A robot with k-bit memory has to explore a tree whose nodes are unlabeled and edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the tree or of its size, and its aim is to traverse all the edges. While O(log ) bits of memory suce to explore any tree of maximum degree if stopping is not required, we show that bounded memory is not sucient to explore with stop all trees of bounded degree (indeed nde log log n) bits of memory are needed for some such trees of size n). For the more demanding task requiring to stop at the starting node after completing exploration, we show a sharper lower bound nd n) on required memory size, and present an algorithm to accomplish this task with O(log n)-bit memory, for all n-node trees.

In this paper we introduce a new notion of traversal sequences that we call exploration sequences. Exploration sequences share many properties with the traversal sequences defined in [AKL+], but they also exhibit some new properties. In particular, they have an ability to backtrack, and their random properties are robust under choice of the probability distribution on labels. Further, we present extremely simple constructions of polynomial length universal exploration sequences for some previously studied classes of graphs (e.g., 2-regular graphs, cliques, expanders), and we also present universal exploration sequences for trees. Our constructions beat previously known lower-bounds on the length of universal traversal sequences. 1

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. 1

Abstract. We consider the problem of perpetual traversal by a single agent in an anonymous undirected graph G. Our requirements are: (1) deterministic algorithm, (2) each node is visited within O(n) moves,(3) the agent uses no memory, it can use only the label of the link via which it arrived to the current node, (4) no marking of the underlying graph is allowed and (5) no additional information is stored in the graph (e.g. routing tables, spanning tree) except the ability to distinguish between the incident edges (called Local Orientation). This problem is unsolvable, as has been proven in [9, 28] even for much less restrictive setting. Our approach is to somewhat relax the requirement (5). We fix the following traversal algorithm: “Start by taking the edge with the smallest label. Afterwards, whenever you come to a node, continue by taking the successor edge (in the local orientation) to the edge via which you arrived ” and ask whether it is for every undirected graph possible to assign the local orientations in such a way that the resulting perpetual traversal visits every node in O(n) moves. We give a positive answer to this question, by showing how to construct such local orientations. This leads to an extremely simple, memoryless, yet efficient traversal algorithm. 1

Motivated by Reingold’s recent deterministic log-space algorithm for UNDIRECTED S-T CONNEC-TIVITY (ECCC TR 04-94), we revisit the general RL vs. L question, obtaining the following results. 1. We exhibit a new complete problem for RL: S-T CONNECTIVITY restricted to directed graphs for which the random walk is promised to have polynomial mixing time. 2. Generalizing Reingold’s techniques, we present a deterministic, log-space algorithm that given a directed graph G that is biregular (i.e., all in-degrees and out-degrees are equal) and two vertices s and t, finds a path between s and t if one exists. 3. Using the same techniques as in Item 2, we give a “pseudorandom generator” for random walks on “consistently labelled” biregular graphs. Roughly speaking, given a random seed of logarithmic length, the generator constructs, in log-space, a “short” pseudorandom walk that ends at an almostuniformly distributed vertex when taken in any consistently labelled biregular graph. 4. We prove that if our pseudorandom generator from Item 3 could be generalized to all biregular graphs (instead of just consistently labelled ones), then our complete problem from Item 1 can be solved in log-space and hence RL = L.

The paper presents a simple construction of polynomial length universal traversal sequences for cycles. These universal traversal sequences are log-space (even NC 1) constructible and are of length O(n 4.03). Our result improves the previously known upper-bound O(n 4.76) for log-space constructible universal traversal sequences for cycles.

Efficient exploration of unknown or unmapped environments has become one of the fundamental problem domains in algorithm design. Its applications range from robot navigation in hazardous environments to rigorous searching, indexing and analysing digital data available on the Internet. A large number of exploration algorithms has been proposed under various assumptions about the capability of mobile (exploring) entities and various characteristics of the environment which are to be explored. This paper considers the graph model, where the environment is represented by a graph of connections in which discrete moves are permitted only along its edges. Designing efficient exploration algorithms in this model has been extensively studied under a diverse set of assumptions, e.g., directed vs undirected graphs, anonymous nodes vs nodes with distinct identities, deterministic vs probabilistic solutions, single vs multiple agent exploration, as well as in the context of different complexity measures including the time complexity, the memory consumption, and the use of other computational resources such as tokens and messages. In this work the emphasis is on memory efficient exploration of anonymous graphs. We discuss in more detail three approaches: random walk, Propp machine and basic walk, reviewing major relevant results, presenting recent developments, and commenting on directions for further research.

A mobile agent (robot), modeled as a finite automaton, has to visit all nodes of a regular graph. How does the memory size of the agent (the number of states of the automaton) influence its exploration capability? In particular, does every increase of the memory size enable an agent to explore more graphs? We give a partial answer to this problem by showing that a strict gain of the exploration power can be obtained by a polynomial increase of the number of states. We also show that, for automata with few states, the increase of memory by even one state results in the capability of exploring more graphs.

Traversal sequences were defined in [AKL+] as a tool for the study of undirected s-t-connectivity. [K1] defines a new variant of traversal sequences, exploration sequences, with certain advantages over the earlier notion of traversal sequences. An exact relationship of these two notions was not known. In this paper we establish a relationship between these two concepts, in particular, we show that universal traversal sequences can be efficiently converted into universal exploration sequences. We also study conversion of universal exploration sequences for d ′-regular graphs into universal exploration sequences for d-regular graphs. Further, we also show certain self-correcting properties of traversal and exploration sequences and we propose a candidate for a universal exploration sequence.

We consider the fundamental problem of exploring an undi-rected and initially unknown graph by an agent with lit-tle memory. The vertices of the graph are unlabeled, and the edges incident to a vertex have locally distinct labels. In this setting, it is known that Θ(logn) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We show that this memory requirement can be decreased significantly by making a part of the mem-ory distributable in the form of pebbles. A pebble is a device that can be dropped to mark a vertex and can be collected when the agent returns to the vertex. We show that for an agent O(log logn) distinguishable pebbles and bits of mem-ory are sufficient to explore any bounded-degree graph with at most n vertices. We match this result with a lower bound exhibiting that for any agent with sub-logarithmic memory, Ω(log logn) distinguishable pebbles are necessary for exploration.