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Syncleus Wiki - Floyd-Warshall Algorithm

Floyd-Warshall Algorithm

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Floyd–Warshall algorithm
Class All-pairs shortest path problem (for weighted graphs)
Data structure Graph
Worst case performance O(|V|^3)
Best case performance \Omega (|V|^3)
Worst case space complexity \Theta(|V|^2)

In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, Roy–Warshall algorithm, Roy–Floyd algorithm, or the WFI algorithm) is a graph analysis algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles, see below) and also for finding transitive closure of a relation R. A single execution of the algorithm will find the lengths (summed weights) of the shortest paths between all pairs of vertices, though it does not return details of the paths themselves.

The Floyd–Warshall algorithm was published in its currently recognized form by Robert Floyd in 1962. However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 and also by Stephen Warshall in 1962 for finding the transitive closure of a graph.[1] The modern formulation of Warshall's algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.

The algorithm is an example of dynamic programming.

Algorithm

The Floyd–Warshall algorithm compares all possible paths through the graph between each pair of vertices. It is able to do this with Θ(|V|3) comparisons in a graph. This is remarkable considering that there may be up to Ω(|V|2) edges in the graph, and every combination of edges is tested. It does so by incrementally improving an estimate on the shortest path between two vertices, until the estimate is optimal.

Consider a graph G with vertices V numbered 1 through N. Further consider a function shortestPath(ijk) that returns the shortest possible path from i to j using vertices only from the set {1,2,...,k} as intermediate points along the way. Now, given this function, our goal is to find the shortest path from each i to each j using only vertices 1 to k + 1.

For each of these pairs of vertices, the true shortest path could be either (1) a path that only uses vertices in the set {1, ..., k} or (2) a path that goes from i to k + 1 and then from k + 1 to j. We know that the best path from i to j that only uses vertices 1 through k is defined by shortestPath(ijk), and it is clear that if there were a better path from i to k + 1 to j, then the length of this path would be the concatenation of the shortest path from i to k + 1 (using vertices in {1, ..., k}) and the shortest path from k + 1 to j (also using vertices in {1, ..., k}).

If w(i, j) is the weight of the edge between vertices i and j, we can define shortestPath(ijk + 1) in terms of the following recursive formula: the base case is

\textrm{shortestPath}(i, j, 0) = w(i, j)

and the recursive case is

\textrm{shortestPath}(i,j,k+1) = \min(\textrm{shortestPath}(i,j,k),\,\textrm{shortestPath}(i,k+1,k) + \textrm{shortestPath}(k+1,j,k))

This formula is the heart of the Floyd–Warshall algorithm. The algorithm works by first computing shortestPath(ijk) for all (ij) pairs for k = 1, then k = 2, etc. This process continues until k = n, and we have found the shortest path for all (ij) pairs using any intermediate vertices. Pseudocode for this basic version follows:

1 let dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)
2 for each vertex v
3    dist[v][v] ← 0
4 for each edge (u,v)
5    dist[u][v] ← w(u,v)  // the weight of the edge (u,v)
6 for k from 1 to |V|
7    for i from 1 to |V|
8       for j from 1 to |V|
9          if dist[i][j] > dist[i][k] + dist[k][j] 
10             dist[i][j] ← dist[i][k] + dist[k][j]
11         end if

Example

The algorithm above is executed on the graph on the left below:

Floyd-Warshall example.svg

Prior to the first iteration of the outer loop, labeled k=0 above, the only known paths correspond to the single edges in the graph. At k=1, paths that go through the vertex 1 are found: in particular, the path 2→1→3 is found, replacing the path 2→3 which has fewer edges but is longer. At k=2, paths going through the vertices {1,2} are found. The red and blue boxes show how the path 4→2→1→3 is assembled from the two known paths 4→2 and 2→1→3 encountered in previous iterations, with 2 in the intersection. The path 4→2→3 is not considered, because 2→1→3 is the shortest path encountered so far from 2 to 3. At k=3, paths going through the vertices {1,2,3} are found. Finally, at k=4, all shortest paths are found.

