Aris Pagourtzis

Oblivious gossiping in ad-hoc radio networks

We study oblivious deterministic and randomized algorithms for gossiping in unknown radio networks. In oblivious algorithms the fact (or probability in case of randomized algorithm) that a processor transmits or not at a given time-step depends solely on its identification number, the total number of processors and the number of the time-step. We distinguish oblivious deterministic algorithms which allow only one processor to transmit in each time-step and term them singleton algorithms. We also distinguish oblivious randomized algorithms where in each time-step all processors have equal probability of transmission, and call them uniform. The merit of oblivious algorithms, especially the singleton and uniform ones, is that they are simple and easy to implement. We observe that gossiping in unknown radio networks on <i>n</i> nodes can be completed in time (<i>n</i> - 1)(<i>n</i> - 2) + 4 by a singleton algorithm. On the other hand, we show that any singleton algorithm takes at least <i>n</i>² - <i>&Ogr;</i>(<i>n</i>^7/4+∈) steps, for any ∈ > 0, whereas any deterministic oblivious algorithm requires at least <i>n</i>²/2 - <i>&Ogr;</i>(<i>n</i>) steps to complete the gossiping. We prove also that there is an oblivious deterministic algorithm for gossiping working in time <i>n</i>² - <i>w</i>(<i>n</i>). Next we show that a uniform oblivious randomized algorithm completes gossiping with high probability in time <i>&Ogr;</i>(min{<i>m, Dd</i>} log² <i>n</i>), where <i>m</i> denotes the number of edges, <i>D</i> is the eccentricity and <i>d</i> the maximum, in-degree in the network. Note that this upper bound is poly-logarithmic in <i>n</i> if <i>D, d</i> = <i>&Ogr;</i>(<i>poly</i> log <i>n</i>). The best related deterministic gossiping algorithm, in terms of performance expressed with respect to <i>n</i>, <i>D</i>, <i>d</i>, has been previously given by Clementi et al. [13], it works in time <i>&Ogr;</i>(<i>Dd</i>² log³ <i>n</i>). We prove also that the upper bound attained by our uniform oblivious randomized algorithm is asymptotically optimal (up to a log-square factor) for a wide range of parameters <i>m</i>, <i>D</i> and <i>d</i> in the class of uniform oblivious randomized algorithms. Finally we observe that in case of symmetric networks the aforementioned oblivious randomized algorithm completes gossiping with high probability in time <i>&Ogr;</i>(<i>n</i> log² <i>n</i>) and that a known deterministic constructive broadcasting algorithm can be adopted to perform oblivious gossiping in time <i>&Ogr;</i>(<i>n</i>^3/2).

Routing and path multicoloring

Abstract In optical networks it is important to make an optimal use of the available bandwidth. Given a set of requests the goal is to satisfy them by using a minimum number of wavelengths. We introduce a variation to this well known problem, by allowing multiple parallel links, in order to be able to satisfy any set of requests even if the available bandwidth is insufficient. In this new approach the goal is to use a minimum number of active links and thus reduce network pricing. In graph-theoretic terms, given a graph, a list of pairs of vertices, and a number of available colors, the goal is to route paths with the given pairs of vertices as endpoints and to find a color assignment to paths that minimizes color collisions over all possible routings and colorings. We present efficient algorithms for simple network topologies. For chains our solutions are optimal; for stars and rings — where it is NP-hard to solve the problem optimally — our solutions are approximate within a factor two of the optimal solution. The key technique involves transformation to edge coloring of bipartite graphs. For rings we also present a 2-approximation algorithm, for a variation of the problem, in which the routing is already prescribed.

Resource Allocation Problems in Multifiber WDM Tree Networks

All-optical networks with multiple fibers lead to several interesting optimization problems. In this paper, we consider the problem of minimizing the total number of fibers necessary to establish a given set of requests with a bounded number w of wavelengths, and the problem of maximizing the number of accepted requests for given fibers and bounded number w of wavelengths. We study both problems in undirected tree networks T=(V,E) and present approximation algorithms with ratio 1 + 4|E|log|V|/OPT and 4 for the former and ratio 2.542 for the latter. Our results can be adapted to directed trees as well.

