A general lower bound for collaborative tree exploration

We consider collaborative graph exploration with a set of $k$ agents. All agents start at a common vertex of an initially unknown graph and need to collectively visit all other vertices. We assume agents are deterministic, vertices are distinguishable, moves are simultaneous, and we allow agents to communicate globally. For this setting, we give the first non-trivial lower bounds that bridge the gap between small ($k \leq \sqrt n$) and large ($k \geq n$) teams of agents. Remarkably, our bounds tightly connect to existing results in both domains. First, we significantly extend a lower bound of $\Omega(\log k / \log\log k)$ by Dynia et al. on the competitive ratio of a collaborative tree exploration strategy to the range $k \leq n \log^c n$ for any $c \in \mathbb{N}$. Second, we provide a tight lower bound on the number of agents needed for any competitive exploration algorithm. In particular, we show that any collaborative tree exploration algorithm with $k = Dn^{1+o(1)}$ agents has a competitive ratio of $\omega(1)$, while Dereniowski et al. gave an algorithm with $k = Dn^{1+\varepsilon}$ agents and competitive ratio $O(1)$, for any $\varepsilon > 0$ and with $D$ denoting the diameter of the graph. Lastly, we show that, for any exploration algorithm using $k = n$ agents, there exist trees of arbitrarily large height $D$ that require $\Omega(D^2)$ rounds, and we provide a simple algorithm that matches this bound for all trees

Building a Nest by an Automaton

A robot modeled as a deterministic finite automaton has to build a structure from material available to it. The robot navigates in the infinite oriented grid Z x Z. Some cells of the grid are full (contain a brick) and others are empty. The subgraph of the grid induced by full cells, called the field, is initially connected. The (Manhattan) distance between the farthest cells of the field is called its span. The robot starts at a full cell. It can carry at most one brick at a time. At each step it can pick a brick from a full cell, move to an adjacent cell and drop a brick at an empty cell. The aim of the robot is to construct the most compact possible structure composed of all bricks, i.e., a nest. That is, the robot has to move all bricks in such a way that the span of the resulting field be the smallest. Our main result is the design of a deterministic finite automaton that accomplishes this task and subsequently stops, for every initially connected field, in time O(sz), where s is the span of the initial field and z is the number of bricks. We show that this complexity is optimal

Shape Recognition by a Finite Automaton Robot

Motivated by the problem of shape recognition by nanoscale computing agents, we investigate the problem of detecting the geometric shape of a structure composed of hexagonal tiles by a finite-state automaton robot. In particular, in this paper we consider the question of recognizing whether the tiles are assembled into a parallelogram whose longer side has length l = f(h), for a given function f(*), where h is the length of the shorter side. To determine the computational power of the finite-state automaton robot, we identify functions that can or cannot be decided when the robot is given a certain number of pebbles. We show that the robot can decide whether l = ah+b for constant integers a and b without any pebbles, but cannot detect whether l = f(h) for any function f(x) = omega(x). For a robot with a single pebble, we present an algorithm to decide whether l = p(h) for a given polynomial p(*) of constant degree. We contrast this result by showing that, for any constant k, any function f(x) = omega(x^(6k + 2)) cannot be decided by a robot with k states and a single pebble. We further present exponential functions that can be decided using two pebbles. Finally, we present a family of functions f_n(*) such that the robot needs more than n pebbles to decide whether l = f_n(h)

Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur

Поведение конечных автоматов в лабиринтах

The paper is devoted to the study of problems on the behavior of finite automata in mazes. For any n, a maze is constructed that can be bypassed with 2n stones but you can’t get around with n stones. The range of tasks is extensive and touches upon key aspects of theoretical Computer Science. Of course, the solution of such problems does not mean the automatic solution of complex problems of complexity theory, however, the consideration of these issues can have a positive impact on the understanding of the essence of theoretical Computer Science. It is hoped that the behavior of automata in mazes is a good model for non-trivial information theoretic problems, and the development of methods and approaches to the study of robot behavior will give more serious results in the future. Problems related to automaton analysis of geometric media have a rather rich history of study. The first work that gave rise to this kind of problems, it is necessary to recognize the work of Shannon [24]. It deals with a model of a mouse in the form of an automaton, which must find a specific target in the maze. Another early work, one way or another affecting our problems, is the work of Fisher [9] on computing systems with external memory in the form of a discrete plane. A serious impetus to the study of the behavior of automata in mazes was the work of Depp [7, 8], in which the following model is proposed: there is a certain configuration of cells from mathbbZ^2 (chess maze), in which finite automata, surveying some neighborhood of the cell in which they are, can move to an adjacent cell in one of four directions. The main question posed in such a model is whether there is an automaton that bypasses all such mazes. In [20], Muller constructed a flat trap for a given automaton (a maze that does not completely bypass) in the form of a 3-graph. Budach [5] constructed a chess trap for any given finite automaton. Note that Budach’s solution was quite complex (the first versions contained 175 pages). More visual solutions to this question are presented here [29, 31, 33, 34]. Antelman [2] estimated the complexity of such a trap by the number of cells, and in [1] Antelman, Budach, and Rollick made a finite trap for any finite automaton system. In the formulation with a chess maze and one automaton, there are a number of results related to the problems of traversability of labyrinths with different numbers of holes, with bundles of labyrinths by the number of States of the automaton, and other issues. An overview of such problems can be found for example here [35]. The impossibility of traversing all flat chess labyrinths with one automaton raised the question of studying the possible amplifications of the automaton model, which will solve the problem of traversal. The main way of strengthening can be the consideration of a collective of automata, instead of one automaton, interacting with each other. A special and widely used case is the consideration of a system of one full-fledged automaton and a certain number of automata of stones, which have no internal state and can move only together with the main automaton. Interaction between machines is a key feature of this gain, it is allowed to have a collective (or one machine with stones) external memory, thereby significantly diversifies its behavior. If you get rid of the interaction of automata, the resulting  independent system will be little better than a single machine. Next, we discuss the known results associated with the collective automata.Работа посвящена исследованию задач о поведении конечных автоматов в лабиринтах. Для любого n строится лабиринт, который можно обойти с помощью 2n камней но нельзя обойти с помощью n камней. Спектр задач обхода обширен и затрагивает ключевые аспекты теоретической Computer Science. Конечно, решение таких задач не означает автоматическое решение сложных проблем теории сложности, тем не менее рассмотрение данных вопросов может положительно сказаться на понимании сути теоретической Computer Science. Есть надежда, что поведение автоматов в лабиринтах является хорошей моделью для нетривиальных теоретико-информационных задач, и отработка методов и подходов к исследованию поведения роботов даст более серьезные результаты с будущем. Задачи связанные c автоматным анализом геометрических сред имеют довольно богатую историю изучения. Первой работой, давшей начало подобного рода задачам, стоит признать работу Шеннона [24]. В ней рассматривается модель мыши в виде автомата, которая должна найти определенную цель в лабиринте. Другая ранняя работа, так или иначе затрагивающая нашу проблематику, это работа Фишера [9] о вычислительных системах с внешней памятью в виде дискретной плоскости. Серьёзным толчком к исследование поведения автоматов в лабиринтах послужила работы Деппа [7, 8], в которых предложена следующая модель: имеется некоторая конфигурация клеток из Z^2 (шахматный лабиринт), в которой конечные автоматы, обозревая некоторую окрестность клетки, в которой они находятся, могут перемещаться в соседнюю клетку в одном из четырёх направлений. Основной вопрос, который ставится в подобной модели, существует ли автомат обходящий все подобные лабиринты. В [20] Мюллер построил для заданного автомата плоскую ловушку (лабиринт который обходится не полностью) в виде 3-графа. Будах [5] построил шахматную ловушку для любого заданного конечного автомата. Отметим, что решение Будаха было довольно сложным (первые варианты содержали 175 страниц). Более наглядные решения данного вопроса представлены здесь [29, 31, 33, 34]. Антельман [2] оценил сложность подобной ловушки по числу клеток, а в [1] Антельман, Будах и Роллик сделали конечную ловушку для любой конечной системы автоматов. В постановке с шахматным лабиринтом и одним автоматом есть ещё ряд результатов, связанных с проблемами обходимости лабиринтов с различными числом дыр, с расслоениями лабиринтов по количеству состояний автомата и другими вопросами. Обзор подобных проблем можно найти например здесь [35]. Невозможность обхода всех плоских шахматных лабиринтов одним автоматом выдвинула вопрос об изучении возможных усилений модели автомата, которая решит задачу обхода. Основным способом усиления может являться рассмотрение коллектива автоматов,вместо одного автомата, взаимодействующих между собой. Частным и широко используемым случаем является рассмотрение системы из одного полноценного автомата и некоторого количества автоматов камней, которые не имеют внутреннего состояние и могут передвигаться только совместно с главным автоматом. Взаимодействие между автоматами является ключевой особенностью данного усиления, оно позволяется иметь коллективу (или одному автомату с камнями) внешнюю память, тем самым существенно разнообразит его поведение. Если от взаимодействия автоматов избавиться, то полученная независимая система будет немногим лучше одного автомата. Далее обсудим известные результаты связанные с коллективом автоматов

Tight bounds for undirected graph exploration with pebbles and multiple agents

We study the problem of deterministically exploring an undirected and initially unknown graph with $n$ vertices either by a single agent equipped with a set of pebbles, or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles $\Theta(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most $n$ vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles, or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory $\Theta(\log \log n)$ pebbles are necessary and sufficient for exploration. We further prove that the same bound holds for the number of collaborating agents needed for exploration. For the upper bound, we devise an algorithm for a single agent with constant memory that explores any $n$-vertex graph using $\mathcal{O}(\log \log n)$ pebbles, even when $n$ is unknown. The algorithm terminates after polynomial time and returns to the starting vertex. Since an additional agent is at least as powerful as a pebble, this implies that $\mathcal{O}(\log \log n)$ agents with constant memory can explore any $n$-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph of size $n$ is already $\Omega(\log \log n)$ when we allow each agent to have at most $\mathcal{O}( \log n ^{1-\varepsilon})$ bits of memory for any $\varepsilon>0$. This also implies that a single agent with sublogarithmic memory needs $\Theta(\log \log n)$ pebbles to explore any $n$-vertex graph