Largest Color Value in a Directed Graph

class Solution {
public:
    int explore(int vertex, string& nodeColors, vector<vector<int>>& graph, vector<vector<int>>& colorCount,
                vector<bool>& visited, vector<bool>& recursionStack) {
        if (recursionStack[vertex]) {
            return INT_MAX;
        }
        if (visited[vertex]) {
            return colorCount[vertex][nodeColors[vertex] - 'a'];
        }
        visited[vertex] = true;
        recursionStack[vertex] = true;

        for (int adjacent : graph[vertex]) {
            if (explore(adjacent, nodeColors, graph, colorCount, visited, recursionStack) == INT_MAX) {
                return INT_MAX;
            }
            for (int color = 0; color < 26; color++) {
                colorCount[vertex][color] = max(colorCount[vertex][color], colorCount[adjacent][color]);
            }
        }

        colorCount[vertex][nodeColors[vertex] - 'a']++;
        recursionStack[vertex] = false;
        return colorCount[vertex][nodeColors[vertex] - 'a'];
    }

    int maximalPathValue(string nodeColors, vector<vector<int>>& edges) {
        int nodes = nodeColors.size();
        vector<vector<int>> graph(nodes);
        for (const auto& edge : edges) {
            graph[edge[0]].push_back(edge[1]);
        }

        vector<vector<int>> colorCount(nodes, vector<int>(26));
        vector<bool> visited(nodes), recursionStack(nodes);
        int maxPathValue = 0;
        for (int v = 0; v < nodes; v++) {
            maxPathValue = max(maxPathValue, explore(v, nodeColors, graph, colorCount, visited, recursionStack));
        }

        return maxPathValue == INT_MAX ? -1 : maxPathValue;
    }
};

The code provided presents a C++ solution designed to determine the largest color value in a directed graph where each vertex has an assigned color. The challenge typically involves detecting cycles in the graph since a cycle may involve infinite loops of color increase, leading to an undefined maximum color value.

• The explore function serves multiple purposes:

  • It detects cycles using a recursionStack.
  • It carries out depth-first search (DFS) to compute the color counts by propagating maximum color counts from child vertices to parent vertices.
  • It manages both a visited array to keep track of visited nodes and colorCount, a two-dimensional vector that tracks color frequencies at each vertex.

• The function returns INT_MAX if a cycle is detected, signaling an indefinite maximum value. Otherwise, it computes and returns the maximum count of the current node’s color post DFS traversal.

• The maximalPathValue function initializes graph structures and loops through each vertex to perform the DFS function calls:

  • It initializes the graph based on given edges, filling the adjacency lists.
  • It initializes other structures necessary for DFS (visited, recursionStack, and colorCount).
  • As it executes DFS from each vertex, it determines the maximum color count obtained from all nodes that were successfully explored (without detecting a cycle).

• The solution evaluates whether the graph contains cycles affecting any node involved in the maximum color path. If any path resulted in the detection of a cycle (colorCount returns INT_MAX), the function returns -1 to indicate the graph's cyclic nature affecting the result. Otherwise, it outputs the highest color count.

This approach efficiently handles graph-based problems concerning path evaluations and cycle detection, accommodating up to 26 distinct colors represented by characters. The correct usage of DFS and cycle detection in graphs is critical in avoiding infinite loops and ensuring the integrity of the computation. The implementation is robust, catering to various graph structures while ensuring optimal performance with proper handling of edge cases, such as cycle detections.

class GraphUtils {
    private int traverse(int node, String paint, Map<Integer, List<Integer>> graph, int[][] tally,
                         boolean[] seen, boolean[] processing) {
        if (processing[node]) return Integer.MAX_VALUE;
        if (seen[node]) return tally[node][paint.charAt(node) - 'a'];
        
        seen[node] = true;
        processing[node] = true;

        if (graph.containsKey(node)) {
            for (int neighbor : graph.get(node)) {
                if (traverse(neighbor, paint, graph, tally, seen, processing) == Integer.MAX_VALUE) {
                    return Integer.MAX_VALUE;
                }
                for (int i = 0; i < 26; i++) {
                    tally[node][i] = Math.max(tally[node][i], tally[neighbor][i]);
                }
            }
        }

        tally[node][paint.charAt(node) - 'a']++;
        processing[node] = false;
        return tally[node][paint.charAt(node) - 'a'];
    }

    public int maxColorValue(String paint, int[][] edges) {
        int n = paint.length();
        Map<Integer, List<Integer>> graph = new HashMap<>();

        for (int[] edge : edges) {
            graph.computeIfAbsent(edge[0], k -> new ArrayList<>()).add(edge[1]);
        }

        int[][] tally = new int[n][26];
        boolean[] seen = new boolean[n];
        boolean[] processing = new boolean[n];
        int maxValue = 0;

        for (int i = 0; i < n; i++) {
            maxValue = Math.max(maxValue, traverse(i, paint, graph, tally, seen, processing));
        }

        return maxValue == Integer.MAX_VALUE ? -1 : maxValue;
    }
}

The provided Java program defines a method to find the largest color value in a directed graph where each node is painted with a color depicted by a character. The program operates in the following manner:

• Initialization: The maxColorValue method accepts a string, representing the colors of the nodes, and a 2D array of edges defining the directed graph. It prepares the graph map and other helper arrays to keep track of visited nodes, nodes in process, and color frequencies.

• Graph Construction: For each edge definition in the edges array, the graph's adjacency list is built using a HashMap, where each node points to its neighboring nodes.

• Cycle Detection & Tallying: The traverse method is recursively called for each node to detect cycles and to tally the count of each color up to that node. Cycle detection is managed using a 'processing' array to check if a node is visited during the current traversal. If a cycle is found (i.e., a node is re-visited before finishing its processing), the function returns Integer.MAX_VALUE.

• Color Counting: Each node increases the count of its color in the tally for itself after all its neighbors have been traversed. It furthers the maximum count of each color among its neighbors to reflect the longest path that reaches that node.

• Result Calculation: After iterating over all nodes, the program calculates the maximum value found in the tally arrays, ignoring cases where a cycle was detected (Integer.MAX_VALUE).

• Output: The maxColorValue function evaluates whether the maximum value is Integer.MAX_VALUE due to a cycle and, if so, returns -1. Otherwise, it returns the maximum color value found, representing the highest frequency of any color along any path in the graph.

This method efficiently handles the complexities of cycle detection and color tallying in directed graphs and provides a robust solution to determine the dominant color based on the paths within the graph.