floyds algorithm

CPSC 413
Assignment 1
Asymptotic Notation and Summations
Sample Analysis

Goal
This document will give a detailed analysis of Floyd-Warshall’s All-Pairs Shortest Path algorithm, which should give you an idea of the detail that is required in your own solution for assignment 1.

Floyd’s Algorithm
• Graph Problem: All-Pairs Shortest Path
• Input: A weighted graph denoted by adjacency matrix W . (The vertices are assumed to be numbered from 1 to n)
• Output: Matrix D containing the length of the paths (or distances) between each vertex in the graph.
• Input Size: matrix W .

1
2
3
4
5
6

7

The number of vertices in the graph, in other words, the dimension of the

Floyd-Warshall(W ) n ← rows(W )
D←W
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]) end for end for end for return D

(This algorithm is from your textbook on page 635.)

1

Analysis
Run-time Function of Floyd-Warshall Algorithm
The run-time function of the Floyd-Warshall algorithm is

 n f (n) = c1 + c2 n2 +

n

 k=1 n

c3 

 i=1 

j=1

for some constants c1 , c2 , c3 >= 1 . This functions can be derived by observing the following:
• The first line in the algorithm can be performed in some constant number of steps, say c1 .
• The second line in the algorithm requires that all elements from matrix W are copied to a new matrix D. There are n2 elements in W (since W is an n × n matrix.) Each copy takes a constant number of steps, say c2 . Hence, the total number of steps performed in line number
2 is c2 n2 . n k=1 .

• The next line is a for loop which can be translated into the sum

• Line number 4 is also a for loop, which can be translated into the sum

n i=1 .

• Line number 5 is another for loop and is similarly translated.
• Line number 6 can be performed in a constant number of steps, say c3 .
• Line numbers 4,5 and 6 appear in the outer most for loop and