Memory Efficient Anonymous Graph Exploration

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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
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  Memory Efficient Anonymous Graph Exploration Leszek G ˛asieniec 1 Tomasz Radzik  2 1 Department of Computer Science, University of Liverpool, Liverpool L69 3BX,United Kingdom. E-mail:  L.A.Gasieniec@liverpool.ac.uk 2 Department of Computer Science, King’s College London, Strand, London WC2R 2LS,United Kingdom. E-mail: Tomasz.Radzik@kcl.ac.uk . Abstract.  Efficient exploration of unknown or unmapped environments has be-come one of the fundamental problem domains in algorithm design. Its applica-tions range from robot navigation in hazardous environments to rigorous search-ing, indexing and analysing digital data available on the Internet. A large numberof exploration algorithms has been proposed under various assumptions about thecapability of mobile (exploring) entities and various characteristics of the envi-ronment which are to be explored. This paper considers the  graph model  , wherethe environment is represented by agraph of connections in which discrete movesare permitted only along its edges. Designing efficient exploration algorithms inthis 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, aswell as in the context of different complexity measures including the time com-plexity, the memory consumption, and the use of other computational resourcessuch as tokens and messages. In this work the emphasis is on memory efficientexploration 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 re-search. 1 Introduction A  graph  is a crucial combinatorial notion used for modeling complex systems in var-ious application domains including communication, transportation and computer net-works, manufacturing,scheduling, molecular biology and peer-to-peer networks. Mod-els based on graphs often involve mobile entities which can move throughout the graphfrom node to node along the edges. We call such entities  agents . An agent can be arobot servicing a hazardous environment, or a software process navigating the Internetin searchforsomeinformation. Graphexploration  referstoproblemsofdesigningalgo-rithms (protocols) for an agent, or a group of agents, to traverse a graph in a systematicand efficient way.In recent years the research on efficient graph exploration gathered a new momen-tum, generated to large extent by the theory and the applications coming ever closertogether. The demand for efficient practical solutions has increased as systems of soft-wareagentsmovingthroughalargenetworkofcomputershavebecomereality.Anotherexample is the relevance of efficient graph exploration algorithms for efficiency of the  Internet search engines. The current applications indicate that the relative importanceof various aspects of graph exploration has been changing, and those changes becomereflected in the theoretical research.In the broad context of algorithmic agent design we distinguish two main models:the  geometricmodel   where the search environmentis representedby two- or higher- di-mensional space (see, e.g., [11,28,59]) and the  graph model  , considered in this paper,where the environment is represented by a finite or infinite graph supported by discretemoves permitted only along its edges. The design of efficient exploration algorithms inthe graph model has been extensively studied under many different assumptions, e.g., directed   vs  undirected graphs ,  anonymous  nodes vs nodes with  distinct identities , or deterministic   vs  probabilistic   solutions, as well as with different performance objec-tives in mind including optimal time complexity, memory consumption, or use of otherresources, see [1,6,9,29–31,34,41,56]. Different studies may also consider differentaims of exploration.The aim of exploration can be to visit each node in the network, oreach edge, and terminate. Alternatively, one may drop the termination requirement andask only for  perpetual   exploration and a guarantee that each node is visited infinitelymanytimes,orperhapsa strongerguaranteethatthenodesarebeingvisitedwithsimilarfrequencies.If nodes have distinct identities (given as  O (log n ) -bit words), the graph stays un-changed (a static graph), and the size of the memory available to the agent (counted inwords) is linear in the number of nodes of the graph, then the depth-first search (DFS)procedure gives linear time exploration. However, in many applications one or moreof these assumptions might not hold. The nodes may not have unique identities; forexample, if the nodes represent very simple devices ( anonymous  graphs). The graphmay keep changing;for example,if it models a growingpeer-to-peernetwork( dynamic  graphs). Finally, the agents may have very limited memory, which may be sufficientfor storing only few node identities. For example, software agents moving through acomputer network from host to host might not be allowed to carry too much data withthem.This surveyis mainlyconcernedwith explorationofanonymousgraphsby agentsequipped with bounded memory.In graph exploration algorithms, the memory utilisation refers actually not only tothe memory of the agents (“carried” by them when they move from node to node), butalso to any extra memory required in the graph environment.The latter may store some(pre-computed)additional informationabout the graph to guide the exploration,or mayallowtheagenttoleavemarksatnodesortomovetokensasit traverses.Thedemandforsimple and cost effective agents as well as the desire to design exploration algorithmsthat are suitable for rigorous mathematical analysis imply the importance of limitingthe local memory of