arXiv subjects

Masafumi Yamashita

Publications and source records attributed to Masafumi Yamashita.

At least 19 recordsLinked to original sources

arXiv2023

Minimum algorithm sizes for self-stabilizing gathering and related problems of autonomous mobile robots

This paper investigates a swarm of autonomous mobile robots in the Euclidean plane, under the semi-synchronous ($\cal SSYNC$) scheduler. Each robot has a target function to determine a destination point from the robots' positions. All robots in the swarm take the same target function conventionally. We allow the robots to take different target functions, and investigate the effects of the number of distinct target functions on the problem-solving ability, regarding target function as a resource to solve a problem like time. Specifically, we are interested in how many distinct target functions are necessary and sufficient to solve a problem $Π$. The number of distinct target functions necessary and sufficient to solve $Π$ is called the minimum algorithm size (MAS) for $Π$. The MAS is defined to be $\infty$, if $Π$ is unsolvable even for the robots with unique target functions. We show that the problems form an infinite hierarchy with respect to their MASs; for each integer $c > 0$ and $\infty$, the set of problems whose MAS is $c$ is not empty, which implies that target function is a resource irreplaceable, e.g., with time. We propose MAS as a natural measure to measure the complexity of a problem. We establish the MASs for solving the gathering and related problems from any initial configuration, i.e., in a self-stabilizing manner. For example, the MAS for the gathering problem is 2. It is 3, for the problem of gathering {\bf all non-faulty} robots at a single point, regardless of the number $(< n)$ of crash failures. It is however $\infty$, for the problem of gathering all robots at a single point, in the presence of at most one crash failure.

cs.DC↗

arXiv2023

Compatibility of convergence algorithms for autonomous mobile robots

We investigate autonomous mobile robots in the Euclidean plane. A robot has a function called target function to decide the destination from the robots' positions. Robots may have different target functions. If the robots whose target functions are chosen from a set $Φ$ of target functions always solve a problem $Π$, we say that $Φ$ is compatible with respect to $Π$. If $Φ$ is compatible with respect to $Π$, every target function $ϕ\in Φ$ is an algorithm for $Π$. Even if both $ϕ$ and $ϕ'$ are algorithms for $Π$, $\{ ϕ, ϕ' \}$ may not be compatible with respect to $Π$. From the view point of compatibility, we investigate the convergence, the fault tolerant ($n,f$)-convergence (FC($f$)), the fault tolerant ($n,f$)-convergence to $f$ points (FC($f$)-PO), the fault tolerant ($n,f$)-convergence to a convex $f$-gon (FC($f$)-CP), and the gathering problems, assuming crash failures. Obtained results classify these problems into three groups: The convergence, FC(1), FC(1)-PO, and FC($f$)-CP compose the first group: Every set of target functions which always shrink the convex hull of a configuration is compatible. The second group is composed of the gathering and FC($f$)-PO for $f \geq 2$: No set of target functions which always shrink the convex hull of a configuration is compatible. The third group, FC($f$) for $f \geq 2$, is placed in between. Thus, FC(1) and FC(2), FC(1)-PO and FC(2)-PO, and FC(2) and FC(2)-PO are respectively in different groups, despite that FC(1) and FC(1)-PO are in the first group.

cs.DC↗

arXiv2020

A Characterization of Semi-Synchrony for Asynchronous Robots with Limited Visibility, and its Application to Luminous Synchronizer Design

A mobile robot system consists of anonymous mobile robots, each of which autonomously performs sensing, computation, and movement according to a common algorithm, so that the robots collectively achieve a given task. There are two main models of time and activation of the robots. In the semi-synchronous model (SSYNC), the robots share a common notion of time; at each time unit, a subset of the robots is activated, and each performs all three actions (sensing, computation, and movement) in that time unit. In the asynchronous model (ASYNC), there is no common notion of time, the robots are activated at arbitrary times, and the duration of each action is arbitrary but finite. In this paper, we investigate the problem of synchronizing ASNYC robots with limited sensing range, i.e., limited visibility. We first present a sufficient condition for an ASYNC execution of a common algorithm ${\cal A}$ to have a corresponding SSYNC execution of ${\cal A}$; our condition imposes timing constraints on the activation schedule of the robots and visibility constraints during movement. Then, we prove that this condition is necessary (with probability $1$) under a randomized ASYNC adversary. Finally, we present a synchronization algorithm for luminous ASYNC robots with limited visibility, each equipped with a light that can take a constant number of colors. Our algorithm enables luminous ASYNC robots to simulate any algorithm ${\cal A}$, designed for the (non-luminous) SSYNC robots and satisfying visibility constraints.

