Algorithmic solutions can help reduce energy consumption in computing environs.

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Abstract.We consider exploration problems where a robot has to construct a complete map of an unknown environment. We assume that the environment is modeled by a directed, strongly connected graph. The robot's task is to visit all nodes and edges of the graph using the minimum number R of edge traversals. Deng and Papadimitriou [Proceedings of the 31st Symposium on the Foundations of Computer Science, 1990, pp. 356-361]

We study a classical problem in online scheduling. A sequence of jobs must be scheduled on m identical parallel machines. As each job arrives, its processing time is known. The goal is to minimize the makespan. Bartal, Fiat, Karloff and Vohra [3] gave a deterministic online algorithm that is 1.986-competitive.~arger, Phillips and Torng [11] generalized the algorithm and proved an upper bound of 1.945. The best lower bound currently known on the competitive ratio that can be achieved by deterministic online algorithms it equal to 1.837. In this paper we present an improved deterministic online scheduling algorithm that is 1.923-competitive, for all m~2. The algorithm is based on a new scheduling strategy, i.e., it is not a generalization of the approach by Bartal et al. Also, the algorithm has a simple structure.Furthermore, we develop a better lower bound.We prove that, for general m, no deterministic online scheduling algorithm can be better than 1.852-competitive.

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1 < α ≤ n/2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for α ≥ 12n log n , every Nash equilibrium forms a tree.Without relying on the tree conjecture, Fabrikant et al. ). Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.

No abstract

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1 < α ≤ n/2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for α ≥ 12n log n , every Nash equilibrium forms a tree.Without relying on the tree conjecture, Fabrikant et al. ). Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.

We show that n integers in the range 1. . n can be stably sorted on an EREW PRAM using O(t) time and o (n(..jlognloglogn+ (logn)2jt)) operations, for arbitrary given t ~ lognloglogn, and on a CREW PRAM using O(t) time and O(n(..jlogn + lognj2 t / 1og ,,)) operations, for arbitrary given t ~ log n. In addition, we are able to sort n arbitrary integers on a randomized CREW PRAM within the same resource bounds with high probability. In each case our algorithm is a factor of almost E>(.y1'Og1i) closer to optimality than an previous algorithms for the stated problem in the stated model, and our third result matches the operation count of the best known sequential algorithm. We also show that n integers in the range 1 .. m can be sorted in O((logn)2) time with O(n) operations on an EREW PRAM using a nonstandard word length of o (log nloglog n log m) bits, thereby greatly improving the upper bound on the word length necessary to sort integers with alinear time-processor product, even sequentially. Our algorithms were inspired by, and in one case directly use, the fusion trees of Ftedman and Willard.