Routing and wavelength assignment in multifiber WDM networks with non-uniform fiber cost

Motivated by the increasing importance of multifiber WDM networks we study a routing and wavelength assignment problem in such networks. In this problem the number of wavelengths per fiber is given and the goal is to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is non-uniform, that is the cost of hiring a fiber may differ from link to link. We consider two variations: undirected, which corresponds to full-duplex communication, and directed, which corresponds to one-way communication. Moreover, for rings we also study the problem in the case of pre-determined routing. We present exact or constant-ratio approximation algorithms for all the above variations in chain, ring and spider networks.

Server Placements, Roman Domination and other Dominating Set Variants

Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed file sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimum dominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies domination problems for two requests. For the problem of placing a minimum number of servers such that two requests at different nodes can be served with two different servers (called win-win), we present a logarithmic approximation, and we prove that nothing better is possible. We show that the same is true for Roman domination, the well studied problem variant that asks for each vertex to either possess its own server or to have a neighbor with two servers. Still the same is true if each idle server can move along one edge while the first of both requests is being served. For planar graphs, we propose a PTAS for Roman domination (and show that nothing better exists), and we get a constant approximation for win-win.

Satisfying a maximum number of pre-routed requests in all-optical rings

We address the problem of maximizing the number of satisfied requests in all-optical networks that use wavelength division multiplexing . We consider the case where requests are pre-routed and formulate it as the maximum path coloring problem. We study the problem for rings and present a (2/3)-approximation algorithm. Along the way we develop a fast matching technique for a special class of bipartite graphs. By using this technique we achieve an O( n + m log L ) time complexity for our approximation algorithm, where n is the number of nodes, m is the number of requests and L is the maximum load of requests on a single link.

Fiber Cost reduction and Wavelength minimization in multifiber WDM networks

Motivated by the increasing importance of multifiber WDM networks we study two routing and wavelength assignment problems in such networks: Fiber Cost Minimization: the number of wavelengths per fiber is given and we want to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is nonuniform, that is the cost of hiring a fiber may differ from link to link. Wavelength Minimization: the number of available parallel fibers on each link is given and we want to minimize the wavelengths per fiber that are needed in order to satisfy a set of communication requests.

Reliable Broadcast with Respect to Topology Knowledge

We study the Reliable Broadcast problem in incomplete networks against a Byzantine adversary. We examine the problem under the locally bounded adversary model of Koo (2004) and the general adversary model of Hirt and Maurer (1997) and explore the tradeoff between the level of topology knowledge and the solvability of the problem.

On a noncooperative model for wavelength assignment in multifiber optical networks

We propose and investigate Selfish Path MultiColoring games as a natural model for noncooperative wavelength assignment in multifiber optical networks. In this setting, we view the wavelength assignment process as a strategic game in which each communication request selfishly chooses a wavelength in an effort to minimize the maximum congestion that it encounters on the chosen wavelength. We measure the cost of a certain wavelength assignment as the maximum, among all physical links, number of parallel fibers employed by this assignment. We start by settling questions related to the existence and computation of and convergence to pure Nash equilibria in these games. Our main contribution is a thorough analysis of the price of anarchy of such games, that is, the worst-case ratio between the cost of a Nash equilibrium and the optimal cost. We first provide upper bounds on the price of anarchy for games defined on general network topologies. Along the way, we obtain an upper bound of 2 for games defined on star networks. We next show that our bounds are tight even in the case of tree networks of maximum degree 3, leading to nonconstant price of anarchy for such topologies. In contrast, for network topologies of maximum degree 2, the quality of the solutions obtained by selfish wavelength assignment is much more satisfactory: We prove that the price of anarchy is bounded by 4 for a large class of practically interesting games defined on ring networks.

On the Complexity of Train Assignment Problems

We consider a problem faced by train companies: How can trains be assigned to satisfy scheduled routes in a cost efficient way? Currently, many railway companies create solutions by hand, a timeconsuming task which is too slow for interaction with the schedule creators. Further, it is difficult to measure how efficient the manual solutions are. We consider several variants of the problem. For some, we give efficient methods to solve them optimally, while for others, we prove hardness results and propose approximation algorithms.