#2203

Hard

LeetCode Problem Workspace

Minimum Weighted Subgraph With the Required Paths

Find the minimum weighted subgraph that connects three specified nodes in a directed graph with constraints.

Problem Statement

You are given an integer n representing the number of nodes in a weighted directed graph, with nodes numbered from 0 to n - 1. The graph also includes a list of directed edges, where each edge has a weight.

Additionally, you are given three distinct nodes: src1, src2, and dest. Your goal is to find the minimum weighted subgraph that contains a path from src1 to dest and from src2 to dest. If no such subgraph exists, return -1.

Examples

Example 1

Input: n = 6, edges = [[0,2,2],[0,5,6],[1,0,3],[1,4,5],[2,1,1],[2,3,3],[2,3,4],[3,4,2],[4,5,1]], src1 = 0, src2 = 1, dest = 5

Output: 9

The above figure represents the input graph. The blue edges represent one of the subgraphs that yield the optimal answer. Note that the subgraph [[1,0,3],[0,5,6]] also yields the optimal answer. It is not possible to get a subgraph with less weight satisfying all the constraints.

Example 2

Input: n = 3, edges = [[0,1,1],[2,1,1]], src1 = 0, src2 = 1, dest = 2

Output: -1

The above figure represents the input graph. It can be seen that there does not exist any path from node 1 to node 2, hence there are no subgraphs satisfying all the constraints.

Constraints

3 <= n <= 105
0 <= edges.length <= 105
edges[i].length == 3
0 <= fromi, toi, src1, src2, dest <= n - 1
fromi != toi
src1, src2, and dest are pairwise distinct.
1 <= weight[i] <= 105

Solution Approach

Shortest Path Calculations

Start by calculating the shortest path from src1 to dest and from src2 to dest using a shortest path algorithm such as Dijkstra's or Bellman-Ford. This ensures that the required paths are part of the solution.

Graph Filtering

Once the shortest paths are found, filter the graph to retain only the edges that are relevant to those paths. The edges that appear in both paths will form the subgraph you need to minimize in terms of weight.

Minimizing the Subgraph Weight

After filtering, calculate the total weight of the subgraph formed by the selected edges. If a valid subgraph exists, return its weight; otherwise, return -1.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the chosen approach. Using Dijkstra’s algorithm, the time complexity is O((E + V) log V), where E is the number of edges and V is the number of nodes. Space complexity can be O(E + V) depending on the graph representation.

What Interviewers Usually Probe

Candidates should discuss how to ensure that both paths (src1 to dest and src2 to dest) are captured in the subgraph.
Watch for optimization techniques in graph filtering, focusing on minimizing unnecessary edges.
Check if the candidate recognizes edge cases, such as no valid paths between the required nodes.

Common Pitfalls or Variants

Common pitfalls

Failing to consider edge cases where no path exists from src1 or src2 to dest.
Choosing the wrong shortest path algorithm for the given constraints.
Not properly minimizing the subgraph or including unnecessary edges in the solution.

Follow-up variants

Consider handling graphs with negative edge weights by using Bellman-Ford instead of Dijkstra’s algorithm.
Challenge candidates with larger graphs, such as increasing the number of nodes and edges, to test algorithm efficiency.
Introduce dynamic graph changes during the process to test how the solution adapts to updates.

FAQ

What is the main pattern to focus on for the 'Minimum Weighted Subgraph With the Required Paths' problem?

The problem combines graph traversal techniques with shortest path algorithms, requiring candidates to efficiently find the minimum weighted subgraph connecting three specific nodes.

Which algorithms are best for solving the 'Minimum Weighted Subgraph With the Required Paths' problem?

Dijkstra’s algorithm or Bellman-Ford are suitable for finding the shortest paths. The choice depends on the presence of negative edge weights in the graph.

What happens if no valid subgraph exists in the 'Minimum Weighted Subgraph With the Required Paths' problem?

If no valid subgraph exists, the correct solution is to return -1, as no paths from src1 to dest or src2 to dest are possible.

How do I optimize my solution for large graphs in the 'Minimum Weighted Subgraph With the Required Paths' problem?

Optimization can be achieved by using efficient shortest path algorithms like Dijkstra's with priority queues, reducing the complexity for large graphs.

What edge cases should I consider when solving the 'Minimum Weighted Subgraph With the Required Paths' problem?

Consider edge cases where no valid paths exist between the nodes, as well as situations where the graph is sparse or contains negative weight edges.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def minimumWeight(
        self, n: int, edges: List[List[int]], src1: int, src2: int, dest: int
    ) -> int:
        def dijkstra(g, u):
            dist = [inf] * n
            dist[u] = 0
            q = [(0, u)]
            while q:
                d, u = heappop(q)
                if d > dist[u]:
                    continue
                for v, w in g[u]:
                    if dist[v] > dist[u] + w:
                        dist[v] = dist[u] + w
                        heappush(q, (dist[v], v))
            return dist

        g = defaultdict(list)
        rg = defaultdict(list)
        for f, t, w in edges:
            g[f].append((t, w))
            rg[t].append((f, w))
        d1 = dijkstra(g, src1)
        d2 = dijkstra(g, src2)
        d3 = dijkstra(rg, dest)
        ans = min(sum(v) for v in zip(d1, d2, d3))
        return -1 if ans >= inf else ans

Continue Practicing

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