1857. Largest Color Value in a Directed Graph

Introduction

In this tutorial, we will explore a problem involving a directed graph composed of colored nodes. Our goal is to determine the largest color value along any valid path in the graph. A valid path is defined as a sequence of nodes connected by directed edges. We will also address the challenge of detecting cycles in the graph, which can complicate our calculations.

Prerequisites

• Basic understanding of graphs and directed graphs
• Familiarity with dynamic programming concepts
• Knowledge of topological sorting and its applications

Problem Statement

We are given a directed graph with n colored nodes and m edges. Each node is represented by an index from 0 to n - 1. The color of each node is represented by a string colors, where colors[i] is a lowercase English letter indicating the color of the ith node.

The edges are represented by a 2D array edges, where edges[j] = [aj, bj] indicates a directed edge from node aj to node bj.

Our task is to return the largest color value of any valid path in the graph, or -1 if the graph contains a cycle.

Examples

Example 1:

• Input: colors = “abaca”, edges = [[0,1],[0,2],[2,3],[3,4]]
• Output: 3
• Explanation: The path 0 -> 2 -> 3 -> 4 contains 3 nodes that are colored "a".

Example 2:

• Input: colors = “a”, edges = [[0,0]]
• Output: -1
• Explanation: There is a cycle from 0 to 0.

Constraints

• n == colors.length
• m == edges.length
• 1 <= n <= 10^5
• 0 <= m <= 10^5
• colors consists of lowercase English letters.
• 0 <= aj, bj < n

Approach

To solve this problem, we will use a combination of topological sorting and dynamic programming. Here’s a step-by-step breakdown of our approach:

1. Cycle Detection and Topological Sort: We will use Kahn’s algorithm to perform a topological sort. If the number of processed nodes is less than the total number of nodes, it indicates that the graph contains a cycle, and we will return -1.
2. Dynamic Programming (DP): For each node processed in topological order, we will maintain a DP array where dp[u][c] represents the maximum count of color c in any path ending at node u.
3. Reverse Adjacency List: We will track predecessors for each node to efficiently update DP values based on incoming edges.
4. Color Count Update: For each node, we will initialize its DP array with its own color count. Then, we will update the DP values by considering all paths from predecessors, incrementing the count if the current node’s color matches.

Implementation

Let’s implement this solution in PHP:

<?php
/**
 * @param String $colors
 * @param Integer[][] $edges
 * @return Integer
 */
function largestPathValue($colors, $edges) {
    // Implementation goes here
}

$colors = "abaca";
$edges = [[0,1],[0,2],[2,3],[3,4]];
echo largestPathValue($colors, $edges);  // Output: 3

$colors = "a";
$edges = [[0,0]];
echo largestPathValue($colors, $edges);  // Output: -1
?>

Explanation

1. Topological Sorting: Using Kahn’s algorithm, we process nodes in a topological order. This ensures that each node is processed only after all its predecessors have been processed, which is crucial for correctly updating DP values.
2. Cycle Detection: If the topological sort processes fewer nodes than the total nodes, a cycle exists, and we return -1.
3. DP Initialization: Each node starts with a DP array initialized to 0 except for its own color, which is set to 1.
4. Updating DP Values: For each node, we update its DP array by considering all incoming edges (predecessors). For each predecessor, we check all color counts and update the current node’s DP values to reflect the maximum possible counts from all paths ending at the current node.
5. Result Calculation: The maximum value in the DP array for each node is tracked to determine the largest color value in any valid path.

This approach efficiently combines topological sorting and dynamic programming to solve the problem in linear time relative to the number of edges and nodes, making it suitable for large input sizes.

Conclusion

In this tutorial, we learned how to find the largest color value in a directed graph using dynamic programming and topological sorting. We also discussed how to detect cycles in the graph, which is crucial for ensuring the validity of our results. By following this approach, you can tackle similar problems involving graphs and paths.

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