It is celebrated that a simple random walk on ℤ and ℤ² returns to the initial vertex v infinitely many times during infinitely many transitions, which is said recurrent, while it returns to v only finite times on ℤ^d for d ≥ 3, which is said transient. It is also known that a simple random walk on a growing region on ℤ^d can be recurrent depending on growing speed for any fixed d. This paper shows that a simple random walk on {0,1,…,N}ⁿ with an increasing n and a fixed N can be recurrent depending on the increasing speed of n. Precisely, we are concerned with a specific model of a random walk on a growing graph (RWoGG) and show a phase transition between the recurrence and transience of the random walk regarding the growth speed of the graph. For the proof, we develop a pausing coupling argument introducing the notion of weakly less homesick as graph growing (weakly LHaGG).

This paper is motivated by a question of whether it is possible to calculate a chaotic sequence efficiently, e.g., is it possible to get the n-th bit of a bit sequence generated by a chaotic map, such as$\beta$-expansion, tent map, and logistic map in$\mathrm{o}(n)\text{time}/\text{space}$? From the viewpoint of theoretical computer science, this paper gives an affirmative answer to the question about the space complexity of a tent map. We prove that a tent code of n-bits with an initial condition uniformly at random is exactly generated in$\mathrm{O}(\log^{2}n)$space in expectation.

Jan-ken, a.k.a. rock-paper-scissors, is a cerebrated example of a non-transitive game with three (pure) strategies, rock, paper and scissors. Interestingly, any Jan-ken generalized to four strategies contains at least one useless strategy unless it allows a tie between distinct pure strategies. Non-transitive dice could be a stochastic analogue of Jan-ken: the stochastic transitivity does not hold on some sets of dice, e.g., Efron's dice. Including the non-transitive dice, this paper is interested in dice sets which do not contain a useless die. In particular, we are concerned with the existence of a strongest (or weakest, symmetrically) die in a dice set under the two conditions that (1) any number appears on at most one die and at most one side, i.e., no tie break between two distinct dice, and (2) the mean pips of dice are the same. We firstly prove that a strongest die never exist if a set of n dice of m-sided is given as a partition of the set of numbers {1,…,mn}. Next, we show some sufficient conditions that a strongest die exists in a dice set which is not a partition of a set of numbers. We also give some algorithms to find a strongest die in a dice set which includes given dice.

Real networks are often dynamic. In response to it, analyses of algorithms on dynamic networks attract more and more attention in network science and engineering. Random walks on dynamic graphs also have been investigated actively in more than a decade, where in most cases the edge set changes but the vertex set is static. The vertex sets are also dynamic in many real networks. Motivated by a new technology of the analysis of random walks on dynamic graphs, this paper introduces a simple model of graphs with an increasing number of vertices and presents an analysis of random walks associated with the cover time on such graphs. In particular, we reveal that a random walk asymptotically covers the vertices all but a constant number if the vertex set grows moderately.

A set function $f$ on a finite set $V$ is submodular if $f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y)$ for any pair $X, Y \subseteq V$. The symmetric difference transformation ( SD-transformation) of $f$ by a canonical set $S \subseteq V$ is a set function $g$ given by $g(X) = f(X \vartriangle S)$ for $X \subseteq V$, where $X \vartriangle S = (X \setminus S) \cup (S \setminus X)$ denotes the symmetric difference between $X$ and $S$. Submodularity and SD-transformations are regarded as the counterparts of convexity and affine transformations in a discrete space, respectively. However, submodularity is not preserved under SD-transformations, in contrast to the fact that convexity is invariant under affine transformations. This paper presents a characterization of SD-transformations preserving submodularity. Then, we are concerned with the problem of discovering a canonical set $S$, given the SD-transformation $g$ of a submodular function $f$ by $S$, provided that $g(X)$ is given by a function value oracle. A submodular function $f$ on $V$ is said to be strict if $f(X) + f(Y) > f(X \cup Y) + f(X \cap Y)$ holds whenever both $X \setminus Y$ and $Y \setminus X$ are nonempty. We show that the problem is solved by using $\mathrm{O}(|V|)$ oracle calls when $f$ is strictly submodular, although it requires exponentially many oracle calls in general.

The rotor-router model is a deterministic process analogous to a simple random walk on a graph, and the discrepancy of token configurations between the rotor-router model and its corresponding random walk has been investigated in some contexts. Motivated by general Markov chains beyond simple random walks, this paper investigates a generalized model which imitates a Markov chain (of multiple tokens) possibly containing irrational transition probabilities. We are concerned with the vertexwise discrepancy of the numbers of tokens between the generalized model and its corresponding Markov chain, and present an upper bound of the discrepancy in terms of the mixing time of the Markov chain.

We consider a distributed system consisting of autonomous mobile computing entities called robots moving in the three-dimensional space (3D-space). The robots are anonymous, oblivious, fully-synchronous and have neither any access to the global coordinate system nor any explicit communication medium. Each robot cooperates with other robots by observing the positions of other robots in its local coordinate system. One of the most fundamental agreement problems in 3D-space is the plane formation problem that requires the robots to land on a common plane, that is not predefined. This problem is not always solvable because of the impossibility of symmetry breaking. While existing results assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. The robots without chirality can never break the symmetry consisting of rotation symmetry and reflection symmetry. Such symmetry in 3D-space is fully described by 17 symmetry types each of which forms a group. We extend the notion of symmetricity [Suzuki and Yamashita, SIAM J. Compt. 1999] [Yamauchi et al., PODC 2016] to cover these 17 symmetry groups. Then we give a characterization of initial configurations from which the fully-synchronous robots without chirality can form a plane in terms of symmetricity.

Creating a swarm of mobile computing entities, frequently called robots, agents, or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three-dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two-dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three-dimensional space . This article presents a necessary and sufficient condition for fully synchronous robots to solve the plane formation problem that does not depend on obliviousness, i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedron.

We investigate pattern formation, i.e., self-organization, by a swarm of mobile robots, which is closely related with the agreement problem in distributed computing. Consider a system of anonymous mobile robots in a 2-dimensional Euclidean space in which each robot repeatedly executes a “Look-Compute-Move” cycle, to observe the positions of all the robots, to compute a route to the next position using an algorithm, and then to trace the route, where the algorithm is common to all robots. The robots are said to be fully synchronous if their Look-Compute-Move cycles are completely synchronized, and the $i$th Look, Compute, and Move of all robots start and end simultaneously. They are said to be asynchronous if no assumptions are made on their synchrony. The robots are said to be oblivious if they have no memory to memorize the execution history and hence behave based only on the robots' positions observed during the immediately preceding Look. We show that the set of geometric patterns formable by oblivious asynchronous robots is exactly the set of those formable by nonoblivious fully synchronous robots, except for a point of multiplicity 2, i.e., gathering for two robots. In short, contrary to our intuition, synchrony and memory do not help in pattern formation, except for gathering. Specifically, we propose an algorithm for oblivious asynchronous robots that, given a geometric pattern as input, forms it, as long as it is formable by nonoblivious fully synchronous robots except for a point of multiplicity 2.

In this paper, we propose the first fully polynomial-time randomized approximation scheme (FPRAS) for closed Jackson networks with single servers. Our algorithm is based on the Markov chain Monte Carlo (MCMC) method, and our scheme returns an approximate solution, for which the size of error satisfies a given error rate. We propose two Markov chains: one is for approximate sampling, and the other is for perfect sampling based on the monotone coupling from the past algorithm.