Advanced Algorithms Minimum spanning tree Generic algorithm Kruskal s

Advanced Algorithms Minimum spanning tree Generic algorithm Kruskal’s algorithm Prim’s algorithm CSE 780 Algorithms

Minimum Spanning Tree Learning outcomes: You should be able to: Write Kruskal’s and Prim’s algorithms Run both algorithms on given graphs Analyze both algorithms CSE 780 Algorithms

Problem A town has a set of houses and a set of roads A road connects 2 and only 2 houses A road connecting houses u and v has a repair cost w(u, v). Goal: repair enough roads such that 1. every house is connected 2. total repair cost is minimum CSE 780 Algorithms

Definition Given a connected graph G = (V, E), with weight function w : E --> R Find min-weight connected subgraph Spanning tree T: A tree that includes all nodes from V T = (V, E’), where E’ E Weight of T: W( T ) = w(e) Minimum spanning tree (MST): A tree with minimum weight among all spanning trees CSE 780 Algorithms

Example CSE 780 Algorithms

MST for given G may not be unique Since MST is a spanning tree: # edges : |V| - 1 If the graph is unweighted: All spanning trees have same weight CSE 780 Algorithms

Generic Algorithm Framework for G = (V, E) : Goal: build a set of edges A E Start with A empty Add edge into A one by one At any moment, A is a subset of some MST for G CSE 780 Algorithms

Finding Safe Edges When A is empty, example of safe edges? The edge with smallest weight Intuition: Suppose S V, --- a cut (S, V-S) – a partition of vertices into sets S and V-S should be connected By the crossing edge with the smallest weight ! That edge also called a light edge crossing the cut (S, V-S) A cut (S, V-S) respects set A of edges If no edges from A crosses the cut (S, V-S) CSE 780 Algorithms

Cut CSE 780 Algorithms

Safe-Edge Theorem: Let A be a subset of some MST, (S, V-S) be a cut that respects A, and (u, v) be a light edge crossing (S, V-S). Then (u, v) is safe for A. Proof Greedy Approach: Based on the Corollary: generic algorithm and the corollary, to compute MST we only need a way to find Let C = (Vc , Ec ) be a connected component in the graph a = (V, A). at each is a light (forest) G safe edge If (u, v) moment. edge connecting C A to some other component in GA , then (u, v) is safe for A. CSE 780 Algorithms

Kruskal’s Algorithm Start with A empty, and each vertex being its own connected component Repeatedly merge two components by connecting them with a light edge crossing them Two issues: Maintain sets of components Choose light edges CSE 780 Algorithms

Pseudo-code CSE 780 Algorithms

Disjoint Set Operations Disjoint set data structure maintains a collection S = {S 1, S 2, …, Sk} of disjoint dynamic sets where each set is identified by a representative (member) of the set. MAKE-SET(x): create a new set whose only member (and representative) is pointed at by x. UNION(x, y): unites dynamic sets containing x and y, Sx and Sy, eliminating Sx and Sy from S. FIND-SET(x): returns a pointer to the representative of the set containing x. CSE 780 Algorithms

Example CSE 780 Algorithms

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Analysis Time complexity: #make-set : |V| #find-set and #union operations: |E| operations O( (|V| + |E|) (|V| )) = O( |E| (|V| )) as |E| >= |V| - 1 is the very slowly growing function (|V| ) = O(lg|V|) = O(lg|E|) Sorting : O(|E| lg |E|) Total: O(|E| lg |E|) = O(|E| lg |V|) Since |E| < |V|2, hence lg|E| = lg|V|. CSE 780 Algorithms

Prim’s Algorithm Start with an arbitrary node from V Instead of maintaining a forest, grow a MST At any time, maintain a MST for V’ V At any moment, find a light edge connecting V’ with (V-V’) I. e. , the edge with smallest weight connecting some vertex in V’ with some vertex in V-V’ ! CSE 780 Algorithms

Prim’s Algorithm cont. Again two issues: Maintain the tree already build at any moment Easy: simply a tree rooted at r : the starting node Find the next light edge efficiently For v V - V’, define key(v) = the min distance between v and some node from V’ (current MST) At any moment, find the node with min key. Use a priority queue ! CSE 780 Algorithms

Pseudo-code (cancel) CSE 780 Algorithms

Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. CSE 780 Algorithms

Example CSE 780 Algorithms

Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. CSE 780 Algorithms

Min Heap CSE 780 Algorithms

MAX HEAP CSE 780 Algorithms

Analysis Time complexity Using binary min-heap for priority queue: Using # initialisation in lines 1 -5: Fibonacci heap: O(|V|) Decrease-Key: While loop is executed |V| times O(1) amortized time Each Extract-min takes O ( log |V| ) => Total Extract-min takes O ( |V| log |V| ) total time complexity Assignment in line 11 involves O(|E| + |V| log |V|) Decrease-Key operation: Each Decrease-Key operation takes O ( log |V| ) The total is O( |E| log |V| ) ) Total time complexity: O (|V| log |V| +|E| log |V|) = O (|E| log |V|) CSE 780 Algorithms

Applications Clustering Euclidean traveling salesman problem CSE 780 Algorithms