FELINE paper as simple example of irreductible false-positive for FELINE labels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# get graph\n",
    "cd=graphs.crown_DAG()\n",
    "cd.pos={1:(1,4),2:(3,6),'u':(2,2),3:(4,1),4:(6,3),'v':(5,5)}\n",
    "\n",
    "# configure plot\n",
    "plt.axis('off')\n",
    "plt.title('Crown DAG $S_0^3$ with the spanning tree overlay '+\n",
    "          '(find_dfs_spanning)')\n",
    "\n",
    "# draw graph and its spanning tree\n",
    "nx.draw_networkx(cd,\n",
    "                 pos=cd.pos,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa')\n",
    "nx.draw_networkx_edges(cd,\n",
    "                       pos=cd.pos,\n",
    "                       edgelist=find_dfs_spanning(cd),\n",
    "                       node_size=500,width=2.0,\n",
    "                       arrowstyle='->')\n",
    "plt.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the last tree in the above spanning forest for $S_0^3$ crown DAG is not visible because if consist of a single vertex (node) $u$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "TODO: in the [Reachability labels for version control graphs](_Reachability_labels_for_version_control_graphs.ipynb) notebook from Google Colaboratory there are a few more examples to be copied, or replaced with interactive visualization (where you can choose the graph):\n",
    "\n",
    "- draw a didactic example of DAG from FELINE\n",
    "- draw DAG from level-filter example from FELINE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing min-post tree intervals (positive-cut filter)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interval labeling (*min-post* strategy) labels each node $u$ with a range $I_u = [s_u, e_u]$, where $e_u$ denotes the rank of the node $u$ in ***some*** post-order traversal of the tree, where the ranks are assumed to begin at 1, and all the children of a node are assumed to be ordered and fixed for that traversal. Further, $s_u$ denotes the lowest rank for any node $x$ in the subtree rooted at $u$ (i.e., including $u$).\n",
    "\n",
    "For more see the section with the definitions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#| export\n",
    "def find_dfs_intervals(DG, attr=None):\n",
    "  \"\"\"Find min-post labeling of all vertices in graph G, using DFS spanning-tree\n",
    "\n",
    "  Min-post interval labeling strategy labels each node 'u' in the directed\n",
    "  spanning tree (DFS tree here) of the acyclic graph `DG` with a range\n",
    "  I_u = [s_u, e_u], where e_u denotes the rank of the node 'u' in post-order\n",
    "  traversal of the tree (ranks assumed to begin at 1), and s_u denotes the\n",
    "  lowest rank of any node 'w' in the subtree rooted at 'u' (including 'u').\n",
    "\n",
    "  Parameters:\n",
    "  -----------\n",
    "  DG : NetworkX DiGraph\n",
    "      Directed acyclic graph.\n",
    "\n",
    "  attr : str, optional (default=None)\n",
    "      If set, name of a node attribute under which store min-post intervals.\n",
    "\n",
    "  Returns:\n",
    "  --------\n",
    "  dict of two-element lists of ints\n",
    "      Dictionary, where keys are node indices, and values are two element\n",
    "      lists storing min-post interval (of integers)\n",
    "  \"\"\"\n",
    "  ivl = {}\n",
    "  # create graph of spanning tree (DFS spanning tree) of DG\n",
    "  span=nx.DiGraph()\n",
    "  span.add_nodes_from(DG)\n",
    "  span.add_edges_from(find_dfs_spanning(DG))\n",
    "  # DEBUG\n",
    "  #print('span=%r' % find_dfs_spanning(DG))\n",
    "\n",
    "  # any post-order ordering would be good\n",
    "  nodelist = nx.dfs_postorder_nodes(DG)\n",
    "  for pos, node in enumerate(nodelist, start=1):\n",
    "    ivl[node] = [pos, pos]\n",
    "    if span.out_degree(node) > 0:\n",
    "      ivl[node][0] = min(pos,\n",
    "                         *[ivl[neigh][0]\n",
    "                           for neigh in span.successors(node)])\n",
    "    # DEBUG\n",
    "    #print('node=%s [%d,%d] -> %r' %\n",
    "    #      (node, ivl[node][0],ivl[node][1],list(span.successors(node))))\n",
    "\n",
    "  if attr is not None:\n",
    "    for node, interval in ivl.items():\n",
    "      DG.nodes[node][attr] = deepcopy(interval)\n",
    "\n",
    "  return ivl"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that there are no runtime errors when running `find_dfs_intervals(graph)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exact results depend on the spanning tree (DFS traversal order)\n",
      "found intervals: {'h': [1, 1], 'c': [1, 2], 'd': [3, 3], 'e': [4, 4], 'a': [1, 5], 'f': [6, 6], 'g': [7, 7], 'b': [6, 8]}\n",
      "provided ones:   {'a': [1, 5], 'b': [7, 9], 'c': [1, 1], 'd': [2, 2], 'e': [3, 4], 'f': [7, 7], 'g': [8, 8], 'h': [3, 3]}\n"
     ]
    }
   ],
   "source": [
    "# DEBUG\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "sd.mpi={\n",
    "    'a':[1,5],'b':[7,9],\n",
    "    'c':[1,1],'d':[2,2],'e':[3,4],'f':[7,7],'g':[8,8],\n",
    "    'h':[3,3],\n",
    "}\n",
    "\n",
    "print('Exact results depend on the spanning tree (DFS traversal order)')\n",
    "print('found intervals: %r' % find_dfs_intervals(sd))\n",
