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Minimum Spanning Tree Algorithms

Minimum Spanning Tree Algorithms. Prim’s Algorithm. What is A Spanning Tree?. A spanning tree for an undirected graph G=(V,E) is a subgraph of G that is a tree and contains all the vertices of G Can a graph have more than one spanning tree?

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Minimum Spanning Tree Algorithms

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  1. Minimum Spanning Tree Algorithms Prim’s Algorithm

  2. What is A Spanning Tree? • A spanning tree for an undirected graph G=(V,E) is a subgraph of G that is a tree and contains all the vertices of G • Can a graph have more than one spanning tree? • Can an unconnected graph have a spanning tree? a b u e c v f d

  3. (u,v)  T w(u,v ) is minimized Mst T: w( T )= Minimal Spanning Tree. • The weight of a subgraph is the sum of the weights of it edges. • A minimum spanning tree for a weighted graph is a spanning tree with minimum weight. • Can a graph have more then one minimum spanning tree? a 4 4 9 3 b u e 14 2 10 15 c v f 3 8 d

  4. Example of a Problem that Translates into a MST The Problem • Several pins of an electronic circuit must be connected using the least amount of wire. Modeling the Problem • The graph is a complete, undirected graph G = ( V, E ,W ), where V is the set of pins, E is the set of all possible interconnections between the pairs of pins and w(e) is the length of the wire needed to connect the pair of vertices. • Find a minimum spanning tree.

  5. Greedy Choice We will show two ways to build a minimum spanning tree. • A MST can be grown from the current spanning tree by adding the nearest vertex and the edge connecting the nearest vertex to the MST. (Prim's algorithm) • A MST can be grown from a forest of spanning trees by adding the smallest edge connecting two spanning trees. (Kruskal's algorithm)

  6. Notation • Tree-vertices: in the tree constructed so far • Non-tree vertices: rest of vertices Prim’s Selection rule • Select the minimum weight edge between a tree-node and a non-tree node and add to the tree

  7. a 4 6 5 b u 14 2 10 c v 3 8 15 d f The Prim algorithm Main Idea Select a vertex to be a tree-node while (there are non-tree vertices) { if there is no edge connecting a tree node with a non-tree node return “no spanning tree” select an edge of minimum weight between a tree node and a non-tree node add the selected edge and its new vertex to the tree} return tree

  8. Implementation Issues • How is the graph implemented? • Assume that we just added node u to the tree. • The distance of the nodes adjacent to u to the tree may now be decreased. • There must be fast access to all the adjacent vertices. • So using adjacency lists seems better • How should the set of non-tree vertices be represented? • The operations are: • build set • delete node closest to tree • decrease the distance of a non-tree node from the tree • check whether a node is a non- tree node

  9. Implementation Issues • How should the set of non-tree vertices be represented? • A priority queue PQ may be used with the priority D[v] equal to the minimum distance of each non-tree vertex v to the tree. • Each item in PQ contains: D[v], the vertex v, and the shortest distance edge (v, u) where u is a tree node • This means: • build a PQ of non-tree nodes with initial values - • fast build heap O (V ) • building an unsorted list O(V) • building a sorted list O(V) (special case)

  10. Implementation Issues • delete node closest to tree (extractMin) • O(lg V ) if heap and • O(V) if unsorted list • O(1) sorted list • decrease the distance of a non-tree node to the tree • We need to find the location i of node v in the priority queue and then execute (decreasePriorityValue(i, p)) where p is the new priority • decreasePriorityValue(i, p) • O(lg V) for heap, • O(1) for unsorted list • O(V ) for sorted list (too slow)

  11. Implementation Issues • What is the location i of node v in a priority queue? • Find in Heaps, and sorted lists O(n) • Unsorted – if the nodes are numbered 1 to n and we use an array where node v is the v item in the array O(1) Extended heap • We will use extended heaps that contain a “handle” to the location of each node in the heap. • When a node is not in PQ the “handle” will indicate that this is the case • This means that we can access a node in the extended heap in O(1), and check v  PQ in O(1) • Note that the “handle” must be updated whenever a heap operation is applied

  12. Implementation Issues 2. Unsorted list • Array implementation where node v can be accesses as PQ[v] in O(1), and the value of PQ[v] indicates when the node is not in PQ.

