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Algorithms -- Paths in graphs

Distances

The distance between two nodes is the length of the shortest path them

Different from DFS, the BFS aims at the shortest path which can be backward constructed if we record the prev node.

procedure bfs(G,s){
// Input: Graph G = (V, E), directed or undirected; From vertex s in V
// Output: For all vertices u reachable from s, dist(u) is the distance from s to u
for all u in V: dist(u) = \infty

dist(s) = 0;
Q = {s};
while Q is not empty:
     u = pop(Q)
     for all edges (u,v) in E:
         if dist(v) = \infty 
              push(Q, v);
              dist(v) = dist(u) + 1;

The queue is FILO

Dijkstra's Algorithm

Assuming the length or weigh is give on edges. We need to deal with it when we try to figure out the shortest route from the source to specific node.

procedure dijkstra(G, l, s){
for all u in V:
    dist(u) = \infty;
    prev(u) = null;

dist(s) = 0;

H = makequeue(V)    \\using dist-values a keys
while H is not empty:
    H = popMin(H);
    for all edges (u,v) in E:
        if dist(v) > dist(u) + l(u,v):
            dist(v) = dist(u) + l(u,v);
            prev(v) = u;
            decreaseKey(H, v);

However, here is an assumption: There is no negative edge in E

Priority queue

The priority queue as a data structure should support such operations:

  1. Insert
  2. getMin
  3. decreaseKey

And there are several implementations:

  1. Array (with sort method)
  2. Binary Heap (it can be realized as an array)
  3. D-ary Heap ( the amount of children is d rather than original 2)
  4. Fibonacci Heap (a little sophisticated)

Bellman-Ford Algorithm

For those containing negative edges, we can update very frequently

procedure update( (u,v) in E)
    dist(v) = min( dist(v), dist(u) + l(u,v) );

procedure bellman-ford(G,l,s){
    for all u in V:
        dist(u) = \infty
        prev(u) = null

    dist(s) = 0;

    repeat |V| -1 times:
        for all e in E:
            update(e)

And actually, for DAG, we can solve the problem in linear time. The difference is that we need to linearize G before updating, and the order of updating is in linearized order.