    "print('provided ones:   %r' % sd.mpi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check visually that min-post tree intervals make sense for an example of small DAG from the FELINE paper"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAEGCAYAAACNaZVuAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8vihELAAAACXBIWXMAAAsTAAALEwEAmpwYAABbEElEQVR4nO2dd3hVRfrHPy9JCF3pShGk2LGiIoIGe/2tIoNdEVGwoYCuKyqKa8FCWIVVirqogMhgWVdBFN2oYFuVJjaKIIL0TgKkzO+PmRtOLvfm3oS0m8znefLknnPmzH3ve+a8Z86U74gxBo/H4/FULaqVtwEej8fjKXt88Pd4PJ4qiA/+Ho/HUwXxwd/j8XiqID74ezweTxXEB3+Px+OpglSY4C8irUXEiEiy284QkT7lbVdxEJHBIvJiIcd7icisfci/gK8qG7H859mbRCkTIrJMRM6Kcmy0iDxY1jYVhxi/Y7yIPBpnPsWOc/saI2MGfxHpIiJfiMgWEdkoIrNF5MTifmFJICIPi0i2iGxzf7+KyCgROTBC2l7upugZ4Vh7EZksIutEZKuILBKRkSLSYl/sM8Y8bozp475jn2/KwgpaZSTov5LGXYt2pZG3Z98wxvQzxvw9nrRFCbCeyBQa/EWkHvAeMBJoADQHhgK7St+0mLxhjKmLtetS4ADguwgPgOuBje5/Pi4AfA2sAo4zxtQDTgWWAF1K2XZPBaWi15wjkYg2lzfeZ4AxJuof0BHYXMjxXsBsYASwGVgKdHb7VwBrgesD6S8E5gBb3fGHA8daAwZIdtsZQJ8o3/swMCFsXxIwD3gmsK8VkAdcBuQATQPHJgD/Kez3R/je5cAJ7vM1zt4j3HYf4J1w+4DfXbrt7u8U559ZwDPAJuA34Pwo3/ma+w1Z7vy/Bnx1vct/PXB/4JxqwN+wD7INwBSgQZT8G2Ef8JuxD8nPgWru2DLgPuBHZ+e/gBruWH133jp37D2gRSDfDODvrnxsAz4EGoVd62j2B/0XK21N4BVnw0/OP39E+a2fubx2OF9eDqQBfwD3Aqudvwv1H9AJ+ML5bB6QVkiZOdz5YjOwEPi/QB6rgaRA2kuB+bGuYcAnNzqffMbe988Nzh/bsPdl38D3/ABcHNhOcX49NoL9xb7O7vi12PtmA3A/tkydFcVX44FH3efQdRmEjSN/Aje4YzcD2cBudx3/4/Y3A950tv4G9A8rU1Ox9/1WYAj2ngpe1+OcH1KAtsAnzu71wERg/0DaeH9HPP57AvgG2AL8mzjLGoEYCbQDPnV5rMdWjguPZzGCXT33418Bzgfqhx3vhQ2qN2CD76PYwvhPIBU4xxWIOoEL2gFbsI8G1gCXlETwd/sfAb4ObD8IfOM+LwAGBo6tBnrFE/QD57wKDHKfx2JvzFsCxwYUErySw/yWDdzk/HYL9g1EonxvgYIWyHMcNvgdg30bO9wdvwv4CmjhrsMY4PUoeT8BjMYW+BSga8gO970/AC2xb1iz2VOoG2IfqrWAuoDGPfwC128JcIizMQMYFqf9kfwXLe0wbKGv737vfKIEf5feAO0C22nYMvyk81XNwvyHffvdAFyALcdnu+3GEb4rBVgMDAaqA2dg74dD3fElwNmB9Br4W6xrGPDJq0BtZ3NoX+j+uRAbwAQ4HcgEjnfH/kogOAB/ARZE8de+XOcjsMH5NPcb0p2v4w3+Odh7OsX5OxMXg4Jp3XY14DtsUK8OtME+9M4NlKls4BKXtiY2uN8UyONpYHQgmJ7t7G6MfcD+I9o9WcjviMd/K4Gj3LV8kz1lv9CyRsHg/zr24VoNqAF0iRnP4gh4h7sf84e7GO/iatDYILYokLYDtgAGa9gbiFCjcMf+AYyIFCQpXvDvF2bPIuAu9/k+YF7gWA5wXmD7duzTdTswLsr33gi86z7/hK3tT3bby9lzc+XbF/67An5bHNiu5dIcEOV7CxS0QJ7BGsQ3wBUB284MHDsQW/CTI+T9CLa20S7K9/YLbF8ALIli47HAprBC/UBg+1bggzjtj+S/aGnzb3C33YeiB//duDeaWP7DviG8FpbnDAJvuIH9XbGVjGqBfa/j3nixlaWX3ee62DeSVnHYEPJJmwhlYq9r7I6/A9zpPjfDPoTque2pwF+j+WwfrvMQ3P3htms7X8cb/LMoeN+sBTqFp3XbJwO/h+V3H/CvQJn6LOx4H+AT91mwrRGnRbHtEmBOtHsy2u+I03/DAttHOB8lxSprFAz+r2IrpC0ifW+kv5gdvsaYn4wxvYwxLbBPp2bYoB1iTeBzljsnfF8dABE5WUT+6zpYt2CDdaNYNhSB5timC0TkVOBgYLI7NgnoICLHuu0N2BsKZ/MoY8z+2N+WEiX/T4GuInIA9uK8AZwqIq2B/YC5RbB1deC7M93HOkU4v0Ae2FpR6PxWwNsisllENmMDSS7QNEIeT2Nrpx+KyFIR+VvY8RWBz8ux1x8RqSUiY0RkuYhsxdaM9heRpDjsi/d4PGmbhdkY/Bwv64wxOwPbhfmvFaBCx9zxLgTKUoBmwApjTF5g33JsOQVbJruLSCrQHfjeGLM8Dhti/lYROV9EvnKDNDZjH9yNAIwxq7BvcZeJyP7Yt/qJUfLZl+tc4NoYY3Zg77t42WCMyYmSdzitgGZh12UwhftrKnCKiDTDvp0YbLMnItLEDQZZ6X73BIoRq+L0X/g9luK+qyhl7a/YB9g3IrJQRHrHsq1IQz2NMT9jn2pHFeW8AJOwbw4tjTH7YZsbpJh5FUBEqgEX4y4eto1YgLkishrbuQtwnfv/MfaGixtjzGJsAeyPrUVswxb8m4FZYTd5/mlF+Y5oX13E9CuwfQj7B/5qGGNW7pWxMduMMYOMMW2w/hsoImcGkrQMfD4I2zwFti32UOBkYzvLT3P7S+R6FoE/sU0jIVpGS1gI4f4tzH8rsLWx4LHaxphhEfJdBbR0ZTPEQdjXfIwxP2Jv9vOBq7D3Rzw2RLMbAPcweRPbp9TUVWqmUfDavILtt1LAl5HKhmNfrvOfBK6HiNTCNoOUBJGu2W9h/qprjLkg2jnGmM3YPoqeWP+/blw1GtscaoCj3e++huKV7Xj8F36PZWPb7eMua8aY1caYm4wxzYC+wPOxRrXFGu1zmIgMCg19FJGWwJXYtsjiUBfYaIzZKSInYR2+T4hIiogcjn2dPgBIF5Ea2At6M/Y1K/R3B3C16+l/GFuLTxeR5i6vRthmrsL4FNtE9KnbzgjbDmcdtsO2TZF/3B7WFPH80cBjItIKQEQai8hfIiUUkYtEpJ2ICLYjLNf9hbhNRFqISANsTeoNt78u9q1uszv2UJF+UckxBbhPROq763h7jPTx+LIw/00ALhaRc0UkSURqiEhalOHBX2Obcv7qymka9gE7OZBmErYycRq2PTgeG2JRHdtWvQ7IEZHzsf1vQd4BjgfuxDYZRGNfrvNU4CKxw8WrY5sYS2puUfh1/AbYKiL3ikhNd22OktjD0idhK4SXUfDhWxfbBLzZlat7imlnPP67RkSOcA/HR4CpxphcilDWREQF9m/CPrhyw9MFiXUhtmHb0r4WkR3YoP8D9mlWHG4FHhGRbdj2wCnFzAfgchHZjm2nfxf7OnmCe6W9BOvwV90TcbUxZjXwEra55jxjzK/YnvQWwDxn02xsba2wiSafYi/oZ1G2C+CadB4DZrtXt07F+K1PAA+48++OI/2zWJ986H7XV9jrGIn2wExsQf8SeN4YkxE4PglbO1rq/kJjq/+B7TRb7/L/oAi/pyR5BNsf9Rv2d0yl8KHIDwOvOF/uNffDEdV/xpgV2A7SwdjgugIbGPa6l4wxu4H/w9bs1wPPA9e5N+gQr2Pbtz8xxqyPx4ZYuDfS/tj7axO2kvVuWJos7NvBwcBbhWT3D4p5nY0xC4HbsGXoT2fLH/GeH4OXgCPcdXzHBcuLsZW835y9L2KbYwvjXew9sMYYMy+wfyj24bgFeJ/CfVQY/yC2/17DtqisxnbW9oeilTXgRGyc3u5+053GmN8KM0z2vOV4PAURkWXYDqWZ5W1LvIjILdjO4NPL25aKjogMAQ4xxlxT3rZ4yp4KI+/g8RQHETlQRE4VkWoicij2rfTt8rarouOaIG7EjhDxVEF88PckOtWxY+C3Ycdt/xvbvOKJgojchG1CmG6Midhc6an8+GYfj8fjqYL4mr/H4/FUQXzwLyYi8oSI3BXYvkVE1ojIdhFp6P4Xa3inFEOqVfaWxG4qIp+JVT0dXhw7KhoicrWIfFhG33WpiKxw1/E4N3EmrZh5FUuBUgIKpG744n/EquvqWOdGya/YZbIsccMZY44KEpH+IhJpfoUnDryyXTEQkcbYscGhGzMFq1vSKTBcrKizdUuam7HDy+qZStK2Z4yZSJSZqKXAM8Dtxph/u+0jy+h7o9EDO1u1Ydis17gxxsRdJkXEAO3dxMaKylhgsYikG2PWlrcxiYav+RePXsA0N1Ya7E1ZA6vaWFFoBfxYWQJ/OdCKinc9fy1u4C9LpIzkkp0kx3T2zNr3FAEf/IvH+bgZvSJyCPCL279ZRD5x+4Ov7ONF5J8i8r5rhvlaRNqGMhORs0XkZ/dKP4o4ppG7GX/PiMh6EVmKVXEMHRuPlbf4q3vVP0tEThKRb8UuWrNGRNJj5F9DRCaIyAY3keZ/ItLUHctwzV7fOJv/7YYOhs7VIrLaHftMRI4MHIvlCyMi/cQurLPJpRV3rMAKaDHSJonIcOef30TkdoljUR0RSRU7USYJO/lvidufv6CO2MWEpojIq+43LBSRjoE8jhOR792xN7AVg5iIyD0i8qeIrJKANouIDMVOirzcXc8bxc7K/tT5eL37nlj5x1UmRSQ0Amie+77L3f6LRGSuKw9fiMjRgbyXiZ1dOx/YISIPiMjUsO9/VkSec59vEJGf3HcvFZG+hdh9r1iNnW0i8osUlB/JIFD2PUUgXgU4/1dAWW8dcGJguzV7K3fmq0diZ+9tBE7CNrVNZI8aaCOsrEIPrKDTAKziaERF00D+/YCf2SO3/N+gDeytevglcK37XAenjlhI/n2B/2AVR5OAE9ijAplBFBlad7w3dtZzKnaG49zAsai+CPjtPWB/rM7JOpz6Km4dhDjT9sOuQ9ACK/c8M/waxfj94eqfy3AqjthZwjuxYmlJ2BnYX7lj1bF6PQPc9eyB1WqJqPIYyP88rGRByKeTwsrQw2E+LrqEb5xlMsrvPx6rqnmy+83XO5+kBvwzF1sea2LfVDIDZSYJO8s3pMpZmOR0Gk6ZFauLswJoFrjX2obZtbG8Y0Ii/vmaf/HYHzuuvCi8ZYz5xtjX9onYaehgA8iPxpipxphsbLBcHTmLAvTE6ouvMMZsxAagwsgG2olII2PMdmNMLH2mbKwIVztjTK4x5jtjzNbA8deMMT8Yq9T4INBTnFKhMeZlYwXjdmGD1jEiEpxmH80XIYYZYzYbY37HPtTCj8eTtifwrDHmD2PMJqzuf0kyyxgzzVhZgdew6wyAlQxJwV6bbGPMVOB/ceTXEys/HPLpwzHSZ+OULI0xO40xxVkTOtZ1CHITMMYY87UrD69gZTSCciXPufKYZaw66fdYqRWwaxlkhsqdMeZ9Y8wSY/kUKyHSNcL35mIrEUeISIoxZpkxZkng+DZiSzh4IuCDf/HYhK3ZFoV4ZW8N8ckSh0sZL4+W0HEjdsGNn10TzkUx0r+G1Q6f7JohnhLbsR0iogyta24ZJiJLxErYLnNpgnK4iSL1XBjh31vDNSk1A1a66xgi1rWBol/PIkv4RqAofm4FDJKC8sItcRLfjnAfT8IKQUKYaqkUIjkdxNgO57uwD8O1YmWWg99ZF6u/4ykiPvgXj/nYQFoShMveCvHJEhc4D9vsERVjzCJjzJVAE+yqVVNFpHYh6bONMUONMUdgl+a8iIIda9FkaK/CilGdha2RtXZpElHqubjf2zzU9+Ao9NoEzivK9SyyhO8+sgJ4zBSUF65ljHk9aFbYORoIqVBeigv+Ep/k9J5MjZlkjOmCfQAZbPkNcTh2eUNPEfHBv3hMw7ZTlgTvA0eKSHdXc+yPlaaOxRSgv1i55frY9V6jIiLXiEhjY9cc2Ox2R5V8FZFuItLBNeVsxQb3YPpoMrR1sc0BG7D9BY/H8VtKgynAnSLSXOyCJfcGD7pO24xS+N4vsX02/UUkWUS6Y9vVYzEF6BXwaaHSyVIMCd8iEi6ZPA7oJ3ZBJhGR2iJyoYhEfQM2xqzD9g/9C6u1/5M7FI/kNAAicqiInOEeGDuxar3B33k6dsSPp4j44F88XgUuEJGa+5qRsTK+CtsmvQErLzs7jlPHYZtl5mHbVmNJzp4HLBQ7kuVZrPLlzkLSH4CVR96KXUXqU6y+eIiIMrRY3yzHdgj/SPHXfthXxmHbkecDc7AP7Bz2BI6WxOfnImGsjHN3bOf0Juwi8THlgI0x07H9PZ9gV1b7JMYpRZbwLSIPE5C+NsZ8i233H4X9XYuxvzEWk7BvgflNPiYOyekAqdh7Yz22rDXBShwjdt2OC7AL03iKiNf2KSYi8jiw1hjzj/K2paxxNeYJxpgXy9uWeHG1y9HGmNDiKHOxa+QWZVlBTwVCRO7Argr41/K2JRHxM3yLiTFmcHnb4ImOeyvrhq39N8U2o+RLPRtjji0fyzwlhTFmZHnbkMj4Zp8KjIiMdpNswv9Gl1D+V0fJvyLNbC0ugl2NaRO22ecn7ESp8jNIZHAUf5dIm7WIdI2S//aSyN9TufDNPh6Px1MF8TV/j8fjqYL44O/xeDxVEB/8PR6Ppwrig7/H4/FUQXzw93g8niqID/4ej8dTBfHB3+PxeKogPvh7PB5PFcTLO1RyMjIkGSurXAOrirgsLa3irwPrqbr4Mls2+Bm+lZCMDGmIXUqxF9AO2A3kYd/0UoFFWEXOl9LSzMbysdLj2YMvs2WPD/6ViIwMqY4VMBuIvXFqFZI8E3tjpQND09LM7tK30OMpiC+z5YcP/pWEjAw5CLtIeXMKv4HCycRq75+VlmZ+Lw3bPJ5I+DJbvvjgXwlwN9G3QAMgqRhZ5AIbgY7+ZvKUBb7Mlj8++Cc47rX5B+ySe8W5iULkAkuBI9PSTHZJ2ObxRMKX2YqBH+qZ+DyEfW3el5sId34zylnz3lMl8GW2AuBr/gmMGyHxB3ZIXEmxE2juR1R4SoN4yuwVV8A998AJJ8SdrS+zxcDX/BOb3tgREiVJnsvX4ykNfJmtIPjgn9j0omijJOKhlsvX4ykNeuHLbIXAB/8Exc2CbBcr3aRJcPXVcMEF0KsXfP55XNm3d/l7PCVGvGUW4OefbXm9+GJ48knYHXtEvy+zRcQH/8SlNXYWZKE0awbPPgvvvQfXXQePPw4bNsTMO9vl7/GUJK2Jo8wCzJwJTz0FEyfCihXw2msxT/Fltoj44J+41CCOttO0NGjUCKpVgzPOgObN4aefYuadS8l2Ins8EGeZBbj0UmjSBOrVg2uugU8+iXmKL7NFxL8mJS47iePhPWMGaA2rV9vtrCzYsiVm3kkuf4+nJImrzAI0brznc9OmsH59zFN8mS0ivuafuCwDqheWYPVqGD4c7rwT/v1v2/Rz8MGxM+7WjdrdujFPRB4rGVP3ICKpIrJdRLJF5NGSzt9ToVlGjDIbYt26PZ/XrrVvr4Xhy2zR8cE/QXESt4sLS7PT1YP228/+nz4dfvst7q84xhhzf2hDRMaKyC8ikicivQo7UUTGi8hud8OE/pIAjDG7jDF1gIlxW+KpFMRTZkO88459AGzdatv909Li+orwMpskIo+KyCoR2SYic0Rk/0gnikgDEXlDRNa7v4kiUg8qb5n1wT+xGY8VuYpI69bQsyfcfjt07w5Ll8JRR8XMM1p+84Bbge/jtO0pY0ydwF9unOd5KjfjKaTMhjjzTDvR66qr4MAD4dprC00eLb+hQGfgFKAecC3Rm4YeBepjJSfaAk2Bh2PZmcj4Nv/E5iXgkcIS9Olj/4pAxAqBMeafACLi21U9+0LMMjt5sv1/9dVx57lXmRWR+sBd2LeB5W73D4XkcTDwjjFmqzv/beD/4rYgAfE1/wQmLc1s3L378BnGpJZIfnl51cnObvdOiWQGt4rIRhH5TkQuK6E8PQmOk2BIJ47af5xkAs9E2N8ByAF6iMhqEflVRG4rJJ9/AheJSH334LgMmF5CNlZIfPBPYLTWJ2/Z8sAFubn1MWbfLqUx1cjLa8DmzY9cXAKmPQe0B5oADwLjReTUEsjXUzkYitXj39emwFyXT6Q3iRbAfsAh2Fp9D+BhETk7Sl7fYzujN7i/XOD5fbSvQuODf4KitW4EaEhJ2bJlCDk5Nckt5q2Umws5ObXYsmUIkFwboH379sWegm+M+d4Ys8EYk2OMmYbtKOte3Pw8lQu3AtdZxrDRGCmWsqQ7byN2QZdIcs5Z7v8jxpgsY8x8YDJwQZQsNfArUBfbP7AEmFAc2xIFH/wTEK11ErZgtgTYurUG/funsmoVZGcXTSU3OzuJVavgr3+tS2Zmnfz9vXv3flRrLSVksgFKKi9PJSAtzfy+efPfx+bmNpW8vKI1W+blpZKb21S2bu1/WyELucx3/+N9uBwDjDHG7DDGbAdGE/1BUSnwwT8xeQA4FyAvL4+RI0fy888befTRNuzceRHGpJCXV/hw6ry86hiTQmbmBdx//4HMnfsn48aNIyTxXbt27Yuxo3sAEJHqIlIDG8RTRKSGiFRzx9JE9tTgRKSHiNQRkWoicg5wDfBuybrAk8horSUn59Aemzalk5V1AdnZ1fKHJkdj507IyUkiK+tCNm1KZ9euLlE7ZI0xS4DPgfvdOP3DgcuB92DvMgv8D+gjIjVFpCZwM3aEW6XFB/8EQ2t9LnYxDADeeecd5syZQ+3atenffyA7d17Nhg1jyMzsvjUnpwXGJJOdncSOHZCdnYwxyeTktCAzs/v2DRvGsHv3tdx1192kpqby+eef89FHHwW/boTW+mT3+UPsq3RnYKz7fJo71hL4MnDendi22M3A08BNxpiMkveGJ4HpDBwKyaxf/xeuvDKJf/0Ldu48AGOSMSYVY1J22//JbN3agH/9C+66qxmZmVfgBir20FrvX8h3XAm0wrbhvw88aIz52B0LL7O9sdpAf2DLbhsquVKoH+qZQGitD8K2nwvAggULeOONNwC44447aNKkCQDG1NmcldX9jays7n0hl//8Zxyff/4J5513PmeccRVuAaUXgb5AzZYtW3LzzTczcuRIxo8fT3JyMvfeey/nn39+yhVXXKG11scbY9IKMe10bCee+37TNVpCEUkF1gApwFPFdoYn0cnX3//iiy/YsCGbhQuPZNu2h9i2LZekpDVbUlM/7bNzZzedl9eY7Ow8pk3ry/btK/jtt9842E5VrwFcgW2i2QV8JyLPGWMeBDDGrATOi/L94WX2NyDiYIfKWmZ9zT9B0FpXB6YADQE2btzIs88+izGG7t27c/zxxweTXws4BZ8ktmypw7JlsHVrXQIr560GbgltdO3alXPOOYecnBzq16/PqFGjuOKKK8DWkCa4foaIGGP6GGNmxPM73GzJ/Y0xtY0xQ2Of4alsaK3rYptgAPjEqbZ169bN7UkiN7fZaxde+NbUvLwDvoEkUlJS6Nq1a4H0jhsBjDE1jDH7hQJ/LHyZ9cE/kRgOnAyQk5PDiBEj2Lp1Kx06dKBnz57BdE8opd6LJ0Ol1CvAuND29ddfT9u2bVm3bh2jRo0iLy9fgPFcbD+Dx1MSKKA2wO+//87ixYupWbMmJ598cjDNS2H/8x8Os2bNYvcegf+OWuujS93iSogP/gmA1voq4PbQ9oQJE/jll19o0KABd955J9Wq5V/G/1L0xaz74yQbUlJSGDhwIHXq1GHOnDm8/fbbwXQPuf4Gj2dfuTH04b///S8AXbp0ITU1f9TPHKXUXPd5Mm7YZuvWrWnTpg07duzgm2++Cebnl3AsBj74V3C01kcQqJ1/+eWXTJs2jaSkJAYOHEi9evVCh/4ErlRK5RQlf6XUTuwEmM0AjRs3pn///ogIU6ZMYf780Ig5BJjk+h08nmKhtT4c29lLTk4On332GRBs8gECtX2l1FbsGPwC6UIPDcc1WuuSmeZehfDBvwLj2kbfxK15umrVKl544QUArr32Wg455JBQ0lygp1JqTXG+Ryn1G7afAIBjjz2Wyy67DGMMzz77LBv2LP3VwJrlbzRPsbkh9OHbb79l27ZtHHTQQbRt2za0excwKeyc/IdBly5dSElJYcGCBaxduza0uyGVXIenNPDBv4LiJliNAw4D2LlzJ8OHD2fnzp2ccsopnH/++cHkf1VKzdqX73P9BE+Etnv06MExxxzDtm3bGDFiBDk5+S8UJ2H7HzyeIqG1TgGuD22Hau/dunVDJH8O4FtKqU1hp36Ok4KuXbt2ft9ARkZGMM2NeIqED/4Vl9txIyKMMYwbN44VK1bQvHlz+vXrF7xZ3gRGlNB3DsH2G1CtWjXuuOMOGjZsyK+//sqECQVmut/m+iE8nqJwAVbviQ0bNjB37lySkpLyR/E4Xgo/SSllgJdD22eccQZgg39gUMI5WuuWpWR3pcQH/wqI1roTgdr1Rx99xOeff05qaioDBw6kZs2aoUOLgN7u5thnXH/BlcAqgHr16jFgwACSkpKYNm0aX3zxRTD5ONcf4fHES37t/NNPP8UYw4knnhjst1qGq3xE4FXc+r9HHHEETZo0Yf369fzwQ75Ks1DJJ2WVND74VzC01o2xHVwpAIsXL2b8+PEA9O3bl5Yt8ys3WcBlrkOsxHD9Bj2xcrgccsghXHfddQCMHj2alStXhpLWAt50/RIeT6ForQ/EaeXk5eXlN/mEavGOfymlIi7wrpRaiZNYrlatGmluaa+wMf83aK19TIsT76gKhJtINRErR8u2bdtIT08nJyeHc845hy5dugST91VKLSgNO5RSs4G/hrbPO+88OnfuzM6dO0lPT2fnHhGWw7BvAF60zROL63AzDH/66SfWrFlDw4YNOfro/CH6BrvKV2HkN/2kpaUhInzzzTds3749tPtgIK0kja7M+OBfsRgCnA22djRq1CjWr19Pu3btuP7664PpxiilXitlW/4BTAUQEfr160fz5s1ZsWIFY8eOzReAw/ZL3B4lD48nNHghfyx+qNZ/+umnB+eofKSUiqbQGeI9YB1Ao0aNOProo8nJyeHzzz8PpvFj/uPEB/8Kgtb6fOzCJwC89dZbzJkzhzp16jBgwABSUlJCh77DLk9Xqrh+hBuxGufUqFGDQYMGkZqayqxZs8IF4Ia7fgqPJxKnYhdVITMzk6+++grYa2z/y3ufVhCl1G5s2z+wp8kobMz/ZTHE3jwOH/wrAFrrVlh9fgGYP38+WmtEhP79+9O4ceNQ0k1ADzcxq9Rx/Qk9cDMsW7RoQd++fQEYP348ixcvDiVNsT9DN46Uj6fKk18bnz17Nrt37+bII4+kadOmod0bgXfizCv/IdGxY0fq1q3LsmXLWLp0aWh3DeygBU8MfPAvZ9yEqanYCVSsX78+X7Dtsssu49hjjw0mv1Yptaws7XP9Cn1D2126dOG8884jJyeH9PR0tm3bFjrUAphYmACcp+rhBgTki0+FOmjDOnonKqV2xZOfUupH4Csgptibp3B88C9/RgAdYY9g27Zt2zjmmGPo0aNHMN1jSqn3y8NA178wJrR93XXX0b59e9avX8/IkSODY63PpujaQp7KTU8CIm5LliyhVq1a0UTc4iW/9h9qOgq9UThO0FofU2yLqwg++JcjWuurCcgqv/baayxatIiGDRtyxx13BDvDPiawgEs5cRe2v4Hk5GQGDBhA3bp1mTt3Lm+99VYw3RDXf+HxQKAWHqqdd+nSherV81ea+14pVdQVs94AMgFatWpF27ZtvdhbMfDBv5zQWh+JXRELsAtaTJ8+PZJg20rgKqVUMZdnLxkCAnCbwI62CAnAaa2ZN6/A/TvB9WN4qjBOxO0UgOzs7PxROdFE3OLFi72VDD74lwPhgm0rV65k9OjRgNXUb9++fShpDlawbW2kfMoa199wTWg71DRljOG5555j/fr1oUMNgKn+5qvy5Ne+v/vuO7Zt20arVq1o06ZNaPcu4PVi5p3/0Dj11FMjib01AP5SzLyrBD74lzFuzPNLwKFQULCtc+fOnHtuAcn8u5VSX0TKp7xQSk0DHg1tX3bZZdEE4DpScppDngTDibhdF9oOrtYV1KWKIOIWL7Ow8ibUrl2bTp3sSGMv9hY/PviXPf2xKxlhjGHMmDH88ccfkQTbpgDPlZeRMXgYmAkFBeAWLVrEq6++Gkx3i+vX8FQ9LsSJuK1fv5558+aRnJwcLuIWc2x/NMLF3oJNP8EBCH79iej44F+GaK07A8+EtmfMmMHs2bNJTU1l0KBB1KhRI3ToF6BPSQm2lTSu/+EqbH8E9erVY9CgQSQlJfHBBx8we/bsYPKxrn/DU7WIKOJWt26+FNRvRBdxi5cCYm9NmzZlw4YNLFiQr3rixd4KwQf/MkJr3QRbm08GWLRoEa+88goA/fr1o0WLFqGkmVjBtm2R8qkoKKXWYd9gcoACEhSjR4/mjz/+CCX1AnBVDK11MwIibqGmmLCO3qgibvGilFoFTAMv9lYcvFPKADfxaRLQHGDr1q2kp6eTm5vLeeedx6mnnhpMfrNSamF52FlUlFJfAneHts8991xOPfVUdu3ald+P4TgUeMkLwFUZrsPFlh9//DGaiNsrJfRd+R2/IbG3//3vf8HJh62BbpFOrOr44F82DAXOBFsTGjlyJBs2bKB9+/b5csmOF5RSE8vFwuLzHPaNBhGhb9++tGjRgpUrVzJmzJigAJzC9nd4KjHhIm6hWnhaWlpw3sqHcYi4xcv7wFqAhg0bcswxx0QSe/MdvxHwwb+U0VpfCNwf2n7zzTeZN28edevWZcCAASQnJ4cO/Q8YUB427guuX6IPtp