Floyd–Warshall algorithm


In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights. A single execution of the algorithm will find the lengths of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation, or widest paths between all pairs of vertices in a weighted graph.

History and naming

The Floyd–Warshall algorithm is an example of dynamic programming, and 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, and is closely related to Kleene's algorithm for converting a deterministic finite automaton into a regular expression. The modern formulation of the algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.

Algorithm

The Floyd–Warshall algorithm compares all possible paths through the graph between each pair of vertices. It is able to do this with comparisons in a graph, even though there may be up to 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 with vertices numbered 1 through . Further consider a function that returns the shortest possible path from to using vertices only from the set as intermediate points along the way. Now, given this function, our goal is to find the shortest path from each to each using any vertex in.
For each of these pairs of vertices, the could be either
or
We know that the best path from to that only uses vertices through is defined by, and it is clear that if there was a better path from to to, then the length of this path would be the concatenation of the shortest path from to and the shortest path from to .
If is the weight of the edge between vertices and, we can define in terms of the following recursive formula: the base case is
and the recursive case is
This formula is the heart of the Floyd–Warshall algorithm. The algorithm works by first computing for all pairs for, then, and so on. This process continues until, and we have found the shortest path for all pairs using any intermediate vertices. Pseudocode for this basic version follows:
let dist be a |V| × |V| array of minimum distances initialized to ∞
for each edge do
dist ← w // The weight of the edge
for each vertex v do
dist ← 0
for k from 1 to |V|
for i from 1 to |V|
for j'' from 1 to |V|
if dist > dist + dist
dist ← dist + dist
end if

Example

The algorithm [|above] is executed on the graph on the left below:
Prior to the first recursion of the outer loop, labeled above, the only known paths correspond to the single edges in the graph. At, paths that go through the vertex 1 are found: in particular, the path is found, replacing the path which has fewer edges but is longer. At, paths going through the vertices are found. The red and blue boxes show how the path is assembled from the two known paths and encountered in previous iterations, with 2 in the intersection. The path is not considered, because is the shortest path encountered so far from 2 to 3. At, paths going through the vertices are found. Finally, at, all shortest paths are found.
The distance matrix at each iteration of, with the updated distances in bold, will be:

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, which form part of a negative cycle, because path-lengths from to can be arbitrarily small. 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:
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.. During the execution of the algorithm, if there is a negative cycle, exponentially large numbers can appear, as large as, where is the largest absolute value of a negative edge in the graph. To avoid overflow/underflow problems one should check for negative numbers on the diagonal of the path matrix within the inner for loop of the algorithm. Obviously, in an undirected graph a negative edge creates a negative cycle involving its incident vertices. Considering all edges of the above example graph as undirected, e.g. the vertex sequence 4 – 2 – 4 is a cycle with weight sum −2.

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. Instead, the shortest-path tree can be calculated for each node in time using memory to store each tree which allows us to efficiently reconstruct a path from any two connected vertices.

Pseudocode https://books.goalkicker.com/AlgorithmsBook/

let dist be a array of minimum distances initialized to
let next be a array of vertex indices initialized to null
procedure FloydWarshallWithPathReconstruction is
for each edge do
dist ← w // The weight of the edge
next ← v
for each vertex v do
dist ← 0
next ← v
for k from 1 to |V| do // standard Floyd-Warshall implementation
for i from 1 to |V|
for j from 1 to |V|
if dist > dist + dist then
dist ← dist + dist
next ← next
procedure Path
if next = null then
return
path =
while u ≠ v
u ← next
path.append
return path

Analysis

Let be, the number of vertices. To find all of
from those of
requires operations. Since we begin with
and compute the sequence of matrices,,,, the total number of operations used is
. Therefore, the complexity of the algorithm is big theta|.

Applications and generalizations

The Floyd–Warshall algorithm can be used to solve the following problems, among others:
Implementations are available for many programming languages.
The Floyd–Warshall algorithm is a good choice for computing paths between all pairs of vertices in dense graphs, in which most or all pairs of vertices are connected by edges. For sparse graphs with non-negative edge weights, a better choice is to use Dijkstra's algorithm from each possible starting vertex, since the running time of repeated Dijkstra is better than the running time of the Floyd–Warshall algorithm when is significantly smaller than. For sparse graphs with negative edges but no negative cycles, Johnson's algorithm can be used, with the same asymptotic running time as the repeated Dijkstra approach.
There are also known algorithms using fast matrix multiplication to speed up all-pairs shortest path computation in dense graphs, but these typically make extra assumptions on the edge weights. In addition, because of the high constant factors in their running time, they would only provide a speedup over the Floyd–Warshall algorithm for very large graphs.