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Kruskal's and Prim's algorithm

by: Robert Notetaker

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Kruskal's and Prim's algorithm

Robert Notetaker
AASU

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Kruskal's algorithm will grow a solution from the cheapest edge by adding the next cheapest edge, provided that it doesn't create a cycle. Prim's algorithm will grow a solution from a random verte...
COURSE
Software Engineering
PROF.
Dr. James
TYPE
Class Notes
PAGES
3
WORDS
CONCEPTS
software engineering
KARMA
25 ?

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This 3 page Class Notes was uploaded by Robert Notetaker on Thursday October 13, 2016. The Class Notes belongs to at Armstrong Atlantic State University taught by Dr. James in Fall 2016. Since its upload, it has received 8 views. For similar materials see Software Engineering in Computer Science and Engineering at Armstrong Atlantic State University.

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Date Created: 10/13/16
Kruskal's algorithm Kruskal's algorithm: sort the edges of G in increasing order by length keep a subgraph S of G, initially empty for each edge e in sorted order if the endpoints of e are disconnected in S add e to S return S Note that, whenever you add an edge (u,v), it's always the smallest connecting the part of S reachable from u with the rest of G, so by the lemma it must be part of the MST. example void kruskal (vertex-set V; edge-set E; edge-set T) int ncomp; /* current number of components */ priority-queue edges /* partially ordered tree */ mfset components; /* merge-find set data structure */ vertex u, v; edge e; int nextcomp; /* name for new component */ int ucomp, vcomp; /* component names */ { makenull (T); makenull (edges); nextcomp = 0; ncomp = n; for (v$\in$V) /* initialize a component to have one vertex of V*/ { nextcomp++ ; initial (nextcomp, v, components); } for (e$\in$E) insert (e, edges); /* initialize priority queue of edges */ while (ncomp > 1) { e = deletemin (edges); let e = (u, v); ucomp = find(u, components); vcomp = find(v, components); if (ucomp! = vcomp) { merge (ucomp, vcomp, components); ncomp = ncomp - 1; } } } Prim's algorithm Prim's algorithm: let T be a single vertex x while (T has fewer than n vertices) { find the smallest edge connecting T to G-T add it to T } Since each edge added is the smallest connecting T to G-T, the lemma we proved shows that we only add edges that should be part of the MST. Again, it looks like the loop has a slow step in it. But again, some data structures can be used to speed this up. The idea is to use a heap to remember, for each vertex, the smallest edge connecting T with that vertex. Prim with heaps: make a heap of values (vertex,edge,weight(edge)) initially (v,-,infinity) for each vertex let tree T be empty while (T has fewer than n vertices) { let (v,e,weight(e)) have the smallest weight in the heap remove (v,e,weight(e)) from the heap add v and e to T for each edge f=(u,v) if u is not already in T find value (u,g,weight(g)) in heap if weight(f) < weight(g) replace (u,g,weight(g)) with (u,f,weight(f)) } example

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