agents and their ability to manipulate the explored environment.One of the most challenging problems in the theory of computation is to look at thefar ends,  border cases , of the considered models. In case of algorithmic agent designsuch a border case may refer to the size of the agent’s memory, where one can limitthe memory of an agent to a constant number of bits. This case is very often modeledas graph exploration by a finite state automaton and it has been extensively studiedalready in the 1970’s [15,54,58,61]. Probably the strongest result in this setting is due to Cook and Rackoff  [19]. They proved that a fixed group of finite automata, that  can permanently cooperate and that can use "teleportation" to move from their currentlocation to the location of any other automaton, cannot explore all graphs. See [42]for some recent results about limits of graph exploration with finite automata. Theseresults imply that we either have to allow the agents to use larger memory, or to divertto randomization,or to providethe agentswith extrastructuralinformationthat restrictsthe set of graphs they have to traverse.Ithasbeenknownforsometimethatifwedonotplacestrictrestrictionsonthelocalmemory, then a single pebble is sufficient to explore an anonymous undirected graph.This result was extended to directed graphs by Bender  et al.  [8]. Note however that agood upper boundon the number of nodes must be known to avoid an exponential-timesolution. Moreover, even if such a bound is known, the time complexity, while polyno-mial, remainsimpracticallyhigh.Anotherpossibilityis to dropdeterminismandto look for randomized solutions. It is known, e.g., that a random walk of length  O ( n 3 log n ) visits all nodes of an arbitrary  n -node graph with high probability [3]. Attempts to re-gain determinism included research on derandomizationof random walks and the mainapproach was  universal traversal sequences  [3] that provide guidance in deterministictraversal of all graphs in a given class. Several important results have been achieved[4,7,46,60] including Reingold’s [60] recent asymptotically optimal  O (log n ) -spacedeterministic algorithm for the undirected  st-connectivity   problem based on a novel O (log n ) − bit navigation mechanism. However, note that the exploration time given bythis algorithm is a polynomial of a rather high degree.Research closely related to the setting adopted in this paper assumes some struc-tural information about the explored environment. Such additional information allowsimprovements in the time or memory complexity of graph traversal. The first results,concerning exploration of a labyrinth using a compass, are due to Blum and Kozen[12]. Later, Flocchini  et al.  [39] introduced a more general notion of   sense of direction and proved that traversal can be performed using  O ( n )  messages/agent moves in thismodel [38]. Fraigniaud  et al.  [40] have shown that interval routing scheme can be usedto achievethe same goal. In fact, givena spanningtree, the graphcan be traversedusing O ( n )  moves. Pelc and Panaite [56] studied the impact of having a map of the graph onthe efficiency of graph exploration. Finally Cohen  et al.  studied efficient navigation ingraphs with nodes marked with a constant number of colors, see [18].Inthispaperwefocusonthreegraphtraversalmethods.Westartwiththeprobabilis-tic  random walk   method and then discuss its recently proposed deterministic counter-part known as the  Propp machine  , which requires some (small) memory at the nodes of the graph. We conclude this survey with presentation of an alternative traversal methodbased on the  basic walk  , in which a small amount of memory is provided to an agentand a certain type of graph preprocessing is permitted. These three methods have sev-eral interesting combinatorial properties and one might say that they have already setcertain standards in agent based anonymous graph exploration. 2 The graph model In this surveywe consider environmentsrepresentedby undirected(symmetric)graphs.We denote by  G  = ( V,E  )  the graph which is to be explored, and assume that it is con-  nected, unless stated otherwise. It will sometimes be convenient to view  G  as a digraph ←→ G  obtained by replacing each undirected edge with two arcs pointing in opposite di-rections. We consider graphs (environments) that are anonymous, i.e., the nodes in thegraphs are neither labeled nor marked in any other way. However, the ends of edgesincident to each node  v  are ordered and often labeled by consecutive integers  1 ,...,d v called  port numbers , where  d v  is the degree of   v .If an agent is in the current step at a node  v , then the standard assumptions are thatit knows  d v , the label of the port through which it has entered  v  and any informationabout the graph that it might have gathered in previous steps and has been able to (hasbeen allowed to) store in its internal limited memory. The agent decides on the basisof this knowledge which port it should take to move in the next step to a neighbourof   v . Agents do not have prior knowledge about the topology of the network. The exactdetails about the resources and abilities of the agents as well as about the objectives of exploration may vary from problem to problem. The number of nodes and the numberofedgesin G are denotedby n and m ,respectively,butnotethat theseparametersmightnot be known to the exploring agents. 