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Analysis of Algorithms

This document provides an in-depth analysis of minimum spanning trees (MSTs), focusing on the main algorithms used to find them, including Kruskal’s, Prim’s, and Borůvka’s algorithms. It covers essential concepts such as MST verification, the unique properties of MSTs with distinct edge weights, and key principles including the cut rule and cycle rule. The information is structured to facilitate understanding of comparison-based MST algorithms, ensuring a clear and comprehensive guide for students and professionals interested in algorithm analysis.

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Analysis of Algorithms

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  1. Analysis of Algorithms Minimum Spanning Trees Uri Zwick February 2014

  2. Find a minimum spanning tree 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  3. Kruskal’s algorithm 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  4. Prim’s algorithm 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  5. Boruvka’s algorithm 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  6. MST verification 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  7. Comparison-based MST algorithms Deterministic Rand.

  8. Assume for simplicity that all edge weights are distinct The MST is then unique

  9. Cut rule S VS The lightest edge in a cut is contained in the MST

  10. Cycle rule C The heaviest edge on a cycle is not contained in the MST

  11. Cuts andcycles The intersection between a cut and a cycle is of even size

  12. Fundamental cycles Tree + non-tree edge  unique cycle The removal of any tree edge on the cyclegenerates a new tree

  13. Cut rule - proof S VS w' w w < w' The lightest edge in a cut is contained in the MST

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