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

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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

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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

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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