[Homework for Introduction to Algorithms, Exercise 23-3]
phoeagon
[Definition]
A bottleneck spanning tree (T) of an undirected graph G is a spanning tree of G whose largest edge weight is the minimum one over all spanning trees of G. We say that the value of the bottleneck spanning tree is the weight of the maximum-weight edge in T.
[Theorem]
A Minimum Spanning Tree (MST) is a Bottleneck Spanning Tree, and the one with the smallest cost.
[Brief Proof]
Consider a Minimum Spanning Tree (A) of Graph G, whose set of sub graphs contain another spanning tree B that is a bottleneck spanning tree, while MST (A) isn't.
First, without lost of generality, suppose e(u, v) in A is the edge with the largest weight in the MST with no ties, whose weight is W(u, v). Then, consider deleting e(u, v) from A, separating the tree into two trees, namely U and V respectively.
Then we construct a set of edges, S, that for each edge (ui, vi), where ui is a vertex in U and vi in V, suffices (ui, vi)∈S. One may notice that there exists at least one edge (uj, vj)∈S which is in the Bottleneck Spanning Tree (B). (Otherwise B is not a spanning tree.)
So, as B is the Bottleneck Spanning Tree while A isn’t, each edge e(x, y) in B suffices e(x, y) < W(u, v). Suppose we have edge e(x,y) in S, replace it with edge e(u,v) in A will make A a Spanning Tree with a even smaller sum of weights. Here we find a contradiction.
Tadalista [url=http://kamagra.rxbill7.com/generic-kamagra-online.php]Generic Kamagra Online[/url] Where To Buy Nizagara Kamagra In Deutschland [url=http://lasix.mdsmeds.com/order-lasix.php]Order Lasix[/url] Wirkung Von Viagra 100 Mg Cialis Generico Barato [url=http://amoxil.mdsmeds.com/amoxil-pills.php]Amoxil Pills[/url] Cephalexin For Veternarian Use Fda Category Listing For Keflex [url=http://lasix.rxbill7.com/comprare-lasix.php]Comprare Lasix[/url] Levitra Ou Cialis Forum Get Macrobid Cod Accepted [url=http://inderal.rxbill7.com/cheapest-inderal.php]Cheapest Inderal[/url] Levitra Flushing Buy Lipitor Online With No Prescription [url=http://priligy.usamedz.com/dapoxetine-priligy-buy.php]Dapoxetine Priligy Buy[/url] Propecia Tuenti Wholesale Viagra 100mg [url=http://cytotec.ccrpdc.com/cytotec-100mcg.php]Cytotec 100mcg[/url] Buy Bactrim Online Overnight Delivery Levitra Sotto Lingua [url=http://propecia.usamedz.com/pharmacy-propecia.php]Pharmacy Propecia[/url] Viagra 50mg Wirkung Buy Prevacid Odt [url=http://prednisone.ccrpdc.com/deltasone-tablet.php]Deltasone Tablet[/url] Cialis Without Prescription Overnight Dapoxetine Approbation De La Fda [url=http://lasix.usamedz.com/lasix-tablets.php]Lasix Tablets[/url] Achat Amoxicillin Pharmacie Vente Achat
connection. Several photographs said to be from Shenzhen, across the border from Hong Kong, showed what appeared to be damaged or overturned cars — most of them Japanese models — as well as several police
the authorities usually keep them contained out of concern they might spiral out of control or turn into popular antigovernment sentiment. Even as the protests began unfolding Sunday morning, a group of conservative
access to avoid inflaming regional tensions. The 10 who landed Sunday did so without permission, and were later questioned by the Japanese Coast Guard. Members of the group said they were responding to the pro-China activists’ landing, and they