8iXwCuRo0azJ49mxkzZgSTP+P6PTyVly5Ae4AdO3bw9ddfA0UXcYsXpVQ2UcTeAhWP7lrr+iX1nZUFH/xLEa11ayBfennu3LlMnToVEeHOO++kUaNGoUMbARWvvklFw/VPXIabdRkauQTwyiuvsGjRolDSZGCK6//wVE4KiLhlZ2dz1FFH0aRJ/iXfCPy7hL8z/2FywgknULduXZYvX85vv/0W2p2KF3vbCx/8SwmtdQ2sYFt9IF8HxxiDUiq8/fMapdTycjK1RHD9FDeHtjt37sz5559Pbm4u6enpbN2av+BYc2CSF4CrfGit6xEQcYuyWteEkq7kKKV+Ar4EL/ZWFHzwLz3+AZwABQXbjj32WLp37x5M96hSanp5GFjSuP6K50Pb1157LYcccggbNmwIF4A7E9sP4qlcXI6btb58+XKWLFlC7dq1Oemkk4JpSqzJJ4y9FnifNWtWUOzteK31saX03QmJD/6lgNb6WgIyyK+++iqLFi2iUaNG4YJtH1H5guBA4BsoKAA3b9483nzzzWC6+11/iKfysNdqXaeeempQxO27Yoi4xUu+2NtBBx1E27ZtyczM9GJvheCDfwmjte5AQP549uzZfPDBB/mCbYFJLn9QAQTbShr3St8T27ZLw4YNufPOOxERpk6dyty5c4PJX3P9Ip4ER2t9BNAJrIjbZ599BuzV5FNkEbd4cf1OU0Lboe8Na/q5xjXHevDBv0RxbZ5vAjUB/vjjj3zBtl69etGuXbtQ0mxsB+/6SPkkOq7/4mpsfwZHH300SimMMYwcOTIoAFcfKwDnb8jEJ79W/b///Y/t27fTqlUrDj744NDunRRfxC1eCoi9Va9enR9++IE1a9aEdtfHi73l44N/CeHGN7+MG+aWlZXF8OHD2bVrF126dOGcc84JJh+klPqqPOwsK5RSHwB/D213796d4447jm3btpGenk52dnbo0AnY/hFPgqK1rk5AxC3U5BMm4vaWUmpzKZsyG7fmdK1atfLF3sKknn3Hr8MH/5LjLuxwx3zBtpUrV9KiRQtuvvnm4E3wBjCqnGwsax4BPgQ7/f7222+ncePGLF68OFwArq/rJ/EkJhcCjcGOaps/f34kEbdSa/IJEU3s7dNPPw0ONjjLrzVh8cG/BNBanwo8Fdr+4IMP+OKLLwpMeHL8TAUWbCtpXH/G1dj+jQIT22bMmMGsWbOCyce4/hJP4pFfm87IyMAYw0knnRQu4pZRRra8CuRCQbG3+fPnh457sTeHD/77iNa6KQHBtl9//TW/VnvLLbfQvHnzUNIdWMG27eVhZ3nh+jUUtp+Ddu3a0atXL4B8OWtHTawAXL0I2XgqKE7E7Xyw0iXBJp8A+yziFi9KqT9xYm8ikm+HF3vbmyrvgH1Ba52M7cRqBlawbcSIEeTm5nLBBRdwyimnBJPfpJT6sTzsLG9c/8ag0PbZZ59N165d8wXgsrKyQofaAy97AbiE4noCIm7r1q2jUaNGdOiQ/xJngPFlbFN+E9Ppp5+eL/YWmGjYCjgj4plVCB/8941HcIqBeXl5PPfcc2zYsIFDDjmEa665Jpjun0qp0h7pUNEZBUwGWyO76aabaNmyZSQBuMuw/SeeCk6cIm4zlFIryti0acAasEONjz32WHJzc8ObGav8mH8f/IuJ1vpi4L7Q9tSpU5k/f34kwbZvCNR6qyqun+Mm4CfYIwBXs2ZNvvjiCz744INg8qe01l3Kw05PkegKtAPYvn07X3/9NSKSr6vvKK0ZvVEJF3sLNv14sbc9+OBfDLTWbQgUrrlz5/Lmm2/mC7Y1bNgwdGgDCSzYVtK4/o7LsP0fNGvWjFtuuQWws6B//fXXUNKQAFzT8rDTEzexRNw2AO+Wi2WBh07Hjh2pW7cuv//+O0uXLg3tTsWuRldl8cG/iLgJSRrYH2DdunU899xzGGPo2bNnuGDb1SWoW14pcCJcfULbnTp14oILLiA3N5cRI0YE22UPBF53/SqeCobrmFeh7SgdvSUu4hYvSqmfgS/AyoycdtppgBd7C+KDf9F5Djge7DT29PR0tm/fznHHHcell14aTPeIUmpGxByqOEqpyQTmOlxzzTUceuihbNiwgeeeey44Jrsbtl/FU/HIF3FbtmwZS5cujSTiVupj+2OQ//2hh9KsWbPYtSv/eXSc1vq4crCrQuCDfxHQWl+PbbcGrFb9kiVLaNy4cbhg2wwCs1s9ERkEfA17BODq1avH/PnzmTp1ajDdfa5/xVOxyK81h2r9Xbp0CYq4fauUWhDhvLJE45oYDzroINq1a0dWVlb+AjOOKtvx64N/nGitjwZGh7Y///xzPvzwQ5KTkxk4cCB16tQJHVqB1eevVIJtJY1SajdWAG4DQIMGDQoIwM2ZMyeY/FXXz+KpAGitjwROBvv2G1oysaxE3OLFib29EdoOrvIV4Oqqqi3lg38caK33wwq21QBYsWIFY8eOBaxgW9u2bUNJs4EelVWwraRx/SFX4QTgOnTowOWXXw7AyJEjWbduXSjp/oCuqjdpBWQvEbfWrVuHi7hNLhfL9ia/47dz585Ur16dhQsXsnr16tDu+sAl5WFYeeODfwwCgm3toKBgW9euXTn77LODyQcopb6JlI8nMkqpDwmsaXDJJZdw/PHHs3379nABuOOx/S2eciSaiFtYrX9qGYi4xcsXuPWlg2JvGRkZwTRVsuPXB//YDAS6gxVse+GFF1i1ahUtW7bkpptuCgq2vU5gFStPkfg7tp+kgADckiVLeOWVV4LpbnL9Lp7y4yKgEewRcUtJSaFLlwLTMsp8bH80wsXeQg+pMLG3M6viuhI++BeC1ror8GRoe/r06Xz11VfUrFkzXLDtJ+DmqiLYVtI43ZdrsP0l1KlTh4EDB5KcnMyHH36Y36bsGO36Xzzlw14ibieeeGKwz2sp8Gm5WBadfLG3ww8/nAMOOMCLveGDf1S01gdgO4uSAH755Rdee+01wAq2NWvWLJS0Sgq2lTSun6QHTgCubdu29O5tm5bHjh3LihX5CgE1sAJw+5WHnVUZrXVz4DwoKOIW1uTzclmJuMWLUmo18D5QYAZyVRd7q1I/Nl7cxKLJ2IlGbNmyJV+w7cILL8xvN3Tc6CYuefYR119yV2j7zDPP5LTTToskANcOLwBXHuSLuC1cuJB169bRuHFjjjrqqNDxPOCVaCeXM/mjj9LS0iKJvR0EnFkulpUTPvhH5lHgdNgj2LZx40YOPfRQrr766mC6kUqpNyLmUI5orZOAk2IkO7mCBs8XgEmwRwDuoIMOYtWqVbzwwgsFtFmw/TGeMsCVlfw5LoWIuP2x99kVgmnAarDDikNib2FNireUi2XlhA/+YWit/w+4N7Q9ZcoUFixYwH777Rcu2PYVcHd52BgHY4G0GGkuBYaVvilFw/Wb3Az8CJCamsrAgQOpWbMmX331FdOnTw8mf1JrfVp52FlV0FpX01r/DdgItAYr4vbNN99UCBG3eFFK5RDQ4wou8B6oUFyqtf5Na9297C0se3zwD6C1bkuggHz//fe89dZb+YJtDRo0CB1aD/R0E5UqFFrrE4l/1uIgrfUhpWlPcVBK7cAKwG0HKwB36623AvDaa6/xyy+/hJImAW+4/hlP6fAU8AROywqsREJ2djYdOnSgcePGod3rKT8Rt3jJfzidcMIJ1KtXjxUrVrBkyZJgmtbYPqUrytq4ssYHf+xqRFrrjsDbwH4Aa9euZeTIkQBcfvnlwXZNA1xVDhrl8dK5CGmTgBNLy5B9wQlz5Y8sOfnkk7nooovyBeC2bNkSOnQA9mY9TmvdJFJenuLh1rq9M3x/FBG3DytiZSiIUuoX3ALvhYi9hXi6sncAV+ofFwutdQOt9X+wa8z+D+gAsHv3btLT09mxYwfHH388l1xySfC0h5VSH5W9tXFT1GuaVCpWlABKqSkEJnZdddVVHHbYYWzcuDFcAK4z8D2wSms9WWtdJ0J2nqLTAbc8aYhly5bx22+/Ubt2bU48sUC9oZPWOqVMrSsiWuuW2I5dYM/Da/bs2UGxtxAtcPMZKitVNvi7TtGZ2EkrBTo+x48fz9KlS2ncuDG33357sENrOrYzuCJT1AdTxGpPBeIe4EuwtbW77rqL/fbbjwULFjBlypTwtElYtcl3K2hndqJRO3xHqJbctWvXoIgbQBvg8LIxq9ici5NoAWjZsiXt27ePJPYWom6knZWFKhv8sUF/LznXzz77jJkzZ5KcnMygQYOCk1d+B66taGOYw1FK/QB8HGdyXYFHZwAFBODWQ0EBuLfeeovvv/8+0mndKFrzlycya4Ibu3fvjibiFqJ5Gdi0LzQL3xGq/YeJvYVYE2lnZaEqB/+9ZIJ///33fMG23r1706ZNvpDkbqxg24ayM2+fGBo7CZAgWvnuAZUvAHfUUUdxxRW2P27kyJGsXbs20mleBnrfWR3c+Pbbb9mxYwcHH3wwrVu3jpT+uzKxqvjsZV8UsTeAnZV94mZVDv7HBzcyMzMZPnw4u3fv5rTTTuPMMwvM9ximlPpf2ZpXfJRSn+PErArhK/eWkBC4fpYXQ9t/+ctfOP7449mxYwfp6ens3r1XX+Px4Ts8RebP4EaoySesozfEKKVUxKdwBWIGbnWvEEGxt7Da/86yM6t8qJLB3ykT7llv0Qm2/fnnnxx00EHhgm0AibgG76M5OTksXLgwfxxzXl4eCxcuDHWU3l+u1hWPbaEPIQG4Jk2asHTpUsaPHx+etpNv999ntuLettatW8eCBQsiibiB7Qu7p6yNKypurP/lhFWMQk1YGRkZwUEEOWVqXDlQJYM/0JHAKJf333+fr7/+mpo1azJw4EBSU1PD01cP31HRUUpNeOeddxYNHTo0f2H0OXPmMHToUKZNm/aVUqqid/RGosB1CAnApaSkMHPmTD777LPg4bq4SUme4qGUMv/+97+3Dh06lI8//hhjDCeddFKwHwxgAvAXpVRC1JRdE2IXIH+1oJDY26ZNm/jkk0+47777mD179m/lZ2XZUFWDf4/Qh59//pmJEycCcOuttwYF24LMibSzovPOO+/0AfInsSxatAiAiRMn3hT9rArN3PAdbdq0KSAA9/vvvwcPXxqe3lM0ZsyYsW7hwoV8/LEdQ9C6dWuWLl0aOvwWcL1SKjva+RURJyJ4GrAIYO7cuRx77LEAfPDBByxZsoTJkyf/Wn4Wlg1VMvi//vrrhw8aNIilS5fmC7ZddNFFnHzyyZGSf0HFn7kYkV27dn0GfJyTY99g3SutzsnJSZi2/jBeBRaG7zzjjDNIS0tj9+7dDB8+nEWLFjFgwAAmT558ZDnYWKlYv379PLDihrVq1WLixImMGjUKYBx2EESFHv0WDdeZe8yOHTtmDRs2jJkzZwLkq8euWbOmvNcfLnUkoGtRZahTp84PO3bsOLJ169YsW7aMww47jCFDhgR1e0JMBG5xa4EmJCLSFQi2h3QwxiRq8Edr3RA7Tf//gvt37drFAw88wPLly2nXrh2LFy+mVq1aGTt27IjYO+mJDxEZCdwe2JXXuHHjO9auXVtpFi6qX7/+S5s3bw6XRLneGPNqxBMqCXtFu8qOiCRhJ6SwbNky6tSpw7HHHkt2dnYw+L8JPKaUSsjmniDGmM9F5DfgYGBuIgd+ADfc9i9a687AEOzEHXbv3s2JJ57I6tWrWbx4MQCZmZkVTrcoAdkU+LwLuGLt2rXvlJMtpcLmzZv7AL9hV5QL8WeU5JWGKlfzF5EjCDQdiAjGGG688UZz7rnn/gxcqZSaV34WljwicjrwLNDLGDO3nM0pUbTWpwCvvfnmm23eeOMNqVatWnDEBsCBxpjVUU73xEBEnsaq1+YAZxpjPotxSsIiIrewZynWU4wxX5WnPaVNpQr+GRmSjB3hUQM7TndZWpopMGRLRO4CRgR25dSrV+/tvLy83tu2bat0kzri8UlloEmTJo0zMzNf3LFjx0UU7Mu60hgzOTx9VfFLUYjkk27dqIvta/mHMSbemeMJi4jcil0L4wZjjKnM5SThg39GhjTEShj3wq7wtBu7olA1IBXboz8eeCktzWwUkbewo0DysAuHPGWM+X3vnBOXovqkfKwsHUSkPXAf9rcLMMYY0w+qtl+i4X2yN1XFJwkb/DMypDrwEHY1pzygViHJM7EXLr1vX0b/+it/A4YbY5YWck7CUVyfAEPT0kyFluMtKiJyJHArMPi//yUL75cC+LKyN1XNJwkZ/DMy5CCsImdzCr9A4WQCK4Gz0tIqXW3f+yQC3i97432yN1XRJwkX/N1F+hZoQPG06HOxS9J1TLSLFQ3vk8h4v+yN98neVFWfJFTwd69lP2CHau7LIiS5wFLgyLQ0k1CzE8PxPomM98veeJ/sTVX2SaLN8H0I+1q2r6tPJWG1vYfss0XlT5F8MmwYvPRSxEOVySfgy0okYvrk99/hppvgggvgzTej5lOlfBInCeeThKn5ux74PwisxFMC7ASaJ2qPfXF8MmwYNG4MN94YNUlC+wR8WYlEvD556imoXRtuuy2ubKuET4pIwvgkkWr+vbE98CVJnss3UfE+iYz3y97E5ZM1ayDyOi0RqRI+KSIJ45NEknfoRRy98GvXwqhRMH8+GANnnAF33hk1eS2X7zMlZWQZ04sYPlm0CJ5+Gv74A04+GSS2wn2i+wTi8Mv69fDcc7ac1KwJPXrAZZcVmmei+6UXMXwycCDMmwcLFth7aOxYaNmy0DwrvU9+/dXePytXwkkn2funRYtC35wTxicJUfN3s+zaxUqXmwv33QdNm8Lrr4PWNvjHoL3LP6GIxyfZ2fDgg3D22fDuu5CWBgUl76OSkD6B+PySlweDB0PbtraMDB9u27e/+SZm9gnpl3jvn/R06NDBVpamT48Z+ENUWp9kZ8OQIXDuufb+OeMMmDUrruwTwicJEfyx06tjTqL4+WfYsAH69bO1uerVbWGOQTaJuehHa2L45McfISfH1mqTk+H00+Gww+LKO1F9AnH45eefYcsWuP56SEmBZs3gwgsh8hreBUhUv7QmjvunmFRan/z4o61QXnaZvX9OO61y3T8V/unkqEEcbXNr19paf1LR+u1zKdkOn7Iipk82bIBGjQo29TRtGlfeieoTiMMva9bYZp+LLtqzLy8Pjj46+jmORPVLXPdPMam0Pol0/zRpElfeCeGTRAn+O4njLaVJE/sAyM0t0gMgicRcrDmmTxo0sEHOmD0FeM0aW9ONQaL6BOLwS5MmcOCBMGFCkfNOVL/Edf8Uk0rrk0j3z9q1lef+SZRmn2XEsY7uYYfZCzZ2LGRlwe7dtvOqMLp1o3a3bswTkcdKxtQ9iEiqiGwXkWwRebSEs19GDJ8ceaR9CL75pn0gfvaZbfKIRQL7BOLwy2GHQa1atl9o1y7rm99+i+2bBPbLMkppHerK7JMjj4Rq1eDtt20ZmTWr/O+fkiQhgr+TUF0cK11SEjz+uO2Zv+IKUCqudlyefvrphVOmTHkltC0iY0XkFxHJE5FehZ0rIs+IyCIR2SYiP4vIdaFjxphdxpg62BXBSpR4fJKSAo88AjNmwMUXW1907Rr3VxxjjLkfQEQaichsEdkgIptF5EsROTXaiSLylIisEJGtIrJcRO4PHStNn0B8fgmVk8WL4cor4ZJL7IiOHTvi+op8vwQRketFxIhIn1gZiEgDEVknIvndh+VdVvaRAj4RkSQReVREVrn7Yo6I7B/tZBE5S0S+F5Edrtz0hPL3Sej+mTbNNhHOnAmnnGL3x0Hw/unqHmLBPyMiEceXichCESl1eflEafYBK6H6MDGGZjVtCo/GWUfIy6sO7CY1NfVEYL7W+iFgODAPeAN4Mo5sdgAXA78CJwIfiMhiY8wX8VmxT4wnhk8OPRTGjStSnpkR8tuOHbu8CDDAX4D/iEgTYyJqm78EDDXG7BCR5sCHIvKTMeatIllSfMYTwy+NGtmRUEUgkl8AEJH6WBnpvdYXjsKTwE+UbeVrPHHcP//4R5HyjOaToUBn4BTgd+BIojSDuMWVJgHXAx8B+wH7F8mK4jOeOO6fF1/cs33LLfYBUAh7+cQY8zlQJ7QtImnAf4APImVgjCmTtacToubveIkStjcvr0DcSgWGAV9PmTJlllu4Ima7nTHmIWPMz8aYPGPM18Dn2EJfFpS4TyLlZ4zZaYz5xRiTh9XIzwXqY4Ww9sKlDdaj84hjqGEJUiZ+CfAE8BywPlYmInIKcBTwrxKyK15eMmafJQzC2csn7kF4F3CTMWa5sfxgjIl2Lz2AXXNhujEmxxizwRizpITtjEbMcjJ3LmzcaJt9PvgAli614/0LIZ5ydz0wNeweKXMSJvi76dLp2CfrPpOVBZMm2c7+5cuXBw8dD3yrtX5UJI4pUQFEpCa29h9vDXCfKGmfuHyiTk4RkfnYB+K7wIvGmLWFpP2be3X9A6iNrd2VCWXpFxE5CegIjI6ViVs/+p/YBdHLVFdl3bopbTIzL87Ly0stqSyj+aQDdsnHHiKyWkR+FZHCxCI6AYjIAhH5U0QmiEjESkVJE085WbEC+vSxzT5aw8MPQ8OGUbMs9P4BEJFaQA/glcLSlQUJE/wdQ7Ha2bn7kokx1TCmATNm2HFbw4cPZ8KECezenT/sNxm4/+CDDz7u6KOPLkqNdTS2yWjGvthXRErEJ+78lcAj0RIYY44G6gFXAYVOdzHGDAPqYh+mrwFb9tG+olLqfnHB/HngDvdWFIv+wNfGmO/20aYiobVuCLyZmXllzby8+hizz7d9YWWlBbbp5hDgYGyge1hEzo6SVwvgWuAyoD1QExi5rwYWgULLycUXw1tv2UlvL71UaJNPzPvHcRn2DfHT4plbciRU8Her5ZyF1c4u1k1tA38dsrIeZdiwPQ/pd999l7vvvpsff/wxf19KSkrNU089dbDW+lmtdZ1I+YVwC10fBfQ0ZaiWVxI+YY8e+Vmx5GhdE9DrwN9E5JgYaY0xZg6Qhb3Jyowy8sutwHxjzJexMhKRZtjgv1dncWmita4GTAAOgmS2bBmCMXX25QEQyydZ7v8jxpgsY8x8YDJwQZT8soB/GWN+NcZsBx4vJG2JU9b3D7bJ59WyjBHRSKjgD+AWS+iI1c4u0mu9MewwpvaGTZueyM3La0SNGnYexsCBA2nRogWrV6/m4YcfZty4cWRm2qxd009/YIHWOmLtRUSGAucD5xhjthb7xxWTffEJtsN6KUVfiCIFq4EeD8lA2yLatc+UgV/OBC51zRursZ2cw0VkVIS0JwEHAj+6tM8CJ7lzS7otPsgDwHmhjby8RmzaNIzc3CYUowkoHp/Md//jDW7zi5C2VCir+0dEWgJpwKvFMLPESbjgD/kX6yhgBLYNOtYFywR2ijCiWrVtB+blNT4Z2zwDQOvWrXnyySfp0aMHSUlJfPTRRwwcOJDt27eTk5PD7t27ycvLaw18eOedd04TkfzCKiL3YZtBzjbGbCjhnxo3xfWJS39kYQVXRDqJSBcRqS4iNUXkXqAp8LU7nhbyiYhUE5G+IlJfLCcBtwEf7+tvLA5Bvxgju+wIr0KJ2y9YAa/DgWPd37fYN5zQEL+0QFmZjp3yH0o7BJgDHGuM2demqYhorc/BjmYpgH0ApJOVdQHGpGBMUqQRW0Hi9onrrP0cuN+N0z8cuBx4D/byCdiO7xtEpI1rD783lLYsKc37J8C1wBfhHdoRfIKItIjX9uKSkMEf7OtaWpp5ALsQw0PYTtbd2CfxVvd/t9s/BKux/WBamslWSn2H7Zh9IJRfSkoKPXv25Mknn6RGjRps3LiRlStXMnbsWK655hp++uknAPLy8s5v27Zttta6uzv1ceAgYFFgDO/gMnFCGPvikxhZp2I7Kjdg2zUvAC40xqxyx1sCwaaPS4ElwDZsk8NIyrYdtwAhv2za9PSdmZk9yclpgTHJGJNKXl5NjEnFmGRycxtupgh+McZsNsasDv1hfbvVGBPq38j3ixuzHky7Bch2n0scrXVLbCd7lEELyWRmXsmGDaO35uXVH0bJlRWAK4FW2PLyPvCgGz0HYWXFGPMytib8NbAc2IV90y5zSvH+CXEdkTt6w+8fsA+KUiVhFnOJB6ek1xqrq7ETWOYmc0RFRHalpqYmXXjhhUlXXHEFALm5uUybNo3JkyeTnZ1NvXr16N27N6eccgpjxoyhU6dOHHvssQBvArcrpSLewCKSCqzBNpE8ZYwp03ZvKLZPdmJvwueMMTFHw4vIi4A2xsTs6C5Pn2itryX/lTuXatXWIbIbY6qTl9cYSJqglIp60yWKX7TW1YHPgJPjSP5/Sqn/gC8rkahoPilJKlXwLy5a6yRs590T2GGJAKxevZoxY8awcKEdudmxY0f69OlDgwYFRqJtAgYAryqlvDMrMAWDf0QKDf6JgtZ6JHY4aSyGKaXuK217PBWThG32KUmUUrlKqZHYNr+PQvsPOOAAHnzwQW6++WZq1qzJt99+y4ABA5g5cyaBh2Z97EzB6VrrVmVuvMcTQGt9BfEF/v8CRZvj7KlU+OAfQCm1DDgXuAHYDFCtWjXOOuss0tPTOf7448nKymLs2LE88sgjrF5doLXnXGCh1vp2N7zO4ylTtNZHAC/GTAh/AlcqpWJ19HoqMT5IhaGUMkqp8cARwNuh/Q0bNuTee++lf//+1K1bl4ULF3L33Xfz3nvvkZeXP7+nNrZj8zOt9aFlbrynyuLmoUwl0GwZhVzgcqXUmtK3ylOR8cE/CkqpP5VS3bEzFNcAiAhdunRhxIgRdOnShd27d/Pqq6/ywAMP8PvvBUZ6nQrM01rfp7WOTwPQ4ykmWmsBxmGHncbiXqXU56VskicB8ME/BkqpN7FvAflDtOrVq0f//v259957adCgAYsXL+bee+9Fa01OTv6bdCp2GOg3Wuvjyt5yTxXiNuCKONK9hdWy8Xj8aJ+ioLU+FxiLHdcPQGZmJhMmTGDmzJkAtGzZkltuuYV27QpIAuUCTwNDlVIVfoWfykplHO2jte6EHdYZ6w1zEdBRKVXmM9A9FRNf8y8CSqkZ2BFBo3BT0mvVqsXNN9/MQw89RNOmTVmxYgX3338/r776Krt27QqdmgT8DdsU1KVcjPdUOrTWjYApxA78WcBlPvB7gvjgX0SUUtuUUncApwG/hPYfeeSRPPPMM1x88cUAvPfee9x999388MMPwdMPAT7XWo/SWtctS7s9lQs3N2UidnZoLPoppWIsaOqpavjgX0yUUrOwGi1P4NQAU1NTufbaa3nsscc46KCDWLNmDY888ghjx47NF4pz3Ab84JqRPJ7i8CBwThzpxiqlKoSQmKdi4YP/PqCU2qmUGozVCZoT2t+uXTuGDRtGz549SUpKYubMmQwYMIBvv/02ePpBwAda61e01mWyeIWncqC1Pg+rLROL74E7S9kcT4Lig38JoJSag9VRuQ+r6UFycjI9evTgqaeeon379mzatImnnnqKZ599lq1bCzS9Xgf8pLXuUfaWexINN4t8IlEF2/LZBPTwAww80fCjfUoYN7nrRSC/YzcvL4/p06czefJkdu3aRd26dbnhhhs49dRTCVsp8m3gNqXUn2VsdpUg0Uf7aK1TsXLJJ8aR/CKl1PulbJIngfE1/xJGKfULcDpWX2U7WImICy+8kGeeeYYOHTqwbds2nnvuOZ588knWry+w5velwI9a6xvcxB2PJ0g68QX+x33g98TCB/9SQCmVp5T6J3ZYaL5Ma9OmTXnggQfo168ftWrV4vvvv2fQoEF89NFHQYmI/YGXgQ+11geXufGeConW+iqs8mwsPiG+/gBPFccH/1JEKbUcu7zj9dg2WESEM844g/T0dDp27EhWVhbjxo3jkUce4c8/C7T2nIUdEdTfDevzVFG01kdi5RtisQor2FYqK4N5Khc++JcyTijuVazuytTQ/gYNGnDPPfdw1113Ua9ePX788Ufuvvtu3n33XXJz8+/dWti1Xj/XWsej2+KpZLj5IG9iy0Jh5AA9lVJrS98qT2XAB/8yQim1RimlgO7AarBvAZ07d2bEiBGcdtppZGdnM2HCBB544AGWL18ePP0UYK7W+n4vFFd1cP0+LwLxKMTeo5SaXcomeSoRPviXMUqpt7FCcS+H9tWtW5fbb7+d++67j4YNG7JkyRL+9re/8cYbb5Cdnb88aHXgUeBbrfUJZW+5pxy4A+gZRzqNfUP0eOLGD/UsR7TWZ2HbcluH9mVmZjJp0iQ+/PBDAFq0aEG/fv045JBDgqfmAsOBh5VSWWVncWKTSEM9tdanYAXbkmMk/RU40ev2eIqKr/mXI0qpmUAHbK0tXyiuT58+PPzwwxx44IH88ccfPPjgg4wfP56dO/Pn6yQBf8UKxZ1WLsZ7Sg2tdWOsYFuswJ+JF2zzFBMf/MsZpdR2pdRd2AVgfgrtP+KII3j66af5y1/+gogwbdo07r77bhYsKKDP1R74VGv9vNa6Xtla7ikN3MiuSUCLOJL3VUr9EDuZx7M3PvhXEJRSXwLHYdv1cwCqV6/O1VdfzeOPP06rVq1Yu3Ytf//73xk9ejQ7duwInn4LdljoBWVvuaeEeQg7zDcWo5VSE0rbGE/lxQf/CoRSapdS6kGgI/BdaH+bNm144oknuOKKK0hOTuaTTz5h4MCBfPPNN8HTWwLva61fczrvngTDPbwfjCPpt8BdpWuNp7Ljg38FRCk1D+iEbdffCVYornv37jz11FMccsghbNq0iWeeeYb09HQ2b94cPP0arERETy8RkThorVsD8dTkNwFKKbUrZkqPpxB88K+gKKVylFJPA8dgR30AdvTPI488wg033EBqaipfffUVAwcO5LPPPiMwcqsx8Abwtta6Wdlb7ykKTrBNA/XjSH6NUmpZ6VrkqQr44F/BUUr9CnTDtutvAysUd/755zN8+HCOPvpotm/fzqhRoxg2bFi4UNxfsG8BffxbQIXmH9imvlg8qpSaVsq2eKoIPvgnAE4objRwJDA9tL9Jkybcf//93HrrrdSuXZs5c+YwcOBAZsyYERSK2w87l2Cm1rpN2VvvKQyt9TVAvziSzgQeLl1rPFUJH/wTCKXUCuBCbLv+BrASEWlpaaSnp3PSSSexc+dOXnrpJYYOHcqqVauCp5+BHRE0wAvFVQy01kcBY+NIuhK4ygu2eUoSH/wTDCcUNxErEfFGaH/9+vW5++67GThwIPvttx8//fQT99xzD++8805QKK4mVhN+tlOK9JQTbl7Gm9hrUhg52A7edaVvlacq4YN/gqKUWquUugK4BCvlC0CnTp0YMWIEp59+OtnZ2UyaNInBgwezbNmy4OknA3O01kO01tXL1HBPSLDtJeCQWGmBQW4OiMdTovjgn+Aopf6N7QvI13uvU6cOt912G4MHD6ZRo0b89ttv/O1vf+P1119n9+7doWQpwFDgO611PKtDeUqOO4F41myeAowsZVs8VRQv7FaJ0FqfgX0I5HfsZmVl8frrrzNjxgyMMTRv3px+/fpx6KEFVILzgBHAEKVUZtlaXXZUBGE3rfWpQAaxdXt+Bk5SSm0rTXs8VRdf869EKKU+AY7GtuvnAdSsWZPevXszdOhQmjVrxsqVKxkyZAgvv/xyUCiuGjAImK+1TisP26sCWusmxC/Y1sMHfk9p4oN/JUMptUMpNQjoDCwM7T/ssMN46qmnuPTSSxERPvjgAwYNGsS8efOCp7cF/qu1HqO13q+MTa/UuBFWrwPxTLq7SSm1MHYyj6f4+OBfSVFKfQ2cgG3XzxeKu/LKKxk2bBgHH3ww69at47HHHuP5559n+/btwdNvBhZqrS8qe8srLY9gh9vG4nml1KTSNsbj8cG/EuOE4h4Gjgf+F9rfunVrHnvsMa666ipSUlLIyMhgwIABfPXVV8HTmwP/0VpPcvrynmLiHqKD40j6DTCwlM3xeAAf/KsESqkF2HWA7waywArFXXLJJTz99NMcdthhbNmyhfT0dIYPHx4uFHcl8JPW+iovEVF0tNYHA6/FkXQjdgF2L9jmKRN88K8iKKVylVLDsR3CGaH9zZo14+GHH6Z3797UqFGDr7/+mgEDBpCRkREUimsITATe1VrHs8iIB9Ba1wCmAvvHSGqAq5VSy0vdKI/H4YN/FUMptRg4E+gLbAUrFHfeeecxfPhwjjnmGHbs2MHzzz/P448/ztq1a4OnX4QViuurtfZlJzbPYpvcYvF3pdQHpW2MxxPE38BVECcUNxY7Oey90P7GjRszePBgbrvtNurUqcO8efMYNGgQ06dPDwrF1QVGAx9rrduVvfWJgdb6OmzHeSw+xHYGezxlig/+VRil1B/A/wFXAevBCsWdfvrppKen06lTJ3bt2sW//vUvHnroIVauXBk8PQ1YoLW+W2sda9x6lUJr3QH7gIzFCmxzjxds85Q5PvhXcZxQ3OtYobj8IYb7778/AwcO5O6772b//ffnl19+4Z577uGtt94iJycnlKwG8DTwpdb66LK3vuLh5kfEI9iWjRVsWx8jncdTKvjg7wFAKbVOKXU1cDFWQhiAk046ifT0dLp160ZOTg6TJ09m8ODBLF26NHh6R6xG0FC3KlWVxI2GehloH0fygW4uhsdTLvjg7ymAUuo9bF/AmNC+OnXqcMstt/DAAw/QuHFjli1bxuDBg5k0aVJQKC4ZGAJ8r7XuVPaWVwgGAN3jSDcZ+Gcp2+LxFIoXdvNExen8vIiVfQBg586dTJ48menTp2OM4cADD+SWW27hsMMOC55qsEsTPqiU2lGWNhdGaQq7aa27YIfQxloo5yesYNv2GOk8nlLF1/w9UVFKZWDnBTyDE4qrUaMGvXr14u9//zvNmzfnzz//ZMiQIbz00ktkZWWFThVsLXiB1vrM8rC9LNFaN8UKtsUK/DuAy3zg91QEfPD3FIpSKlMpdQ/QCVgQ2n/IIYfw1FNPcdlll5GUlMSMGTMYOHAgc+bMCZ5+MHbt4Be11vuXreVlgxvpNBk4MI7kfZRSP5WySR5PXPjg74kLpdT/sB27D2FHqpCSksLll1/OE088QZs2bdiwYQNPPPEEo0aNYtu2AmrEN2Inh/2l7C0vdf6OHfYai1FKqcmlbIvHEze+zd9TZNz6vy9hl4MEIDc3l/fff5833niD7Oxs9ttvP3r37k2nTp0QKSAJNAXor5RaU8Y2J2MllQtbQWs50DHe4Zda6/8D/h1H0q+B05RSu2Om9HjKCB/8PcXC6dPfATwG1ArtX7VqFWPGjOGnn2zrxoknnkifPn2oX79+8PSN2KUMJyqlyqQAaq2fxgrbxWIOtkM2p7BEWus2wPdArHUPNgDHK6V+j8tQj6eM8MHfs0+4IDgWqxcEQF5eHjNnzmTixIlkZWVRq1YtrrvuOrp16xb+FjAd6FfagdGJ0S0jdodsiOuUUlGVOJ1g2xfAcTHyMcB5SqkP4/xej6fM8MHfs8+4yU29geEEasLr169n3Lhx+Z3AHTp0oG/fvjRp0iR4+nbgXmC0UiqPUkBrfR72QRMv6W41tPB8WgMnATcA58WRz0NKKa/b46mQ+ODvKTG01s2A54H8jl1jDLNmzWL8+PFs27aN1NRUrrzySs477zyqVSsw3uBz7GiYX0vBri4u/3j5u1JqSOD8FGAccH0R8vgAuLC0Hmgez77iR/t4Sgyl1CrgUuByYB1YobiuXbuSnp5O586d2bVrF+PHj+fBBx/kjz/+CJ7eFbuA/L2lIBT3NbCpCOnD3xKGU7TA/ztwjQ/8noqMr/l7SgWtdUNgBFBgxuy3337LuHHj2LRpE8nJyXTv3p1LLrmE5OQC8f574Eal1NwStOdB4pNOnoUdmWPcefsDq4gt1BYiG+iilPqmOHZ6PGWFD/6eUkVrfT5WJ6hlaN+OHTuYMGECH3/8MQCtWrWiX79+tG3bNnhqLvAktglmZwnYsR82iNeKkfQspdTHgfMuAN4vwlfdppR6vhgmejxlim/28ZQqSqnpwFHYvgAAateuTd++fRkyZAhNmzZl+fLlDB48mAkTJgSF4pKwi57P0Vp3LgE7thB7Ld3lwCdh++JZiSvEL8ALRbHL4ykvfM3fU2ZorU/DCsXlSx7v2rWLN954g/fffx9jDAcccAD9+vXjiCOOCJ5qgP8A1YEDsJLTzxV1CKVrilpL9EpPX7fCWfCcT4BucX7FEqWUX93MkxD44O8pU7TWNbESEXcTGHe/aNEiRo8ezYoVKwA4++yzufrqq6lVq9BWmiI3sWitbyYgVx1ghlJqr+GbWutM4m/v/1EpdWRR7PF4ygsf/D3lgtb6BKxExDGhfTk5Obz11lu8/fbb5Obm0rBhQ2666SaOPz5qy4vByjF8X8Tv7gKMwoqx7QSejPQQcfMXijJiZ4xSql9RbPF4ygsf/D3lhhs//1fsIjDVQ/t///13XnjhBZYsWQJAly5d6NWrF/Xq1YuUzRfY0TWlUpC11vHmuwY4Tin1Z2nY4fGUND74e8odrfXh2LeAU0L78vLymDZtGpMnT2b37t3UrVuX3r1707lz53CJCICblFIvlpJtf2L7GQpjGnaCmg/8noTBB39PhcAJxd0KPAHUDu1fvXo1Y8aMYeHChQB07NiRPn360KBBg+Dpu4GWSqm1pWDXJcBb2AVqwpmBXa3sfyX9vR5PaeODv6dC4fRzxgJnh/YZY/j444957bXXyMrKombNmlx77bWceeaZiAiZmZls2bLl9f79+19VSjZ1xspBN8f2ASzGvm3MLo3v83jKAh/8PRUO19GaAZwW3L9x40bGjRvHd999B8CRRx5J3759GTt2LIsWLTK7du06zBgTlzZQRoYkA62BGthO32VpaaZQGWePpzLhg7+nQqK1fgh4OHy/MYYvvviCl19+mW3btlG9enVatWrFokWLAKYZYy6MlmdGhjTEqo/2Atphm4vysOP+U4FFwHjgpbQ0s7Fkf5HHU7Hwwd9TIdFaHwAsJcoY+61btzJ+/HhmzZoFgIgYY4wAFxtj3gumzciQ6ti5BQOxwb6wyQOZ2IdBOjA0Lc341bc8lRIv7+CpkCilVmM18yOqcdatWxeARo0akZycjAv8AK+ISI1QuowMOQj4AbgL28QTS9unlkt3F/CDO9/jqXT44O+psCilPsMKwvUAHgVewSp+bjfG5C1atMisX7+enJwCTfUNcMs1usD9LdCG2EE/nFruvG/9A8BTGfHNPp6ERURqAocCB2EfEp2BgwH13/+yDlvjb0P8yzdGIhfb/HRkWprJ3jeLPZ6Kg6/5exIWY0yWMWauMeZdY8w/jTFXG2M6G2NWYtv4mxMh8F9xBbgBQ/GQBDTDzkL2eCoNPvh7Kh1uVM9Ait7UE43awN0ZGdIgZkqPJ0Hwwd9TGelN0QTZ4iHP5evxVAp88PdURnoRo9a/eDHceCNcdBEMHQq7Yw/orOXy9XgqBT74eyoVbuZuzAVVMjLgySdh0iRYuhQ++CCu7Nu7/D2ehMcHf09lozV25m6hdO8OjRpBvXpwyin2TSAOsl3+Hk/C44O/p7JRgzja+4OioDVqQFZWXHnnuvw9noTHB39PZWMnpVSuu3Wj3hln8I2IPFbSeYtIqohsF5FsEXm0pPP3eMLxwd9T2VhGYFWwkma//TjOGHN/aFtExorILyKSJyK9CjtXRHqKyBcikikiGcFjxphdxpg6wMRSMdzjCcMHf0+lwskyx9eCXww2byY3bNc87CI08awjvBH4BzCshM3yeIqMH7ngqYyMx8pBRxzuOXlywe1eveLKMzNSfsaYfwKIyM5YGRhjZrq0feL6Ro+nFPE1f09l5CVKvmz7e8VTqfAF2lPpcAuxpGNr6yVBJvBMCeXl8VQIfPD3VFaGAithrzb6opLr8nlkny3yeCoQPvh7KiVuBa6zsJ2sxX0A5Lrzz/Jyzp7Khg/+nkpLWpr5HeiI1eMvahPQDndeR5dPRESkuls5TIAUEakhItXcsTQRMYG0SS5tMlDNpU0pol0eT4ngg7+nUuMC91HACOwEsFgPgUyXbgR2AZeogd/xIZCFXUhmrPt8mjvWEvgykPZad/wFoKv7PC7e3+LxlCR+JS9PlcHp8ffGqnO2x2r15GIXbEkBFgH/Av7lOo0L4IZz7gKeM8Y8GOv7RORFQBtjZsSRNhVY4+x4yhgzNM6f5fEUCx/8PVUSp87ZGqvVsxNY5iaIeTxVAh/8PR6Ppwri2/w9Ho+nCuKDv8fj8VRBfPD3eDyeKogP/h6Px1MF8cHf4/F4qiA++Hs8Hk8VxAd/j8fjqYL44O/xeDxVEB/8PR6Ppwrig7/H4/FUQXzw93g8niqID/4ej8dTBfl/6s6jIC9myrQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# get graph\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "# TODO: move setting original positions to `small_DAG_FELINE()` to avoid repetitions\n",
