Prefetching constitutes a valuable tool toward efficient Web surfing. As a result, estimating the amount of resources that need to be preloaded during a surfer's browsing becomes an important task. In this regard, prefetching can be modeled as a two-player combinatorial game [Fomin et al., Theoretical Computer Science 2014], where a surfer and a marker alternately play on a given graph (representing the Web graph). During its turn, the marker chooses a set of $k$ nodes to mark (prefetch), whereas the surfer, represented as a token resting on graph nodes, moves to a neighboring node (Web resource). The surfer's objective is to reach an unmarked node before all nodes become marked and the marker wins. Intuitively, since the surfer is step-by-step traversing a subset of nodes in the Web graph, a satisfactory prefetching procedure would load in cache all resources lying in the neighborhood of this growing subset. Motivated by the above, we consider the following problem to which we refer to as the Maximum Rooted Connected Expansion (MRCE) problem. Given a graph $G$ and a root node $v_0$, we wish to find a subset of vertices $S$ such that $S$ is connected, $S$ contains $v_0$ and the ratio $|N[S]|/|S|$ is maximized, where $N[S]$ denotes the closed neighborhood of $S$, that is, $N[S]$ contains all nodes in $S$ and all nodes with at least one neighbor in $S$. We prove that the problem is NP-hard even when the input graph $G$ is restricted to be a split graph. On the positive side, we demonstrate a polynomial time approximation scheme for split graphs. Furthermore, we present a $\frac{1}{6}(1-\frac{1}{e})$-approximation algorithm for general graphs based on techniques for the Budgeted Connected Domination problem [Khuller et al., SODA 2014]. Finally, we provide a polynomial-time algorithm for the special case of interval graphs.

In this paper we study space-efficient deterministic population protocols for several variants of the majority problem including plurality consensus. We focus on space efficient majority protocols in populations with an arbitrary number of colours C represented by k-bit labels, where k = ceiling (log C). In particular, we present asymptotically space-optimal (with respect to the adopted k-bit representation of colours) protocols for (1) the absolute majority problem, i.e., a protocol which decides whether a single colour dominates all other colours considered together, and (2) the relative majority problem, also known in the literature as plurality consensus, in which colours declare their volume superiority versus other individual colours. The new population protocols proposed in this paper rely on a dynamic formulation of the majority problem in which the colours originally present in the population can be changed by an external force during the communication process. The considered dynamic formulation is based on the concepts studied by D. Angluin et al. and O. Michail et al. about stabilizing inputs and composition of population protocols. Also, the protocols presented in this paper use a composition of some known protocols for static and dynamic majority.

A mechanism for identifying bandings in large "zero-one" N-dimensional data sets, using a sampling technique, is presented. The challenge of identifying bandings in data is the large number of potential permutations that need to be considered. To circumvent this a banding score mechanism is proposed that avoids the need to consider large numbers of permutations. This has been incorporated into a proposed banded pattern mining algorithm, the Exact ND Banded Pattern Mining (END BPM) algorithm. Although this operates well on reasonably sized datasets, there is still a challenge with respect to large N-dimensional data sets that cannot be held in primary storage. To this end a sampling technique is also proposed. The approach is fully described and evaluated using the GB cattle movement database, a "real life" database that records all movements of cattle in GB.

We study a distributed coordination mechanism for uniform agents located on a circle. The agents perform their actions in synchronised rounds. At the beginning of each round an agent chooses the direction of its movement from clockwise, anticlockwise, or idle, and moves at unit speed during this round. Agents are not allowed to overpass, i.e., When an agent collides with another it instantly starts moving with the same speed in the opposite direction (without exchanging any information with the other agent). However, at the end of each round each agent has access to limited information regarding its trajectory of movement during this round. We assume that n mobile agents are initially located on a circle unit circumference at arbitrary but distinct positions unknown to other agents. The agents are equipped with unique identifiers from a fixed range. The location discovery task to be performed by each agent is to determine the initial position of every other agent. Our main result states that, if the only available information about movement in a round is limited to distance between the initial and the final position, then there is a superlinear lower bound on time needed to solve the location discovery problem. Interestingly, this result corresponds to a combinatorial symmetry breaking problem, which might be of independent interest. If, on the other hand, an agent has access to the distance to its first collision with another agent in a round, we design an asymptotically efficient and close to optimal solution for the location discovery problem.

A set of mobile robots is deployed on a simple curve of finite length, composed of a finite set of vital segments separated by neutral segments. The robots have to patrol the vital segments by perpetually moving on the curve, without exceeding their uniform maximum speeds. The quality of patrolling is measured by the idleness, i.e., the longest time period during which any vital point on the curve is not visited by any robot. Given a configuration of vital segments, our goal is to provide algorithms describing the movement of the robots along the curve so as to minimize the idleness.

