Floyd – Warshall Alogorithm
Presented by-Kapil Kumar
Cse-iii rd year
[email protected]
Graph - Basis
• A Non linear data structure
• Set of vertices and edges
• Classification According to Direction of edges
1. Directed Graph
2. Undirected graph
• Classification According to weight on edges
1. Weighted graph
2. Unweighted graph
vertex
edge
Graph – Basis (continue..)
• Representation
1.Incident matrix
2.Adjacency matrix
3.Adjacency list
4.Path matrix
What is Floyd warshall algorithm?
• An example of dynamic programming
• An algorithm for finding shortest path in a weighted graph with
positive or negative edge weights
• No negative weight cycle
• Find the length of shortest path between all pair of vertices
History
 Bernard Roy in 1959
 Robert Floyd in 1962
 Stephen Warshall in 1962
Naming
The Floyd's algorithm
 the Roy–Warshall algorithm
 the Roy–Floyd algorithm, or
 the WFI algorithm
What is shortest path?
There are several paths
between A and D:
7
2
Path 1: A -> B -> D
=9
Path 2: A -> C -> D
=7
Path 3: A -> B -> C -> D = 6
1
4
3
Floyd Warshall Algorithm
function Floyd(L[l.. n, 1..n]): array [1.. n,1. n]
array D[L..n,1 ..n]
DL
for k  1 to n do
for i  1 to n do
for j  1 to n do
D[i,j] min(D[i, j],D[i, k]+D[k,j])
return D
Time
0(1)
0(n)
0(n)
0(n)
Floyd Warshall Algorithm
• The running time is O(n3).
• The space requirements are
O(n2)
Example
1
2
3
4
5
1
2
3
4
5
1
2
3
4
1
2
3
4
5
1
2
3
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1
2 3
4 5
1 2 3
4 5
1 2 3
1
2
3
4
5
4 5
5
Application
 Fast computation of path finder networks.
 Optimal routing.
 Detecting negative-weight cycles in graph.
 Shortest paths in directed graphs
 Transitive closure of directed graphs.
 Inversion of real matrices
 Optimal routing.
 Maximum bandwidth paths
Any Query ?
Thank you