cs.DC↗

arXiv2019

Oblivious Permutations on the Plane

We consider a distributed system of n identical mobile robots operating in the two dimensional Euclidian plane. As in the previous studies, we consider the robots to be anonymous, oblivious, dis-oriented, and without any communication capabilities, operating based on the Look-Compute-Move model where the next location of a robot depends only on its view of the current configuration. Even in this seemingly weak model, most formation problems which require constructing specific configurations, can be solved quite easily when the robots are fully synchronized with each other. In this paper we introduce and study a new class of problems which, unlike the formation problems so far, cannot always be solved even in the fully synchronous model with atomic and rigid moves. This class of problems requires the robots to permute their locations in the plane. In particular, we are interested in implementing two special types of permutations -- permutations without any fixed points and permutations of order $n$. The former (called MOVE-ALL) requires each robot to visit at least two of the initial locations, while the latter (called VISIT-ALL) requires every robot to visit each of the initial locations in a periodic manner. We provide a characterization of the solvability of these problems, showing the main challenges in solving this class of problems for mobile robots. We also provide algorithms for the feasible cases, in particular distinguishing between one-step algorithms (where each configuration must be a permutation of the original configuration) and multi-step algorithms (which allow intermediate configurations). These results open a new research direction in mobile distributed robotics which has not been investigated before.

cs.DC↗

arXiv2019

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $σ$ (where $σ=1$ corresponds to no rotational symmetry), we prove that $k=σ+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.

cs.DC↗

arXiv2019

Exploration of Finite 2D Square Grid by a Metamorphic Robotic System

We consider exploration of finite 2D square grid by a metamorphic robotic system consisting of anonymous oblivious modules. The number of possible shapes of a metamorphic robotic system grows as the number of modules increases. The shape of the system serves as its memory and shows its functionality. We consider the effect of global compass on the minimum number of modules necessary to explore a finite 2D square grid. We show that if the modules agree on the directions (north, south, east, and west), three modules are necessary and sufficient for exploration from an arbitrary initial configuration, otherwise five modules are necessary and sufficient for restricted initial configurations.

cs.DC↗

arXiv2019

Coding theory for noiseless channels realized by anonymous oblivious mobile robots

We propose an information transmission scheme by a swarm of anonymous oblivious mobile robots on a graph. The swarm of robots travel from a sender vertex to a receiver vertex to transmit a symbol generated at the sender. The codeword for a symbol is a pair of an initial configuration at the sender and a set of terminal configurations at the receiver. The set of such codewords forms a code. We analyze the performance of the proposed scheme in terms of its code size and transmission delay. We first demonstrate that a lower bound of the transmission delay depends on the size of the swarm, and the code size is upper bounded by an exponent of the size of the swarm. We then give two algorithms for a swarm of a fixed size. The first algorithm realizes a near optimal code size with a large transmission delay. The second algorithm realizes an optimal transmission delay with a smaller code size. We then consider information transmission by swarms of different sizes and present upper bounds of the expected swarm size by the two algorithms. We also present lower bounds by Shannon's lemma and noiseless coding theorem.

cs.DC↗

arXiv2019

Finding Submodularity Hidden in Symmetric Difference

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-stransformations 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 ${\rm O}(|V|)$ oracle calls when $f$ is strictly submodular, although it requires exponentially many oracle calls in general.

cs.DM↗

arXiv2019

Can Walker Localize The Middle Point of A Line-segment?

This paper poses a question about a simple localization problem. The question is if an {\em oblivious} walker on a line-segment can localize the middle point of the line-segment in {\em finite} steps observing the direction (i.e., Left or Right) and the distance to the nearest end point. This problem is arisen from {\em self-stabilizing} location problems by {\em autonomous mobile robots} with {\em limited visibility}, that is a widely interested abstract model in distributed computing. Contrary to appearances, it is far from trivial if this simple problem is solvable or not, and unsettled yet. This paper is concerned with three variants of the problem with a minimal relaxation, and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.

