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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,

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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)

slide2

1988 – before studying algorithms

2000 – after studying algorithms

slide3

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?

np hard the barrier
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

the npc theorem
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

slide7
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
slide8
艱困而逐漸褪色
  • Life finds the ways
    • Approximation
    • Online
    • Distributed
    • Mobile
    • New models
      • Quantum computing
      • Bio-computing
approximation algorithms1
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
polynomial time approximation scheme
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,
the first ptas not sure
The first PTAS (Not sure)
  • In Ronald L. Graham’s 1969 paper for scheduling problem (Contribution also due to Knuth and another)
an example tsp
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

the doubling tree algorithm
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

the error ratio
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
oct definition
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)
slide18

First studied by T.C. Hu 1974 SICOMP

First approximation appeared in Wong 1980

a way to a ptas

A way to a PTAS

A case study of the MRCT problem Optimum Communication Spanning Trees

minimum routing cost spanning trees
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
approximation comparing with a trivial lower bound
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)
slide23
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

to find an approx
To find an approx.
  • A lower bound of the optimum
  • An algorithm
  • Analyze the worst-case ratio
solution decomposition
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
metric mrct
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
metric mrct1
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

slide28
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?
slide29

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
slide30
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

slide31
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

to a k star
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)
solution decomposition1
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」
slide34
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
slide35
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
slide37
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
slide38
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
slide39
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
slide40
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