Shortest Path in a Binary Weight Graph

February 18, 2021

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;
     pq.erase(pq.begin());

     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];
```