Prim’s Algorithmand an MST Speed-UpBERJ CHILINGIRIAN

Today’s Plan

(1) Prim’s Algorithm (2) Implementation (3) Runtime Analysis (4) Speed-Up

Key Idea

Boruvka’s Algorithm

Prim’s Algorithm

Improved MST Algorithm

Minimum Spanning Tree

Given: undirected, connected, weighted graph

Return: tree with all vertices and of minimum weight

n verticesm edges

Prim’s Algorithm: The Idea

Algorithm

Correctness

Cut Lemma: At every step, all edges in are in the edges of the MST.

Simple Implementation

Better Implementation

Runtime Analysis

Procedure Number of Calls

Time per OperationArray

Time per OperationBinary Heap

Time per OperationFibonacci Heap

Insert

Extract Min

Decrease Key

Prim’s -

Achieving Time

Yao, 1975:

use linear median finding algorithm to approx. sorting of edges

Boruvka’s + Prim’s:

run Boruvka’s for iterations to produce

run Prim’s on

Boruvka’s + Prim’s

After iterations of Boruvka’s Algorithm# of Edges# of Vertices

Boruvka’s + Prim’s

Suppose , then

# of vertices is

Boruvka’s + Prim’s

Boruvka’s AlgorithmFor iterations

Prim’s Algorithm𝐺∗

𝑂 ¿

𝑂 (𝑚+𝑛∗ log𝑛)=𝑂 (𝑚+ 𝑛log𝑛

log𝑛)=𝑂 (𝑚+𝑛)

Boruvka’s + Prim’s

𝑂 (𝑚 log log𝑛)+𝑂(𝑚+ 𝑛log𝑛

log𝑛)=𝑂 (𝑚 log log𝑛)

Boruvka’s Algorithm Prim’s Algorithm

Questions?

Similarity to Dijkstra’sDIJKSTRA’S ALGORITHM

PRIM’S ALGORITHM