EE512 Graphical Models Fall 2009

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1 EE512 Graphical Models Fall 2009 Prof. Jeff Bilmes University of Washington, Seattle Department of Electrical Engineering Fall Quarter, Lecture 13 - Nov 12th, 2009 Last updated $Id: lec13.tex,v /11/18 10:16:30 bilmes Exp $ Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 1 / 45

2 Class Road Map L1 (10/1): intro, cond. indep., ex. GMs L2 (10/6): GMs and MRFs L3 (10/8): MRFs, mobius, FGs L4 (10/13): Sem BNs L5 (10/15): Sem BNs 2. L6 (10/20): Evidence L7 (10/22): Inf. Trees 1 L8 (10/27): Inf. Trees 2 L9 (10/29): Inf. Trees 3 L10 (11/03): Inf. Trees 4 L11 (11/05): Inf. Trees 5 L12 (11/10): Inf. Trees 6 L13 (11/12): L14 (11/17): L15 (11/19): L16 (11/24): L17 (12/01): L18 (12/03): L19 (12/08): (video lecture) L20 (12/10): (video lecture) L21 (12/18): Friday, 2:30-4:20pm, final presentations Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 2 / 45

3 Readings Read tree inference.pdf Read evidence.pdf Read dgms.pdf Read ugms.pdf. Read intro.pdf. Optionally (but encouraged): read chapters 1 through 10 in Jordan text. Read relevant chapters in Koller/Friedman text. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 3 / 45

4 Homework/Project Project proposals due in 1 week from now (Thursday, Nov 19th, 11:59pm). What to turn in: One page proposals that describe what you plan to do for a project, again me PDF files (only PDF accepted). I ll comment on the PDF and mail it back to you. Final project will be a 4-page conference-style research paper due on Dec. 17th. Project should ideally be on some aspect of the material we have learnt. It could also be an implementation (i.e., a fast implementation of the JT algorithm, or loopy BP, and some reporting and experiences that you have had in doing this). While it ideally should be research-oriented, it is not acceptable to propose whatever machine learning task you are currently working on (e.g., An application of SVMs to protein folding would not be acceptable). Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 4 / 45

5 the MAXCLIQUE NP-COMPLETE problem There are a class of related problems that equivalently indicate the difficulty were are in. Theorem Given an arbitrary graph G = (V, E), find the largest clique C V (G), where large is measured in terms of C is an NP-complete optimization problem. Approximation algorithms - possible to do no worse than O((log V ) 2 / V ) times size of true maximum size clique. Inapproximable V 1/2 ɛ for any ɛ > 0. If we could find the smallest k such that it could be embedded it a k tree, we could identify the maximum clique in the graph. How? Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 5 / 45

6 Review Maximum-Cardinality search (MCS) (repeatedly select next node that has greatest number of previously selected neighbors). O( V + E ). Key insight - try to generate perfect elimination order in reverse. Want to do multiple queries, clique queries vs. non-clique queries, do it cheaply re-use work. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 6 / 45

7 Decomposition of G and Decomposable graphs Definition (Decomposition of G) A decomposition of a graph G = (V, E) (if it exists) is a partition (A, B, C) of V such that: C separates A from B in G. C is a clique. if A and B are both non-empty, then the decomposition is called proper. Definition A graph G = (V, E) is decomposable if either: 1) G is a clique, or 2) G possesses a proper decomposition (A, B, C) s.t. both subgraphs G[A C] and G[B C] are decomposable. Note part 2. It says possesses. Bottom of tree might affect top. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 7 / 45

8 Decomposable models and decomposition trees CDF A ACDF CH DK B C F D E BH DI EIJ EI DI BH BGH CK G H I J BGH BCH EI CK DIK DEI CH CDF K EIJ DEI CHK DK BCH CHK ACDF CDFK DIK CDFK p(x) = C C(G) p(x C ) S S(G) p(x S) d(s) 1 (1) Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 8 / 45

9 Decomposable models Proposition All of the maxcliques in a graph lie on the leaf nodes of the binary decomposition tree Proposition The set of all minimal separators of graph constitute the unique non-leaf nodes of the binary decomposition tree, with d(s) 1 being the number of times the minimal separator appears as a given non-leaf node. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 9 / 45

10 Triangulated vs. decomposable Theorem A given graph G = (V, E) is triangulated iff it is decomposable. So the following graph classes are identical: chordal (triangulated) graphs Those that have a perfect elimination order (one where elimination can be run w/o any fill-in) Those that are decomposable Those that MCS algorithm outputs chordal Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 10 / 45

