CSE 780 Algorithms Advanced Algorithms Minimum spanning tree Generic algorithm Kruskal’s algorithm Prim’s algorithm.

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 thatA 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 = (Vc ,Ec ) be a connected component in the graph

(forest) GA= (V, A). If (u, v) is a light edge connecting C to some other component in GA

, 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 finda 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 edgesDisjoint 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 = {S1,S2, …,Sk} 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, Sx and Sy, eliminating Sx and Sy

from S. FIND-SET(x): returns a pointer to the

representative of the set containing x.

CSE 780 Algorithms

Example

CSE 780 Algorithms

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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

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CSE 780 Algorithms

Example

CSE 780 Algorithms

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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 complexityO(|E| + |V| log |V|)

CSE 780 Algorithms

Applications

Clustering

Euclidean traveling salesman problem