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 symmetric-key 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 IIS-0325345 and CNS-06274488 from the

We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected 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 3-connected 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.

We prove that every n-vertex graph G with pathwidth pw(G) has a three-dimensional straight-line grid drawing with O(pw(G) n) volume. Thus for

We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossing-free. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossings-free, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixed-edges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for tree-path pairs with at most one bend per tree-edge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).

The set of all orientations of a planar graph with prescribed outdegrees carries the structure of a distributive lattice. This general theorem is proven in the first part of the paper. In the second part the theorem is applied to show that interesting combinatorial sets related to a planar graph have lattice structure: Eulerian orientations, spanning trees and Schnyder woods. For the Schnyder wood application some additional theory has to be developed. In particular it is shown that a Schnyder wood for a planar graph induces a Schnyder wood for the dual.

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

This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3-connected planar map M, ordered by inclusion, and showed that the order dimension of PM is always equal to 4. In this paper, we show that if M is any planar map, then the order dimension of PM is still at most 4.

Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a oorplan for any n-node plane triangulation. In comparison with previous oor-planning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces oor-plans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to t the oor-plan area in a rectangle of size (n 1) . Lower bounds on the worst-case area for oor-planning any plane triangulation are also provided in the paper.

Abstract. A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z 3 and the edges by non-crossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every n-vertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queue-number. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree. 1

A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.