In an interesting amalgam between the exploration problem and the illumination problem, [153] considère the problem ofasingle robot starting at an arbitrary vertex of a polygon and trying to find the shortest path inside the polygon that reaches a different arbitrary vertex, and provide an algorithm that is linear in the shortest distance between those two vertices, while also providing a 1.41 compétitive ratio as a lower bound (the quotient between the distance travelled by the algorithm and the shortest path for their problem given knowledge ofthe environment).
A very similar problem was studied in [82], where an agent is tasked with the création of the map of a room filled with obstacles. The proposed algorithm is mea-sured by the compétitive ratios between the distance required by the algorithm and the optimaldistance, givenknowledge ofthe environment. They showthat there are no algorithms guaranteeing acompétitive ratio for general rooms with obstacles, yet provide an algorithm for a roomwith a bounded numberofobstacles. This bound is further improved in [154], which also provides an algorithm that is within aconstant factor of optimal in the worse-case for arbitrary rooms with unbounded number of polygons.
[64] explores a similar problem: a robot bas to explore a terrain with obstacles, both represented as polygons. Two scénarios are considered: in the first one, the robot has unlimited vision (though blocked by obstacles) whereas in the second one, the robot has aunitary vision range. The robot is taskedwith the full exploration of the environment, and the efficiency ofthe algorithm used is measured based on the lengthofthe trajectory ofthe robot. Forboth cases, an upperbound and tightlower bound are provided.
An interesting twistto this problemisconsideredin [155], where arobotattempts tolocaliseitselfinagéométrieenvironmentpriorly known. Theyreferto thisproblem as the location problem, and builds upon the algorithms provided in [11]. The article alsoraises the question oflandmark placementin an environmentin ordertosimplify further attempts at localisation. A landmark is a point in the environment that can be used by a robot to localise itself. The algorithm provided in the paper is asymptotically better than the one discussed in [11].
The concept ofunknownenvironment with landmarks isreferred to as the bound-aryplace graph in [194], Using analgorithm thatfocusesonincrémental construction oftheenvironment,the authors manage toidentify aillandmarks in the environment.
The algorithm described in the paper has been implemented in a mobile robots plat-form, thus proving the possibility to correctly explore an environment without the need for GPS-like Systems or metric représentation oftheenvironment.
The search for a target located at an unknown location in a one-dimensional environment (the line) is known as the cow-path problem, or linear search problem, as introduced in [19], which proves that a minimum distance of fi(9d), with d the original distance between the robot and a treasure, is required for a robot on a line to find the treasure.
The problem of exploration on the line by a single robot is considered in [81]
with an additional assumption: whereas most algorithms allow the robot to change its direction without cost, this paper adds a cost of d to each change ofdirection of the robot. An algorithm solving the problem in cost 9 ■ OPT +2d, with OPT the optimaltime with knowledge ofthe location ofthe target. The notion ofturn cost is also applied to exploration on the star graph, and an algorithm solvingthis problem in optimal time is provided, with matching lower bound. Randomised strategies are discussed.
[149] considers randomised solutions to this problem, and conclude that the ran-dom algorithm they provide is optimal, using the total distancecovered by the robot as measurement forthe performance oftheiralgorithm. The authors furthermore ob
serve that their algorithm is almost twice as efficient as any deterministic algorithm.
A lower bound of fl(v/n)-competitive is proven for both deterministic and ran-domisedalgorithms inpolygonalenvironmentsin[5], andisgeneralisedtotri-dimensional rectilinear polyhedra without obstacles.
Simple greedy algorithms are used to explore unknown terrainsin [156], and even though they do not yield optimal results (asis to be expected), they offer acceptable results. Theauthors discussapplications andadvantagesoftheusage ofsuchanalgo
rithm, and observe thatit cantake advantageofprior knowledgeofthe environment, and offer the possibility to be applied to multiple robots acting in coopération.
