1 / 25

Classifying Customer-Provider Relationships in the Internet

Classifying Customer-Provider Relationships in the Internet. Thomas Erlebach,Alexander Hall Computer Engineering and Networks Lab.,ETH Z űrich Thomas Schank Dep.of Computer & Information Science,Universit ä t Konstanz. jp. us. AS AS peer peer.

jaimin
Download Presentation

Classifying Customer-Provider Relationships in the Internet

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. ClassifyingCustomer-Provider Relationships in the Internet Thomas Erlebach,Alexander Hall Computer Engineering and Networks Lab.,ETH Zűrich Thomas Schank Dep.of Computer & Information Science,Universität Konstanz

  2. jp us AS AS peer peer wang@dais.is.tohoku.ac.jpから 中村　先生（nakamura@umunhum.stanford.edu.us）にメールを送信する場合では ac edu tohoku berkeley stanford is riec umunhum AS AS customer provider dais

  3. Preliminaries Given a simple,undirected graph G=(V,E) and a set P of simple ,undirected paths in G. • if it starts with zero or more customer-provider edges • followed by zero or one peer-peeredge • followed by zero or more provider-customer edges

  4. Examples × ×

  5. Lemma2 • for a given edge classification , is vaild no source node in it • peer-peer edges can be completely disregarded. ×

  6. The problems in this paper • Type-of-Relationship problem(ToR) given a graph G=(V,E) and a set P of simple,undirected paths in G,classifying the edges of the graph into customer-provider relationships such that as many of the paths in P as possible are valid. • ALLToR decide of G s.t. are valid,and compute it if it exists. • MAXToR compute an orientation of G s.t. ,

  7. The ALLToR Problem ---------solve ALLToR in linear time by reducing it to 2SAT Lemma 3: • of length k can be split up into k-1 paths , of length 2 and • Of G , p is valid all are valid. Note:edge appears negated if it is pointing away from the internal node of path and not negated if it is pointing towards

  8. The ALLToR Problem ---------what is 2SAT problem • SAT problem variables: clauses: a literal is a variable or the negation of a variable and a clause is a set of literals. A clause is true is one of the literals in the clause is true. the input to SAT is a collection of clauses the output is the answer to: Is there an assignment of true/false to the variables so that every clause is satisfied (satisfied means the clause is true)? when all clauses ,we call it 2SAT problem.

  9. The ALLToR Problem ---------solve ALLToR in linear time by reducing it to 2SAT 1.Orient the edges of G=(V,E)arbitrarily. 2.Split all paths into paths of length 2,where is the length of path . 3.Construct a 2SAT instance where each edge corresponds to a variable and each path (of length 2) corresponds to a clause . 4.Solve the resulting 2SAT instance. 5. If the 2SAT instance is not satisfiablethe ALLToR is not solvable. otherwise,flip :whose corresponding variable has been assigned false by 2SAT

  10. The MAXToR Problem---------prove that MAXToR is NP-hard • To prove that MAXToR is NP-hard,we wil give an reduction from the well known NP-hard maximum independent set problem(MAXIS) to MAXToR. Lemma 4 with two paths s.t.同時にvalidできない

  11. The MAXToR Problem---------prove that MAXToR is NP-hard • What is MAXIS problem? ・INSTANCE: Graph . ・SOLUTION: An independent set of vertices s.t. arenot joined by an edge max( )

  12. The MAXToR Problem---------prove that MAXToR is NP-hard MAXISのinstance 　　　　　　　 からMAXToRのinstance　　　　を作る ・ ー> と　　はedge-disjoint. ・ ー> と　　は同時にvalidにしないように。

  13. The MAXToR Problem---------conclusion もしMAXToRは多項式時間で解ける或いは近似解を求められるならば、MAXISに対しても同じことを言える。 しかし、MAXISはNP-困難である、そして任意の　　　　に対して近似率は　　　の近似アルゴリズムは存在しないことは既に知られている。

  14. Approximating MAXToR instances with bounded path length • a simple approximation algorithm 各edgeに対して勝手に方向をつける。 a path of length is valid with probability

  15. Approximating MAXToR instances with bounded path length 　　　　　　　　　　　　　　　　　　　　　　　　　ならば 　　　：近似解、　　　：最適解

  16. Approximating MAXToR instances with bounded path length • MAX2SATを用い、もっと良い近似率を得られる。 方針： 符号の定義：

  17. Approximating MAXToR instances with bounded path length ①パスの長さ＝２のとき， MAXToRの近似率＝0.931(MAX2SATの近似率)

  18. Approximating MAXToR instances with bounded path length ②パスの長さ＝３のとき，

  19. Approximating MAXToR instances with bounded path length ②パスの長さ＝３のとき， 同様に 既存研究によって

  20. Approximating MAXToR instances with bounded path length ③パスの長さ＝４のとき， ②の場合と同じ方法で評価する ④パスの長さ＞４のとき，以上の方法は助からない。

  21. Approximating MAXToR instances with bounded path length

  22. MAXToRはAPX-完全 ・すべてのパス長＝２の場合でも、MAXToRはある定数を超える近似率を 　得られない。 方針：

  23. MAXToRはAPX-完全 • ． • ． • ． 作られたMAXToRのinstanceの解に対して、

  24. MAXToRはAPX-完全 パス長＝２の場合に対応するため、

  25. MAXToRはAPX-完全 同様に 既存研究によって

More Related