Showing papers by "Evangelos Kranakis published in 1999"

Compass routing on geometric networks.

Abstract: Suppose that a traveler arrives to the City of Toronto, and wants to walk to the famous CN-Tower, one of the tallest free-standing structures in the world. Assume now that our visitor, lacking a map of Toronto, is standing at a crossing from which he can see the CN-tower, and several streets S1, . . . , Sm that he can choose to start his walk. A natural (and most likely safe assumption), is that our visitor must choose to walk first along the road that points closest in the direction of the CN-tower, see Figure 1. A close look at maps of numerous cities around the world, show us that the previous way to explore a new, and unknown city will in general yield walks that will be close enough to the optimal ones to travel from one location to another. In mathematical terms, we can model the map of many cities by geometric graphs in which street intersections are represented by the vertices of our graphs, and streets by straight line segments. Compass routing on geometric networks, in its most elemental form yields the following algorithm:

The Impact of Knowledge on Broadcasting Time in Radio Networks

Abstract: We consider the problem of distributed deterministic broadcasting in radio networks. Nodes send messages in synchronous time-slots. Each node v has a given transmission range. All nodes located within this range can receive messages from v. However, a node situated in the range of two or more nodes that send messages simultaneously, cannot receive these messages and hears only noise. Each node knows only its own position and range, as well as the maximum of all ranges. Broadcasting is adaptive: Nodes can decide on the action to take on the basis of previously received messages, silence or noise.We prove a lower bound on broadcasting time in this model and construct a broadcasting protocol whose performance matches this bound for the simplest case when nodes are situated on a line and the network has constant depth.We also show that if nodes do not even know their own range, every broadcasting protocol must be hopelessly slow. While distributed randomized broadcasting algorithms, and, on the other hand, deterministic off-line broadcasting algorithms assuming full knowledge of the radio network, have been extensively studied in the literature, ours are the first results concerning broadcasting algorithms that are distributed and deterministic at the same time.We show that in this case the amount of knowledge available to nodes influences the efficiency of broadcasting in a significant way.

Searching with Uncertainty.

Abstract: We consider the problem of searching for an item in a distributed network in the presence of uncertainty. Although the location of the item in the network is unknown, information about its whereabouts can be obtained by querying the nodes of the network. The nodes have databases providing the rst edge on a shortest path to the item sought. This information is correct with some bounded probability. In this paper we give algorithms for searching in a distributed network that has the topology of a ring or torus under various models.

Bubbles: Adaptive Routing Scheme for High-Speed Dynamic Networks

Abstract: This paper presents the first dynamic routing scheme for high-speed networks. The scheme is based on a hierarchical bubbles partition of the underlying communication graph. Dynamic routing schemes are ranked by their adaptability, i.e., the maximum number of sites to be updated upon a topology change. An advantage of our scheme is that it implies a small number of updates upon a topology change. In particular, for the case of a bounded degree network it is proved that our scheme is optimal in its adaptability by presenting a matching tight lower bound. Our bubble routing scheme is a combination of a distributed routing database, a routing strategy, and a routing database update. It is shown how to perform the routing database update on a dynamic network in a distributed manner.

Isomorphic triangulations with small number of Steiner points

Abstract: Assume that an isomorphism between two n-vertex simple polygons, P,Q (with k,l reflex vertices, respectively) is given. We present two algorithms for constructing isomorphic (i.e. adjacency preserving) triangulations of P and Q, respectively. The first algorithm computes isomorphic triangulations of P and Q by introducing at most O((k+l)2) Steiner points and has running time O(n+(k+l)2). The second algorithm computes isomorphic traingulations of P and Q by introducing at most O(kl) Steiner points and has running time O(n+kllog n). The number of Steiner points introduced by the second algorithm is also worst-case optimal. Unlike the O(n2) algorithm of Aronov, Seidel and Souvaine1 our algorithms are sensitive to the number of reflex vertices of the polygons. In particular, our algorithms have linear running time when for the first algorithm, and kl≤n/log n for the second algorithm.

