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;
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];