## 1 Introduction

Search problems are concerned with the exploration of a domain, aiming to identify the location of a hidden object. More particularly, in evacuation-type problems where the domain is the unit disk, introduced recently by Czyzowicz et al. in [16], a group of mobile agents collectively search for a hidden item (the exit) placed on the perimeter of the disk, attempting to expedite the time it takes for the last agent to evacuate, i.e. reach the exit. As it was the case in [16], as well as in a series of follow-up improvements and problem variations, the main objective was the design of evacuation algorithms that minimize the worst case performance. In contrast, real-life search-and-rescue operations, in which current problems find applications, are mostly concerned with good average performance

. Keeping also in mind that, in realistic search tasks, mobile agents do not have unbounded resources and at the same time it is imperative that the search terminates successfully with probability 1, one is motivated to study average case - worst case trade-offs for evacuation search problems.

In this direction, we initiate the study of the traditional evacuation problem first introduced in [16] from the perspective of average case analysis, which in our case is equivalent to designing efficient randomized algorithms. More specifically, we introduce problem Evac which, at a high level, asks for efficient evacuation algorithms that perform well on average, given that their worst case performance does not exceed (which can be thought as the maximum time robots can operate, e.g. due to energy restrictions). The problem seems particularly challenging given that the worst case performance analysis of all known evacuation algorithms require tedious analysis, tailored to robots’ trajectories, and followed by intense, computer-assisted calculations, which are always numerical. Our results pertain to new families of evacuation algorithms, whose worst case performance analysis can be done rigorously, and whose average case analysis requires again intense computer-assisted calculations, achieving average case - worst case trade-offs for a wide spectrum of values. Our computer-assisted calculations rely on a novel theoretical and unified approach to compute the cost of any evacuation algorithm and for any placement of the hidden item without relying on tedious analysis specific to robots’ trajectories . Equipped with these techniques, we also verify, somehow surprisingly, that the best evacuation algorithms known prior to this work, designed to perform well in the worst case, do not perform well for Evac, adding this way to the motivation of our problem.

### 1.1 Related Work

In search problems, mobile agents, commonly referred as robots, aim to locate efficiently a hidden item placed in some geometric domain. Numerous search-types problems have been introduced and studied since the 60’s, when two seminal papers on probabilistic search, [8] and [9], were concerned with minimizing the expected time

to locate the item. The number of search-type variants, along with the difficulty of the underlying mathematical problems and the elegance of many results soon gave rise to what is known nowadays as Search Theory. Many of the variants have been classified in surveys, e.g.

[24] [10], while a number of books provide a comprehensive study for similar problems, e.g. see [36, 1, 4] and the most recent [5].Search-type problems have also been studied under the perspective of exploration in [2, 3, 23, 30] by a single robot, and in [38, 37, 12] by multiple robots. Terrain mapping has been the main search task even in problems where exploration is not the primary objective, e.g. [31, 33, 35]. Numerous other search-type problems have been introduced and classified as hide-and-seek and pursuit-evasion games, e.g. see [14, 26, 32, 34]. Overall the list of search-type problems is enormous, and having given a representative list above, in what follows we refer only to the most relevant ones.

The perception of a search-type problem as an evacuation problem, from a theoretical perspective, appeared almost a decade ago, e.g. in [7, 25]. The problem we study here is a direct follow-up to the evacuation problem Evac (a search-type problem) first introduced in [16], which included many variants based on the number of robots and the communication model between them. In the variant Evac which is relevant to our work, two robots start from the center of the unit disk, while an exit is hidden somewhere on the perimeter. The robots move at speed 1, their perception of their environment is restricted to their location and they can exchange information only by meeting. The goal is to minimize the worst case evacuation time, i.e. the time it takes the last robot to reach the exit, over all exit placements. The upper bound of 5.73906 in [16] was later improved to 5.628 in [21], and further to the currently best known 5.625 in [11], while the best lower bound known for the problem is 5.255 due to [21].

