Collision-Free Network Exploration

International audienceA set of mobile agents is placed at different nodes of a $n$-node network. The agents synchronously move along the network edges in a {\em collision-free} way, i.e., in no round may two agents occupy the same node. In each round, an agent may choose to stay at its currently occupied node or to move to one of its neighbors. An agent has no knowledge of the number and initial positions of other agents. We are looking for the shortest possible time required to complete the collision-free {\em network exploration}, i.e., to reach a configuration in which each agent is guaranteed to have visited all network nodes and has returned to its starting location. We first consider the scenario when each mobile agent knows the map of the network, as well as its own initial position. We establish a connection between the number of rounds required for collision-free exploration and the degree of the minimum-degree spanning tree of the graph. We provide tight (up to a constant factor) lower and upper bounds on the collision-free exploration time in general graphs, and the exact value of this parameter for trees. For our second scenario, in which the network is unknown to the agents, we propose collision-free exploration strategies running in $O(n^2)$ rounds for tree networks and in $O(n^5\log n)$ rounds for general networks

Derandomized Squaring of Graphs

We introduce a “derandomized ” analogue of graph squaring. This op-eration increases the connectivity of the graph (as measured by the second eigenvalue) almost as well as squaring the graph does, yet only increases the degree of the graph by a constant factor, instead of squaring the degree. One application of this product is an alternative proof of Reingold’s re-cent breakthrough result that S-T Connectivity in Undirected Graphs can be solved in deterministic logspace.

Online competition : examining competition between online and dual-channel retailers in the Norwegian electronics market

The purpose of this thesis is to explore competition among retailers online. Online retailing has grown significantly in recent years and is expected to keep growing. Simultaneously, traditional brick-and-mortar retailers are experiencing less growth. Many traditional brickand-mortar retailers have chosen to adapt by entering the online channel, becoming dualchannel retailers. It is generally anticipated that the online channel is more competitive than traditional brick-and-mortar retailing, due to lower search costs, technology and barriers to entry. As dual-channel retailers operate in both markets, we wish to examine how this will affect their prices. We test the hypothesis of online efficiency through looking at the price levels and dispersion of online and dual-channel retailers. We use a quantitative research methodology to examine whether online and dual-channel retailers prices are different. Using price data from various retailers within the electronics industry, we look at price levels and price dispersion to study competition between the retailer types. The results show that the prices of online and dual-channel retailers are significantly different. However, the results were the opposite of what we predicted, given the theory. We found that dual-channel retailers have significantly lower prices and lower price dispersion compared to online retailers. The results do not therefore support the existence of an online disutility cost, or the notion of online channel efficiency.nhhma

The cover time of random digraphs

d log

How Well Do Random Walks Parallelize?

A random walk on a graph is a process that explores the graph in a random way: at each step the walk is at a vertex of the graph, and at each step it moves to a uniformly selected neighbor of this vertex. Random walks are extremely useful in computer science and in other fields. A very natural problem that was recently raised by Alon, Avin, Koucky, Kozma, Lotker, and Tuttle (though it was implicit in several previous papers) is to analyze the behavior of k independent walks in comparison with the behavior of a single walk. In particular, Alon et al. showed that in various settings (e.g., for expander graphs), k random walks cover the graph (i.e., visit all its nodes), Ω(k)-times faster (in expectation) than a single walk. In other words, in such cases k random walks efficiently “parallelize ” a single random walk. Alon et al. also demonstrated that, depending on the specific setting, this “speedup ” can vary from logarithmic to exponential in k. In this paper we initiate a more systematic study of multiple random walks. We give lower and upper bounds both on the cover time and on the hitting time (the time it takes to hit one specific node) of multiple random walks. Our study revolves over three alternatives for th