Improve crossword puzzle solver implementation (#14568)

* Improve crossword puzzle solver implementation

The original `is_valid` returned `False` for any non-empty cell, which meant two words sharing a letter could never be placed together. Since crossword grids are built on exactly those intersections, this broke the core use case. Related to that, `remove_word` during backtracking blindly blanked every cell of the removed word, wiping out letters that belonged to already-placed crossing words. The fix snapshots the grid before placement and only clears cells that were empty beforehand. There was also a mutation bug: `words.remove(word)` does an O(n) scan and modifies state shared across the call stack, replaced here with a local slice per frame. On top of the fixes, the solver now tries the longest word first at each level, a standard "most-constrained variable" heuristic that cuts down backtracks significantly on larger inputs.

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* Refactor character check in crossword solver

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

Author: 3 people

backtracking/crossword_puzzle_solver.py

@@ -6,29 +6,28 @@ def is_valid(
 ) -> bool:
     """
     Check if a word can be placed at the given position.
+    A cell is valid if it is empty or already contains the correct letter
+    (enabling crossing/intersection between words).
 
-    >>> puzzle = [
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', '']
-    ... ]
+    >>> puzzle = [['', '', '', ''], ['', '', '', ''],
+    ...           ['', '', '', ''], ['', '', '', '']]
     >>> is_valid(puzzle, 'word', 0, 0, True)
     True
-    >>> puzzle = [
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', '']
-    ... ]
     >>> is_valid(puzzle, 'word', 0, 0, False)
     True
+    >>> puzzle2 = [['w', '', ''], ['o', '', ''], ['r', '', ''], ['d', '', '']]
+    >>> is_valid(puzzle2, 'word', 0, 0, True)
+    True
+    >>> is_valid(puzzle2, 'cat', 0, 0, True)
+    False
     """
-    for i in range(len(word)):
-        if vertical:
-            if row + i >= len(puzzle) or puzzle[row + i][col] != "":
-                return False
-        elif col + i >= len(puzzle[0]) or puzzle[row][col + i] != "":
+    rows, cols = len(puzzle), len(puzzle[0])
+    for i, ch in enumerate(word):
+        r, c = (row + i, col) if vertical else (row, col + i)
+        if r >= rows or c >= cols:
+            return False
+        cell = puzzle[r][c]
+        if cell not in ("", ch):
             return False
     return True
 
@@ -37,95 +36,87 @@ def place_word(
     puzzle: list[list[str]], word: str, row: int, col: int, vertical: bool
 ) -> None:
     """
-    Place a word at the given position.
-
-    >>> puzzle = [
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', '']
-    ... ]
+    Place a word at the given position in the puzzle.
+
+    >>> puzzle = [['', '', '', ''], ['', '', '', ''],
+    ...           ['', '', '', ''], ['', '', '', '']]
     >>> place_word(puzzle, 'word', 0, 0, True)
     >>> puzzle
     [['w', '', '', ''], ['o', '', '', ''], ['r', '', '', ''], ['d', '', '', '']]
     """
-    for i, char in enumerate(word):
+    for i, ch in enumerate(word):
         if vertical:
-            puzzle[row + i][col] = char
+            puzzle[row + i][col] = ch
         else:
-            puzzle[row][col + i] = char
+            puzzle[row][col + i] = ch
 
 
 def remove_word(
-    puzzle: list[list[str]], word: str, row: int, col: int, vertical: bool
+    puzzle: list[list[str]],
+    word: str,
+    row: int,
+    col: int,
+    vertical: bool,
+    snapshot: list[list[str]],
 ) -> None:
     """
-    Remove a word from the given position.
-
-    >>> puzzle = [
-    ...     ['w', '', '', ''],
-    ...     ['o', '', '', ''],
-    ...     ['r', '', '', ''],
-    ...     ['d', '', '', '']
-    ... ]
-    >>> remove_word(puzzle, 'word', 0, 0, True)
+    Remove a word from the puzzle, restoring only cells that were empty
+    before placement. Cells shared with crossing words are preserved.
+
+    >>> puzzle = [['w', 'o', 'r', 'd'], ['', '', '', ''],
+    ...           ['', '', '', ''], ['', '', '', '']]
+    >>> snap = [['', 'o', 'r', 'd'], ['', '', '', ''],
+    ...         ['', '', '', ''], ['', '', '', '']]
+    >>> remove_word(puzzle, 'word', 0, 0, False, snap)
     >>> puzzle
-    [['', '', '', ''], ['', '', '', ''], ['', '', '', ''], ['', '', '', '']]
+    [['', 'o', 'r', 'd'], ['', '', '', ''], ['', '', '', ''], ['', '', '', '']]
     """
     for i in range(len(word)):
-        if vertical:
-            puzzle[row + i][col] = ""
-        else:
-            puzzle[row][col + i] = ""
+        r, c = (row + i, col) if vertical else (row, col + i)
+        if snapshot[r][c] == "":
+            puzzle[r][c] = ""
 
 
 def solve_crossword(puzzle: list[list[str]], words: list[str]) -> bool:
     """
     Solve the crossword puzzle using backtracking.
+    Words are tried longest-first to prune the search space early.
+    Intersections between words (shared letters) are supported.
 
-    >>> puzzle = [
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', '']
-    ... ]
-
-    >>> words = ['word', 'four', 'more', 'last']
-    >>> solve_crossword(puzzle, words)
+    >>> puzzle = [['', '', '', ''], ['', '', '', ''],
+    ...           ['', '', '', ''], ['', '', '', '']]
+    >>> solve_crossword(puzzle, ['word', 'four', 'more', 'last'])
     True
-    >>> puzzle = [
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', ''],
-    ...     ['', '', '', '']
-    ... ]
-    >>> words = ['word', 'four', 'more', 'paragraphs']
-    >>> solve_crossword(puzzle, words)
+    >>> puzzle2 = [['', '', '', ''], ['', '', '', ''],
+    ...            ['', '', '', ''], ['', '', '', '']]
+    >>> solve_crossword(puzzle2, ['word', 'four', 'more', 'paragraphs'])
     False
     """
+    if not words:
+        return True
+
+    remaining = sorted(words, key=len, reverse=True)
+    word, rest = remaining[0], remaining[1:]
+
     for row in range(len(puzzle)):
         for col in range(len(puzzle[0])):
-            if puzzle[row][col] == "":
-                for word in words:
-                    for vertical in [True, False]:
-                        if is_valid(puzzle, word, row, col, vertical):
-                            place_word(puzzle, word, row, col, vertical)
-                            words.remove(word)
-                            if solve_crossword(puzzle, words):
-                                return True
-                            words.append(word)
-                            remove_word(puzzle, word, row, col, vertical)
-                return False
-    return True
+            for vertical in (True, False):
+                if is_valid(puzzle, word, row, col, vertical):
+                    snapshot = [r[:] for r in puzzle]
+                    place_word(puzzle, word, row, col, vertical)
+                    if solve_crossword(puzzle, rest):
+                        return True
+                    remove_word(puzzle, word, row, col, vertical, snapshot)
+
+    return False
 
 
 if __name__ == "__main__":
     PUZZLE = [[""] * 3 for _ in range(3)]
     WORDS = ["cat", "dog", "car"]
-
     if solve_crossword(PUZZLE, WORDS):
         print("Solution found:")
         for row in PUZZLE:
-            print(" ".join(row))
+            print(" ".join(cell or "." for cell in row))
     else:
-        print("No solution found:")
+        print("No solution found.")