    "sd.tree={\n",
    "  'a':(2,3),'b':(4.5,3),\n",
    "  'c':(1,2),'d':(2,2),'e':(3,2),'f':(4,2),'g':(5,2),\n",
    "  'h':(3,1)\n",
    "}\n",
    "\n",
    "# find min-post intervals, and make labels\n",
    "intervals=find_dfs_intervals(sd)\n",
    "intervals_repr={\n",
    "  v:('[%d,%d]'%(i[0],i[1]))\n",
    "  for (v,i) in intervals.items()\n",
    "}\n",
    "\n",
    "# configure plot\n",
    "plt.axis('off')\n",
    "plt.title('Small DAG with the spanning tree overlay and interval labels\\n'+\n",
    "          '(find_dfs_spanning, find_dfs_intervals)')\n",
    "\n",
    "# draw graph, its spanning tree and min-post tree interval labels\n",
    "nx.draw_networkx(sd,\n",
    "                 pos=sd.tree,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa',node_color='y')\n",
    "nx.draw_networkx_edges(sd,\n",
    "                       pos=sd.tree,\n",
    "                       edgelist=find_dfs_spanning(sd),\n",
    "                       node_size=500,width=2.0,\n",
    "                       arrowstyle='->')\n",
    "nx.draw_networkx_labels(sd,\n",
    "                        pos={u:(x+0.35,y-0.08) for (u,(x,y)) in sd.tree.items()},\n",
    "                        labels=intervals_repr)\n",
    "plt.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> TODO: \n",
    "> - interactive visualization of spanning tree and min-post tree intervals shown as labels\n",
    "> - interactive visualization of post(v) vs level(v) drawn graph, with min-post intervals as labels"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing min-post graph intervals (negative-cut filter) (TODO)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing PReaCH extensions to min-post intervals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Code taken from [section \"DFS spanning-tree extra data\" in Google Colab version](https://colab.research.google.com/drive/1V-U7_slu5Z3s5iEEMFKhLXtaxSu5xyzg#scrollTo=4l0Ld0Jklq1o) of the [Reachability labels for version control graphs](_Reachability_labels_for_version_control_graphs.ipynb) notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#| export\n",
    "def find_dfs_intervals_extra(DG, extra=False):\n",
    "    \"\"\"Find DFS-derived data of all vertices in graph G\n",
    "\n",
    "    Min-post interval labeling strategy labels each node 'u' in the directed\n",
    "    spanning tree (DFS tree here) of the acyclic graph `DG` with a range\n",
    "    I_u = [s_u, e_u], where e_u denotes the rank of the node 'u' in post-order\n",
    "    traversal of the tree (ranks assumed to begin at 1), and s_u denotes the\n",
    "    lowest rank of any node 'w' in the subtree rooted at 'u' (including 'u').\n",
    "    This interval is stored under 'min' and 'post' keys in returned dict.\n",
    "\n",
    "    For depth first-search (DFS) all nodes reachable from node 'u' have their\n",
    "    topological ordering in DFS tree, i.e. 'post', smaller than the 'post'\n",
    "    value for 'u'.  Additionally we can define 'f_min' as minimum 'post'\n",
    "    number for each node reachable from 'u', and 'f_gap' as maximum 'post'\n",
    "    number of nodes reachable from 'u' that are not in DFS subtree rooted\n",
    "    at 'u' or None if 'f_gap' would be equal to 'u' node's 'min' number.\n",
    "\n",
    "    We can also define 'p_tree' to be node reachable from 'u', but not in\n",
    "    subtree rooted at 'u', that ('p_tree') has largest subtree.  If such\n",
    "    node does not exist it is set to None.  For DFS spanning tree we can\n",
    "    calculate this value effectively.\n",
    "\n",
    "    This is based on *modified* algorithms from PReaCH paper [1];\n",
    "    note that the original version in the paper [1] uses\n",
    "    preorder not postorder numbers (pre-max not min-post).\n",
    "\n",
    "    References:\n",
    "    -----------\n",
    "    [1] Florian Merz, Peter Sanders \"PReaCH: A Fast Lightweight\n",
    "        Reachability Index using Pruning and Contraction Hierarchies\" (2014)\n",
    "        In: Schulz A.S., Wagner D. (eds) Algorithms - ESA 2014. ESA 2014.\n",
    "        Lecture Notes in Computer Science, vol 8737. Springer, Berlin,\n",
    "        Heidelberg (Conference Paper: European Symposium on Algorithms)\n",
    "        https://doi.org/10.1007/978-3-662-44777-2_58\n",
    "        http://arxiv.org/abs/1404.4465\n",
    "\n",
    "    Parameters:\n",
    "    -----------\n",
    "    DG : NetworkX DiGraph\n",
    "        Directed acyclic graph.\n",
    "\n",
    "    extra : bool, optional (default=False)\n",
    "        Whether to compute extra DFS data in addition to min-post interval.\n",
    "\n",
    "    Returns:\n",
    "    --------\n",
    "    dict of dicts\n",
    "        Dictionary, where keys are node indices, and values are dictionaries\n",
    "        with various DFS spanning-tree derived data.\n",
    "    \"\"\"\n",
    "    data = {}\n",
    "    # create graph of spanning tree (DFS spanning tree) of DG\n",
    "    span = nx.DiGraph()\n",
    "    span.add_nodes_from(DG)\n",
    "    span.add_edges_from(find_dfs_spanning(DG))\n",
    "\n",
    "    # needs depth-first search for some of indices to actually work\n",
    "    nodelist = nx.dfs_postorder_nodes(DG)\n",
    "    for pos, node in enumerate(nodelist, start=1):\n",
    "        data[node] = {}\n",
    "\n",
    "        # DEBUG\n",
    "        # print('node=%s\\n -> %r\\n => %r' %\n",
    "        #      (node,\n",
    "        #       list(span.successors(node)),\n",
    "        #       list(DG.successors(node))))\n",
    "\n",
    "        data[node]['post'] = pos\n",
    "        if span.out_degree(node) == 0:\n",
    "            data[node]['min'] = pos\n",
    "        else:\n",
    "            data[node]['min'] = min(pos,\n",
    "                                    *[data[neigh]['min']\n",
    "                                      for neigh in span.successors(node)])\n",
    "\n",
