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CSE 421 Algorithms

This lecture discusses minimum spanning trees, the greedy algorithms that compute them (Prim's and Kruskal's algorithms), and the reverse-delete algorithm. It also explores the proofs of optimality and tie-breaking rules for edge costs.

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CSE 421 Algorithms

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  1. CSE 421Algorithms Richard Anderson Lecture 11 Minimum Spanning Trees

  2. Announcements • Monday – Class in EE1 003 (no tablets)

  3. Foreign Exchange Arbitrage USD 1.2 1.2 EUR CAD 0.6 USD 0.8 0.8 EUR CAD 1.6

  4. Minimum Spanning Tree 15 t a 6 9 14 3 4 e 10 13 c s 11 20 5 17 2 7 g b 8 f 22 u 12 1 16 v Temporary Assumption: Edge costs distinct

  5. Why do the greedy algorithms work? • For simplicity, assume all edge costs are distinct • Let S be a subset of V, and suppose e = (u, v) be the minimum cost edge of E, with u in S and v in V-S • e is in every minimum spanning tree • Or equivalently, if e is not in T, then T is not a minimum spanning tree e S V - S

  6. Proof • Suppose T is a spanning tree that does not contain e • Add e to T, this creates a cycle • The cycle must have some edge e1 = (u1, v1) with u1 in S and v1 in V-S • T1 = T – {e1} + {e} is a spanning tree with lower cost • Hence, T is not a minimum spanning tree S V - S e Why is e lower cost than e1?

  7. Optimality Proofs • Prim’s Algorithm computes a MST • Kruskal’s Algorithm computes a MST

  8. Reverse-Delete Algorithm • Lemma: The most expensive edge on a cycle is never in a minimum spanning tree

  9. Dealing with the distinct cost assumption • Force the edge weights to be distinct • Add small quantities to the weights • Give a tie breaking rule for equal weight edges

  10. MST Fun Facts • The minimum spanning tree is determined only by the order of the edges – not by their magnitude • Finding a maximum spanning tree is just like finding a minimum spanning tree

  11. Divide and Conquer Array Mergesort(Array a){ n = a.Length; if (n <= 1) return a; b = Mergesort(a[0..n/2]); c = Mergesort(a[n/2+1 .. n-1]); return Merge(b, c);

  12. Algorithm Analysis • Cost of Merge • Cost of Mergesort

  13. T(n) = 2T(n/2) + cn; T(1) = c;

  14. Recurrence Analysis • Solution methods • Unrolling recurrence • Guess and verify • Plugging in to a “Master Theorem”

  15. A better mergesort (?) • Divide into 3 subarrays and recursively sort • Apply 3-way merge

  16. T(n) = aT(n/b) + f(n)

  17. T(n) = T(n/2) + cn

  18. T(n) = 4T(n/2) + cn

  19. T(n) = 2T(n/2) + n2

  20. T(n) = 2T(n/2) + n1/2

  21. Recurrences • Three basic behaviors • Dominated by initial case • Dominated by base case • All cases equal – we care about the depth

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