cs.DC↗

arXiv2017

Plane Formation by Synchronous Mobile Robots without Chirality

We consider a distributed system consisting of autonomous mobile computing entities, called robots, moving in a specified space. The robots are anonymous, oblivious, and have neither any access to the global coordinate system nor any explicit communication medium. Each robot observes the positions of other robots and moves in terms of its local coordinate system. To investigate the self-organization power of robot systems, formation problems in the two dimensional space (2D-space) have been extensively studied. Yamauchi et al. (DISC 2015) introduced robot systems in the three dimensional space (3D-space). While existing results for 3D-space assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. 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. It has been shown that the solvability of the plane formation problem by robots with chirality is determined by the rotation symmetry of their initial local coordinate systems because the robots cannot break it. We show that when the robots lack chirality, the combination of rotation symmetry and reflection symmetry determines the solvability of the plane formation problem because a set of symmetric local coordinate systems without chirality is obtained by rotations and reflections. This richer symmetry results in the increase of unsolvable instances compared with robots with chirality and a flaw of existing plane formation algorithm. In this paper, we give a characterization of initial configurations from which the robots without chirality can form a plane and a new plane formation algorithm for solvable instances.

cs.DC↗

arXiv2016

The Parity Hamiltonian Cycle Problem

Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that the walk may use an edge more than once. First, we give a complete characterization of the graphs which have PHCs, and give a linear time algorithm to find a PHC, in which every edge appears at most four times, in fact. In contrast, we show that finding a PHC is NP-hard if a closed walk is allowed to use each edge at most z times for each z=1,2,3 (PHCz for short), even when a given graph is two-edge connected. We then further investigate the PHC3 problem, and show that the problem is in P when an input graph is four-edge connected. Finally, we are concerned with three (or two)-edge connected graphs, and show that the PHC3 is in P for any C_>=5-free or P6-free graphs. Note that the Hamiltonian cycle problem is known to be NP-hard for those graph classes.

cs.CC↗

arXiv2016

Universal Systems of Oblivious Mobile Robots

An oblivious mobile robot is a stateless computational entity located in a spatial universe, capable of moving in that universe. When activated, the robot observes the universe and the location of the other robots, chooses a destination, and moves there. The computation of the destination is made by executing an algorithm, the same for all robots, whose sole input is the current observation. No memory of all these actions is retained after the move. When the universe is a graph, distributed computations by oblivious mobile robots have been intensively studied focusing on the conditions for feasibility of basic problems (e.g., gathering, exploration) in specific classes of graphs under different schedulers. In this paper, we embark on a different, more general, type of investigation. With their movements from vertices to neighboring vertices, the robots make the system transition from one configuration to another. Viewing this transition as the computation of an abstract function, we ask which functions are computed by which systems. Our main interest is on identifying sets of systems that are "universal", in the sense that they can collectively compute all finite functions. We are able to identify several such classes of fully synchronous systems. In particular, among other results, we prove the universality of the set of all graphs with at least one robot, of any set of graphs with at least two robots whose quotient graphs contain arbitrarily long paths, and of any set of graphs with at least three robots and arbitrarily large finite girths. We then focus on the minimum size that a network must have for the robots to be able to compute all functions on a given finite set. We are able to approximate the minimum size of such a network up to a factor that tends to 2 as $n$ goes to infinity.

cs.DC↗

arXiv2016

Pattern Formation Problem for Synchronous Mobile Robots in the Three Dimensional Euclidean Space

We consider a swarm of autonomous mobile robots each of which is an anonymous point in the three-dimensional Euclidean space (3D-space) and synchronously executes a common distributed algorithm. We investigate the pattern formation problem that requires the robots to form a given target pattern from an initial configuration and characterize the problem by showing a necessary and sufficient condition for the robots to form a given target pattern. The pattern formation problem in the two dimensional Euclidean space (2D-space) has been investigated by Suzuki and Yamashita (SICOMP 1999, TCS 2010), and Fujinaga et al. (SICOMP 2015). The symmetricity $ρ(P)$ of a configuration (i.e., the positions of robots) $P$ is intuitively the order of the cyclic group that acts on $P$. It has been shown that fully-synchronous (FSYNC) robots can form a target pattern $F$ from an initial configuration $P$ if and only if $ρ(P)$ divides $ρ(F)$. We extend the notion of symmetricity to 3D-space by using the rotation groups each of which is defined by a set of rotation axes and their arrangement. We define the symmetricity $\varrho(P)$ of configuration $P$ in 3D-space as the set of rotation groups that acts on $P$ and whose rotation axes do not contain any robot. We show the following necessary and sufficient condition for the pattern formation problem which is a natural extension of the existing results of the pattern formation problem in 2D-space: FSYNC robots in 3D-space can form a target pattern $F$ from an initial configuration $P$ if and only if $\varrho(P) \subseteq \varrho(F)$. For solvable instances, we present a pattern formation algorithm for oblivious FSYNC robots. The insight of this paper is that symmetry of mobile robots in 3D-space is sometimes lower than the symmetry of their positions and the robots can show their symmetry by their movement.

