Floyd-Warshall: All-Pairs Shortest Path

By wiki, Floyd-Warshall is a graph analysis algorithm for finding shortest paths in a weighted, directed graph. A single execution of the algorithm will find the shortest paths between all pairs of vertices. The Floyd–Warshall algorithm is an example of dynamic programming.

 1 /* Assume a function edgeCost(i,j) which returns the cost of the edge from i to j
2    (infinity if there is none).
3    Also assume that n is the number of vertices and edgeCost(i,i)=0
4 */
5
6 int path[][];
7 /* A 2-dimensional matrix. At each step in the algorithm, path[i][j] is the shortest path
8    from i to j using intermediate values in (1..k-1).  Each path[i][j] is initialized to
9    edgeCost(i,j).
10 */
11
12 procedure FloydWarshall ()
13    for k: = 0 to n − 1
14       for each (i,j) in (0..n − 1)
15          path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );

For numerically meaningful output, Floyd-Warshall assumes that there are no negative cycles (in fact, between any pair of vertices which form part of a negative cycle, the shortest path is not well-defined because the path can be infinitely small). Nevertheless, if there are negative cycles, Floyd–Warshall can be used to detect them. A negative cycle can be detected if the path matrix contains a negative number along the diagonal. If path[i][i] is negative for some vertex i, then this vertex belongs to at least one negative cycle.

Applications:

Shortest paths in directed graphs (Floyd's algorithm).

Transitive closure of directed graphs (Warshall's algorithm). In Warshall's original formulation of the algorithm, the graph is unweighted and represented by a Boolean adjacency matrix. Then the addition operation is replaced by logical conjunction (AND) and the minimum operation by logical disjunction (OR).

Optimal routing. In this application one is interested in finding the path with the maximum flow between two vertices. This means that, rather than taking minima as in the pseudocode above, one instead takes maxima. The edge weights represent fixed constraints on flow. Path weights represent bottlenecks; so the addition operation above is replaced by the minimum operation.

Testing whether an undirected graph is bipartite

Minmax / Maxmin Distance

Shortest Path Problems:

int Floyd_Warshall (int n) {
for(int k = 0; k < n; ++k)
 for(int i = 0; i < n; ++i)
   for(int j = 0; j < n; ++ j)
     G[i][j] = min(G[i][j], G[i][k] + G[k][j]);
}     

• topCoder: MoneyExchange
  
  
• UVa 104: Arbitrage
  
  
• UVa 436: Arbitrage (II)
  
  
• UVa 423: MPI Maelstrom
  
  
• UVa 567: Risk
  

Transitive Closure/Hull

for(int k = 0; k < N; ++k)
for(int i = 0; i < N; ++i)
 for(int j = 0; j < N; ++j)
   G[i][j] = G[i][j] || (G[i][k] && G[k][j]);

• UVa 334: Identifying concurrent events

Minmax/ Maxmin distance

void minmax(int n) {

for(int k = 0; k < n; ++k)
for(int i = 0; i < n; ++i)
for(int j = 0; j < n; ++j)
G[i][j] = min(G[i][j], max(G[i][k], G[k][j]));
}

void maxmin(int n) {
for(int k = 0; k < n; ++k)
for(int i = 0; i < n; ++i)
for(int j = 0; j < n; ++j)
G[i][j] = max(G[i][j], min(G[i][k], G[k][j]));
}



• UVa 534: Frogger
  
  
• UVa 10048: Audiophobia
  
  
• UVa 544: Heavy Cargo
  
  
• UVa 10099: The Tourist Guide