automata: Fix broken universal start states with sparse DFA

The state IDs were not remapped, which will usually result in an index
out of range error.

Add a test based on is_special_state's doctest, which will validate the
start state's behavior with a custom searcher.

regex-automata/src/dfa/sparse.rs

@@ -1846,6 +1846,14 @@ impl StartTable<Vec<u8>> {
             let new_start_id = remap[dfa.to_index(old_start_id)];
             sl.set_start(anchored, sty, new_start_id);
         }
+        for ustart in [
+            &mut sl.universal_start_unanchored,
+            &mut sl.universal_start_anchored,
+        ] {
+            if let Some(id) = ustart {
+                *id = remap[dfa.to_index(*id)];
+            }
+        }
         Ok(sl)
     }
 }

regex-automata/tests/dfa/api.rs

@@ -3,7 +3,7 @@ use std::error::Error;
 use regex_automata::{
     dfa::{dense, Automaton, OverlappingState},
     nfa::thompson,
-    HalfMatch, Input, MatchError,
+    Anchored, HalfMatch, Input, MatchError,
 };
 
 // Tests that quit bytes in the forward direction work correctly.
@@ -67,3 +67,93 @@ fn unicode_word_implicitly_works() -> Result<(), Box<dyn Error>> {
     assert_eq!(Ok(Some(expected)), dfa.try_search_fwd(&Input::new(b" a")));
     Ok(())
 }
+
+// A variant of [`Automaton::is_special_state`]'s doctest, but with universal start states.
+#[test]
+fn universal_start_search() -> Result<(), Box<dyn Error>> {
+    fn find<A: Automaton>(
+        dfa: &A,
+        haystack: &[u8],
+    ) -> Result<Option<HalfMatch>, MatchError> {
+        let mut state = dfa
+            .universal_start_state(Anchored::No)
+            .expect("regex should not require lookbehind");
+        let mut last_match = None;
+        // Walk all the bytes in the haystack. We can quit early if we see
+        // a dead or a quit state. The former means the automaton will
+        // never transition to any other state. The latter means that the
+        // automaton entered a condition in which its search failed.
+        for (i, &b) in haystack.iter().enumerate() {
+            state = dfa.next_state(state, b);
+            if dfa.is_special_state(state) {
+                if dfa.is_match_state(state) {
+                    last_match =
+                        Some(HalfMatch::new(dfa.match_pattern(state, 0), i));
+                } else if dfa.is_dead_state(state) {
+                    return Ok(last_match);
+                } else if dfa.is_quit_state(state) {
+                    // It is possible to enter into a quit state after
+                    // observing a match has occurred. In that case, we
+                    // should return the match instead of an error.
+                    if last_match.is_some() {
+                        return Ok(last_match);
+                    }
+                    return Err(MatchError::quit(b, i));
+                }
+                // Implementors may also want to check for start or accel
+                // states and handle them differently for performance
+                // reasons. But it is not necessary for correctness.
+            }
+        }
+        // Matches are always delayed by 1 byte, so we must explicitly walk
+        // the special "EOI" transition at the end of the search.
+        state = dfa.next_eoi_state(state);
+        if dfa.is_match_state(state) {
+            last_match = Some(HalfMatch::new(
+                dfa.match_pattern(state, 0),
+                haystack.len(),
+            ));
+        }
+        Ok(last_match)
+    }
+
+    fn check_impl(
+        dfa: impl Automaton,
+        haystack: &str,
+        pat: usize,
+        offset: usize,
+    ) -> Result<(), Box<dyn Error>> {
+        let haystack = haystack.as_bytes();
+        let mat = find(&dfa, haystack)?.unwrap();
+        assert_eq!(mat.pattern().as_usize(), pat);
+        assert_eq!(mat.offset(), offset);
+        Ok(())
+    }
+
+    fn check(
+        dfa: &dense::DFA<Vec<u32>>,
+        haystack: &str,
+        pat: usize,
+        offset: usize,
+    ) -> Result<(), Box<dyn Error>> {
+        check_impl(dfa, haystack, pat, offset)?;
+        check_impl(dfa.to_sparse()?, haystack, pat, offset)?;
+        Ok(())
+    }
+
+    let dfa = dense::DFA::new(r"[a-z]+")?;
+    let haystack = "123 foobar 4567";
+    check(&dfa, haystack, 0, 10)?;
+
+    let dfa = dense::DFA::new(r"[0-9]{4}")?;
+    let haystack = "123 foobar 4567";
+    check(&dfa, haystack, 0, 15)?;
+
+    let dfa = dense::DFA::new_many(&[r"[a-z]+", r"[0-9]+"])?;
+    let haystack = "123 foobar 4567";
+    check(&dfa, haystack, 1, 3)?;
+    check(&dfa, &haystack[3..], 0, 7)?;
+    check(&dfa, &haystack[10..], 1, 5)?;
+
+    Ok(())
+}