{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

07shortestpathslabelcorrectingalgo

# 07shortestpathslabelcorrectingalgo - 15.082J 6.855J The...

This preview shows pages 1–9. Sign up to view the full content.

15.082J / 6.855J February 27, 2003 The Label Correcting Algorithm

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Overview of the Lecture A generic algorithm for solving shortest path problems z negative costs permitted z but no negative cost cycle (at least for now) The use of reduced costs All pair shortest path problem INPUT G = (N, A) with costs c Node 1 is the source node There is no negative cost cycle z We will relax that assumption later
3 Optimality Conditions Lemma. Let d*(j) be the shortest path length from node 1 to node j, for each j. Let d( ) be node labels with the following properties: d(j) d(i) + c ij for i N for j 1. (1) d(1) = 0. (2) Then d(j) d*(j) for each j. Proof . Let P be any path from node 1 to node j, with length c(P), and suppose P has k arcs. Claim : d(j) c(P). Note: if P is the shortest path from 1 to j, then d(j) c(P) = d*(j), which is what we want to prove.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document