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Algorithm Design and Analysis

L ECTURE 8 Greedy Graph Alg’s II Implementing Dijkstra MST. Algorithm Design and Analysis. CSE 565. Adam Smith. Rough algorithm ( Dijkstra ). Maintain a set of explored nodes S whose shortest path distance d(u) from s to u is known. Initialize S = { s }, d(s) = 0.

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Algorithm Design and Analysis

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  1. LECTURE 8 • Greedy Graph Alg’s II • Implementing Dijkstra • MST Algorithm Design and Analysis CSE 565 Adam Smith A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  2. Rough algorithm (Dijkstra) • Maintain a set of explored nodes S whose shortest path distance d(u) from s to u is known. • Initialize S = {s}, d(s) = 0. • Repeatedly choose unexplored node v which minimizes • add v to S, and set d(v) = (v). shortest path to some u in explored part, followed by a single edge (u, v) (e) v d(u) u S s A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  3. Review Question • Is Dijsktra’s algorithm correct with negative edge weights? A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  4. Proof of Correctness (Greedy Stays Ahead) Invariant. For each node u S, d(u) is the length of the shortest path from s to u. Proof:(by induction on |S|) • Base case: |S| = 1 is trivial. • Inductive hypothesis: Assume for |S| = k 1. • Let v be next node added to S, and let (u,v) be the chosen edge. • The shortest s-u path plus (u,v) is an s-v path of length (v). • Consider any s-v path P. We'll see that it's no shorter than (v). • Let (x,y) be the first edge in P that leaves S,and let P' be the subpath to x. • P + (x,y) has length ·d(x)+ (x,y)·(y)·(v) P y x P' s S u v A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  5. Implementation • For unexplored nodes, maintain • Next node to explore = node with minimum (v). • When exploring v, for each edge e = (v,w), update • Efficient implementation: Maintain a priority queue of unexplored nodes, prioritized by (v). Priority Queue A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  6. Priority queues • Maintain a set of items with priorities (= “keys”) • Example: jobs to be performed • Operations: • Insert • Increase key • Decrease key • Extract-min: find and remove item with least key • Common data structure: heap • Time: O(logn) per operation A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  7. Pseudocode for Dijkstra(G, ) • d[s]  0 • for each vÎV – {s} • dod[v]  ¥; p[v]  ¥ • S   • Q  V ⊳Q is a priority queue maintainingV – S,keyed on [v] • whileQ ¹ • dou EXTRACT-MIN(Q) • SSÈ {u}; d[u] p[u] • for each vÎAdjacency-list[u] • doif[v] > [u] + (u, v) • then [v] d[u] + (u, v) explore edges leaving v Implicit DECREASE-KEY A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  8. n times degree(u) times Handshaking Lemma  ·mimplicit DECREASE-KEY’s. Analysis of Dijkstra • whileQ ¹ • dou EXTRACT-MIN(Q) • SSÈ {u} • for each vÎAdj[u] • doif[v] >[u] + w(u, v) • then[v] [u] + w(u, v) PQ Operation Dijkstra Array Binary heap d-way Heap Fib heap † ExtractMin n n log n HW3 log n DecreaseKey m 1 log n HW3 1 Total n2 m log n m log m/n n m + n log n † Individual ops are amortized bounds A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  9. Output: A spanning treeT — a tree that connects all vertices — of minimum weight: . Minimum spanning tree (MST) • Input: A connected undirected graph G = (V, E) with weight function w : E R. • For now, assume all edge weights are distinct. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  10. Example of MST 6 12 9 5 7 14 15 8 10 3 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  11. Example of MST 6 12 9 5 7 14 15 8 10 3 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  12. Greedy Algorithms for MST • Kruskal's: Start with T = . Consider edges in ascending order of weights. Insert edge e in T unless doing so would create a cycle. • Reverse-Delete: Start with T = E. Consider edges in descending order of weights. Delete edge e from T unless doing so would disconnect T. • Prim's: Start with some root node s. Grow a tree T from s outward. At each step, add to T the cheapest edge e with exactly one endpoint in T. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  13. Cycles and Cuts • Cycle: Set of edges the form (a,b),(b,c),(c,d),…,(y,z),(z,a). • Cut: a subset of nodes S. The corresponding cutset D is the subset of edges with exactly one endpoint in S. 2 3 1 6 4 Cycle C = (1,2),(2,3),(3,4),(4,5),(5,6),(6,1) 5 8 7 2 3 1 Cut S = { 4, 5, 8 } Cutset D = (5,6), (5,7), (3,4), (3,5), (7,8) 6 4 S 5 8 7 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  14. Cycle-Cut Intersection • Claim. A cycle and a cutset intersect in an even number of edges. • Proof: A cycle has to leave and enter the cut the same number of times. C S V - S A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  15. C f S e e is in the MST f is not in the MST Cut and Cycle Properties • Cut property. Let S be a subset of nodes. Let e be the min weight edge with exactly one endpoint in S. Then the MST contains e. • Cycle property. Let C be a cycle, and let f be the max weight edge in C. Then the MST does not contain f. