Maximum Weighted K-Edge Path - Solution & Explanation

Problem Overview: You are given a weighted graph and an integer k. The task is to compute the maximum possible total weight of a path that uses exactly k edges. Nodes can appear multiple times unless restricted by the problem statement, so the focus is purely on maximizing weight while respecting the edge count constraint.

Approach 1: Brute Force DFS Enumeration (Exponential Time)

The most direct idea is to start a depth‑first search from every node and explore all paths of length k. At each step, recursively visit every outgoing edge while tracking the current weight and number of edges used. When the recursion depth reaches k, update the global maximum. This approach touches every possible path combination and quickly explodes in size as the branching factor grows. Time complexity is roughly O(V * d^k) where d is the average degree, and space complexity is O(k) for the recursion stack. It demonstrates the problem structure but is impractical for larger graphs.

Approach 2: Dynamic Programming on Edge Count (O(K * E))

A more scalable solution uses dynamic programming over the number of edges used. Define dp[i][v] as the maximum weight achievable when reaching node v using exactly i edges. Initialize the base case for zero edges, then iteratively build results for 1..k. For each iteration, scan every edge (u → v, w) and update dp[i][v] = max(dp[i][v], dp[i-1][u] + w). This resembles the relaxation step in Bellman‑Ford but limited to exactly k steps. The algorithm processes each edge for each edge-count layer, giving O(K * E) time and O(K * V) space.

To reduce memory, you can keep only two layers: the previous edge count and the current one. That drops space to O(V) while maintaining the same runtime. The graph itself is usually represented with adjacency lists from a graph structure, and intermediate values can be stored in arrays or a hash table if node IDs are sparse.

Recommended for interviews: The dynamic programming approach with edge-count transitions is what interviewers typically expect. Brute force DFS shows you understand the path enumeration aspect, but the DP formulation demonstrates algorithmic maturity. The key insight is recognizing that the number of edges used forms a natural DP dimension, allowing you to reuse results from k‑1 edges instead of recomputing paths repeatedly.