Since the introduction of Evac in [16], a number of variants emerged, focusing on different geometric domains, different number of robots and robots’ specifications, different communication models etc. Examples include evacuation from the disk with more than 1 exits in the wireless model [15], evacuation of a group of robots on a line [13] (generalization of the celebrated Cow-Path problem [6]), evacuation in the presence of faulty robots in a line [20] and in a disk [17], evacuation with advice [29] while more recently evacuation with combinatorial requirements on the robots that need to evacuate, e.g. [27, 28, 18, 19].

### 1.2 Outline of Our Results & Paper Organization

We initiate the study of evacuating 2 robots from the disk in the face-to-face model from an average case complexity perspective. In particular we introduce problem Evac in which one tries to minimize the expected performance of randomized evacuation algorithms, subject to that the worst case performance does not exceed . The problem is particularly challenging given that existing positive results, from a worst case complexity perspective, rely on tedious theoretical analysis tailored to algorithmic solutions, and supported by intense computer-assisted calculations. One of our main contributions is a unified and simple approach to quantify the performance of any evacuation algorithm and for any input. Equipped with this technique, we first verify that none of the previously known evacuation algorithms has good average case performance. Then, we introduce families of evacuation algorithms that have competitive average case performance, given that their worst case performance does not exceed , for a wide range of ’s. Our results rely on rigorous and technical worst case performance analysis for the newly proposed algorithms. Building upon our new technique for efficiently evaluating the cost of evacuation algorithms for any input, we are able to numerically compute the average case performance of our algorithms, as well as to quantify formally the induced average case -worst case trade-offs.

In Section 2.1 we formally define Evac and we give a high-level outline of the results we establish. Section 2.2 contains one of our main contributions, which is a systematic process to compute the performance of any evacuation algorithm, given that robots’ trajectories have convenient representations, described in Section 2.3. In Section 3 we analyze two benchmark algorithms for Evac, as well as we motivate further the problem for certain values of , among others showing, somehow surprisingly, that none of the previously proposed evacuation algorithms is efficient for our problem. Section 4 describes our main contributions in the form of new families of evacuation algorithms. Then, in Section 5 we perform rigorous worst case analysis for all new algorithms and in Section 6 we perform average case analysis, using our results from Section 2.2 along with heavy computer-assisted calculations. In the same section, we also quantify formally all our results for Evac. Finally, in Section 7 we conclude with some open problems.

## 2 Preliminaries

### 2.1 Problem Definition & Main Results

In Evac, two searchers (robots) start from the center of the unit disk. Moving at maximum speed 1, the two robots can move anywhere on the plane. Somewhere on the perimeter of the disk there is a hidden object (exit) that can be located by any of the robots only if the robot is co-located with the exit.

The two robots do not see each other from distance, neither can they exchange messages unless they meet (face-to-face model), but they can agree in advance on each other’s trajectories. A feasible evacuation algorithm is determined by the trajectories of the robots, in which eventually both robots reach the exit. For simplicity, we also require, w.l.o.g. that eventually any robot stays idle. For convenience, we think that the center of the unit disk lies at the origin of a Cartesian system, and we denote by the point , which will be referred to as an instance of Evac when the exit is placed at . Given instance , we define the evacuation time of the feasible evacuation algorithm as the time it takes the last robot to reach the exit.

In this work we are concerned with determining tradeoffs between the worst case and the average case performance (of uniform placements of the exit) of evacuation algorithms for Evac. More specifically, we say that an evacuation algorithm with evacuation cost on instance is -efficient if

where the expectation is with respect to the uniform distribution over

. Special to our problem is that can also be interpreted as the expected performance of a randomized algorithm based on . Indeed, consider an algorithm which first performs a random rotation of the disk around the origin of angle , where is chosen uniformly at random from , and then simulates . This random step is equivalent to choosing a deployment point uniformly at random on the disk. Due to the symmetry of the domain, it is irrelevant where the adversary will place the unique exit, and hence the expected performance of this randomized algorithm equals .For algorithms parameterized by parameter(s) , the pair will correspond to a subset of (and a curve if is only one parameter), that we will refer to as the Efficient Frontier. We also adopt an optimization perspective of the problem, and we introduce the following optimization problem Evac on parameter :

