1 / 34

Efficient Triangulation Algorithm for Junction Trees Explained

This presentation delves into a triangulation algorithm for junction trees based on a paper by Ann Becker and Dan Geiger. It covers the natural approach, properties of decomposition, and the algorithm's correctness. Includes proofs, algorithms, complexity analysis, and formal definitions.

brendahahn
Download Presentation

Efficient Triangulation Algorithm for Junction Trees Explained

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. Graph Triangulation by Dmitry Pidan Based on the paper “A sufficiently fast algorithm for finding close to optimal junction tree” by Ann Becker and Dan Geiger

  2. Definition: junction tree

  3. The natural approach

  4. Example ==> ==>

  5. The natural approach • X is called a “minimum vertex cut” • The main disadvantage – there is no guarantee on the size of the maximal clique in an output triangulated graph

  6. Graph decomposition

  7. Graph decomposition (cont.)

  8. Example a b c d e

  9. Properties of decomposition - Lemma 1

  10. Properties of decomposition - Lemma 2

  11. Triangulation algorithm

  12. Example (one-level recursive call) h a d f c g b e i j k

  13. Trialgulation algorithm - intuition • We use a set W as a “balance factor” between the decomposition sets A, B and C – we are interested that a largest set will be as small as possible. • At every iteration a produced clique is kept small (due to the guarantees of the decomposition)

  14. Triangulation algorithm (cont.)

  15. Proof of correctness • Termination • Validity of the failure statement – follows immediately from Lemma 2 • An output in the case of success is a triangulated graph • Cliquewidth in the case of success is as guaranteed

  16. Proof of correctness (cont.)

  17. Proof of correctness (cont.)

  18. Proof of correctness (cont.)

  19. Finding a decomposition

  20. Finding a decomposition (cont.) • The existence of W-decomposition is checked as follows: • First, a decomposition of graph into disconnected components is found, using approximation algorithm for weighted minimal vertex cut problem • Next, A, B and C components of the decomposition are constructed by unifying the components that contain an appropriate subsets of W

  21. Finding a decomposition (cont.) • Finally, X is constructed from an initial common subset of W and X unified with the vertex cut found. If X stands for the size requirements then the decomposition is a required one. • More formally – in the next 3 slides

  22. The 3-way vertex cut problem • Definition: given a weighted undirected graph and three vertices, find a set of vertices of minimum weight whose removal leaves each of the three vertices disconnected from other two. • Known to be NP-hard • Polynomial approximation algorithms: • A simple 2-approximation algorithm • 4/3-approximation algorithm • Garg N. et al, “Multiway cuts in directed and node-weigthed graphs”

  23. Finding a decomposition: Procedure I

  24. Finding a decomposition: Procedure II

  25. Complexity

  26. Backup slides – proofs and some formalism

  27. Proof of Lemma 1

  28. Proof of Lemma 1 (cont.)

  29. Proof of Lemma 1 (cont.)

  30. Proof of Lemma 2

  31. Proof of Lemma 2 (cont.)

  32. Theorem 1 (formal definition of algorithm correctness)

  33. Finding a decomposition - proof of correctness

  34. Finding a decomposition - proof of correctness (cont.)

More Related