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2714. Find Shortest Path with K Hops πŸ”’

Description

You are given a positive integer n which is the number of nodes of a 0-indexed undirected weighted connected graph and a 0-indexed 2D array edges where edges[i] = [ui, vi, wi] indicates that there is an edge between nodes ui and vi with weight wi.

You are also given two nodes s and d, and a positive integer k, your task is to find the shortest path from s to d, but you can hop over at most k edges. In other words, make the weight of at most k edges 0 and then find the shortest path from s to d.

Return the length of the shortest path from s to d with the given condition.

 

Example 1:

Input: n = 4, edges = [[0,1,4],[0,2,2],[2,3,6]], s = 1, d = 3, k = 2
Output: 2
Explanation: In this example there is only one path from node 1 (the green node) to node 3 (the red node), which is (1->0->2->3) and the length of it is 4 + 2 + 6 = 12. Now we can make weight of two edges 0, we make weight of the blue edges 0, then we have 0 + 2 + 0 = 2. It can be shown that 2 is the minimum length of a path we can achieve with the given condition.

Example 2:

Input: n = 7, edges = [[3,1,9],[3,2,4],[4,0,9],[0,5,6],[3,6,2],[6,0,4],[1,2,4]], s = 4, d = 1, k = 2
Output: 6
Explanation: In this example there are 2 paths from node 4 (the green node) to node 1 (the red node), which are (4->0->6->3->2->1) and (4->0->6->3->1). The first one has the length 9 + 4 + 2 + 4 + 4 = 23, and the second one has the length 9 + 4 + 2 + 9 = 24. Now if we make weight of the blue edges 0, we get the shortest path with the length 0 + 4 + 2 + 0 = 6. It can be shown that 6 is the minimum length of a path we can achieve with the given condition.

Example 3:

Input: n = 5, edges = [[0,4,2],[0,1,3],[0,2,1],[2,1,4],[1,3,4],[3,4,7]], s = 2, d = 3, k = 1
Output: 3
Explanation: In this example there are 4 paths from node 2 (the green node) to node 3 (the red node), which are (2->1->3), (2->0->1->3), (2->1->0->4->3) and (2->0->4->3). The first two have the length 4 + 4 = 1 + 3 + 4 = 8, the third one has the length 4 + 3 + 2 + 7 = 16 and the last one has the length 1 + 2 + 7 = 10. Now if we make weight of the blue edge 0, we get the shortest path with the length 1 + 2 + 0 = 3. It can be shown that 3 is the minimum length of a path we can achieve with the given condition.

 

Constraints:

  • 2 <= n <= 500
  • n - 1 <= edges.length <= min(104, n * (n - 1) / 2)
  • edges[i].length = 3
  • 0 <= edges[i][0], edges[i][1] <= n - 1
  • 1 <= edges[i][2] <= 106
  • 0 <= s, d, k <= n - 1
  • s != d
  • The input is generated such that the graph is connected and has no repeated edges or self-loops

Solutions

Solution 1: Dijkstra Algorithm

First, we construct a graph $g$ based on the given edges, where $g[u]$ represents all neighboring nodes of node $u$ and their corresponding edge weights.

Then, we use Dijkstra's algorithm to find the shortest path from node $s$ to node $d$. However, we need to make some modifications to Dijkstra's algorithm:

  • We need to record the shortest path length from each node $u$ to node $d$, but since we can cross at most $k$ edges, we need to record the shortest path length from each node $u$ to node $d$ and the number of edges crossed $t$, i.e., $dist[u][t]$ represents the shortest path length from node $u$ to node $d$ and the number of edges crossed is $t$.
  • We need to use a priority queue to maintain the current shortest path, but since we need to record the number of edges crossed, we need to use a triple $(dis, u, t)$ to represent the current shortest path, where $dis$ represents the current shortest path length, and $u$ and $t$ represent the current node and the number of edges crossed, respectively.

Finally, we only need to return the minimum value in $dist[d][0..k]$.

The time complexity is $O(n^2 \times \log n)$, and the space complexity is $O(n \times k)$, where $n$ represents the number of nodes and $k$ represents the maximum number of edges crossed.

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class Solution:
    def shortestPathWithHops(
        self, n: int, edges: List[List[int]], s: int, d: int, k: int
    ) -> int:
        g = [[] for _ in range(n)]
        for u, v, w in edges:
            g[u].append((v, w))
            g[v].append((u, w))
        dist = [[inf] * (k + 1) for _ in range(n)]
        dist[s][0] = 0
        pq = [(0, s, 0)]
        while pq:
            dis, u, t = heappop(pq)
            for v, w in g[u]:
                if t + 1 <= k and dist[v][t + 1] > dis:
                    dist[v][t + 1] = dis
                    heappush(pq, (dis, v, t + 1))
                if dist[v][t] > dis + w:
                    dist[v][t] = dis + w
                    heappush(pq, (dis + w, v, t))
        return int(min(dist[d]))

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class Solution {
    public int shortestPathWithHops(int n, int[][] edges, int s, int d, int k) {
        List<int[]>[] g = new List[n];
        Arrays.setAll(g, i -> new ArrayList<>());
        for (int[] e : edges) {
            int u = e[0], v = e[1], w = e[2];
            g[u].add(new int[] {v, w});
            g[v].add(new int[] {u, w});
        }
        PriorityQueue<int[]> pq = new PriorityQueue<>((a, b) -> a[0] - b[0]);
        pq.offer(new int[] {0, s, 0});
        int[][] dist = new int[n][k + 1];
        final int inf = 1 << 30;
        for (int[] e : dist) {
            Arrays.fill(e, inf);
        }
        dist[s][0] = 0;
        while (!pq.isEmpty()) {
            int[] p =