11 Cluster graphs Definition (Cluster graph) Consider forming a new graph based on G where the new graph has nodes that correspond to clusters in the original G, and has edges existing between two (cluster) nodes only when the corresponding clusters have a non-zero intersection. That is, let C(G) = { C 1, C 2,..., C I } = be a set of I clusters of nodes V (G), where C i V (G), i I. Consider a new graph J = (I, E) where each node in J corresponds to a set of nodes in G, and where edge (i, j) E if C i C j. We will also use S ij = C i C j as notation. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 11 / 45

12 Cluster Trees If the graph is a tree, then we have what is called a cluster tree. Definition (Cluster Tree) Let C = { C 1, C 2,..., C I } be a set of node clusters of graph G = (V, E). A cluster tree is a tree T = (I, E T ) with vertices corresponding to clusters in C and edges corresponding to pairs of clusters C 1, C 2 C. We can label each vertex in i I by the set of graph nodes in the corresponding cluster in G, and we label each edge (i, j) E T by the cluster intersection, i.e., S ij = C i C j. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 12 / 45

13 Cluster Graphs/Trees F G F,G,A,B H,F,A,K,C H,C,I H A B C K D E ABK C,D,K D,K,E,J I J Left: a graph. Right: A cluster graph with I = 6 clusters, where C 1 = {F, G, A, B}, C 2 = {H, F, A, K, C},.... There is an edge (1, 2) since C 1 C 2 = {F, A}. If we remove all but the blue edges, then we get a cluster tree. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 13 / 45

14 Cluster Intersection Property (c.i.p.) Definition (Cluster Intersection Property) We are given a cluster tree T = (I, E T ), and let C 1, C 2 be any two clusters in the tree. Then the cluster intersection property states that C 1 C 2 C i for all C i on the (by definition, necessarily) unique path between C 1 and C 2 in the tree T. A given cluster tree might or might not have that property. Example Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 14 / 45

15 Running Intersection Property (r.i.p.) Definition (Running Intersection Property (r.i.p.)) Let C 1, C 2,..., C l be an ordered sequence of subsets of V (G). Then the ordering obeys the running intersection property (r.i.p.) property if for all i > 1, there exists j < i such that C i ( k<i C k ) = C i C j. r.i.p. is defined in terms of clusters of nodes in a graph. r.i.p. holds if such an ordering can be found. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 15 / 45

16 Running Intersection Property (r.i.p.) Given sequence of clusters C 1, C 2,..., C l. Define the history (accumulation) of sequence at position i: H i = C 1 C 2 C i. (2) Innovation (residual) or new nodes in C i not encountered in the previous history, as: R i = C i \ H i 1. (3) Lastly, define the non-innovation, commonality, or separation elements between new and previous history: S i = C i H i 1 (4) Note C i = R i S i, i th clusters consists of the innovation R i and the commonality S i. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 16 / 45

17 Running Intersection Property (r.i.p.) H i C j S i C i S i = (C 1 C 2 C i 1 ) C i = C j C i Clusters are in r.i.p. order if the commonality S i between new and history is fully contained in one element of history. I.e., there exists an j < i such that S i C j. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 17 / 45

18 First Two Properties Lemma The cluster intersection and running intersection properties are identical. Proof. Starting with clusters in r.i.p. order, construct cluster tree by connecting each i to its corresponding j node. This is a tree. Also, take any C i, C k with i > k. S i summarizes everything between C i and H i 1 so C i C k S i. Apply recursively on unique path between C i and C j. Conversely, perform traversal (depth or breadth first search) on cluster tree. That order will satisfy r.i.p. since any possible intersection between C i, C j on unique path, it must be fully contained in neighbor. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 18 / 45

19 First Two Properties BCE BE BEH BC ABC AB ABD BD BDF Example of a set of node clusters (within the cloud-like shapes) arranged in a tree that satisfies the r.i.p. and also the cluster intersection property. The intersections between neighboring node clusters are shown in the figure as square boxes. Consider the path or {B, E, H} {B, D, F } = {B}. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 19 / 45

20 Induced sub-tree property (i.s.p.) Definition (Induced Sub-tree Property) Given a a cluster tree T for graph G, the induced sub-tree property holds for T if for all v V, the set of clusters C C such that v C induces a sub-tree T (v) of T. Note, by definition the sub-tree is necessarily connected. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 20 / 45

