Given a connected graph with N vertices and M edges. The task is to find the shortest path from source to the destination vertex such that the difference between adjacent edge weights in the shortest path change from positive to negative and vice versa ( Weight(E1) > Weight(E2) < Weight(E3) …. ). If no such path exists then print -1.
Examples:
Input: source = 4, destination = 3
Output: 19
4 – 2 – 1 – 3 (Edge Weights: 8, 3, 10) and 4 – 1 – 2 – 3 (Edge Weights: 6, 3, 10) are the only valid paths.
Second path takes the minimum cost i.e. 19.Input: source = 2, destination = 4
Output: -1
No such path exists.
Approach: Here, We need to keep two copies of adjacent lists one for positive difference and other for negative difference. Take a Priority Queue as in Dijkstras Algorithm and keep four variables at a time i.e.,
- cost: To store the cost of the path till current node.
- stage: An integer variable to tell what element needs to be taken next, if the previous value was negative then a positive value needs to be taken else take negative.
- weight: Weight of the last visited node.
- vertex: Last visited vertex.
For every vertex push the adjacent vertices based on the required condition (value of stage). See the code for better understanding.
Below is the implementation of the above approach:
// C++ implementation of the approach #include <bits/stdc++.h> using namespace std; #define N 100005 // To store the graph vector<pair<int, int> > incr[N], decr[N]; int _incr[N], _decr[N], shortest[N]; int n, m, src, dest, MAXI = 1LL << 30; // Function to add edges void Add_edge(int x, int y, int w) { incr[x].push_back({ w, y }); incr[y].push_back({ w, x }); decr[x].push_back({ -w, y }); decr[y].push_back({ -w, x }); } // Function to find the shortest distance from // source to destination int Modified_Dijkstra() { // Total cost, stage, weight of previous, vertex priority_queue<pair<pair<int, int>, pair<int, int> > > q; // Sort the edges for (int i = 1; i <= n; i++) { sort(incr[i].begin(), incr[i].end()); sort(decr[i].begin(), decr[i].end()); } for (int i = 1; i <= n; i++) shortest[i] = MAXI; // Push the source vertex q.push({ { 0, 0 }, { 0, src } }); while (!q.empty()) { // Take the top element in the queue pair<pair<int, int>, pair<int, int> > FRONT = q.top(); // Remove it from the queue q.pop(); // Store all the values int cost = -FRONT.first.first; int stage = FRONT.first.second; int weight = FRONT.second.first; int v = FRONT.second.second; // Take the minimum cost for the vertex shortest[v] = min(shortest[v], cost); // If destination vertex has already been visited if (shortest[dest] != MAXI) break; // To make difference negative if (stage) { // Start from last not visited vertex for (int i = _incr[v]; i < incr[v].size(); i++) { // If we can take the ith vertex if (weight > incr[v][i].first) q.push({ { -(cost + incr[v][i].first), 0 }, { incr[v][i].first, incr[v][i].second } }); else { // To keep the last not visited vertex _incr[v] = i; break; } } } // To make difference positive else { // Start from last not visited vertex for (int i = _decr[v]; i < decr[v].size(); i++) { // If we can take the ith vertex if (weight < -decr[v][i].first) q.push({ { -(cost - decr[v][i].first), 1 }, { -decr[v][i].first, decr[v][i].second } }); else { // To keep the last not visited vertex _decr[v] = i; break; } } } } if (shortest[dest] == MAXI) return -1; return shortest[dest]; } // Driver code int main() { n = 5, src = 4, dest = 3; // Adding edges Add_edge(4, 2, 8); Add_edge(1, 4, 6); Add_edge(2, 3, 10); Add_edge(3, 1, 10); Add_edge(1, 2, 3); Add_edge(3, 5, 3); cout << Modified_Dijkstra(); return 0; } |
19
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