(Evac) | ||||

s.t. |

Due to an analysis we perform later, Evac is interesting as long as . At a high level, values above are obtained from two benchmark algorithms, , where , hence being efficient in worst case and inefficient in average case, while being efficient in average case and inefficient in worst case. As it is common for Evac (and many follow-up variation problems) closed forms for the cost of best-solutions known do not exist, and upper and lower bounds are given numerically. Our results involve upper bounds for a continuous spectrum of parameters for problem Evac. In particular we propose families of algorithms (over some parameters) so that, as their parameters vary, we obtain and , for each . The curve summarizing our results is depicted in Figure 1, and it is later quantified in Theorem 6.2 (see Section 5).

Note that an -efficient algorithm gives a solution of value for Evac. Our approach to prove Theorem 6.2 is to define families of evacuations algorithms parameterized by parameter(s) . We will prove that these algorithms are -efficient for some functions , and in particular the evaluation of the worst case performance will be exact and monotone in , while the computation of will be computer-assisted. Then we will set , and will be able to describe the average case performance as a function of as .

### 2.2 Computing Evacuation Times

For any feasible evacuation algorithm, we define by , the first time that is visited by any robot. Clearly, when a robot, say locates the exit at , it may attempt to catch while moving along ’s trajectory along the shortest line segment, say of length . Once robots meet, they return together to , inducing total evacuation cost

All existing results for Evac

, from a worst case complexity perspective, rely on numerical computer-assisted estimation of

, after identifying properties of the maximizer. In this section, we elevate existing arguments, and we propose a generalized and unified approach for computing , for any and for any robots’ trajectories. For the sake of formality, as well as for practical purposes, robots’ trajectories will be defined by parametric functions , where are continuous and piecewise differentiable. In particular, search protocols for the two robots will be given by trajectories , where will denote the position of robot at time . Therefore, any evacuation algorithm will be identified by a tuple. To simplify notation, we will only determine the trajectories from the moment the two robots reach the perimeter of the circle, and until the entire circle is searched, and we will silently assume that robots stay put after exploration is over.

###### Lemma 1

Consider instance of Evac, and suppose that for a feasible evacuation algorithm , robot 1 is the first robot that finds the exit. Then , where is the smallest root, no less than , of function

(1) |

###### Proof

First observe that is continuous, and assuming that the two robots are not co-located when the exit is found, we have . At the same time, since the evacuation algorithm is feasible, is eventually a constant, and hence for big enough we have that becomes eventually negative. By the mean value theorem, there is for which .

Now consider the smallest positive root of , no less than . At time , is located at point , and it is away from the location of the discovered exit. At the same time, moves with speed 1 along the shortest path to catch in her trajectory. Hence it takes some extra time from the moment the exit is found until she reaches point . By definition we have , and therefore as claimed. ∎

For some special trajectories, admits a simpler description that we describe next. Before that, we introduce some notation pertaining to a function , which we widely use in the remaining of the paper:

(2) |

To simplify notation, we will also abbreviate by . The fact that is well defined follows easily from the monotonicity of in .

###### Lemma 2

For some instance of Evac, suppose that for a feasible evacuation algorithm , is the founder of the exit, say at time . Assume that both lie on the circle at arc distance , and suppose that ’s movement is along the perimeter of the circle toward the complementary arc of length . Then, .

###### Proof

The lemma follows by applying transformation in the definition of in Lemma 1, so that . ∎

We are ready to conclude with a corollary that will be handy for computing evacuation times numerically, and without relying on excessive case analysis, as it was the case before.

###### Corollary 1

Consider feasible evacuation algorithm for Evac. For any instance for which is the exit founder, the evacuation cost can be computed as where is the smallest root, at least , of .