An A* approximation algorithm is used in [172] to détermine the path followed by a robot with limited visibility in an unknown environment discretised as a two-dimensional grid. Experimental results are provided to discuss the efhciency of the algorithm.
2.3 Collective Search and Exploration
The search and exploration problems were studied under many variations, and some of those involve multiple robots. The classical problem of exploration can be mod-ified by the addition of one or more robots. Sometimes, a graph that could not be entirely mapped with the use ofa single robot can now be completely recognised by a team of collaborating robots. The presence of multiple robots allows for varions strategies,such astheintroduction ofimmovable robotsthatserveas markers tohelp another robots orientate in a given environment. Those immovable robots are often referred to as pebbles, or sometimes, when they can store and share information, as whiteboards. The presence of multiple robots introduces a new variable for the studied problem, that is, the communication between autonomous robots. The four most studied variations of a communication model are the face-to-face communica
tion, where robots can only share information when they are located at the exact same location; the wireless communication, where robots can freely exchange com
munication with no regard to their respective location; the pebble communication, where robots are unable to communicate, but may leave markers on the ground that contain no information; and finally, the whiteboard communication, where robots may leave markers on the ground that do contain a certain amount of information that can be read by other robots.
The ringenvironment isdiscussed in [114],where agroupof k identicalrobots are tasked with the exploration ofail nodes ofa ring ofsize n. In this model, robots are equipped with sensors, but are unable to communicate. Each node must be visited by at least one robot, and ail robots must décidé to remain iclle for the algorithm to be completed. It is shown that this problem is impossible if k and n are co-prirne (the only common divisor between k and n is 1). The authors also show that the
minimum number ofrobots required to explore aring ofsize n is O(logn).
[22] considerstheexplorationofagraphenvironmentwithtwo robots, and provide an algorithm, called the homing-sequence algorithm, to accomplish this in a time that ispolynomial in n the number ofnodes. Theycompare the performances ofthis algorithm to a random walk algorithm, and conclude that the random algorithm has better performances than the homing-sequence algorithm.
In [99], a group of robots equipped with wireless communication are tasked with the exploration ofailnodes in a graph, and must thenreturn to their initialposition.
The robots hâve no prior knowledge of the graph. The efficiency ofthe algorithm is given by the compétitive ratio between the totaltime required by the algorithm and thesmallest amountoftimerequired,given knowledgeofthe graph. A lowerbound of Q(log/c/loglogfc) with k the numberofrobots isintroduced forthe compétitiveratio of any given deterministic algorithms. Using the number of edges traversed instead of the elapsed time, the authors présent an algorithm with a compétitive ratio of (4 — 2/k) on trees.
A linear algorithm for the exploration of ail nodes of a graph by a group of robots is provided in [84]. More precisely, the paper considers a scénario in which a group ofrobots can explore a n-node graph located at a distance at most D of the starting point of the robots in O(D), if the number of agents k is polynomial in D and n (k = Dn1+e < n2+e for any e > 0). This algorithm works even with a local communication model in which robots can only exchange information when they are located at the sanie node.
[132] classifies decisionproblems for multiplerobots in a graphenvironment using deterministic algorithms. It is proven that the class of ail problems that can be verified with a certificate is much wider than the class of ail decidable problems in this context, and thepapershows the existence ofa Mobile-Agent Vérifiable-complété
problem.
Analternativecommunication modelisdiscussed in [189], where robotswith wire-less communication hâve a limited range ofcommunication. An algorithm based on a distributed bidding model is introduced, using two measures: (1) the distance be-tween robots, and (2) amap synchronisation mechanism; both ofwhich are meant to reduce the amount of information exchanged between robots. Simulations are used to measure the efficiency of the algorithm.
A similarbidding model is used in [165], to the différencethat in this paper, com
munication is impossible between robots. More so, [165] is concerned with collision between robots, and the bidding value function used to détermine the path of ail robots also aims at avoidingcollisions.