CSE 780 Algorithms 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 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 An edge is safe if adding it to A still maintains that A is a subset of a MST

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 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 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 Corollary: Let C = (V c,E c ) be a connected component in the graph (forest) G A = (V, A). If (u, v) is a light edge connecting C to some other component in G A, then (u, v) is safe for A. Greedy Approach: Based on the generic algorithm and the corollary, to compute MST we only need a way to find a safe edge at each moment.

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 Disjoint set data structure Scan edges from low to high weight

CSE 780 Algorithms Pseudo-code

CSE 780 Algorithms Disjoint Set Operations Disjoint set data structure maintains a collection S = {S 1,S 2, …,S k } 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, S x and S y, eliminating S x and S y from S. FIND-SET(x): returns a pointer to the representative of the set containing x.

CSE 780 Algorithms Example

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

CSE 780 Algorithms 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 McGraw-Hill Companies, Inc. Permission required for reproduction or display.

CSE 780 Algorithms Example

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

CSE 780 Algorithms Applications Clustering Euclidean traveling salesman problem