  13. Lines 1-5 initialize the priority queue PQ to contain all Vertices. Ds for all vertices except r, are set to infinity. r is the starting vertex of the TThe T so far is empty Add closest vertex and edge to current T Get all adjacent vertices v of u,update D of each non-tree vertex adjacent to u Store the current minimum weight edge, and updated distance in the priority queue 1. for each uV2. doD [u ]  3. D[ r ] 0 4. PQ make-heap(D,V, {})//No edges5. T 6.7. whilePQ  do8. (u,e )  PQ.extractMin() 9. add (u,e) to T10. for each v  Adjacent (u ) // execute relaxation11. do if v PQ && w( u, v ) < D [ v ]12. then D [ v ]  w (u, v) 13. PQ.decreasePriorityValue ( D[v], v, (u,v )) 14. return T // T is a mst. Prim’s Algorithm

  14. Prim’s AlgorithmInitialization Prim (G ) 1. for each uV2. doD [u ]  3. D[ r ] 0 4. PQ make-heap(D,V, {})//No edges5. T 

  15. Building the MST // solution check 7. whilePQ  do//Selection and feasibility 8. (u,e )  PQ.extractMin() // T contains the solution so far . 9. add (u,e) to T10. for each v  Adjacent (u )11. do if v  PQ && w( u, v ) < D [ v ]12. then D [ v ]  w (u, v) 13. PQ.decreasePriorityValue (D[v], v, (u,v) ) 14. return T

  16. 1. for each uV2. doD [u ]  3. D[ r ] 0 4. PQ make-PQ(D,V, {})//No edges5. T 6. 7. whilePQ  do8. (u,e )  PQ.extractMin() 9. add (u,e) to T10. for each v  Adjacent (u )11. do if v PQ && w( u, v ) < D [ v ]12. then D [ v ]  w (u, v) 13. PQ.decreasePriorityValue (D[v], v, (u,v)) 15. return T // T is a mst. Assume a node in PQ can be accessed in O(1) ** Decrease key for v requires O(lgV ) provided the node in heap with v’s data can be accessed in O(1) Time Analysis Using Extended Heap implementation Lines 1 -6 run in O (V ) Max Size of PQ is | V | Count7 =O (V ) Count7(8) = O (V )  O( lg V ) Count7(10) = O(deg(u ) ) =O( E ) Count7(10(11)) = O(1)O( E ) Count7(10(11(12))) = O(1) O( E ) Count7(10(13)) = O( lg V)  O( E ) Decrease- Key operation on the extended heap can be implemented in O( lg V) So total time for Prim's Algorithm is O ( V lg V + E lg V ) What is O(E ) ? Sparse Graph, E =O(V) , O (E lg V)=O(V lg V ) Dense Graph, E=O(V2), O (E lg V)=O(V2 lg V)

  17. Time Analysis 1. for each uV2. doD [u ]  3. D[ r ] 0 4. PQ make-PQ(D,V, {})//No edges5. T 6.7. whilePQ  do8. (u,e )  PQ.extractMin() 9. add (u,e) to T10. for each v  Adjacent (u )11. do if v PQ && w( u, v ) < D [ v ]12. then D [ v ]  w (u, v) 13. PQ.decreasePriorityValue (D[v], v, (u,v)) 15. return T // T is a mst. Using unsorted PQ Lines 1 - 6 run in O (V ) Max Size of PQ is | V | Count7 = O (V ) Count7(8) = O (V )  O(V ) Count7(10) = O(deg(u ) ) =O( E ) Count7(10(11)) = O(1)O( E ) Count7(10(11(12))) = O(1) O( E ) Count7(10(13)) =O( 1)  O( E ) So total time for Prim's Algorithm is O (V + V2 +E) = O (V2 ) For Sparse/Dense graph : O( V2 ) Note growth rate unchanged for adjacency matrix graph representation

  18. handle A B C 1 2 3 Prim - extended HeapAfter Initialization PQ T A 1 0, (A, {}) 2 6 B 5 C G , (B, {}) , (C, {}) 2 3 Prim (G, r)1. for each uV2. do D [u ]  3. D[ r ] 0 4. PQ make-heap(D,V, { })5. T  G A B C C 6 B 2 A 2 C 5 A 6 B 5

  19. handle A B C Null 2 1 Prim - extended Heap Build tree - after PQ.extractMin PQ A T (A, {}) 2 1 , (C, {}) 6 B 5 C G , (B, {}) 7. whilePQ  do 8. (u,e)  PQ.extractMin()9. add (u,e) to T 2

  20. handle A B C Null 1 2 Update B adjacent to A PQ T (A, {}) A 1 2, (B, {A, B}) 2 6 B 5 C G , (C, {}) 2 10. for each v  Adjacent (u )11. // relaxation operation // relaxation 11. do if v PQ && w( u, v ) < D [ v ]12. then D [ v ]  w (u, v) 13. PQ.decreasePriorityValue ( D[v], v, (u,v))

  21. handle A B C Null 1 2 Update C adjacent to A PQ T (A, {}) 1 2, (B, {A, B}) 6, (C, {A, C}) A 2 2 6 B 5 C G

  22. handle A B C Null Null 1 Build tree - after PQ.extractMin PQ T (A, {}) (B, {A, B}) 6, (C, {A, C}) 1 A 7. whilePQ  do 8. (u,e)  PQ.extractMin()9. add (u,e) to T 2 6 B 5 C G

