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Largest Color Value in a Directed Graph

Difficulty: Hard

Problem Description

There is a directed graph of n colored nodes and m edges. The nodes are numbered from 0 to n - 1. You are given a string colors where colors[i] is a lowercase English letter representing the color of the i-th node in this graph (0-indexed). You are also given a 2D array edges where edges[j] = [aj, bj] indicates that there is a directed edge from node aj to node bj. A valid path in the graph is a sequence of nodes x1 -> x2 -> x3 -> ... -> xk such that there is a directed edge from xi to xi+1 for every 1 <= i < k. The color value of the path is the number of nodes that are colored the most frequently occurring color along that path. Return the largest color value of any valid path in the given graph, or -1 if the graph contains a cycle.

Key Insights

  • We need to identify the largest color value along valid paths in a directed graph.
  • The problem becomes complex due to the presence of cycles, which must be detected.
  • Topological sorting can be employed to handle the directed acyclic graph (DAG) part.
  • We can use a dynamic programming approach to calculate the maximum color value along paths.

Space and Time Complexity

Time Complexity: O(n + m) - where n is the number of nodes and m is the number of edges, as we traverse the graph to build adjacency lists and perform topological sorting.

Space Complexity: O(n + m) - for storing the graph representation (adjacency list) and additional structures for tracking in-degrees and color values.

Solution

To solve this problem, we can use the following approach:

  1. Build an adjacency list from the edge list to represent the directed graph.
  2. Use Kahn's algorithm for topological sorting to detect cycles in the graph. If any nodes remain with non-zero in-degrees after processing, a cycle exists.
  3. While performing the topological sort, maintain a count of the color occurrences for each node.
  4. For each node processed, update the color counts for its neighbors based on the current path.
  5. The maximum color value across all nodes will yield the result.

Code Solutions

from collections import defaultdict, deque

def largestColorValue(colors, edges):
    n = len(colors)
    graph = defaultdict(list)
    in_degree = [0] * n

    # Build the graph and calculate in-degrees
    for u, v in edges:
        graph[u].append(v)
        in_degree[v] += 1

    # Topological sorting using Kahn's algorithm
    queue = deque()
    for i in range(n):
        if in_degree[i] == 0:
            queue.append(i)

    color_count = [[0] * 26 for _ in range(n)]  # Count of each color

    max_color_value = 0
    processed_nodes = 0

    while queue:
        node = queue.popleft()
        processed_nodes += 1

        # Update color counts for the current node
        color_index = ord(colors[node]) - ord('a')
        for i in range(26):
            color_count[node][i] = (color_count[node][i] + (1 if i == color_index else 0))

        # Propagate to neighbors
        for neighbor in graph[node]:
            for i in range(26):
                color_count[neighbor][i] = max(color_count[neighbor][i], color_count[node][i])

            in_degree[neighbor] -= 1
            if in_degree[neighbor] == 0:
                queue.append(neighbor)

    # Check if there's a cycle
    if processed_nodes != n:
        return -1

    # Find the maximum color value
    for i in range(n):
        max_color_value = max(max_color_value, max(color_count[i]))

    return max_color_value
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