2019-04-29/2019-05-01 Graphs-cont'd

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def Dijkstra(v, G[1..n]):
    # O(n)
    dist[1..n] filled with inf
    dist[v] = 0
    finalized[1..n] filled with False
    while something is not finalized:
        # O(n)
        u = not finalized with the smallest distance
        # O(1)
        finalized[u] = True
    # O(m)
        for (w, l) in G[u]:
            # O(1)
            dist[w] = min(dist[w], dist[u] + l)
    return dist

Complexity
O(n+1)n+O(n)+O(m) = O(n^2)

failed if negative edges

def bellmanford(v, G[1..n]):
    # O(n)
    dist[1..n] filled with inf
    dist[v] = 0
    for i from 1 to n-1:
        # O(n)
        ndist = dist.deepcopy()
        # O(n+m)
        for u from 1 to n:
            for (w,l) in G[u]:
                ndist[w] = min(ndist[w], dist[u]+l)
        # O(n)
        dist = ndist.deepcopy()
    return dist

Complexity
O(n) + n(O(m+n)) = O(nm+n^2)

Floyd-Warshall
In: Graph G
Out: dist[1..n][1..n]
dist[u][w] from u to w

Look[k,u,v] = # the length of the shortest path from u to w s.t the inner vertices on the path are from 1 to k (path中间的vertices)

Look[n,,] -- answer
Look[0,u,v] = l if uv are neighbors or inf otherwise

Look[k+1,u,v] = min(Look[k,u,v], Look[k,u,k+1]+Look[k,k+1,v])