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Research in Algorithms -- Spanning Trees and Optimization Problems

Research in Algorithms -- Spanning Trees and Optimization Problems. 樹德科技大學 資訊工程系 吳邦一 (B. Y. Wu). 他們在做些什麼. 做 Algorithm 的是不食人間煙火的 從實驗室設備來觀察 … Algorithm 的研究僅需要一枝筆與一張紙. What the computer scientists work for?. Solving problems by computation Efficiently and correctly

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Research in Algorithms -- Spanning Trees and Optimization Problems

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  1. Research in Algorithms-- Spanning Trees and Optimization Problems 樹德科技大學 資訊工程系 吳邦一 (B. Y. Wu)

  2. 他們在做些什麼 • 做Algorithm的是不食人間煙火的 • 從實驗室設備來觀察 … • Algorithm的研究僅需要一枝筆與一張紙

  3. What the computer scientists work for? • Solving problems by computation • Efficiently and correctly • more powerful computing machines • more efficient algorithms • The problems • In computer science • In other fields

  4. Research in algorithms Model Motivations Problem Data

  5. Research in algorithms • Algorithm design • Algorithm analysis • Problem analysis • Data analysis

  6. Algorithm analysis • How good an algorithm is • Mathematics • Experiments (simulation by programs) • The objectives vary with models • Time • Space • Message passing

  7. Measurement of the goodness • Asymptotically • The Big-O notation • Tractable vs. intractable • Polynomial vs. exponential • NP-complete theory

  8. Algorithms and Graphs • Graph: • A mathematic model for relations • Usually the binary relation • 許許多多重要的問題可以用GRAPH來描述 • 對GRAPH的研究有助於問題的解

  9. Algorithm researchers vs. Mathematicians • M: • 在一graph中與其它點之最遠距離最小的稱為center, 一個tree中必然存在center但可能有一個或兩個center • A: given a tree Twhile T has more than two nodes do Remove all leaves in T; output the remaining nodes as the centers of T • The centers of a tree can be found in linear time.

  10. Special graphs • DATA對問題難度的影響 • Find the distance in a graph • Find the distance in a tree • A solution for all data • We hope but not always possible • 退而求其次 • Motivation

  11. Special graphs • 根據graph的特性所定義 • 太多的graph class已被定義出來 • Perfect graphs • Triangulated graphs • Comparability graphs • Permutation graphs • Interval graphs • Circular- Arc graphs

  12. Approximation algorithms

  13. NP-hard: the barrier • Since 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

  14. 艱困而逐漸褪色 • 逐水草而居 • Life finds the ways • Approximation • Online • Distributed • Mobile • New models • Quantum computing • Bio-computing

  15. 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

  16. 最高境界: 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

  17. An example -- TSP • 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

  18. 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

  19. The error ratio • MST<=TSP • 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

  20. Spanning Trees & Optimization Problems

  21. Two classical problems • Minimum spanning trees (MST) • Minimum building cost • Prim’s & Kruskal’s algorithm • Shortest-paths trees (SPT) • Shortest paths from one node to all the others • Dijkstra’s algorithm

  22. Minimum routing cost spanning trees • A spanning tree with minimum all-to-all distance • NP-hard in the strong sense • Tree with short edges may have large routing cost

  23. Approximation– comparing with a trivial lower bound • A lower bound • d(T,u,v)>=d(G,u,v) • Opt>=Σd(G,u,v) • The median of G: a node m min Σvd(G,m,v) • Since min<=mean, Σvd(G,m,v)<=(1/n) Σd(G,u,v)

  24. Y : a shortest path tree rooted at m • d(Y,i,j)<=d(Y,i,m)+d(Y,m,j) • Σd(G,u,v)<=2nΣvd(G,m,v)<=2*OPT • A shortest path tree rooted at the median is a 2-approximation of the MRCT.

  25. Solution decomposition • 假設T是一個OPT, 我們將T做一些處理, 得到另一個解Y, 使得 • Y的cost不至於與T相差太多 • Y屬於某一種特殊類別的解, 而這類別中的最佳解是可以在polynomial time 求得的 • 注意: 我們無法得知Y, Y並不會出現在algorithm中, 只在分析中扮演一箇中計的角色

  26. Metric MRCT • 假設T是OPT, m是T的centroid • 一個tree的centroid是去掉它的話, 剩下的subtree均不會超過一半的node • 在計算cost時, d(T,m,v)至少被計算n次 • opt>=nΣvd(T,m,v) • Let Y: the star centered at m • Y is a 2-approximation m >=n/2 v

  27. 利用solution decomposition証得 • 存在一個star是2-approximation • 已窮舉法嘗試所有的star (n個)並取出最好的, 必然是一個2-approximatin • Can we do better?

  28. 3-star =>1.5-approximation • K-star => (k+3)/(k+1)-approxiamtion • The best k-star for fixed k can be found in polynomial time • We have a PTAS

  29. Optimum Communication spanning Tree • A generalized version of MRCT • Min Σλ(u,v)d(T,u,v), λ(u,v) is the requirement • Product-requirement OCT (PROCT) • λ(u,v)=r(u)r(v) • Sum-requirement OCT (SROCT) • λ(u,v)=r(u)+r(v)

  30. Multiple sources • Single source: shortest-paths tree • All sources: MRCT • Multiple source: k-MRCT • 2-MRCT is also NP-hard • 2-MRCT admits a PTAS

  31. Steiner tree

  32. Spanning Trees and Optimization Problems S T O P ! Spanning Trees Optimization Problems & B.Y. Wu and K.M. Chao CRC press, 2004 即將出版

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