    "        # print(\"data[%s]: %r\" % (node, data[node]))\n",
    "\n",
    "        # min-post graph interval (min over whole graph)\n",
    "        if DG.out_degree(node) == 0:\n",
    "            data[node]['f_min'] = pos\n",
    "        else:\n",
    "            data[node]['f_min'] = min(pos,\n",
    "                                      *[data[neigh]['f_min']\n",
    "                                        for neigh in DG.successors(node)])\n",
    "\n",
    "        # PReaCH specific extensions\n",
    "        if not extra:\n",
    "            continue\n",
    "\n",
    "        if DG.out_degree(node) == 0:\n",
    "            data[node]['f_gap'] = None\n",
    "            data[node]['p_tree'] = None\n",
    "        else:\n",
    "            f_gap_candidates = \\\n",
    "                [data[neigh]['post']\n",
    "                 for neigh in DG.successors(node)\n",
    "                 if data[neigh]['post'] < data[node]['min']] + \\\n",
    "                [data[neigh]['f_gap']\n",
    "                 for neigh in DG.successors(node)\n",
    "                 if data[neigh]['f_gap'] is not None]\n",
    "            if f_gap_candidates:\n",
    "                data[node]['f_gap'] = max(f_gap_candidates)\n",
    "            else:\n",
    "                data[node]['f_gap'] = None\n",
    "            # print(\"f_gap[%s]: %r\" % (node, data[node]['f_gap']))\n",
    "\n",
    "            # print(\"p_tree: %r (candidates, part 1)\" %\n",
    "            #      [data[neigh]['p_tree']\n",
    "            #       for neigh in DG.successors(node)])\n",
    "            p_tree_candidates = \\\n",
    "                [data[neigh]['p_tree']\n",
    "                 for neigh in DG.successors(node)\n",
    "                 if data[neigh]['p_tree'] is not None and\n",
    "                    data[neigh]['post'] < data[node]['min']] + \\\n",
    "                [neigh\n",
    "                 for neigh in DG.successors(node)\n",
    "                 if data[neigh]['post'] < data[node]['min']]\n",
    "            if p_tree_candidates:\n",
    "                data[node]['p_tree'] = max(\n",
    "                    p_tree_candidates,\n",
    "                    key=lambda n: data[n]['post'] - data[n]['min']\n",
    "                )\n",
    "                # print(\"p_tree[%s]: %r\" % (node, data[node]['p_tree']))\n",
    "            else:\n",
    "                data[node]['p_tree'] = None\n",
    "\n",
    "        # print(\"\\n\")\n",
    "\n",
    "    return data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'h': {'post': 1, 'min': 1, 'f_min': 1, 'f_gap': None, 'p_tree': None},\n",
       " 'c': {'post': 2, 'min': 1, 'f_min': 1, 'f_gap': None, 'p_tree': None},\n",
       " 'd': {'post': 3, 'min': 3, 'f_min': 3, 'f_gap': None, 'p_tree': None},\n",
       " 'e': {'post': 4, 'min': 4, 'f_min': 1, 'f_gap': 1, 'p_tree': 'h'},\n",
       " 'a': {'post': 5, 'min': 1, 'f_min': 1, 'f_gap': 1, 'p_tree': None},\n",
       " 'f': {'post': 6, 'min': 6, 'f_min': 1, 'f_gap': 1, 'p_tree': 'h'},\n",
       " 'g': {'post': 7, 'min': 7, 'f_min': 7, 'f_gap': None, 'p_tree': None},\n",
       " 'b': {'post': 8, 'min': 6, 'f_min': 1, 'f_gap': 1, 'p_tree': None}}"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# DEBUG\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "find_dfs_intervals_extra(sd, extra=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# get graph\n",
    "sd=graphs.small_DAG_FELINE()\n",
    "# TODO: move setting original positions to `small_DAG_FELINE()` to avoid repetitions\n",
    "sd.tree={\n",
    "  'a':(2,3),'b':(4.5,3),\n",
    "  'c':(1,2),'d':(2,2),'e':(3,2),'f':(4,2),'g':(5,2),\n",
    "  'h':(3,1)\n",
    "}\n",
    "\n",
    "# find min-post intervals, and make labels\n",
    "intervals=find_dfs_intervals_extra(sd)\n",
    "intervals_repr={\n",
    "  v:('[%d:%d,%d]'%(i['f_min'],i['min'],i['post']))\n",
    "  for (v,i) in intervals.items()\n",
    "}\n",
    "\n",
    "# configure plot\n",
    "plt.axis('off')\n",
    "plt.title('Small DAG with the spanning tree overlay and extended interval labels\\n'+\n",
    "          '(find_dfs_spanning, find_dfs_intervals_extra)')\n",
    "\n",
    "# draw graph, its spanning tree and min-post tree interval labels\n",
    "nx.draw_networkx(sd,\n",
    "                 pos=sd.tree,\n",
    "                 node_size=500,width=8.0,\n",
    "                 edge_color='#aaaaaa',node_color='y')\n",
    "nx.draw_networkx_edges(sd,\n",
    "                       pos=sd.tree,\n",
    "                       edgelist=find_dfs_spanning(sd),\n",
    "                       node_size=500,width=2.0,\n",
    "                       arrowstyle='->')\n",
    "nx.draw_networkx_labels(sd,\n",
    "                        pos={u:(x+0.25,y-0.12) for (u,(x,y)) in sd.tree.items()},\n",
    "                        labels=intervals_repr)\n",
    "plt.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- **\\[PReaCH\\]** \n",
    "  Florian Merz, Peter Sanders:\n",
    "  \"_PReaCH: A Fast Lightweight Reachability Index using Pruning and Contraction Hierarchies_\" (2014)  \n",
    "  In: Schulz A.S., Wagner D. (eds) Algorithms - ESA 2014. ESA 2014.\n",
    "  Lecture Notes in Computer Science, vol 8737.  \n",
    "  Springer, Berlin, Heidelberg (Conference Paper: European Symposium on Algorithms)  \n",
    "  <https://doi.org/10.1007/978-3-662-44777-2_58>\n",
    "  \n",
    "  arXiv:1404.4465v1 \\[cs.DS\\], 17 Apr 2014,  \n",
    "  <http://arxiv.org/abs/1404.4465>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Converted 01_tools.ipynb.\n",
      "Converted 02_related.ipynb.\n",
      "Converted 03_example_graphs.ipynb.\n",
      "Converted 05_reachability_index.ipynb.\n",
      "Converted 06_levels.ipynb.\n",
      "Converted 07_interval_labels.ipynb.\n",
      "Converted 08_reach.ipynb.\n",
      "Converted index.ipynb.\n"
     ]
    }
   ],
   "source": [
    "#| include: false\n",
    "# this should be the last cell of the notebook\n",
    "from nbdev import nbdev_export\n",
    "nbdev_export()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}