### 2.3 Trajectories’ Description

Robots’ trajectories will be described in phases. We will always omit the “deployment phase”, i.e. the movement from the circle center to its perimeter, and we will only describe the trajectories from the moment robots start searching the circle. In each phase, robot , will be moving between two explicit points, either along an arc, or along a line segment (chord of an arc), see Observations 2.1 and 2.2 below. We will summarize robot’s trajectories in tables of the following format.

Robot | Phase # | Trajectory | Duration |
---|---|---|---|

1 | |||

2 | |||

In order to ease notation, trajectory of phase will be described with parametric equations as if the time is reset to 0 after time , where (this is the time that robots reach the circle). The two fundamental trajectory components are movements along arcs and movements along line segments.

###### Observation 2.1

Let and . The trajectory of an object moving at speed 1 on the perimeter of a unit circle with initial location is given by the parametric equation If the movement is counter-clockwise (ccw), and clockwise (cw) otherwise.

###### Observation 2.2

Consider distinct points in . The trajectory of a speed 1 object moving along the line passing through and with initial position is given by the parametric equation

Finally, the analysis of our algorithms’ trajectories will give rise to a number of constants. For the reader’s convenience, we list here the numerical values of the most common constants that will be encountered later; All constants are formally defined when they are first introduced.

## 3 Two Benchmark Algorithms & Motivation

In this section we describe two benchmark algorithms for Evac, as well as perform average case analysis to algorithms previously proposed in the literature. The reader may consult Figure 2 for the algorithms analyzed in this section.

Czyzowicz et al. [16] were the first to introduce an evacuation algorithm for Evac, which we denote here by (see Figure 2 on the left).

###### Definition 1 (Benchmark Algorithm )

For all , and .

###### Observation 3.1

Benchmark Algorithm is -efficient.

###### Proof (Observation 3.1)

Note that it takes time to search the entire circle, and that the two trajectories are symmetric with respect to horizontal axis. Therefore, we may assume that the instance satisfies .

Clearly, for any such , we have that . By Lemma 2, we have that . Numerical calculations (software assisted) show that

∎

should be understood as being efficient in the worst case, but inefficient on average. The claim becomes transparent by introducing the following naive algorithm for Evac that we depict in the middle of Figure 2.

###### Definition 2 (Benchmark Algorithm )

For each , .

###### Observation 3.2

Benchmark Algorithm is -efficient.

###### Proof (Observation 3.2)

It is easy to see that for all we have and . Therefore , and hence

∎

should be understood as highly efficient on average, but inefficient in the worst case. Moreover, it should be clear that are feasible solutions to Evac, for and , respectively. We conjecture that is indeed the optimal evacuation algorithm among all algorithms with worst case performance no more than . At the same time, below we show that is the best algorithm for Evac, when , among those previously used to improve upon the worst case performance. The importance of this observation is twofold; first we are motivated to study Evac for the entire spectrum of , and second we deduce that in order to perform well on average, we need to devise and analyze new evacuation algorithms.

Upper bounds for the worst case performance of were later improved in [22, 11], first to 5.628, and then to 5.625, using refined algorithms, respectively. The main idea behind the improvement is to understand the monoticity of for algorithm . Indeed, the following lemma was implicit in both [22, 11], and can be obtained numerically.

###### Lemma 3

There is , where , so that evacuation cost of for Evac on instance is strictly increasing for , and strictly decreasing in . In particular, .

Consider now an execution of in which one of the robots, say continues searching on the circle and is close to approach a location that would be the meeting point if the instance was . In an attempt to help expedite a potential meeting (in case is approaching) and effectively reducing the cost of the worst case, would make a minor detour toward the interior of the disk, before returning back to the exploration of the circle. This simple idea was explored in [22] where the following family of algorithms were introduced, parameterized by and point within the unit disk, see also right of Figure 2.

###### Definition 3 (1-Detour Algorithm )

For all , the trajectory of is defined as

Robot | Phase # | Trajectory | Duration |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 |

The trajectory of is symmetric with respect to the horizontal axis.

