Minimum Spanning Trees Kun-Mao Chao (趙坤茂) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

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.

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 work efficiently.

Its minimum spanning trees?

Some of its minimum spanning trees

Bor\r{u}vka's Algorithm (1926)

Kruskal's Algorithm (1956)

Prim’s Algorithm (1957)

A simple hybrid algorithm Bor\r{u}vka's Algorithm: O(m log n) Prim’s Algorithm: O(m + n log n) Kruskal’s Algorithm: O(m log m) A hybrid algorithm of Bor\r{u}vka's Algorithm and Prim’s Algorithm: O(m loglog n). How? Hint: First apply the contraction step in Bor\r{u}vka's Algorithm for O(loglog n) time.

Approximation algorithms

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 Heuristic or for special data

Too hard to survive? Life finds the ways Approximation Online Distributed New models Quantum computing Bio-computing

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

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). If (1/k) not in the exponent, FPTAS

An example – Minimum tour (MT) Starting at a node, find a tour of min distance traveling all nodes and back to the starting node. 6 8 2 15 10 5 3 10 2

A doubling tree algorithm Find a minimum spanning tree Output the Euler tour in the doubling tree of MST 6 6 8 8 2 2 15 15 10 10 5 5 3 3 10 10 2 2

The error ratio MST<=MT It is a 2-approximation MST is the minimum cost of any spanning tree. A tour must contain a spanning tree since it is connected. It is a 2-approximation Triangle inequality => 2-approximation for TSP (visiting each city only once) Why?

All exact science is dominated by the idea of approximation. -- Bertrand Russell (1872 - 1970)