| Class | All-pairs shortest path problem (for weighted graphs) |
|---|---|
| Data structure | Graph |
| Worst-caseperformance | |
| Best-caseperformance | |
| Averageperformance | |
| Worst-casespace complexity |
Incomputer science, theFloyd–Warshall algorithm (also known asFloyd's algorithm, theRoy–Warshall algorithm, theRoy–Floyd algorithm, or theWFI algorithm) is analgorithm for findingshortest paths in a directedweighted graph with positive or negative edge weights (but with no negative cycles).[1][2] A single execution of the algorithm will find the lengths (summed weights) 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 thetransitive closure of a relation, or (in connection with theSchulze voting system)widest paths between all pairs of vertices in a weighted graph.
The Floyd–Warshall algorithm is an example ofdynamic programming, and was published in its currently recognized form byRobert Floyd in 1962.[3] However, it is essentially the same as algorithms previously published byBernard Roy in 1959[4] and also byStephen Warshall in 1962[5] for finding the transitive closure of a graph,[6] and is closely related toKleene's algorithm (published in 1956) for converting adeterministic finite automaton into aregular expression, with the difference being the use of a min-plussemiring.[7] The modern formulation of the algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.[8]
The Floyd–Warshall algorithm compares many possible paths through the graph between each pair of vertices. It is guaranteed to find all shortest paths and is able to do this with comparisons in a graph,[1][9] even though there may be edges in the graph. 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 length of the shortest possible path (if one exists) from to using vertices only from the set as intermediate points along the way. Now, given this function, our goal is to find the length of the shortest path from each to each usingany vertex in. By definition, this is the value, which we will findrecursively.
Observe that must be less than or equal to: we havemore flexibility if we are allowed to use the vertex. If is in fact less than, then there must be a path from to using the vertices that is shorter than any such path that does not use the vertex. Since there are no negative cycles this path can be decomposed as:
And of course, these must be ashortest such path (or several of them), otherwise we could further decrease the length. In other words, we have arrived at the recursive formula:
The base case is given by
where denotes the weight of the edge from to if one exists and ∞ (infinity) otherwise.
These formulas are the heart of the Floyd–Warshall algorithm. The algorithm works by first computing for all pairs for, then, 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 ∞ (infinity)for each edge (u,v)do dist[u][v] = w(u,v)// The weight of the edge (u,v)for each vertexvdo dist[v][v] = 0forkfrom 1to |V|forifrom 1to |V|forjfrom 1to |V|if dist[i][j] > dist[i][k] + dist[k][j] dist[i][j] = dist[i][k] + dist[k][j]end if
Note: A common mistake in implementing the Floyd–Warshall algorithm is to misorder the triply nested loops (The correct order isKIJ). The incorrectIJK andIKJ algorithms do not give correct solutions for some instance. However, we can prove that if these are repeated three times, we obtain the correct solutions.[10]
The algorithm above is executed on the graph on the left below:
Prior to the first recursion of the outer loop, labeledk = 0 above, the only known paths correspond to the single edges in the graph. Atk = 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 (in terms of weight). Atk = 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. Atk = 3, paths going through the vertices {1,2,3} are found. Finally, atk = 4, all shortest paths are found.
The distance matrix at each iteration ofk, with the updated distances inbold, will be:
| k = 0 | j | ||||
| 1 | 2 | 3 | 4 | ||
|---|---|---|---|---|---|
| i | 1 | 0 | ∞ | −2 | ∞ |
| 2 | 4 | 0 | 3 | ∞ | |
| 3 | ∞ | ∞ | 0 | 2 | |
| 4 | ∞ | −1 | ∞ | 0 | |
| k = 1 | j | ||||
| 1 | 2 | 3 | 4 | ||
|---|---|---|---|---|---|
| i | 1 | 0 | ∞ | −2 | ∞ |
| 2 | 4 | 0 | 2 | ∞ | |
| 3 | ∞ | ∞ | 0 | 2 | |
| 4 | ∞ | −1 | ∞ | 0 | |
| k = 2 | j | ||||
| 1 | 2 | 3 | 4 | ||
|---|---|---|---|---|---|
| i | 1 | 0 | ∞ | −2 | ∞ |
| 2 | 4 | 0 | 2 | ∞ | |
| 3 | ∞ | ∞ | 0 | 2 | |
| 4 | 3 | −1 | 1 | 0 | |
| k = 3 | j | ||||
| 1 | 2 | 3 | 4 | ||
|---|---|---|---|---|---|
| i | 1 | 0 | ∞ | −2 | 0 |
| 2 | 4 | 0 | 2 | 4 | |
| 3 | ∞ | ∞ | 0 | 2 | |
| 4 | 3 | −1 | 1 | 0 | |
| k = 4 | j | ||||
| 1 | 2 | 3 | 4 | ||
|---|---|---|---|---|---|
| i | 1 | 0 | −1 | −2 | 0 |
| 2 | 4 | 0 | 2 | 4 | |
| 3 | 5 | 1 | 0 | 2 | |
| 4 | 3 | −1 | 1 | 0 | |
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 (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:
Hence, to detect negativecycles 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.[9] However, when a negative cycle is present, during the execution of the algorithm exponentially large numbers on the order of can appear, where is the largest absolute value edge weight in the graph. To avoid integer underflow problems, one should check for a negative cycle within the innermost for loop of the algorithm.[11]
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, we can use theshortest-path tree, which can be calculated for each node in time using memory, and allows us to efficiently reconstruct a directed path between any two connected vertices.
The arrayprev[u][v] holds the penultimate vertex on the path fromu tov (except in the case ofprev[v][v], where it always containsv even if there is no self-loop onv):[12]
let dist be a array of minimum distances initialized to (infinity)let prev be a array of vertex indices initialized tonullprocedureFloydWarshallWithPathReconstruction()isfor each edge (u, v)do dist[u][v] = w(u, v)// The weight of the edge (u, v) prev[u][v] = ufor each vertex vdo dist[v][v] = 0 prev[v][v] = vfor kfrom 1to |V|do// standard Floyd-Warshall implementationfor ifrom 1to |V|for jfrom 1to |V|if dist[i][j] > dist[i][k] + dist[k][j]then dist[i][j] = dist[i][k] + dist[k][j] prev[i][j] = prev[k][j]
procedurePath(u, v)isif prev[u][v] = nullthenreturn [] path = [v]whileu ≠vdo v = prev[u][v] path.prepend(v)return path
Let be, the number of vertices. To find all of (for all and) from those of requires operations. Since we begin with and compute the sequence of matrices,,,, each having a cost of,the totaltime complexity of the algorithm is.[9][13]
The Floyd–Warshall algorithm can be used to solve the following problems, among others:
Implementations are available for manyprogramming languages.
For graphs with non-negative edge weights,Dijkstra's algorithm can be used to find all shortest paths from asingle vertex with running time. Thus, running Dijkstra starting ateach vertex takes time. Since, this yields a worst-case running time of repeated Dijkstra of. While this matches the asymptotic worst-case running time of the Floyd-Warshall algorithm, the constants involved matter quite a lot. When agraph is dense (i.e.,), the Floyd-Warshall algorithm tends to perform better in practice. When the graph is sparse (i.e., is significantly smaller than), Dijkstra tends to dominate.
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 usingfast matrix multiplication to speed up all-pairs shortest path computation in dense graphs, but these typically make extra assumptions on the edge weights (such as requiring them to be small integers).[17][18] 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.
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