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CSE 421 Algorithms

CSE 421 Algorithms. Richard Anderson Lecture 21 Shortest Path Network Flow Introduction. Announcements. Friday, 11/18, Class will meet in CSE 305 Reading 7.1-7.3, 7.5-7.6 Section 7.4 will not be covered. Find the shortest paths from v with exactly k edges. 7. y. v. 5. 1. -2. -2.

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CSE 421 Algorithms

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  1. CSE 421Algorithms Richard Anderson Lecture 21 Shortest Path Network Flow Introduction

  2. Announcements • Friday, 11/18, Class will meet in CSE 305 • Reading 7.1-7.3, 7.5-7.6 • Section 7.4 will not be covered

  3. Find the shortest paths from v with exactly k edges 7 y v 5 1 -2 -2 3 1 x 3 z

  4. Express as a recurrence • Optk(w) = minx [Optk-1(x) + cxw] • Opt0(w) = 0 if v=w and infinity otherwise

  5. Algorithm, Version 1 foreach w M[0, w] = infinity; M[0, v] = 0; for i = 1 to n-1 foreach w M[i, w] = minx(M[i-1,x] + cost[x,w]);

  6. Algorithm, Version 2 foreach w M[0, w] = infinity; M[0, v] = 0; for i = 1 to n-1 foreach w M[i, w] = min(M[i-1, w], minx(M[i-1,x] + cost[x,w]))

  7. Algorithm, Version 3 foreach w M[w] = infinity; M[v] = 0; for i = 1 to n-1 foreach w M[w] = min(M[w], minx(M[x] + cost[x,w]))

  8. Algorithm 2 vs Algorithm 3 7 y v 5 1 -2 -2 3 1 x 3 z

  9. Correctness Proof for Algorithm 3 • Key lemma – at the end of iteration i, for all w, M[w] <= M[i, w]; • Reconstructing the path: • Set P[w] = x, whenever M[w] is updated from vertex x 7 y v 5 1 -2 -2 3 1 x 3 z

  10. Negative Cost Cycle example 7 y v 3 1 -2 -2 3 3 x -2 z

  11. If the pointer graph has a cycle, then the graph has a negative cost cycle • If P[w] = x then M[w] >= M[x] + cost(x,w) • Equal after update, then M[x] could be reduced • Let v1, v2,…vk be a cycle in the pointer graph with (vk,v1) the last edge added • Just before the update • M[vj] >= M[vj+1] + cost(vj+1, vj) for j < k • M[vk] > M[v1] + cost(v1, vk) • Adding everything up • 0 > cost(v1,v2) + cost(v2,v3) + … + cost(vk, v1) v1 v4 v2 v3

  12. Negative Cycles • If the pointer graph has a cycle, then the graph has a negative cycle • Therefore: if the graph has no negative cycles, then the pointer graph has no negative cycles

  13. Finding negative cost cycles • What if you want to find negative cost cycles?

  14. Network Flow

  15. Network Flow Definitions • Capacity • Source, Sink • Capacity Condition • Conservation Condition • Value of a flow

  16. Flow Example u 20 10 30 s t 10 20 v

  17. Residual Graph u u 15/20 0/10 5 10 15 15/30 s t 15 15 s t 5 5/10 20/20 5 20 v v

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