1 / 41

Approximation algorithm Design a case study of MRCT

Approximation algorithm Design a case study of MRCT. 樹德科技大學 資訊工程系 吳邦一 (B. Y. Wu). 1988 – before studying algorithms. 2000 – after studying algorithms. Ron Rivest. Leonard Adleman. Adi Shamir. RSA. Last year, after Prof. Chang went to NSYSU for a speech,

carlos-ruiz
Télécharger la présentation

Approximation algorithm Design a case study of MRCT

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Approximation algorithm Design a case study of MRCT 樹德科技大學 資訊工程系 吳邦一 (B. Y. Wu)

  2. 1988 – before studying algorithms 2000 – after studying algorithms

  3. Ron Rivest Leonard Adleman Adi Shamir RSA Last year, after Prof. Chang went to NSYSU for a speech, A student asked me for a question: 為何做演算法的人皆白髮而做security的易禿頭？

  4. Algorithm research and NP-Complete Theorem

  5. NP-hard: the barrier • Since the results of Cook (1971) , Levin(1973) & Karp (1972), many important problems have been shown to be NP-hard. Karp 1985 Turing Award Cook 1982 Turing Award Levin

  6. The NPC Theorem • The name “NP-Complete” is due to Knuth（高德納） • Garey and Johnson在1979年所著的Computers and Intractability: A Guide to the Theory of NP-Completeness書中蒐列了數以百計的重要NPC問題，到今天，NPC的問題已經列不勝列了。 • According to Wikipedia（維基百科）, 在2002 年的一項調查中，一百位研究者裡面有61位相信NP不等於P，9位相信NP=P，22位不確定，而有8位研究者認為此問題在目前的假設基礎下是無法證明的。 Knuth 1974 Turing Award Johnson

  7. For an NP-Complete or NP-hard problem, it is not expected to find an efficient algorithm. Or maybe you need the 1,000,000 USD award • In 70s, the life-cycle of a problem • Defined • NP-hard • Heuristic or for special data

  8. 艱困而逐漸褪色 • Life finds the ways • Approximation • Online • Distributed • Mobile • New models • Quantum computing • Bio-computing

  9. Approximation algorithms

  10. Approximation algorithms • For optimization (min/max) problem • Heuristic vs. approximation algorithms • Ensuring the worst-case quality • The error • Relative and Absolute • A function k of input size n. A k-approximation: • minimization: sol/opt<=k;maximization: opt/sol<=k • The ratio is always >1

  11. 最高境界: Polynomial time approximation scheme • Some algorithms are of fixed ratio • Approximation scheme: allow us to make trade-off between time and quality • The more time, the better quality • PTAS: 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). • FPTAS if (1/k) not in the exponent,

  12. The first PTAS (Not sure) • In Ronald L. Graham’s 1969 paper for scheduling problem (Contribution also due to Knuth and another)

  13. 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

  14. The 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

  15. 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

  16. Optimum communication spanning tree Problems

  17. OCT: definition • Input: • an undirected graph with nonnegative edge lengths • a nonnegative requirement for each pair of vertices • Output: • a spanning tree minimizing the total communication cost summed over all pairs of vertices, in which the cost of a vertex pair is the distance multiplied by their requirement, that is, we want to minimize Σ λi,j dT(i,j)

  18. First studied by T.C. Hu 1974 SICOMP First approximation appeared in Wong 1980

  19. A way to a PTAS A case study of the MRCT problem Optimum Communication Spanning Trees

  20. 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

  21. Approximation– comparing with a trivial lower bound • A lower bound • d(T,u,v)>=d(G,u,v) (樹上距離<=原圖最短路徑) • 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)

  22. 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. m j j i

  23. To find an approx. • A lower bound of the optimum • An algorithm • Analyze the worst-case ratio

  24. 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, 只在分析中扮演一個中繼的角色

  25. Metric MRCT • For easy to understand, we consider only the metric case • The input is a metric graph: a complete graph with edge length satisfying the triangle inequality

  26. Metric MRCT • 假設T是OPT, r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,r,v)至少被計算n次 • opt>=nΣvd(T,r,v) • Let Y: the star centered at r • C(Y)= 2(n-1)Σvd(Y,r,v) • Y is a 2-approximation r >=n/2 v

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

  28. δ-separator • Separator of a tree: • Centroid is a ½ separator • How the 2-approx. algorithm works? • Guess (try all possible) the separator • Connect the others greedily • Distance increases only for nodes in the same branch -- we don’t pay too much

  29. To get better result, we try to generalize the centroid to general δ-separator • Indeed, when δ↘, the error↘ • But it costs too much to obtain the exact δ-separator for δ<1/2. • For example, a 1/3-separator may have n/3 nodes 1/3-separator n/3 n/3

  30. 屬下犧牲了上司也該犧牲 • We don’t need a perfect separator • Only some critical nodes are necessary • Leaves of the separator (確保下屬有個好的依歸) • Branch nodes of the separator(確保結構) δ-separator

  31. To a k-Star • k-star: a tree with at most k internal nodes • Need some other work to show the ratio(通常這樣的話代表了背後有慘不忍睹的內容)

  32. Solution decomposition • 從一個OPT開始，我們設法將他改造成一個k-star，並證明此k-star是一個不錯的approximation • 設計一個演算法可以求得最好的k-star，既然他是最好，當然不比那個改造的差 • 精緻的分析是重要的，「好，要說的出口」

  33. 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

  34. 一些經驗之談 • Evolutionary tree reconstruction • 給一個n個物種的距離矩陣，找一個tree以此n個物種為leaves, 使得兩兩物種之間在樹上的距離>=給定的距離且最小化距離總合 • 這個問題比較難，因為樹的中間節點是可以任意給的 • Steiner tree vs. Spanning tree

  35. 花了不少時間study Steiner tree • 先做做Spanning 的case • MRCT • 找到separator的方法 • (15/8)-approx => 1.577 =>1.5 =>4/3+ • 兩種extension • 這個方法在general graph上不可能做到比4/3+更好了

  36. 困難點在於受限於shortest path tree • 如果是metric graph就有可能做到更好 • 但是metric graph的case還不知是不是NP-hard • 對於證明NPC實在是很厭煩了 • 把Garey & Johnson的書翻了又翻 • 遠在天邊 近在眼前 • 把general case transform 到metric case • 不只解決NP-hard的疑問, 證明了metric上的approx. 可以用在general case 上

  37. 找到k-star的方法 • 意外的插曲 • 研究是很競爭的 • 提心吊膽 難以入眠 • 謎底揭曉的那一刻 • 1997年，我做到了兩年來作夢都夢不到的事 • 更多的extension

  38. 做研究是在千百次失敗中期待一次成功 • 做行政是在千百次成功中等待一次失敗 • 研究之路很迷人，如果有人結伴而行則更加美好（當學生很幸福啊！） • 李老師告訴我說：沒有計畫，只有方向 • 研究如此，人生何嘗不是

  39. Thanks Q&A

More Related