21 Three properties Lemma Induced sub-tree property holds whenever cluster intersection property holds Proof. Assume induced subtree holds. Take all v C i C j, then each such v induces a sub-tree of T, and all of these sub-trees overlap on the unique path between C i and C j in T. Conversely, when cluster intersection property holds, given v V, consider all clusters that contain v, C(v) = {C C : v C}. For any pair C 1, C 2 C(v), we have that C 1 C 2 exists on the unique path between C 1 and C 2 in T, and since v C 1 C 2, v always exists on each of these paths. These paths, considered as a union together, cannot form a cycle (since they are paths on a tree). Moreover, these paths unioned together form a tree (they re connected). Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 21 / 45

22 Tree decomposition A tree decomposition is a cluster tree that satisfies induced sub-tree property (e.g., r.i.p. and c.i.p. as well). That is: Definition (tree decomposition) Given a graph G = (V, E), a tree-decomposition of a graph is a pair ({C i : i I }, T ) where T = (I, E T ) is a tree with node index set I, edge set E T, and {C i } i (one for each i I ) is a collection of clusters (subsets) of V (G) such that: 1 i I C i = V 2 for any edge (u, v) E(G), there exists i I with u, v C i 3 (r.i.p.) for any v V, the set {i I : v C i } forms a (nec. connected) subtree of T Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 22 / 45

23 Tree decomposition is also hard The tree-width of the tree-decomposition is the size of the largest C i minus one (i.e., max i I C i 1. Theorem Given graph G = (V, E), finding the tree decomposition T = (I, F ) of G that minimizes the tree width (max i I C i 1) is an NP-complete optimization problem. Note that this is approximable within O(log V ) but it is not possible to better than V 1 ɛ for any ɛ > 0. How does this relate to our problem though? Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 23 / 45

24 Junction Tree Definition Given a graph G = (V, E), a junction tree corresponding to G (if it exists) is a cluster tree T = (C, E T ) having the r.i.p. over the clusters, and where thoe nodes u, v adjacent to every edge (u, v) E(G) are together in at least one cluster. If clusters correspond to the cliques (resp. maxcliques) in G, it is a junction tree of cliques (resp. maxcliques). Junction tree could be defined on the cliques or maxcliques. Can get a junction tree of cliques by taking a chain of subsets of every maxclique and appending the junction tree with that chain, and r.i.p. still holds. Can do the other direction by merging cliques into neighboring superset cliques until we have only maxcliques left. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 24 / 45

25 Examples junction trees and not A ABD ABD B C D E BCD D CD CDE BD BCD CD CDE not all trees of maxcliques satisfy the r.i.p. not all trees of maxcliques are junction trees no r.i.p. order in middle case right case is ok Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 25 / 45

26 Examples junction trees and not F G F,G,A,B H,F,A,K,C H,C,I F,G,A,B H,F,A,K,C H,C,I H A B C K ABK C,D,K D,K,E,J D E ABK C,D,K D,K,E,J I Given middle graph, we have two examples (left and right) of cluster trees. Are any of them junction trees for this graph? Are any of them junction trees for some graph? Why/why not? J Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 26 / 45

27 Examples junction trees and not F G F,G,A,B H,F,A,K,C H,C,I H A B C K D E A,B,K,F C,D,K D,K,E,J I J Given left graph, is the right example a junction tree for left graph? Is it a junction tree for some graph? Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 27 / 45

28 Examples junction trees and not F G F,G,A,B H,F,A,K,C H,C,I H A B C K D E A,B,K,F C,D,K D,K,E,J I J Given left graph, is the right example a junction tree for left graph? Is it a junction tree for some graph? Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 28 / 45

29 Examples junction trees and not CDF A ACDF CH DK B C F D E BH DI EIJ EI DI BH BGH CK G H I J BGH BCH EI CK DIK DEI CH CDF K EIJ DEI CHK DK BCH CHK ACDF CDFK Junction tree of cliques for above graph. Note that r.i.p. holds. DIK CDFK BGH CKH ACDF DKI IEJ BH CH CK CDF DK DI EI CBH DKFC EID Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 29 / 45

30 Key theorem: Junction tree of maxcliques vs. triangulated graphs Theorem A graph G = (V, E) is decomposable iff a junction tree of maxcliques for G exists. Proof. Induction on the number of maxcliques. If G has one maxclique, it is both a junction tree and decomposable. Assume true for k maxcliques and show it for k Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 30 / 45

