{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Lecture10 - CSE 421 Algorithms Richard Anderson Lecture 10...

This preview shows pages 1–7. Sign up to view the full content.

CSE 421 Algorithms Richard Anderson Lecture 10 Minimum Spanning Trees

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

View Full Document
Minimum Spanning Tree a b c s e g f 9 2 13 6 4 11 5 7 20 14 t u v 15 10 1 8 12 16 22 17 3 Undirected Graph G=(V,E) with edge weights
Greedy Algorithms for Minimum Spanning Tree [Prim] Extend a tree by including the cheapest out going edge [Kruskal] Add the cheapest edge that joins disjoint components [ReverseDelete] Delete the most expensive edge that does not disconnect the graph 4 11 5 7 20 8 22 a b c d e

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

View Full Document
Why do the greedy algorithms work? For simplicity, assume all edge costs are distinct
Edge inclusion lemma Let S be a subset of V, and suppose e = (u, v) is the minimum cost edge of E, with u in S and v in V-S e is in every minimum spanning tree of G Or equivalently, if e is not in T, then T is not a minimum spanning tree S V - S e

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

View Full Document
Proof Suppose T is a spanning tree that does not contain e Add e to T, this creates a cycle The cycle must have some edge e 1 = (u 1 , v 1 ) with u 1 in S and v 1 in V-S T 1 = T – {e 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}