Optimal adaptive fault diagnosis for simple multiprocessor systems

Abstract: We studied adaptive system-level fault diagnosis for multiprocessor systems. Processors can test each other and future tests can be selected on the basis of previous test results. Fault-free testers give always correct test results, while faulty testers are completely unreliable. The aim of diagnosis is to determine correctly the fault status of all processors. We present adaptive diagnosis algorithms for systems modeled by trees, rings, and tori. These algorithms use the smallest possible number of tests in each case. Our results also imply optimal diagnosis for more general systems, assuming a small number of faults. The cost of adaptive diagnosis were found to be significantly smaller than that of classical (one-step) diagnosis.

Dissections, cuts and triangulations.

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Near optimal-partitioning of rectangles and prisms.

Abstract: Several variants of this problem have appeared in the literature. Overmars and Welzl [OW85] studied the problem of cutting a polygon drawn on a piece of paper in the cheapest possible way. Croft, Falconer and Guy [CFG91] studied problems related to tiling and dissection of circles and squares. Bose et al. [BCK+98, BCL98] studied the problem of cutting squares and circles into equal area pieces. Kong et al. [KMW87] and [KMR88] addressed a variant of Problem 1 in the context of parallel computing, where they were concerned with partitioning a rectangle equitably among a set of processors. However, their objective was to minimize the maximum perimeter of the rectangles in their decomposition.

Station Layouts in the Presence of Location Constraints

Abstract: In wireless communication, the signal of a typical broadcast station is transmited from a broadcast center p and reaches objects at a distance, say, R from it. In addition there is a radius r, r < R, such that the signal originating from the center of the station is so strong that human habitation within distance r from the center p should be avoided. Thus every station determines a region which is an "annulus of permissible habitation". We consider the following station layout (SL) problem: Cover a given (say, rectangular) planar region which includes a collection of orthogonal buildings with a minimum number of stations so that every point in the region is within the reach of a station, while at the same time no building is within the dangerous range of a station. We give algorithms for computing such station layouts in both the one-and two-dimensional cases.

Minimizing congestion of layouts for atm networks with faulty links

Abstract: We consider the problem of constructing virtual path layouts for an ATM network consisting of a complete network Kn of n processors in which a certain number of links may fail. Our main goal is to construct layouts which tolerate any configuration of up to f faults and have the least possible congestion. First, we study the minimal congestion of 1-hop f-tolerant layouts in Kn. For any positive integer f we give upper and lower bounds on this minimal congestion and construct f-tolerant layouts with congestion corresponding to the upper bounds. Our results are based on a precise analysis of the diameter of the network Kn[ℱ] which results from Kn by deleting links from a set ℱ of bounded size. Next we study the minimal congestion of h-hop f-tolerant layouts in Kn, for larger values of the number h of hops. We give upper and lower bounds on the order of magnitude of this congestion, based on results for 1-hop layouts. Finally, we consider a random, rather than worst case, fault distribution where links fail independently with constant probability p<1. Our goal now is to construct layouts with low congestion that tolerate the existing faults with high probability. For any p<1, we show the existence of 1-hop layouts in Kn, with congestion O(log n).

Domino tilings of orthogonal polygons.

Abstract: We consider orthogonal polygons with vertices located at integer lattice points. We show that if all of the sides of a simple orthogonal polygon without holes have odd lengths, then it cannot be tiled by dominoes. We provide similar characterizations for orthogonal polygons with sides of arbitrary length. We also give some generalizations for polygons with holes and polytopes in 3 dimensions.

On Tilable Orthogonal Polygons

Abstract: We consider rectangular tilings of orthogonal polygons with vertices located at integer lattice points. Let G be a set of reals closed under the usual addition operation. A G-rectangle is a rectangle at least one of whose sides is in G. We show that if an orthogonal polygon without holes can be tiled with G-rectangles then one of the sides of the polygon must be in G. As a special case this solves the conjecture that domino tilable orthogonal polygons must have at least one side of even length. We also explore separately the case of othogonal polygons placed in a chessboard. We establish a condition which determines the number of black minus white squares of the chessboard occupied by the polygon. This number depends exclusively on the parity sequence of the lengths of the sides of the orthogonal polygon. This approach produces a dierent proof of the conjecture of the non domino-tilability of of orthogonal polygons without even length sides. We also give some generalizations for polygons with holes and polytopes in 3 dimensions.