The crux of the contribution of [22] was to prove that there exists for which the worst case performance is no more than 5.644 (and a delicate refinement is needed to achieve 5.628). Notably, their analysis is tedious and lengthy, whereas we can obtain the same result, relying again on numerical calculations, with minimal effort. Then, [11] introduced variations of in which each robot performs more than 1 detours (see Phases 2,3 of ). Hence, -detour algorithms are parameterized by a sequence , where and , and points in the disk. Even 2-detour algorithms achieve worst case performance 5.625, while for each , t-detour algorithms do induce strictly improved performance (for appropriate choices of the parameters) but the improvement is negligible.

Motivated by the results in [22, 11], one is tempted to ask whether any algorithm in the family improves upon with respect to the average case analysis. The next claim is due to exhaustive, computer-assisted numerical calculations, see also Figure 3.

###### Theorem 3.3

For every and for every in the unit disk .

Theorem 3.3 provides strong motivation for studying problem Evac, since it shows that in oder to establish good upper bounds, i.e. our main results depicted in Figure 1 and quantified later in Theorem 6.2, one needs to employ new evacuation algorithms. Recall that even that was first calculated in [16], or first calculated in [22] for various , were all estimated with computer-assisted calculations. Due to the nature of the problem, we are bound to rely on computer-assisted calculations as well. Notably, our much more intense computational work is feasible only because we employ the new method for computing evacuation times due to Corollary 1 and Definition 3 of trajectories. Overall, in order to verify Theorem 3.3 we compute pairs for more than 500,000 different parameter values and we depict them in Figure 3.

## 4 New Evacuation Algorithms

In this section we propose families of evacuation algorithms for problem Evac, for the entire spectrum of . Our algorithms are summarized in Figure 4.

First we define families of evacuation algorithms that, as we show next, perform well for Evac in the “neighborhood of ”, i.e. for close to . Our algorithms are parameterized by , and their circle exploration lasts .

###### Definition 4 (Algorithm )

For all , the trajectory of is defined as

Robot | Phase # | Trajectory | Duration |
---|---|---|---|

1 | |||

2 | |||

3 |

where is defined in (2). The trajectory of is defined as , for all .

is depicted in Figure 4 on the left. At a high level is a modification of that is based on the following idea. The execution of is the same as in until each robot searches an arc of length (and hence coincides with ). After time , abandons her trajectory and catches , on the perimeter of the circle resembling a trajectory as if the exit was located at . It is not difficult to see that the definition of above satisfies .

Next we define a family of algorithms which, as we show later, perform well in the “neighborhood of ”, i.e. for close to . For this recall definition (2) of . We let be the root of . For every we define a family of algorithms on parameter whose circle exploration lasts .

###### Definition 5 (Algorithm )

For all , the trajectory of is defined as

Robot | Phase # | Trajectory | Duration |
---|---|---|---|

1 | |||

2 | |||

3 |

The trajectory of is defined as , for all .

is depicted in the middle of Figure 4. The condition that is added for simplicity to ensure that the latest catching point occurs while the other robot is still searching, and is not mandatory. At a high level is a generalization of (note that ). For the first time units, robots search in the same direction until arrives at the deployment point of . Then, catches on the circle, as if the exit was located at (which by Lemma 2 happens in extra time).

Finally we introduce a family of evacuation algorithms which will perform well for Evac for intermediate values of . For this we generalize family so that the two robots perform two alternating jumps, with parameters satisfying , see right of Figure 4.

###### Definition 6 (Algorithm )

For notational convenience, we set . For all , the trajectories of are defined as follows

Robot | Phase # | Trajectory | Duration |
---|---|---|---|

1 | |||

2 | |||

3 | |||

1 | |||

2 | |||

3 |

Robots’ trajectories have the following meaning. As in the family of algorithms , parameter represents the arc distance the two robots have before the one preceding decides to jump ahead. In the two robots meet again once the jumper reaches the perimeter of the circle. In the jumper deploys a little further away on the circle so that when the other robot reaches the deployment point of the jumper, the two robots are at arc distance . As a result, the time it takes both robots to complete searching the entire circle is , as well as coincides with . Finally, note that even though will be eventually invoked for seemingly restricted values of (), the deviation in the performance will be significant enough (e.g. ) to account for its utilization in our upper bounds.

