The Floyd-Warshall algorithm finds the shortest paths between all pairs of vertices in a weighted graph. It works by iteratively updating the shortest path between all pairs through intermediate vertices. The algorithm takes as input the adjacency matrix representing the graph and outputs the distance matrix containing the shortest paths between all pairs of vertices.

1. Module 4
Dynamic programming
Design and Analysis of Algorithm(18CS42)
Lecture Presentation
Divya K S
Dept. of CSE

2. Contents
1. Definition of Dynamic Programming
2. Transitive closure: Warshall’s Algorithm
3. All pair shortest paths: Warshall’s and Floyd’s Algorithm

3. Dynamic Programming
1. It is a technique for solving problems with overlapping subproblems.
2. Example: Floyd’s all pair shortest path problem.

5. Transitive closure -----Graph
E1=(a,b) and E2=(b,c) then P= (a,c)
A B
C

6. Example for Warshall’s algorithm
Given digraph:
Step 1: Adjacency matrix of the given graph
a b c d
a 0 1 0 0
b 0 0 0 1
c 0 0 0 0
d 1 0 1 0
R(0)=
(a,b),(d,a)
(d,a)-(a,b)---(d,b)

7. a b c d
a 0 1 0 0
b 0 0 0 1
c 0 0 0 0
d 1 1 1 0
R(1)=
a b c d
a 0 1 0 1
b 0 0 0 1
c 0 0 0 0
d 1 1 1 1
R(2)=
(a,b) (b,d) (d,b)
1. (a,b) (b,d) --(a,d)
2. (d,b) (b,d) ---(d,d)
a b c d
a 0 1 0 1
b 0 0 0 1
c 0 0 0 0
d 1 1 1 1
R(3)=
(d,c)
a b c d
a 1 1 1 1
b 1 1 1 1
c 0 0 0 0
d 1 1 1 1
R(4)=
(a,d) (b,d) (d,a)(d,b)(d,c)(d,d)
1. (a,d)(d,a)--(a,a)
2. (b,d)(d,a)-(b,a)
3. (a,d)(d,c)-(a,c)
4. (b,d)(d,c)-(b,c)
5. (b,d)(d,b)-(b,b)

8. Pseudocode of warshall’s algorithm

9. Assignment

10. Dijkstra’s----Single source shortest path
All pair shortest path--- Floyd’s
A B
C
D
A—B
A—C
A—D
B—A
B—C
B—D
C—A
C—B
C—D
D—A
D—B
D—C
--Dis

11. All pair shortest path
Problem Definition:
Given a weighted connected graph(undirected or directed), the all-pair shortest-paths problem
asks to find the distances from each vertex to all other vertices.
Distance Matrix:
It is a n-by-n matrix which records the lengths of the shortest paths such that : the element dij
In the ith row and jith column of this matrix indicates the length of the shortest path from the
ith vertex to the jth vertex(1<=I,j<=n).

12. Floyd’s algorithm
 It is applicable to both directed and undirected graph.
 Input graph is connected weighted graph.
 Weights can be of both positive and negative weights.
 But graph should not contain negative-length cycle.
 Used to solve all pair shortest path.
 Uses warshall’s algorithm idea of transitive closure.
2
-1
-3

13. Example
Step 1: Distance Matrix
a b c d
a 0 in 3 in
b 2 0 in in
c in 7 0 1
d 6 in in 0
a b c d
a 0 in 3 4
b 2 0 5 in
c in 7 0 1
d 6 in 9 0
Step 2:
D(1)=
D(0)=
D(1)[b,c]:
D(0)[b,c] D(0)[b,a]+D(0)[a,c]
Infini 2+3=5
D(1)[b,d]:
D(0)[b,d] D(0)[b,a]+D(0)[a,d]
Infini 2+infini
D(1)[c,b]:
D(0)[c,b] D(0)[c,a]+D(0)[a,b]
7 ini+ini
D(1)[c,d]:
D(0)[c,d] D(
1
D(1)[d,b]:
D(0)[d,b] D
Ini in
D(1)[d,c]:
D(0)[d,c] D
Ini

14. a b c d
a 0 10 3 4
b 2 0 5 6
c 7 7 0 1
d 6 16 9 0
a b c d
a 0 in 3
b 2 0 5 in
c 7 0
d in 0
a b c d
a
b
c
d
D(2)=
D(3)=
D(4)=
D(2)[a,c]
D(1)[a,c] D(1)[a,b]+D(1)[b,c]
3 ini+5

15. Pseudo code

16. Assignment
1. Write an algorithm for all pair shortest path.