Shortest Paths with negative weights

Dynamic Programming

Pseudocode

// Shortest paths with negative edges
Shortest-Path(G, t) {
        foreach node v ∈ V
                M[0, v] <- infinity
        M[0, t] <- 0

        for i = 1 to n-1
                foreach node v ∈ V
                        M[i ,v] <- M[i-1, v]
                foreach edge(v,w) ∈ E
                        M[i,v] <- min{ M[i,v], M[i-1, w] + Cvw}
}
// big theta(mn) time big theta(n square) space
// Finding the shortest paths
// Maintain a "successor" for each table entry

Bellman-Ford 算法

//Bellman-Ford algorithm
d[s] <- 0
        for each v ∈ V-{s}
                do d[v] <- infinity
for i <- 1 to |V| - 1
        do for each edge(u, v) ∈ E
                do if d[v] > d[u] + w(u,v)
                       then d[v] <- d[u] + w(u,v)    // relaxation step

for each edge(u,v) ∈ E
        do if d[v] > d[u] + w(u,v)
                then report that a negative-weight cycle exists
At the end, d[v] = delta(s, v), if no negative-weight cycles
Time = O(VE)

Dynamic Programming Summary