Minimum Spanning Trees

Minimum Spanning Trees

Kun-Mao Chao ( 趙坤茂 )

Department of Computer Science an d Information Engineering

National Taiwan University, Taiwan

E-mail: kmchao@csie.ntu.edu.tw

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Minimum Spanning Trees

• A minimum spanning tree (MST) of a weighted graph G is a spanning tree of G whose edges sum to minimum weight.

• In other words, a minimum spanning tree is a tree formed from a subset of the edges in a given

undirected graph, with two properties:

– it spans the graph, i.e., it includes every vertex in the graph, and

– it is a minimum, i.e., the total weight of all the edges is as low as possible.

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Minimum Spanning Trees

The minimum spanning tree problem is always included in algorithm textbooks since

• it arises in many applications,

• it is an important example where greedy

algorithms always deliver an optimal solution, and • clever data structures are necessary to make it

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Bor\r{u}vka's

Algorithm

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Kruskal's

Algorithm

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Prim’s Alg

orithm

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NP-hard: the barrier

• Since Levin & Cook (1971) & Karp (1972),

many important problems have been shown

to be NP-hard.

• The life-cycle of a problem

– Defined – NP-hard

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Too hard to survive?

Life finds the ways

– Approximation – Online – Distributed – Mobile – New models • Quantum computing • Bio-computing

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Approximation algorithms

• Heuristic vs. approximation algorithms

– Ensuring the worst-case quality

• The error ratio

– Relative and Absolute – A k-approximation:

minimization: sol/opt<=k; maximization: opt/sol<=k – The ratio is always >1

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Polynomial time approximation

scheme

• For any fixed k>0, it finds a

(1+k)-approximation in polynomial time.

– Usually (1/k) appears in the time complexity, e.q. O(n/k), O(n1/k).

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An example – Minimum tour (MT)

• Starting at a node, find a tour of min

distance traveling all nodes and back to the

starting node.

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A doubling tree algorithm

• Find a minimum spanning tree

• Output the Euler tour in the doubling tree of

MST

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The error ratio

• MST<=MT

– MST is the minimum cost of any spanning tree. – A tour must contain a spanning tree since it is c

onnected.

• It is a 2-approximation

• Triangle inequality => 2-approximation for

TSP (visiting each city only once) Why?

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All exact science is dominated by the

idea of approximation.