Given an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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Given an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)

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In the following algorithm suppose we have the following image attached. If we look for only neighbors of node u we get that the modified edge is included in the new MST using this algorithm if the modified edge has the smallest weight amongst the neighbors of u in the other set (but this is wrong since there is another edge in the set that contains u that has less weight than the modified edge). 

G
и эрон
this edges mod
Current MST
• modified odje
MST
edge
new
Transcribed Image Text:G и эрон this edges mod Current MST • modified odje MST edge new
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If we run Prim's again in step 2, that would merit a mlogn runtime not m runtime. Is there an alternative to this algorithm?

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