  23. handle A B C Null Null 1 Update C adjacent to B PQ T (A, {}) T (A, {}) (B, {A, B}) 5, (C, {B, C}) 1 A 10. for each v  Adjacent (u )11. // relaxation operation 2 6 B 5 C G

  24. handle A B C Null Null Null Build tree - after PQ.extractMin PQ T (A, {}) T (A, {}) (B, {A, B}) (C, {B, C}) A 7. whilePQ  do 8. (u,e)  PQ.extractMin()9. add (u,e) to T 2 6 B 5 C G

  25. Prim - unsorted listAfter Initialization PQ T A B A C 12 0, (A, {}) , (B, {}) , (C, {}) 4 B 5 C G G A B C C 4 B 12 Prim (G, r)1. for each uV2. do D [u ]  3. D[ r ] 0 4. PQ make-PQ(D,V, { })5. T  A 12 C 5 A 4 B 5

  26. Build tree - after PQ.extractMin PQ T (A, {}) A B A C 12 Null , (B, {}) , (C, {}) 4 B 5 C G 7. whilePQ  do 8. (u,e)  PQ.extractMin()9. add (u,e) to T

  27. Update B, C adjacent to A PQ T (A, {}) A A B C 12 Null 12, (B, {A, B}) 4, (C, {A, C}) 4 B 5 C G 10. for each v  Adjacent (u )11. // relaxation operation

  28. Build tree - after PQ.extractMin T (A, {}) (C, {A, C}) A 12 PQ 4 B 5 A B C C Null Null 12, (B, {A, B}) G 7. whilePQ  do 8. (u,e)  PQ.extractMin()9. add (u,e) to T

  29. Update B adjacent to C PQ T (A, {}) T (A, {}) (C, {A, C}) A 12 4 B 5 A B C C Null Null 5, (B, {C, B}) G 10. for each v  Adjacent (u )11. // relaxation operation

  30. Build tree - after PQ.extractMin PQ T (A, {}) T (A, {}) (C, {A, C}) (B, {C, B}) A 12 4 B 5 A B C C Null Null Null G 7. whilePQ  do 8. (u,e)  PQ.extractMin()9. add (u,e) to T

  31. Lemma 1 Let G = ( V, E) be a connected, weighted undirected graph. Let T be a promising subset of E. Let Y be the set of vertices connected by the edges in T. If e is a minimum weight edge that connects a vertex in Y to a vertex in V - Y, then T { e } is promising. Note: A feasible set is promising if it can be extended to produce not only a solution , but an optimal solution. In this algorithm: A feasible set of edges is promising if it is a subset of a Minimum Spanning Tree for the connected graph.

  32. Outline of Proof of Correctness of Lemma 1 T is the promising subset, and e the minimum cost edge of Lemma 1 Let T ' be the MST such that T T ' We will show that if e T' then there must be another MST T" such that T {e}  T". Proof has 4 stages: 1. Adding e to T', closes a cycle in T' {e}. 2. Cycle contains another edge e' T' but e' T 3. T"=T'  {e} - {e’ }is a spanning tree 4. T" is a MST

  33. Lemma 1 T  Y X X  V -Y e X X X X X X X X X X X X X The Promising Set of Edges Selected by Prim MST T' but e  T'

  34. Lemma 1  Y  V -Y X X X X X X X X X X X X X X X Since T' is a spanning tree, it is connected. Adding the edge, e, creates a cycle. T'  {e} Stage 1 u v e In T' there is a path from uY to v  V -Y. Therefore the path must include another edge e' with one vertex in Y and the other in V-Y. Stage 2 e'

  35. Lemma 1  Y  V -Y X T " e X X X X X X X • If we remove e’ from T’ { e } the cycle disappears. • T”=T’ {e } - {e’} is connected. T’ is connected. Every pair of vertices connected by a path that does not include e’ is still connected in T”. Every pair of vertices connected by a path that included e’ , is still connected in T” because there is a path in T" =T’  { e } - { e’ } connecting the vertices of e’. Stage 3 w( e )  w( e' )By the way Prim picks the next edge

  36. Lemma 1  Y  V -Y X T" e X X X X X X X • w( e )  w( e' ) by the way Prim picks the next edge. • The weight of T", w(T") = w (T' ) + w( e ) - w( e' )  w(T'). • But w (T' )  w(T")because T' is a MST. • So w (T' ) = w(T")and T" is a MST Stage 4 Conclusion T { e } is promising

  37. Theorem : Prim's Algorithm always produces a minimum spanning tree. Proof by induction on the set T of promising edges.Base case: Initially, T = is promising.Induction hypothesis: The current set of edges T selected by Prim is promising.Induction step: After Prim adds the edge e,T {e } is promising.Proof: T {e } is promising by Lemma 1. Conclusion: When G is connected, Prim terminates with |T | = |V | -1, and T is a MST.

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