Inthe graph environment, theexploration problemis usually resolvedwhen therobot has explored ail edges of the graph. The common measure of efficiency of a given algorithm isits compétitiveratio: the ratiobetween theamount ofedge traversais re
quired byan agentfollowingthe algorithmand theminimalamount ofedge traversais required by an robot withperfect knowledge ofthe environment.
In [83], asinglerobothasforgoal theexplorationofadirected, stronglyconnected graph. The robot is able to differentiate a new point fromone it has already visited, and can correctly identify the degree of each node it visits. The efficiency of the algorithm is measured by the ratio ofedge traversed without prior knowledge ofthe graph compared to the minimal amount ofedge traversais required with knowledge of the graph. The authors of [83] provide a tight bound of 2 for Eulerian graphs, and prove that this ratio is unbounded when the deficiency (the smallest number of edgesthat hâve to be addedto thegraph in orderto make it Eulerian) ofthe graph is
unbounded. The authors also provide an algorithmwith anumber ofedge traversais that is exponential in the deficiency ofthe graph.
A different measure of the efficiency ofthe algorithm is used in [112], where the efficiency ofarobotexploringailedgesofadirected,stronglyconnectedgraphis mea-sured using a compétitive ratio between the number ofedges traversed by the robot and the minimal number ofedge traversais required by a robot with full knowledge ofthe graph. The authors claim to introduce the first deterministic online algorithm with a compétitive ratio polynomial in d the deficiency ofthe graph.
[88| considers the task ofexploring ail edges ofan undirected connected graph by a single robot traversing as few edges as possible. The quality of a given algorithm is measured by the ratio between the number of edges traversed and the minimal number ofedge traversais given knowledge ofthe graph. Such a ratio is referred to as the overhead ofthe algorithm for a givenclass ofgraphswhen maximised over ail possiblestartingnodesofthegraph forthisgivenclassofgraphs. The paperconsiders three possible scénarios: in the first one, the robot knows nothing about the graph;
in the second, the robot has an isométrie map of the graph without spécification about thestarting point; and in the thirdone, the robot is also aware ofits starting point. For various classes ofgraphs, an algorithm is provided, often with an optimal overhead.
The first sub-exponential algorithm to explore an unknown environment repre-sented as adirected, strongly connected graph is provided in [4], where analgorithm with a bound ofd°(logd^m is provided, with d the deficiency ofthe graph and m the number ofedges. The authors also show that theiralgorithm is optimal by proving a matching lower bound for this variation ofthe exploration problem.
Simple Depth-First search (DFS) algorithms hâve been used to explore graphs in [166], with the number of edge traversais used as a measure for the efficiency of the
algorithm. They establish an upperbound ofmin(mn,dri2+m) edge traversais, with d being the defieiency ofthe graph, m edges and n nodes.
DFS algorithme are also used in [128], where a robot has to solve the exploration problem (traverse ail edges) in an unlabelled graph with no prior knowledge of the graph. In particular, there is no knowledge about the size of the graph. It is shown that a robot with constant memory K is unable to explore certain graphs with a maximumdegreed\d > 3andK+1 nodes. Moreover, arobotneedsat least Q(Dlogd) bits of memory to explore agraph ofdiameter D and maximum degree d, and this is a tight bound.
In [93], the taskofexploringa tree ofmaximumdegree A is given to a robot that has a limited memory. It is observed that 0(logA) bits of memory are required to exploresuchtreeifstoppingis not required (that is, the robotdoesnot hâvetostop at the completion ofthe exploration). Ifstopping is required, it is shown that bounded memory is not sufficient. Moreover, the authors show that ül(logloglogn) bits of memory are required for some trees with n vertices. Finally, the paper considers a variation of the problem where the robot has to return to its starting node after exploration,and showthat at least fî(logn) bits ofmemory are required todo so. An algorithm that matches this bound is provided in [138], thus solvingthe exploration oftrees with O(logn) bits ofmemory.
The exploration problem is presented in adifferent light in [195], under the naine of Chinese Postman Problem (CPP), and avariation for directed graph, the Directed Chinese Postman Problem (DPP). The standard définition of this problem requires for the agent to return to its original position after exploring ail of the edges; the paper also introduces the Open Chinese Postman Problem, where the requirement of returning to the original position is dropped. The paper offers algorithms that résolves ail those problems, but without optimising the amount ofmemory used.
Memory concerns are considered in [127], where it is shown, for a directed graph with n nodesofmaximumout-degree d, thataminimum offi(nlogd) bitsofmemory arerequiredby anagenttocomplététheexplorationofthegraph. Thepaperprovides an algorithm that solves the problem with 0(nd\ogn) bits.
In [39], a robot in an undirected graph has to visit ail edges, then return to its starting point. The efficiency of the algorithm is measured by the number of edges traversed. Alternatively, the efficiency is measured by the minimal amount of memory required by the robot. In this paper, the robot knows a bound n on the size ofthe graph, and a bound d on the maximal degree ofa node. The paper offers two algorithms, both aiming to improve the upper bound established in [65]. The first algorithm aims at a minimal number ofedge traversais while the second tries to minimise the required amount of memory for the robot. Both algorithms represent an improvement compared to the upper bound established in [65].
[86] explores the information required by a robot in order to draw a complété or partialmapofagraph. A complétémap ofagraphisunderstoocl tobeanisomorphic copy ofthemap including its port numbers, and apartial map is understood to be a spanningtreewithportnumbers. Therobotisforcedtouseadeterministicalgorithm, andis unableto marknodesinany way. Itisproven thatthis mapdrawingis possible without,furtherinformationifthegraphisatree. Otherwise, somebitsofinformation, called advice, are required for the robot to construct either the complété or partial map. Thepaperestablishesthattheminimalsizeoftheadviceislinkedtothenumber ofnodes n ofthe graph, the number ofedges m ofthe graph, and the multiplicity //.
ofthe graph, that is, the number of nodes that hâve an identical view of the graph.
Bounds on the minimal size of the advice for both the construction ofa partial and complété map are provided. Tight bounds are provided for fj,= 1.
In [177], a robot in a connected weighted unlabelled undirected unknown graph
has to visit ail nodes ofthe graph, without necessarily having to visit ail edges. The robot only learns the weight of ail edges when it is located in a node adjacent to those edges. The paper builds upon the work of [148], that presented an improved DFS algorithm that was 16-competitive on planar graphs. [177] prove that the afore-mentioned algorithm does not hâve constant compétitive ratio on general graphs.
Furthermore, the paperprovides aconstantcompétitive algorithm for general graphs with a bounded number ofdistinct weights.