31 Junction tree of maxcliques triangulated graphs JT implies Decomposable... proof continued. a junction tree exists decomposable: Let T be a junction tree of maxcliques C, and let C 1, C 2 be adjacent in T. The edge C 1, C 2 in the tree separates T into two sub-trees T 1 and T 2, with V i being the nodes in T i, G i = G[V i ] being the subgraph of G corresponding to T i, and C i being the set of maxcliques in T i, for i = 1, 2. Thus V (G) = V 1 V 2, and C = C 1 C 2. Note that C 1 C 2 =. We also let S = V 1 V 2 which is the intersection of all the nodes in each of the two trees. C 1 C 2 Tree T 1 with nodes V 1 forming Tree T 2 with nodes V 2 forming graph G 1 = G[V 1] and maxcliques CGraphical 1. Models Fall 2009 cliques Lecture C Nov 12th, 2009 page 31 / graph G 2 = G[V 2] and max- Prof. Jeff Bilmes EE512 45

32 Junction tree of maxcliques triangulated graphs JT implies Decomposable... proof continued. Also, the nodes in T i are maxcliques in G i and T i is a junction tree for G i since r.i.p. still holds in the subtrees of a junction tree. Therefore, by induction, G i is decomposable. To show that G is decomposable, we need to show that: 1) S = V 1 V 2 is complete, and 2) that S separates G[V 1 \ S] from G[V 2 \ S]. If v S, then for each G i (i = 1, 2), there exists a clique C i with v C i, and the path in T joining C 1 and C 2 passes through both C 1 and C 2. Because of the r.i.p., we thus have that v C 1 and v C 2 and so v C 1 C 2. This means that V 1 V 2 C 1 C 2. But C i V i since C i is a clique in the corresponding tree T i. Therefore C 1 C 2 V 1 V 2 = S, so that S = C 1 C 2. This means that S contains all nodes that are common among the two subgraphs and moreover that S is complete as desired.... Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 32 / 45

33 Junction tree of maxcliques triangulated graphs JT implies Decomposable... proof continued. Next, to show that S is a separator, we take u V 1 \ S and v V 2 \ S (note that such choices mean u V 2 and v V 1 due to the commonality property of S). Suppose the contrary that S does not separate V 1 from V 2, which means there exists a path u, w 1, w 2,..., w k, v for the given u, v with w i S for all i. Therefore, there is a clique C C containing the set {u, w 1 }. We must have C C 2 since u V 2, which means C C 1 or C V 1 implying that w 1 V 1 and moreover that w 1 V 1 \ S. We repeat this argument with w 1 taking the place of u and w 2 taking the place of w 1 in the path, and so on until we end up with v V 1 \ S which is a contradiction. Therefore, S must separate V 1 from V 2. We have thus formed a decomposition of G as (V 1 \ S, S, V 2 \ S) and since G i is decomposable (by induction), we have that G is decomposable.... Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 33 / 45

34 Junction tree of maxcliques triangulated graphs Decomposable implies JT... proof continued. decomposable a junction tree exists: Since G is decomposable, let (W 1, W 2, S) be a proper decomposition of G into decomposable subsets G 1 = G[V 1 ] and G 2 = G[V 2 ] with V i = W i S. By induction, since G 1 and G 2 are decomposable, there exits a junction tree T 1 and T 2 corresponding to maxcliques in G 1 and G 2. Since this is a decomposition, with separator S, we can form all maxcliques C = C 1 C 2 with C i maxcliques of V i for tree T i. Choose C 1 C 1 and C 2 C 2 such that S C 1 and S C 2 which is possible since S is complete, and must be contained in some maxclique in both T 1 and T 2. We form a new tree T by linking C 1 T 1 with C 2 T 2. We need next to ensure that this new junction tree satisfies r.i.p.... Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 34 / 45

35 Junction tree of maxcliques triangulated graphs Decomposable implies JT... proof continued. Let v V. If v V 2, then all cliques containing v are in C 1 and those cliques form a connected tree by the junction tree property since T 1 is a junction tree. The same is true if v V 1. Otherwise, if v S (meaning that v V 1 V 2 ), then the cliques in C i containing v are connected in T i including C i for i = 1, 2. But by forming T by connecting C 1 andc 2, and since v is arbitrary, we have retained the junction tree property. Thus, T is a junction tree. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 35 / 45

36 Cliques or Maxcliques Lemma A junction tree of maxcliques for graph G = (V, E) exists iff a junction tree of cliques for graph G = (V, E) exists. Corollary How can we get from one to the other? A graph G is triangulated iff a junction tree for G exists. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 36 / 45

37 How to build a junction tree Maximum cardinality search algorithm can do this. If graph is triangulated, it produces a list of cliques in r.i.p. order. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 37 / 45