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  16. Proof of Cut Property Cut property: Let S be a subset of nodes. Let e be the min weight edge with exactly one endpoint in S. Then the MST T* contains e. • Proof:(exchange argument) • Suppose e does not belong to T*. • Adding e to T* creates a cycle C in T*. • Edge e is both in the cycle C and in the cutset D corresponding to S  there exists another edge, say f, that is in both C and D. • T' = T*  {e} - {f} is also a spanning tree. • Since ce < cf, cost(T') < cost(T*). Contradiction. ▪ f S e T* A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  17. Proof of Cycle Property Cycle property: Let C be a cycle in G. Let f be the max weight edge in C. Then the MST T* does not contain f. • Proof:(exchange argument) • Suppose f belongs to T*. • Deleting f from T* creates a cut S in T*. • Edge f is both in the cycle C and in the cutset D corresponding to S  there exists another edge, say e, that is in both C and D. • T' = T*  {e} - {f} is also a spanning tree. • Since ce < cf, cost(T') < cost(T*). Contradiction. ▪ f S e T* A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  18. Greedy Algorithms for MST • Kruskal's: Start with T = . Consider edges in ascending order of weights. Insert edge e in T unless doing so would create a cycle. • Reverse-Delete: Start with T = E. Consider edges in descending order of weights. Delete edge e from T unless doing so would disconnect T. • Prim's: Start with some root node s. Grow a tree T from s outward. At each step, add to T the cheapest edge e with exactly one endpoint in T. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  19. Prim's Algorithm: Correctness • Prim's algorithm. [Jarník 1930, Prim 1959] • Apply cut property to S. • When edge weights are distinct, every edge that isadded must be in the MST • Thus, Prim’s alg. outputs the MST S A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  20. Correctness of Kruskal • [Kruskal, 1956]: Consider edges in ascending order of weight. • Case 1: If adding e to T creates a cycle, discard e according to cycle property. • Case 2: Otherwise, insert e = (u, v) into T according to cut property where S = set of nodes in u's connected component. e Case 1 S v e u Case 2 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  21. Review Questions Let G be a connected undirected graph with distinct edge weights. Answer true or false: • Let e be the cheapest edge in G. Some MST of G contains e? • Let e be the most expensive edge in G. No MST of G contains e? A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  22. Non-distinct edges? A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  23. Implementation of Prim(G,w) IDEA: Maintain V – Sas a priority queue Q(as in Dijkstra). Key each vertex in Q with the weight of the least-weight edge connecting it to a vertex in S. • Q V • key[v]  ¥ for all vÎV • key[s]  0 for some arbitrary sÎV • whileQ¹  • dou  EXTRACT-MIN(Q) • for each vÎAdjacency-list[u] • do ifvÎQ and w(u, v) < key[v] • thenkey[v]  w(u, v) ⊳DECREASE-KEY • p[v]  u At the end, {(v, p[v])} forms the MST. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  24. Q(n) total n times degree(u) times Analysis of Prim • Q V • key[v]  ¥ for all vÎV • key[s]  0 for some arbitrary sÎV • whileQ¹  • dou  EXTRACT-MIN(Q) • for each vÎAdj[u] • do ifvÎQ and w(u, v) < key[v] • thenkey[v]  w(u, v) • p[v]  u Handshaking Lemma  Q(m)implicit DECREASE-KEY’s. Time: as in Dijkstra A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  25. n times degree(u) times Analysis of Prim • whileQ¹  • dou  EXTRACT-MIN(Q) • for each vÎAdj[u] • do ifvÎQ and w(u, v) < key[v] • thenkey[v]  w(u, v) • p[v]  u Handshaking Lemma  Q(m)implicit DECREASE-KEY’s. PQ Operation Prim Array Binary heap d-way Heap Fib heap † ExtractMin n n log n HW3 log n DecreaseKey m 1 log n HW3 1 Total n2 m log n m log m/n n m + n log n † Individual ops are amortized bounds A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  26. Greedy Algorithms for MST • Kruskal's:Start with T = . Consider edges in ascending order of weights. Insert edge e in T unless doing so would create a cycle. • Reverse-Delete: Start with T = E. Consider edges in descending order of weights. Delete edge e from T unless doing so would disconnect T. • Prim's: Start with some root node s. Grow a tree T from s outward. At each step, add to T the cheapest edge e with exactly one endpoint in T. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  27. Union-Find Data Structures • With modifications, amortized time for tree structure is O(nAck(n)), where Ack(n), the Ackerman function grows much more slowly than log n. • See KT Chapter 4.6 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

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