Shortest Paths

Floyd's Algorithm


















How do you store a weighted graph in a matrix?
(A weighted graph has edge labels that represent distance or cost.)
(graph6.pdf)

What's the length of a path in a weighted graph?

What's the distance matrix of a weighted graph?

	d[i][j] is the length of the shortest path from node i to node j


How do you compute the distance matrix?

If d[i][k] + d[k][j] is less than d[i][j],
what should happen to the distance matrix?

What if you compare d[i][k] + d[k][j] with d[i][j]
for all possible triples of nodes (k, i, j)?

	for (k = 0; k < n; k++)
	  for (i = 0; i < n; i++)
	    for (j = 0; j < n; j++)
	      if (d[i][k] + d[k][j] < d[i][j])
	        d[i][j] = d[i][k] + d[k][j];


















Show the steps of Floyd's algorithm.

Give the distance matrix after using each node as the pivot.
(after each iteration of the outside loop)
(graph4.pdf)

  a b c d e
a 0 1 - 1 -
b 1 0 1 - -
c - - 0 - 1
d 1 - - 0 1
e - 1 - - 0


















Classwork

You may work with a partner.
Show the steps of Floyd's algorithm.
Give the distance matrix after pivoting with each node.
(graph6.pdf)

  a b c d e
a 0 2 4 - 1
b 4 0 - 1 -
c 1 - 0 - -
d - 5 1 0 1
e 2 2 - 1 0