hw_06 - CS 473: Fundamental Algorithms, Spring 2012...

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CS 473: Fundamental Algorithms, Spring 2012 Homework 6 (due Tuesday, 23:55:00, March 13, 2012) See homework 1. Each student individually have to also do quiz 6 online. Version: 1.11 1. ( 30 pts. ) An Improved Algorithm for MST. Consider Bor˚ uvka’s algorithm for MST as seen in the discussion. Consider a modified al- gorithms for MST that works as follows. It runs Bor˚ uvka’s algorithm for some number of iterations and then switches to Prim’s algorithm (with Fibonacci’s heaps). Show that by appropriately choosing the switching point, one can obtain an algorithm for MST with run- ning time O (( m + n ) log log n ). Describe and analyze the algorithm. Don’t forget to prove correctness. 2. ( 30 pts. ) Amortized Analysis Recall from class the idea of Path-Compression in the Union-Find data structure for maintain- ing disjoint sets. Prove that if Path-Compression is used then the total time for k operations in which all the union operations precede all the
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