Suppose that a set of m tasks are to be shared as equally as possible among a set of n resources. A game-theoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent” and require each agent to select a resource, with the cost of a resource being the number of agents that select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For $m \gg n$, the system becomes approximately balanced (an $\epsilon$-Nash equilibrium) in expected time $O(\log \log m)$. We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time $O(\log \log m + n^4)$. We also give a lower bound of $\Omega(\max\{\log \log m, n\})$ for the convergence time.

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. Cryan and Dyer [J. Comput. System Sci., 67 (2003), pp. 291-310] have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which employs Markov chain methods indirectly. They leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we show that the "2 x 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris [Random Walks in Convex Sets, Ph.D. thesis, Department of Statistics, University of California, Berkeley, CA, 2000] and Morris and Sinclair [SIAM J. Comput., 34 (2004), pp. 195-226] for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste [Ann. Appl. Probab., 3 (1993), pp. 696-730] to show that the 2 x 2 chain is also rapidly mixing.

Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontree-like graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some $\degree \geq 3$, the maximum degree of G is at most $\degree$. We show that the spin system consisting of q-colorings of G has strong spatial mixing, provided $q > \alpha \degree-\gamma$, where $\alpha\approx 1.76322$ is the solution to $\alpha^\alpha=e$, and $\gamma = \frac{4\alpha^3-6\alpha^2-3\alpha+4}{2(\alpha^2-1)}\approx 0.47031$. Note that we have no additional lower bound on q or $\degree$. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics, such as the integer lattice $\zset^d$ and the triangular lattice. For these graphs (in fact, for any graph in which the distance-k neighborhood of a vertex grows subexponentially in k), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. Our general result gives, for example, a ``hand proof' of strong spatial mixing for 7-colorings of triangle-free 4-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp.\ 551--579] (for the rectangular lattice) and by Bubley, Dyer, Greenhill, and Jerrum [\textit{SIAM J. Comput.}, 29 (1999), pp.\ 387--400].) It also gives a hand proof of strong spatial mixing for 5-colorings of triangle-free 3-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided earlier by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp.\ 551--579].) Toward the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for $q=10$ on the lattice $\zset^3$. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us a (machine-assisted) proof of strong spatial mixing for $10$-colorings of the triangular lattice. (Such a proof for $11$ colors was given by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp.\ 551--579].) For completeness, we also show that our strong spatial mixing proof implies rapid mixing of Glauber dynamics for sampling proper colorings of neighborhood-amenable graphs. (It is known that strong spatial mixing often implies rapid mixing, but existing proofs seem to be written for $\zset^d$.) Thus our strong spatial mixing results give rapid

Recursively-constructed couplings have been used in the past for mixing on trees. We show for the first time how to extend this technique to nontree-like graphs such as the integer lattice. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some /spl Delta/ /spl ges/ 3, the maximum degree of G is at most /spl Delta/. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q > /spl alpha//spl Delta/, where /spl alpha/ /spl ap/ 1.76322 is the solution to /spl alpha//sup /spl alpha// = e. Note that we have no additional lower bound on q or /spl Delta/. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics such as the integer lattice /spl Zopf//sup d/ and the triangular lattice. For these graphs (in fact, for any graph in which the distance-k neighbourhood of a vertex grows sub-exponentially in k), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. We extend our general result, obtaining, for example, the first "hand proof" of strong spatial mixing for 7-colourings of triangle-free 4-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal (1997) for the rectangular lattice and by Bubley et al. (1999)). The extension also gives the first hand proof of strong spatial mixing for 5-colourings of triangle-free 3-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided by Salas and Sokal). Towards the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for q = 10 on the lattice /spl Zopf//sup 3/. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us the first (machine-assisted) proof of strong spatial mixing for 10-colourings of the triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.).

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. (2002) have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which only employs Markov chain methods indirectly. But they leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we answer this question in the affirmative, and hence provide a very different proof of the main result of Cryan and Dyer. We show that the "2 /spl times/ 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris and Sinclair (2002) (see also Morris (2002)) for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste (1993) to show that the 2 /spl times/ 2 chain is rapidly mixing. As part of our analysis, we give the first proof that the 2 /spl times/ 2 chain mixes in time polynomial in the input size when both the number of rows and the number of columns is constant.

Staircase walks are lattice paths from (0,0) to (2n,0) which take diagonal steps and which never fall below the x-axis. A path hitting the x-axis /spl kappa/ times is assigned a weight of /spl lambda//sup /spl kappa//, where /spl lambda/>0. A simple local Markov chain, which connects the state space and converges to the Gibbs measure (which normalizes these weights) is known to be rapidly mixing when /spl lambda/=1, and can easily be shown to be rapidly mixing when /spl lambda/1, known in the statistical physics community as adsorbing staircase walks. The main new ingredient is a decomposition technique which allows us to analyze the Markov chain in pieces, applying different arguments to analyze each piece.