cs.DC↗

arXiv2016

Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space

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 paper 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 polyhedrdon.

cs.DC↗

arXiv2015

Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates deterministic random walks, which is a deterministic process analogous to a random walk. While there are several progresses on the analysis of the vertex-wise discrepancy (i.e., $L_\infty$ discrepancy), little is known about the {\em total variation discrepancy} (i.e., $L_1$ discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of the $L_1$ discrepancy between the expected number of tokens in a Markov chain and the number of tokens in its corresponding deterministic random walk. First, we give a simple but nontrivial upper bound ${\rm O}(mt^*)$ of the $L_1$ discrepancy for any ergodic Markov chains, where $m$ is the number of edges of the transition diagram and $t^*$ is the mixing time of the Markov chain. Then, we give a better upper bound ${\rm O}(m\sqrt{t^*\log t^*})$ for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.

cs.DM↗

arXiv2015

Deterministic Random Walks for Rapidly Mixing Chains

The rotor-router model is a deterministic process analogous to a simple random walk on a graph. This paper is concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational transition probabilities. We investigate the discrepancy of the number of tokens at a single vertex between the functional-router model and its corresponding Markov chain, and give an upper bound in terms of the mixing time of the Markov chain.

cs.DM↗

arXiv2013

Rendezvous of Two Robots with Constant Memory

We study the impact that persistent memory has on the classical rendezvous problem of two mobile computational entities, called robots, in the plane. It is well known that, without additional assumptions, rendezvous is impossible if the entities are oblivious (i.e., have no persistent memory) even if the system is semi-synchronous (SSynch). It has been recently shown that rendezvous is possible even if the system is asynchronous (ASynch) if each robot is endowed with O(1) bits of persistent memory, can transmit O(1) bits in each cycle, and can remember (i.e., can persistently store) the last received transmission. This setting is overly powerful. In this paper we weaken that setting in two different ways: (1) by maintaining the O(1) bits of persistent memory but removing the communication capabilities; and (2) by maintaining the O(1) transmission capability and the ability to remember the last received transmission, but removing the ability of an agent to remember its previous activities. We call the former setting finite-state (FState) and the latter finite-communication (FComm). Note that, even though its use is very different, in both settings, the amount of persistent memory of a robot is constant. We investigate the rendezvous problem in these two weaker settings. We model both settings as a system of robots endowed with visible lights: in FState, a robot can only see its own light, while in FComm a robot can only see the other robot's light. We prove, among other things, that finite-state robots can rendezvous in SSynch, and that finite-communication robots are able to rendezvous even in ASynch. All proofs are constructive: in each setting, we present a protocol that allows the two robots to rendezvous in finite time.

cs.MA↗

arXiv2012

Expressivity of Time-Varying Graphs and the Power of Waiting in Dynamic Networks

In infrastructure-less highly dynamic networks, computing and performing even basic tasks (such as routing and broadcasting) is a very challenging activity due to the fact that connectivity does not necessarily hold, and the network may actually be disconnected at every time instant. Clearly the task of designing protocols for these networks is less difficult if the environment allows waiting (i.e., it provides the nodes with store-carry-forward-like mechanisms such as local buffering) than if waiting is not feasible. No quantitative corroborations of this fact exist (e.g., no answer to the question: how much easier?). In this paper, we consider these qualitative questions about dynamic networks, modeled as time-varying (or evolving) graphs, where edges exist only at some times. We examine the difficulty of the environment in terms of the expressivity of the corresponding time-varying graph; that is in terms of the language generated by the feasible journeys in the graph. We prove that the set of languages $L_{nowait}$ when no waiting is allowed contains all computable languages. On the other end, using algebraic properties of quasi-orders, we prove that $L_{wait}$ is just the family of regular languages. In other words, we prove that, when waiting is no longer forbidden, the power of the accepting automaton (difficulty of the environment) drops drastically from being as powerful as a Turing machine, to becoming that of a Finite-State machine. This (perhaps surprisingly large) gap is a measure of the computational power of waiting. We also study bounded waiting; that is when waiting is allowed at a node only for at most $d$ time units. We prove the negative result that $L_{wait[d]} = L_{nowait}$; that is, the expressivity decreases only if the waiting is finite but unpredictable (i.e., under the control of the protocol designer and not of the environment).

cs.DC↗