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Dynamic and efficient key management for access hierarchies
 In Proceedings of the ACM Conference on Computer and Communications Security
, 2005
"... Hierarchies arise in the context of access control whenever the user population can be modeled as a set of partially ordered classes (represented as a directed graph). A user with access privileges for a class obtains access to objects stored at that class and all descendant classes in the hierarchy ..."
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Cited by 119 (7 self)
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Hierarchies arise in the context of access control whenever the user population can be modeled as a set of partially ordered classes (represented as a directed graph). A user with access privileges for a class obtains access to objects stored at that class and all descendant classes in the hierarchy. The problem of key management for such hierarchies then consists of assigning a key to each class in the hierarchy so that keys for descendant classes can be obtained via efficient key derivation. We propose a solution to this problem with the following properties: (1) the space complexity of the public information is the same as that of storing the hierarchy; (2) the private information at a class consists of a single key associated with that class; (3) updates (i.e., revocations and additions) are handled locally in the hierarchy; (4) the scheme is provably secure against collusion; and (5) each node can derive the key of any of its descendant with a number of symmetrickey operations bounded by the length of the path between the nodes. Whereas many previous schemes had some of these properties, ours is the first that satisfies all of them. The security of our scheme is based on pseudorandom functions, without reliance on the Random Oracle Model. 18 Portions of this work were supported by Grants IIS0325345 and CNS06274488 from the
The LCMlattice in monomial resolutions
, 1999
"... Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: ..."
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Cited by 60 (7 self)
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Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were:
Convex drawings of Planar Graphs and the Order Dimension of 3Polytopes
 ORDER
, 2000
"... We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. ..."
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Cited by 44 (14 self)
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We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.
Trapezoid Graphs and Generalizations, Geometry and Algorithms
 DISCRETE APPLIED MATHEMATICS
, 1993
"... Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan, Golumbic and Pinter [DGP]. They propose an O(n²) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs ..."
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Cited by 33 (0 self)
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Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan, Golumbic and Pinter [DGP]. They propose an O(n²) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs. Based on a geometric representation of trapezoid graphs by boxes in the plane we design optimal, i.e., O(n log n), algorithms for chromatic number, weighted independent set, clique cover and maximum weighted clique on such graphs. We also propose generalizations of trapezoid graphs called ktrapezoidal graphs. The ideas behind the clique cover and weighted independent set algorithms for trapezoid graphs carry over to higher dimensions. This leads to O(n log k\Gamma1 n) algorithms for ktrapezoidal graphs. We also propose a new class of graphs called circle trapezoid graphs. This class contains trapezoid graphs, circle graphs and circulararc graphs as subclasses. We show that cli...
Vertex decomposable graphs and obstructions to shellability
, 2009
"... Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMac ..."
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Cited by 30 (2 self)
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Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMacaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.
Inapproximability results for sparsest cut, optimal linear arrangement, and precedence constrained scheduling
 IN PROCEEDINGS OF 48TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2007
"... This paper is published in the proceedings of FOCS 2007 and is subject to some copyright restrictions. We consider (Uniform) Sparsest Cut, Optimal Linear Arrangement and the precedence constrained scheduling problem 1prec  � wjCj. So far, these three notorious NPhard problems have resisted all at ..."
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Cited by 27 (9 self)
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This paper is published in the proceedings of FOCS 2007 and is subject to some copyright restrictions. We consider (Uniform) Sparsest Cut, Optimal Linear Arrangement and the precedence constrained scheduling problem 1prec  � wjCj. So far, these three notorious NPhard problems have resisted all attempts to prove inapproximability results. We show that they have no Polynomial Time Approximation Scheme (PTAS), unless NPcomplete problems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as hard to approximate as Vertex Cover when the socalled fixed cost, that is present in all feasible solutions, is subtracted from the objective function.
Order dimension, strong Bruhat order and lattice properties for posets
 ORDER
, 2002
"... We determine the order dimension of the strong Bruhat order on finite Coxeter groups of types A, B and H. The order dimension is determined using a generalization of a theorem of Dilworth: dim(P) = width(Irr(P)), whenever P satisfies a simple ordertheoretic condition called here the dissective pro ..."
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Cited by 19 (6 self)
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We determine the order dimension of the strong Bruhat order on finite Coxeter groups of types A, B and H. The order dimension is determined using a generalization of a theorem of Dilworth: dim(P) = width(Irr(P)), whenever P satisfies a simple ordertheoretic condition called here the dissective property (or “clivage ” in [16, 21]). The result for dissective posets follows from an upper bound and lower bound on the dimension of any finite poset. The dissective property is related, via MacNeille completion, to the distributive property of lattices. We show a similar connection between quotients of the strong Bruhat order with respect to parabolic subgroups and lattice quotients.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 18 (3 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
KEY MANAGEMENT FOR NONTREE ACCESS HIERARCHIES
, 2006
"... Access hierarchies are useful in many applications and are modeled as a set of access classes organized by a partial order. A user who obtains access to a class in such a hierarchy is entitled to access objects stored at that class, as well as objects stored at its descendant classes. Efficient sche ..."
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Cited by 18 (6 self)
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Access hierarchies are useful in many applications and are modeled as a set of access classes organized by a partial order. A user who obtains access to a class in such a hierarchy is entitled to access objects stored at that class, as well as objects stored at its descendant classes. Efficient schemes for this framework assign only one key to a class and use key derivation to permit access to descendant classes. Ideally, the key derivation uses simple primitives such as cryptographic hash computations and modular additions. A straightforward key derivation time is then linear in the length of the path between the user’s class and the class of the object that the user wants to access. Recently, work presented in [2] has given an efficient solution that significantly lowers this key derivation time, while