feat(backtracking): add m-coloring algorithm with type hints and doctests

backtracking/m_coloring_problem.py

@@ -0,0 +1,48 @@
+from typing import List
+
+
+def is_safe(
+    node: int, color: int, graph: List[List[int]], n: int, col: List[int]
+) -> bool:
+    """
+    Check if it is safe to assign a color to a node.
+
+    >>> is_safe(0, 1, [[0,1],[1,0]], 2, [0,1])
+    False
+    >>> is_safe(0, 2, [[0,1],[1,0]], 2, [0,1])
+    True
+    """
+    return all(not (graph[node][k] == 1 and col[k] == color) for k in range(n))
+
+
+def solve(node: int, col: List[int], m: int, n: int, graph: List[List[int]]) -> bool:
+    """
+    Recursively try to color the graph using at most m colors.
+
+    >>> solve(0, [0]*3, 3, 3, [[0,1,0],[1,0,1],[0,1,0]])
+    True
+    >>> solve(0, [0]*3, 2, 3, [[0,1,0],[1,0,1],[0,1,0]])
+    False
+    """
+    if node == n:
+        return True
+    for c in range(1, m + 1):
+        if is_safe(node, c, graph, n, col):
+            col[node] = c
+            if solve(node + 1, col, m, n, graph):
+                return True
+            col[node] = 0
+    return False
+
+
+def graph_coloring(graph: List[List[int]], m: int, n: int) -> bool:
+    """
+    Determine if the graph can be colored with at most m colors.
+
+    >>> graph_coloring([[0,1,1],[1,0,1],[1,1,0]], 3, 3)
+    True
+    >>> graph_coloring([[0,1,1],[1,0,1],[1,1,0]], 2, 3)
+    False
+    """
+    col = [0] * n
+    return solve(0, col, m, n, graph)