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Analysis of Algorithms

This document provides a comprehensive analysis of directed minimum spanning trees (MST) as presented by Uri Zwick. It discusses the equivalence of conditions defining a directed spanning tree rooted at a specified vertex, the method of selecting cheapest entering edges, and strategies for growing and contracting paths. The paper explores key lemmas regarding cycles formed by cheapest edges and their implications for finding optimal spanning trees. This analysis is essential for understanding advanced algorithms in graph theory.

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Analysis of Algorithms

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

  2. Directed minimum spanning trees 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  3. Directed minimum spanning trees 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  4. Directed minimum spanning trees 11 16 22 17 5 8 1 13 3 18 30 12 25 9 2 15

  5. Directed spanning trees Lemma:The following conditions are equivalent: (i) T is a directed spanning tree of G rooted at r. (ii) The indegree of r in T is 0, the indegree of every other vertex of G in T is 1, and T is acyclic, i.e., contains no directed cycles. (iii) The indegree of r in T is 0, the indegree of every other vertex of G in T is 1, and there are directed paths in T from r to all other vertices.

  6. Directed spanning trees

  7. Cheapest entering edges Select the cheapest edge entering each vertex, other than the specified root r. If, magically, we get a directed spanning tree rooted at r, then it is of course an optimal tree

  8. Cheapest entering edges Select the cheapest edge entering each vertex, other than the specified root r. We may, of course, get cycles…

  9. Cheapest entering edges Lemma: If C is a cycle composed of cheapest entering edges, then there is a MDST T that contains all edges of C, except for one.

  10. Cheapest entering edges Lemma: If C is a cycle composed of cheapest entering edges, then there is a MDST T that contains all edges of C, except for one.

  11. Growing a path and contracting

  12. Contraction phase – An example

  13. Expansion phase – An example

  14. Expansion phase – Another example

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