MCA 202: Discrete Structures Instructor Neelima Gupta

Table of Contents Prim’s MST Algorithm Kruskal MST Algorithm

Minimum Spanning Tree

THANKS Saumya Agarwal (Roll No 35) & Saurabh Garg(Roll No 36) (MCA 2012)

Prim’s Algorithm for Minimum Spanning Tree THANKS Saumya Agarwal (Roll No 35) & Saurabh Garg(Roll No 36) (MCA 2012)

Example a b c d ef h g THANKS Saumya Agarwal (Roll No 35) & Saurabh Garg(Roll No 36) (MCA 2012) T Edges from T to V-T Selected edge

Idea of Correctness THANKS Saumya Agarwal (Roll No 35) & Saurabh Garg(Roll No 36) (MCA 2012)

Cut A cut is defined as a collection of edges which separates one collection of vertices from another. THANKS Saumya Agarwal (Roll No 35) & Saurabh Garg(Roll No 36) (MCA 2012) T V-T

Thanks: Shammi-37 and Shivangi-38 (MCA 202) Step 1: Sort the edges in the increasing order of weights. Step 2: Pick the minimum weighted edge and include it in your partial constructed forest if it does not form a cycle. Step 3: Repeat step 2 until you include all the vertices (n) or exactly (n-1) edges. Kruskal’s algorithm

Thanks: Shammi-37 and Shivangi-38 (MCA 202) a b c d h e g f Minimum Spanning Tree Traversing edges: be df ah ef, de  The edge ‘de’ is traversed but not included because it forms cycle. fg cd, bc  Similarly, ‘bc’ is traversed but not included. gh, ab  The edge ‘ab’ is not traversed at all as we have included n-1 edges. a b c h e g f  Now we have completed the tree, with minimized weights. d

Thanks: Shammi-37 and Shivangi-38 (MCA 202)  How to check cycle: If both the vertices are of same component it will form cycle.  Data structure used: Linked list. How much time to check for cycles? How much time to merge two components?  Only constant amount of time is needed to perform step 2, as edges are already sorted before step 1.