minimum spanning trees kun-mao chao ( 趙坤茂 ) department of computer science and information...
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Minimum Spanning Trees
Kun-Mao Chao (趙坤茂 )Department of Computer Science an
d Information EngineeringNational Taiwan University, Taiwan
E-mail: [email protected]
WWW: http://www.csie.ntu.edu.tw/~kmchao
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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
work efficiently.
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Its minimum spanning trees?
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Some of its minimum spanning trees
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(a) (b)
(a, b, c, d, e, f)
(g, h)
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(c)
(a, b, c, d, e, f)
(g, h)
2(a, b, c, d, e, f, g, h)
(d)
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(e)
Bor\r{u}vka's Algorithm
(1926)
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(a) (d, e) is added.
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(b) (g, h) is added.
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(c) (e, f) is added.
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(d) (b, d) is added.
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(e) (e, g) is added.
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(f) (b, c) is added.
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(g) (a, b) is added.
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(h) A minimum spanning tree.
Kruskal's Algorithm
(1956)
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(a) (a, b) is added.
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(b) (b, d) is added.
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(c) (d, e) is added.
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(d) (d, f) is added.
4 3
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1 1
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(e) (f, h) is added.
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1 1
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(f) (g, h) is added.
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(g) (b, c) is added.
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(h) A minimum spanning tree.
Prim’s Algorithm
(1957)
Approximation algorithms
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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– Heuristic or for special data
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Too hard to survive?
Life finds the ways– Approximation – Online – Distributed– 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).– If (1/k) not in the exponent, FPTAS
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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
connected.
• 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 theidea of approximation.
-- Bertrand Russell (1872 - 1970)