Create DP4_GuitarFingering_Tetris_SuperMarioBros.java

Author: 17tanya

DynamicProgramming/MIT-6.006-IntroToAlgosNotes/DP4_GuitarFingering_Tetris_SuperMarioBros.java

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+/*
+In the recurrence, we usually take the min of a bunch of options or the max of a bunch of options.
+And those options correspond to a guessed feature. We don't know whether the go left or go right, so we try them both.
+That's guessing. But there's another way to guess. So two kinds of guessing. 
+
+In the previous problems we GUESSED which subproblems to use to solve the bigger problem.
+
+In KNAPSACK problem, we added MORE SUBPROBLEMS to GUESS/REMEMBER more features of the solution.
+  1. If we look at the suffix - we don't know the capacity we have used up - we don't remember details about the prefix
+  2. That is why we add MORE SUBPROBLEMS - to REMEMBER how much capacity has been used in the prefix.
+  3. We generated MORE SUBPROBLEMS by - MULTIPLYING every sub-problem by S different choices, which is how many units of knapsack   still remain.
+  4. Simply put - we are REMEMBERING MORE about the prefix(past).
+  
+  
+//PIANO OR GUITAR FINGERING - Single Hand, Single Note
+Given sequence of n notes, find fingering for each note(single).
+  1. Fingers are labelled as - 1,2,....,F
+  2. Difficulty Measure - d(p,f,q,g)
+     where, f and g denote finger labels
+            p and q denote notes
+            finger f plays the p note and finger g plays the q note
+            
+            
+DP Approach 1(WRONG APPROACH)- 
+1. Subproblems - How to play notes[i:] onwards(suffix)
+2. Guess - Which finger to use for note[i]
+3. Recurrence - 
+      DP(i) = min{ DP(i+1) + d(i,f,i+1.??????) for each f in 1...F}
+      d() is the difficulty measure, can be thought of as the transition cost
+The above approach is WRONG beause the transition function requires us to know the finger used for the transition.
+We need more information about what we are going to do next.
+
+We need to add MORE SUBPROBLEMS.
+
+
+DP approach 2(Correct) - 
+1. Subproblems - How to play notes[i:], when we use finger f for notes[i]
+2. Guess - finger g for notes[i+1]
+3. Recurrence - 
+      DP(i,f) = min{ DP(i+1,g) + d(i,f,i+1,g) for g in 1...F }
+4. Topological Order - 
+      Bottom up approach 
+      for i in reversed(range(n)): //go from right to left
+        for f in 1...F:
+5. Original Problem - 
+   Here the subproblems we are solving require us to provide to first finger label, but we don't know that.
+   We don't know which finger to start with, we consider them all and take the min.
+   min(DP(0,f) for f in 1...F)
+  Time = (num of subproblems)*(time/subproblem) = theta(nF)*theta(F) = theta(n . F^2) 
+          when F in small ~ linear time 
+   
+DAG - complete bipartite graph
+               notes
+      0 ..........i............. n-1
+    1 x1 y1 z1 ....................
+    2 x2 y2 z2 .................... 
+f   . .............................
+    . .............................
+    F .............................
+   
+   
+Each cell in the matrix is a node in the BIPARTITE GRAPH ==> 
+  each x in column 0 is connected to each y in column 1
+  x1 --> (y1,y2,.....yF)
+  .
+  .
+  xF --> (y1,y2,.....yF)
+  
+  each y in column 1 is connected to each z in column 2
+  y1 --> (z1,z2,.....zF)
+  .
+  .
+  yF --> (z1,z2,.....zF)
+  and so on...
+  
+We need to find the shortest path in this DAG, the current formulation is not a single source shortest path problem.
+We are looking at all sources to find the shortest path.
+
+This can be converted to single source shortest path by using a dummy node and connecting it with all column 0 nodes with weight 0. Now, the source is this dummy node.
+*/