Download - Minimum Spanning Tree (MST)

Minimum Spanning Tree (MST)

Motivation

For an electrical circuit certain pins have to be grounded. Finding the arrangement of wires (connecting those pins) that uses the least amount of wire.Let a G(V,E) be a graph such that (u, v) E and a weight w(u,v) corresponding to wire needed to join u and v.We are looking for an acyclic subset T E that connects all vertices and whose total weight w(T) is minimized.

MotivationSince T is acyclic and connects all the vertices, it must form a tree.

Since it spans the graph, it is called a spanning tree.

MST or actually means Minimum Weight Spanning Tree.

Minimum Spanning TreeProblem: given a connected, undirected, weighted graph:1410364529158

Minimum Spanning TreeProblem: given a connected, undirected, weighted graph, find a spanning tree using edges that minimize the total weight

Minimum Spanning TreeWhich edges form the minimum spanning tree (MST) of the following graph?HBCGEDFA1410364529158

Minimum Spanning TreeAnswer:HBCGEDFA1410364529158

Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);

Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);1410364529158Run on example graph

Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);01410364529158Pick a start vertex rr

Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);01410364529158Black vertices have been removed from Qu

Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);031410364529158Black arrows indicate parent pointersu

Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);14031410364529158u

Review: Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);

What is the hidden cost in this code?

Review: Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v));

Review: Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v));

How often is ExtractMin() called?How often is DecreaseKey() called?

Review: Prims AlgorithmMST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);What will be the running time? A: Depends on queue binary heap: O(E lg V) Fibonacci heap: O(V lg V + E)

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