Add graph colouring

dynamic_programming/egg_drop.py

@@ -0,0 +1,46 @@
+"""
+Egg Dropping Problem is a well-known problem in computer science.
+The task is to find the minimum number of attempts required in the
+worst case to find the highest floor from which an egg can be dropped
+without breaking, given a certain number of floors and eggs.
+
+Wikipedia: https://en.wikipedia.org/wiki/Dynamic_programming#Egg_dropping_puzzle
+"""
+
+
+def egg_drop(eggs: int, floors: int) -> int:
+    """
+    Calculate the minimum number of attempts required in the worst case
+    to determine the highest floor from which an egg can be dropped
+    without breaking it.
+
+    Parameters:
+    eggs (int): Number of eggs available.
+    floors (int): Number of floors to test.
+
+    Returns:
+    int: Minimum number of attempts required in the worst case.
+
+    Example:
+    >>> egg_drop(2, 10)
+    4
+
+    >>> egg_drop(3, 14)
+    4
+    """
+    # Initialize dp table with integers
+    dp = [[0 for _ in range(floors + 1)] for _ in range(eggs + 1)]
+
+    # Fill dp table for one egg (we have to try all floors)
+    for i in range(1, floors + 1):
+        dp[1][i] = i
+
+    # Fill the rest of the dp table
+    for e in range(2, eggs + 1):
+        for f in range(1, floors + 1):
+            dp[e][f] = floors + 1  # Start with an arbitrary large integer
+            for k in range(1, f + 1):
+                res = 1 + max(dp[e - 1][k - 1], dp[e][f - k])
+                dp[e][f] = min(dp[e][f], res)
+
+    return dp[eggs][floors]

graphs/graph_colouring.py

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+"""
+Graph Coloring Problem is a classic problem in graph theory.
+The task is to assign colors to the vertices of a graph so that no two adjacent vertices
+share the same color, and the number of colors used is minimized.
+
+Wikipedia: https://en.wikipedia.org/wiki/Graph_coloring
+"""
+
+
+def is_safe(
+    graph: list[list[int]], color: list[int], vertex: int, color_choice: int
+) -> bool:
+    """
+    Helper function to check if it is safe to color a vertex with a specific color.
+
+    Parameters:
+    graph (list[list[int]]): The adjacency matrix of the graph.
+    color (list[int]): The list of colors assigned to each vertex.
+    vertex (int): The vertex to check.
+    color_choice (int): The color to be assigned.
+
+    Returns:
+    bool: True if it's safe to assign the color, otherwise False.
+
+    Example:
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
+    >>> color = [-1, -1, -1]
+    >>> is_safe(graph, color, 0, 1)
+    True
+    """
+    return all(
+        not (graph[vertex][i] == 1 and color[i] == color_choice)
+        for i in range(len(graph))
+    )
+
+
+def graph_coloring_util(
+    graph: list[list[int]], num_colors: int, color: list[int], vertex: int
+) -> bool:
+    """
+    Utility function that uses backtracking to solve the m-coloring problem.
+
+    Parameters:
+    graph (list[list[int]]): The adjacency matrix of the graph.
+    num_colors (int): The maximum number of colors.
+    color (list[int]): The list of colors assigned to each vertex.
+    vertex (int): The current vertex to be colored.
+
+    Returns:
+    bool: True if all vertices are successfully colored, otherwise False.
+
+    Example:
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
+    >>> color = [-1, -1, -1]
+    >>> graph_coloring_util(graph, 3, color, 0)
+    True
+    """
+    if vertex == len(graph):
+        return True
+
+    for color_choice in range(1, num_colors + 1):
+        if is_safe(graph, color, vertex, color_choice):
+            color[vertex] = color_choice
+            if graph_coloring_util(graph, num_colors, color, vertex + 1):
+                return True
+            color[vertex] = -1  # Backtrack
+
+    return False
+
+
+def graph_coloring(graph: list[list[int]], num_colors: int) -> bool:
+    """
+    Solves the m-coloring problem using backtracking.
+
+    Parameters:
+    graph (list[list[int]]): The adjacency matrix of the graph.
+    num_colors (int): The maximum number of colors.
+
+    Returns:
+    bool: True if the graph can be colored with `num_colors` colors, otherwise False.
+
+    Example:
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
+    >>> graph_coloring(graph, 3)
+    True
+    """
+    color = [-1] * len(graph)
+    return graph_coloring_util(graph, num_colors, color, 0)