Behavior with negative cycles

A negative cycle is a cycle whose edges sum to a negative value. There is no shortest path between any pair of vertices i, j which form part of a negative cycle, because path-lengths from i to j can be arbitrarily small (negative). For numerically meaningful output, the Floyd–Warshall algorithm assumes that there are no negative cycles. Nevertheless, if there are negative cycles, the Floyd–Warshall algorithm can be used to detect them. The intuition is as follows:

  • The Floyd–Warshall algorithm iteratively revises path lengths between all pairs of vertices (ij), including where i = j;
  • Initially, the length of the path (i,i) is zero;
  • A path {(i,k), (k,i)} can only improve upon this if it has length less than zero, i.e. denotes a negative cycle;
  • Thus, after the algorithm, (i,i) will be negative if there exists a negative-length path from i back to i.

Hence, to detect negative cycles using the Floyd–Warshall algorithm, one can inspect the diagonal of the path matrix, and the presence of a negative number indicates that the graph contains at least one negative cycle.[2] Obviously, in an undirected graph a negative edge creates a negative cycle (i.e., a closed walk) involving its incident vertices.

Path reconstruction

The Floyd–Warshall algorithm typically only provides the lengths of the paths between all pairs of vertices. With simple modifications, it is possible to create a method to reconstruct the actual path between any two endpoint vertices. While one may be inclined to store the actual path from each vertex to each other vertex, this is not necessary, and in fact, is very costly in terms of memory. For each vertex, one need only store the information about the highest index intermediate vertex one must pass through if one wishes to arrive at any given vertex. Therefore, information to reconstruct all paths can be stored in a single |V| × |V| matrix next where next[i][j] represents the highest index vertex one must travel through if one intends to take the shortest path from i to j.

To implement this, when a new shortest path is found between two vertices, the matrix containing the paths is updated. The next matrix is updated along with the matrix of minimum distances dist, so at completion both tables are complete and accurate, and any entries which are infinite in the dist table will be null in the next table. The path from i to j is the path from i to next[i][j], followed by the path from next[i][j] to j. These two shorter paths are determined recursively. This modified algorithm runs with the same time and space complexity as the unmodified algorithm.

let dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)
let next be a |V| × |V| array of vertex indices initialized to null

procedure initNextArray ()
   for i from 1 to |V|
      for j from 1 to |V|
         if i = j or dist[i][j] = ∞ (infinity) then
            next[i][j] ← 0
         else
            next[i][j] ← i

procedure FloydWarshallWithPathReconstruction ()
   for each vertex v
      dist[v][v] ← 0
   for each edge (u,v)
      dist[u][v] ← w(u,v)  // the weight of the edge (u,v)
   initNextArray()
   for k from 1 to |V|
      for i from 1 to |V|
         for j from 1 to |V|
            if dist[i][k] + dist[k][j] < dist[i][j] then
               dist[i][j] ← dist[i][k] + dist[k][j]
               next[i][j] ← k

function Path (i, j)
   if dist[i][j] = ∞ then
     return "no path"
   var intermediate ← next[i][j]
   if intermediate = i then
     return " "   // the direct edge from i to j gives the shortest path
   else
     return Path(i, intermediate) + intermediate + Path(intermediate, j)

Analysis

Let n be |V|, the number of vertices. To find all n2 of shortestPath(i,j,k) (for all i and j) from those of shortestPath(i,j,k−1) requires 2n2 operations. Since we begin with shortestPath(i,j,0) = edgeCost(i,j) and compute the sequence of n matrices shortestPath(i,j,1), shortestPath(i,j,2), …, shortestPath(i,j,n), the total number of operations used is n · 2n2 = 2n3. Therefore, the complexity of the algorithm is Θ(n3).

Applications and generalizations

The Floyd–Warshall algorithm can be used to solve the following problems, among others:

Implementations

Implementations are available for many programming languages.

See also

  • Dijkstra's algorithm, an algorithm for finding single-source shortest paths in a more restrictive class of inputs, graphs in which all edge weights are non-negative
  • Johnson's algorithm, an algorithm for solving the same problem as the Floyd–Warshall algorithm, all pairs shortest paths in graphs with some edge weights negative. Compared to the Floyd–Warshall algorithm, Johnson's algorithm is more efficient for sparse graphs.

References

  1. ^ Weisstein, Eric. "Floyd-Warshall Algorithm". Wolfram MathWorld. Retrieved 13 November 2009. 
  2. ^ "Lecture 12: Shortest paths (continued)" (PDF). Network Flows and Graphs. Department of Industrial Engineering and Operations Research, University of California, Berkeley. 7 October 2008. 
  3. ^ Penaloza, Rafael. "Algebraic Structures for Transitive Closure". 

External links