38 Maximum Cardinality Search with maxclique order Algorithm 1: Maximum Cardinality Search: Determines if a graph G is triangulated. Input: An undirected graph G = (V, E) with n = V. Result: is triangulated?, if so MCS ordering σ = (v 1,..., v n ), and maxcliques in r.i.p. order. L ; i 1 ; C ; while V \ L > 0 do Choose v i argmax u V \L δ(u) L ; /* v i s previously labeled neighbors has max cardinality. */ c i δ(v i ) L ; /* c i is v i s neighbors in the reverse elimination order. */ if {v i } c i is not complete in G then return not triangulated ; if c i c i 1 then C (C, {c i 1 {v i 1 }}) ; /* Append the next maxclique to list C. */ if i = n then C (C, {c i {v i }}) ; ; /* Append the last maxclique to list C. */ L L {v i } i i + 1 ; return triangulated, the ordering σ, and the set of maxcliques C which are in r.i.p. order. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 38 / 45

39 Note: graph must be triangulated. I.e., maximum spanning tree of a cluster graph where the clusters are maxcliques but the graph is not triangulated will clearly not produce a Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 39 / 45 Logistics Review Junction Trees Intersection Graphs How to build a junction tree Alternatively, we can construct the maxcliques in any form (say by running elimination) and find a maximal spanning tree over the edge-weighted cluster graph, where clusters correspond to maxcliques, and edge weights correspond to the size of the intersection of the two adjacent maxcliques. Prim s algorithm can run in O( E + V log V ), much better than V 2 for sparse graphs. Theorem A tree of maxcliques T is a junction tree iff it is a maximum spanning tree on the maxclique graph, with edge weights set according to the cardinality of the separator between the two maxcliques.

40 Other aspects of JTs There can be multiple JTs for a given triangulated graph (e.g., consider any graph where d(s) 3 for some separator S). JTs are not binary decomposition trees (BDTs), but they are related. Leaf nodes of BDTs correspond to nodes in a JT. Non-leaf nodes in a BDTs correspond to edges in a JT. Therefore, edges in a JT correspond to all minimal separators in triangulated graph G. Set of maxcliques unique Different JTs always have same set of nodes and separators, just different configurations. Again, JT can be over not just maxcliques. JT can exist over all cliques, or over some cliques (if they contain all maxcliques) Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 40 / 45

41 Edge Clique Covers Set cover - sets must cover the ground/universal set (ground set cover) Vertex cover - vertices must cover the edges (edge vertex cover) Edge cover - edges must cover the vertices (vertex edge cover) clique cover - cliques cover the edges (edge clique cover) The maxcliques of a junction tree constitute an edge clique cover for triangulated graph G start with set of nodes V = C C C. Add edge between u, v V if exists a C C such that u, v C. Going from G to JT and back to the graph yields the same graph. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 41 / 45

42 Intersection Graphs Definition (Intersection Graph) An intersection graph is a graph G = (V, E) where each vertex v V (G) corresponds to a set U v and each edge (u, v) E(G) exists only if U u U v. some underlying set of objects U and a multiset of subsets of U of the form U = {U 1, U 2,..., U n } with U i U might have some i, j where U i = U j. Theorem Every graph is an intersection graph. This can be seen informally by consider an arbitrary graph, create a U i for every node, and construct the subsets so that the edges will exist when taking intersection. Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 42 / 45

43 Interval Graphs Intersection graphs where there subsets are intervals/segments [a, b] in R Any graph that can be constructed this way is an interval graph a b c d e Are all graphs interval graphs? a b c e d Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 43 / 45

44 Interval Graphs Intersection graphs where there subsets are intervals/segments [a, b] in R Any graph that can be constructed this way is an interval graph a b c d e Are all graphs interval graphs? 4-cycle a b c e d Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 43 / 45

45 Sub-tree intersection Graphs T 1 T 2 T 3 T Intersection graph, where subsets are (nec. connected) subtrees of some ground tree. Intersection exists if there are any nodes in common amongst the two corresponding trees. T 4 T 5 T 7 T8 T 6 T 9 T 10 T T 12 T 1 T 2 T T 4 T 5 T 6 T 12 T 9 T 8 T 11 T 7 T 10 Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 44 / 45

46 Sub-tree intersection graphs Theorem A graph G = (V, E) is triangulated iff it corresponds to a sub-tree graph (i.e., an intersection graph on subtrees of some tree). Prof. Jeff Bilmes EE512 Graphical Models Fall 2009 Lecture 13 - Nov 12th, 2009 page 45 / 45