3 The random walk A  random walk   on an (undirected, connected) graph  G  starting at a node  v 0  is an (infi-nite) random sequence  ( v 0 ,v 1 ,v 2 ,... )  of nodes in  G  such that for each  i ≥  1 , node  v i is selected randomly and uniformly from all neighbours of node  v i − 1 . Using the graphexploration terminology, we say that an agent moves in step  i  from node  v i − 1  to itsrandom neighbour  v i . To implement such random exploration of an arbitrary  n -nodegraph, the agent has to be able to select a random neighbour of the current node, so itneeds  O (log n ) -bit memory and access to  log n  random bits per step.The  (node) cover time of graph  G  from a node   v  is the expected number of steps C  v ( G )  the random walk starting from node  v  takes to visit all nodes of the graph.The  cover time   C  ( G )  of graph  G  is defined as the maximum  C  v ( C  )  over all nodes v  ∈  V   . Thus the cover time is the worst-case (over all starting nodes) expected timeof exploring the whole graph. For graphs of some special types, the cover time canbe easily estimated, or even calculated exactly. For example, it is easy to show that thecovertimeofthegraph P  n  whichisan n -nodesimplepathisequalto C  ( P  n ) = ( n − 1) 2 ,by solving a simple recurrence relation for the number expected number of steps  H  i requiredto reach the end of the path starting from its  i -th node. Calculation of the covertime of the  n -nodeclique  K  n  is the  couponcollector   problem:at step  i  select randomlyand uniformly one of the  n  coupons/nodes and wait until all coupons have been seen,with a slightmodificationthat thenextcoupon/nodehas to bedifferentfromthe onejustselected. If we have already seen exactly i distinct nodes,then the probabilitythat in thenext step we will see a new node is equal to  ( n − i ) / ( n − 1) ,so the expected number of steps before a new node is encountered is equal to  ( n − 1) / ( n − i )  and the cover time C  ( K  n )  is equal to  n − 1 i =1  ( n − 1) / ( n − i ) = ( n − 1)  n − 1 i =1  1 /i  =  n (ln n  +  O (1)) .Random walks became an important tool in algorithm design and complexity the-ory when Aleliunas  et al.  [3] showed in 1979 that the cover time of   every   graph ispolynomial, or more specifically, at most  2 m ( n − 1) . Since then general techniques for  bounding the cover times have been developed and bounds for the cover times for vari-ous families of graphs have been derived. An agent using a random walk on an  n -nodegraph with the cover time bounded by  T  ( n ) =  poly ( n )  should have an  O (log n ) -bitcounter to count  T  ( n )  steps to terminate with the knowledge that the whole graph hasbeen exploredwith constant probability,or to count T  ( n )  p log n  steps to terminate withthe knowledge the whole graph has been explored with the high probability of at least 1 − 1 /n  p . Thus a good bound  T  ( n )  on the cover time is required not to expand unnec-essarily the exploration time, and we review below the main known bounds. Observethat an agent implementing a random walk needs memory for two purposes. It needs O (log ∆ )  bits to implement individual moves from the current node to a random neigh-bour, where  ∆  ≤  n  is a bound on the degree of a node, and  O (log n )  bits to count themoves. If we do not require that the agent terminates, than the total of   O (log ∆ )  bitssuffices. In particular, constant-size memory is sufficient for the randomized  perpetual  exploration of any constant degree graph.Feige [36,35] showed the following tight bounds on the range of the cover times of  n -node graphs: (1 − o (1)) n ln n ≤ C  ( G ) ≤ (1 +  o (1)) 427 n 3 . Thus  K  n  is an example of a graph with the cover time achieving the lower bound,while it can be shown that the  n -node  lollipop   graph given in Figure 1 has the covertime achieving the upper bound. A quadratic  O ( n 2 )  upper bound on the cover timeof regular graphs was first shown by Kahn, Linial, Nisan and Saks in 1989, and thebest know bound of   2 n 2 is due to Feige [37]. This worst-case upper bound for regulargraphs should be contrasted with Rubinfeld’s [62]  O ( n log n )  bound on the cover timeof regular  expander   graphs, and with Cooper and Frieze’s [20] recent result showingthat the cover time of a  random   d -regular graph is  (1 +  o (1)) d − 1 d − 2  n ln n  with highprobability. The cover time of the 2-dimensional √  n ×√  n  grid is  Θ ( n log 2 n ) , withthe upper bound due to Chandra  et al.  [17] and the lower bounddue to Zuckerman[65]. Aldous [2] showed that the cover time of the  n -node  k -dimensional grid is  Θ ( n log n ) ,for  k ≥ 3 .Aleliunas’  et al.  [3] polynomialupper boundon the cover time of an arbitrary graphwas an important result for the computational complexity theory as it showed that theundirected  s - t  connectivity problem can be solved by a randomised log-space algo-rithm. This started a vast body of research with the goal to settle the conjecture thatthis problem can be solved by a  deterministic   log-space algorithm, and, ultimately, themore general conjecture that  any   problem solvable by a randomised log-space algo-rithm can be solved by a deterministic log-space algorithm. A natural approachto settlethe first conjecture was to de-randomise random walks. In this framework the objec-tive was to produce an explicit universal traversal sequence (UTS), i.e., a sequence  p 1 ,...,p k  of port labels, such that the path guided by this sequence visits all edgesof any graph of a given size. It is known that, with high probability, a sequence of length O ( n 3 d 2 log n ) , chosenuniformlyatrandom,guidesa walkinany d -regular(con-nected)  n -node graph [3]. Unfortunately,explicit short UTS are known only for specialfamilies of graphs, including 2-regular graphs [7,13,16,50,53], 3-regular graphs [49], cliques [51], and expanders [46]. Some of these sequences can be constructed in log-
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