1. Minimum Spanning Trees
Kun-Mao Chao (趙坤茂)
Department of Computer Science and Information Engineering
National Taiwan University, Taiwan
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2. 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.

3. 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.

4. Its minimum spanning trees?

5. Some of its minimum spanning trees

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

7. Kruskal's Algorithm
(1956)

8. Prim’s Algorithm
(1957)

9. 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.

10. Approximation algorithms

11. 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

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

13. 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

14. 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

15. 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

16. 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

17. 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?

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