## 5 Worst Case Performance Analysis

In this section we perform worst case analysis for all algorithmic families with respect to their parameters. Notably, results in this section are quantified formally and exactly by closed formulas. At a high level, each of will be invoked to solve Evac for different values of , and each of them will have competitive average case performance for the corresponding worst case performance . As an easy warm-up, we analyze .

###### Lemma 4 (Worst Case Analysis for )

Let , where . Then, for all , we have that

###### Proof (Lemma 4)

First it is easy to show that the worst case evacuation time is induced either when finds the exit while moving from to , or while are exploring the circle together (after having met). By Lemma 2, the cost in the first case would be

where the values of the piecewise function above follow from Lemma 3. In the other case, the worst placement of exit is obtained using instances for arbitrary small values of in which case the evacuation cost becomes .

Overall, is is easy to see that showing that the dominant evacuation cost when is . For the evacuation cost becomes equal to . ∎

In a similar fashion, we can easily analyze .

###### Lemma 5 (Worst Case Analysis for )

For all , we have .

###### Proof (Lemma 5)

We distinguish three cases as to where the exit is. If , then the worst instance is when for arbitrarily small , and the cost is . In the second case and it is not difficult to see that the worst case induced cost in this case is not more than that of the first case. Finally, in the third case , and the two robots move together, so the total cost, in the worst case, is , when for arbitrarily small . It is not difficult to see that the dominant case is actually the third one, and in fact the two cases induce the same cost when . By the definition of we know that . Hence the costs become equal when . ∎

Next, we analyze , which requires more technical arguments. For this we will invoke only for special parameters, whose choice is motivated by the following observation pertaining to the performance of (whose generalization is ). From the proof of Lemma 5, it follows that among all algorithms , where (see discussion before Definition 5), the one with minimum worst case evacuation cost is , and the cost becomes . In fact, for all there are two different values of for which , and we restrict so that we obtain evacuation algorithms with minimum average case cost. Moreover, is the only parameter for which and as a byproduct, it is the algorithm in the family that minimizes the worst case.

By Lemma 5 we know that as , the value of that minimizes approaches . That value of is what made the evacuation cost of attain the same value in two different (worst case) exit placements. Motivated by this, and for values of not too big, we still find the optimal choices of that minimize the worst case performance.

###### Lemma 6 (Worst Case Analysis for )

Let , and set . Then for all we have

###### Proof (Lemma 6)

Let . First we show that is the worst case performance of for two specific placements of the exit.

We proceed by describing evacuation cost assuming two arbitrary for two different instances . Using Lemma 2, we see that

(3) |

Since the total search time is , we also see that

(4) |

Now we claim that (3), (4) are equal when . Indeed, equating (3), (4) gives

(5) |

But then, using (2), we see that

(6) |

Substituting (6) into (5), we see that the value of for which (3), (4) are equal satisfies , as promised. Substituting this special value of either in (3) or in (4) induces evacuation cost .

Next we show that as long as is not too big, is indeed the worst case evacuation cost. We consider the following cases , for possible instances ; Clearly, (3), (4) demonstrate the worst case evacuation costs for instances in , respectively, and the cost in both cases, for is equal to .

If then . It is easy to see that both are monotone in , so the worst case evacuation in this case is

(7) |

Denote satisfying (2) by . Using (2) and the definition of , we see that

For simplicity, we denote that satisfies the equation above by . Then, continuing from (7), the worst case evacuation cost when becomes , an expression that depends exclusivey on . The latter cost is no more than if and only if , and numerically we verify that this is satisfied as long as (see also Figure 5).

Finally, it is easy to verify that and are increasing and decreasing respectively for and that (for ). As a result, the worst case evacuation cost of case cannot exceed that of case , and hence the lemma follows. ∎

## 6 Average Case Performance Analysis & the Efficient Frontier

In this section we perform average case analysis for all algorithmic families , with respect to their parameters. For the sake of exposition of our results, we set and for , as in Lemma 6, we set

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