How can you implement a function to compute the longest path in a given DAG?

Question Analysis

To solve the problem of finding the longest path in a Directed Acyclic Graph (DAG), we need to understand a few key concepts:

1. DAG Characteristics: A DAG is a graph that is directed and has no cycles, meaning there is no way to start at a node and return to it by following the directed edges. This property ensures that there is a topological ordering of its vertices.

2. Longest Path Problem: Unlike in undirected graphs, where finding the longest path is NP-hard, in a DAG, it is feasible to do so efficiently due to the absence of cycles. The longest path in a DAG can be determined using topological sorting.

3. Graph Representation: The graph can be represented using adjacency lists or adjacency matrices.

4. Algorithm Approach: The problem typically involves performing a topological sort of the graph and then relaxing the edges in the order of the topological sort. This method ensures that we consider all possible paths in a DAG and determine the longest one.

Answer

To compute the longest path in a given DAG, we can use the following algorithmic approach:

1. Perform Topological Sorting:

   • Use a Depth-First Search (DFS) to perform topological sorting of the vertices. This will give us a linear ordering of vertices such that for every directed edge u -> v, vertex u comes before v in the ordering.

2. Initialize Distances:

   • Create an array dist to keep track of the longest distance from the source. Initialize all distances to negative infinity, except for the starting node, which should be set to 0.

3. Relax the Edges:

   • Process each vertex in topological order, and for each vertex u, update the distance to its adjacent vertices v using the formula:
     [
     \text{if } \text{dist}[v] < \text{dist}[u] + \text{weight}(u, v) \text{ then update } \text{dist}[v] = \text{dist}[u] + \text{weight}(u, v)
     ]

4. Extract the Longest Path:

   • The maximum value in the dist array represents the length of the longest path in the DAG.

Here is a sample Python function implementing the above algorithm:

def longestPathDAG(graph, start):
    from collections import defaultdict, deque

    # Step 1: Topological Sort using DFS
    def topologicalSort(v, visited, stack):
        visited[v] = True
        for neighbor, weight in graph[v]:
            if not visited[neighbor]:
                topologicalSort(neighbor, visited, stack)
        stack.append(v)

    # Step 2: Initialize distances and perform topological sorting
    num_vertices = len(graph)
    visited = [False] * num_vertices
    stack = []

    for i in range(num_vertices):
        if not visited[i]:
            topologicalSort(i, visited, stack)

    # Step 3: Initialize distances
    dist = [-float('inf')] * num_vertices
    dist[start] = 0

    # Step 4: Relax edges in topological order
    while stack:
        u = stack.pop()
        if dist[u] != -float('inf'):
            for v, weight in graph[u]:
                if dist[v] < dist[u] + weight:
                    dist[v] = dist[u] + weight

    # Step 5: Return the longest path
    return max(dist)

# Example usage:
# graph = {
#     0: [(1, 5), (2, 3)],
#     1: [(3, 6), (2, 2)],
#     2: [(4, 4), (5, 2), (3, 7)],
#     3: [(5, 1)],
#     4: [(5, -2)],
#     5: []
# }
# print(longestPathDAG(graph, 1))  # Example to find the longest path starting from node 1

This function computes the longest path length from a starting node in a DAG using topological sorting and edge relaxation.