UVic Programming Club

Shortest Path in a Binary Weight Graph

  1. Bridges
    Simple BFS does not work because there can be a longer path that has a smaller total weight.

    Method 1: 0-1 BFS (\(O(V + E)\))
    Use double ended queue (deque) to store a node
    Performing BFS, if an edge has weight=0, then push the node at the front of the deque; if an edge has weight=1, then push back
    While pushing to the deque, we update the distance from the current node to its neighbours (relax each neighbour), which is similar to Dijkstra.
     dist[neighbour] = min(dist[neighbour], dist[current] + edge_weight)

    Method 2: Dijkstra (\(O(VlogV + E)\))
    This is a more general algorithm to find the shortest path in a non-negative weighted graph.
    Set distance from the source to other vertices to \(\infty\) and dist[source]=0
    Use a priority queue (pq) to dynamically sort the pair (dist[vertex], vertext) by non-decreasing distance
    While pq is not empty, Dijkstra’s algorithm tries to relax each neighbour.

     // Reference: Competitive Programming 4
     vector<int> dist(n, INF);
     dist[0] = 0;
     set<pair<int, int>> pq;
     for (int i = 0; i < n; i++)
         pq.emplace(dist[i], i);
     while (!pq.empty()) {
         auto front = *pq.begin();
         int d = front.first, cur = front.second;
         for (auto item : graph[cur]) {
             int v = item.first, w = item.second;
             if (dist[cur] + w < dist[v]) {
                 pq.erase(pq.find({dist[v], v}));
                 dist[v] = dist[cur] + w;
                 pq.emplace(dist[v], v);
     cout << dist[n - 1];