Keywords

1 Introduction

In the realm of mobile agent computing, search-type problems are concerned with the design of searchers’ (robots’) trajectories in some known search space to locate a hidden object. Single searcher problems have been introduced and studied as early as the 60’s by the mathematics community [11, 12], and later in the late 80’s and early 90’s by the theoretical computer science community [8]. The previously studied variations focused mainly on the type of search domain, e.g. line or plane or a graph, and the type of computation, e.g. deterministic or randomized. Since search was also conducted primarily by single searchers, termination was defined as the first time the searcher hit the hidden object. In the last decade with the advent of robotics, search-type problems have been rejuvenated within the theoretical computer science community, which is now concerned with novel variations including the number of searchers (mobile agents), the communication model, e.g. face-to-face or wireless, and robots’ specifications, e.g. speeds or faults, including crash-faults or byzantine faults. As a result of the multi-searcher setup, termination criteria are now subject to variations too, and these include the number or the type of searchers that need to reach the hidden item (for a more extended discussion with proper citations, see Sect. 1.1).

One of the most studied search domains, along with the line, is that of a circle, or a disk. In a typical search-type problem in the disk, the hidden item is located on the perimeter of the unit circle, and searchers start in its center. Depending on the variation considered, and combining all specs mentioned above, a number of ingenious search trajectories have been considered, often with counter-intuitive properties. Alongside the hunt for upper bounds (as the objective is always to minimize some form of cost, e.g. time or traversed space or energy) comes also the study of lower bounds, which are traditionally much more challenging to prove (and which rarely match the best known positive results).

Search on the unbounded plane as well as in other 2-dimensional domains, e.g. triangles or squares, has been considered too, giving rise to a long list of treatments, often with fewer tight (optimal) results. While the list of variations for searching on the plane keeps growing, there is one attribute that is common to all previous results where robots’ trajectories lie in \(\mathbb {R}^2\), which is the underlying Euclidean metric space. In other words, distances and trajectory lengths are all measured with respect to the Euclidean \(\ell _2\) norm. Not only the underlying geometric space is well understood, but it also enjoys symmetries, and admits standard and elementary analytic tools from trigonometry, calculus, and analytic geometry.

We deviate from previous results, and to the best of our knowledge, we initiate the study of a search-type problem with mobile agents in \(\mathbb {R}^2\) where the underlying metric space is induced by any \(\ell _p\) norm, \(p\ge 1\). The problem is particularly challenging since even “highly symmetric” shapes, such as the unit circle, enjoy fewer symmetries in non-Euclidean spaces. Even more, robot trajectories are measured with respect to the underlying metric, giving rise to technical mathematical expressions for measuring the performance of an algorithm. In particular, we consider the problem of reaching (evacuating from) a hidden object (the exit) placed on the perimeter of the \(\ell _p\) unit circle. Our unit-speed searchers start from the center of the circle, placed at the origin of the Cartesian plane \(\mathbb {R}^2\), and are controlled by a centralized algorithm that allows them to communicate their findings instantaneously. Termination is determined by the moment that the last searcher reaches the exit, and the performance analysis is evaluated against a deterministic worst case adversary. For this problem we provide optimal evacuation algorithms. Apart from the novelty of the problem, our contributions pertain to (a) a technical analysis of search (optimal) algorithms that have to vary with p, giving rise to our upper bounds, and to (b) an involved geometric argument that also uses, to the best of our knowledge, a novel observation on convex geometry that relates a given \(\ell _p\) unit circle’s arcs to its chords, giving rise to our matching lower bounds.

1.1 Related Work

Our contributions make progress in Search-Theory, a term that was coined after several decades of celebrated results in the area, and which have been summarized in books [3, 5, 6, 53]. The main focus in that area pertains to the study of (optimal) searchers’ strategies who compete against (possibly hidden) hider(s) in some search domain. An even wider family of similar problems relates to exploration [4], terrain mapping, [48], and hide-and-seek and pursuit-evasion [49].

The traditional problem of searching with one robot on the line [8] has been generalized with respect to the number of searchers, the type of searchers, the search domain, and the objective, among others. When there are multiple searchers and the objective is that all of them reach the hidden object, the problem is called an evacuation problem, with the first treatments dating back to over a decade ago [10, 35]. The evacuation problem that we study is a generalization of a problem introduced by Czyzowicz et al. [21] and that was solved optimally. In that problem, a hidden item is placed on the (Euclidean) unit disk, and is to be reached by two searchers that communicate their findings instantaneously (wireless model). Variations of the problem with multiple searchers, as well as of another communication model (face-to-face) was considered too, giving rise to a series of follow-up papers [15, 25, 32]. Searching the boundary of the disc is also relevant to so-called Ruckle-type games, and closely related to our problem is a variation mentioned in [9] as an open problem, in which the underlying metric space is any \(\ell _p\)-induced space, \(p\ge 1\), as in our work.

The search domain of the unit circle that we consider is maybe one of the most well studied, together with the line [18]. Other topologies that have been considered include multi-rays [16], triangles [20, 27], and graphs [7, 14]. Search for a hidden object on an unbounded plane was studied in [47], later in [34, 46], and more recently in [1, 33].

Search and evacuation problems with faulty robots have been studied in [22, 39, 50] and with probabilistically faulty robots in [13]. Variations pertaining to the searcher’s speeds appeared in [36, 38] (immobile agents), in [45] (speed bounds) and in [26] (terrain dependent speeds). Search for multiple exits was considered in [28, 51], while variation of searching with advice appeared in [41]. Some variations of the objective include the so-called priority evacuation problem [23, 30] and its generalization of weighted searchers [40]. Randomized search strategies have been considered in [11, 12] and later in [42] for the line, and more recently in [19] for the disk. Finally, turning costs have been studied in [31] and an objective of minimizing a notion pertaining to energy (instead of time) was studied in [29, 44], just to name a few of the developments related to our problem. The reader may also see recent survey [24] that elaborates more on selected topics.

1.2 High Level of New Contributions and Motivation

The algorithmic problem of searching in arbitrary metric spaces has a long history [17], but the focus has been mainly touching on database management. In our work, we extend results of a search-type problem in mobile agent computing first appeared in [21]. More specifically, we provide optimal algorithms for the search-type problem of evacuating two robots in the wireless model from the \(\ell _p\) unit disk, for \(p\ge 1\) (previously considered only for the Euclidean space \(p=2\)). The novelty of our results is multi-fold. First, to the best of our knowledge, this is the first result in mobile agent computing in which a search problem is studied and optimally solved in \(\ell _p\) metric spaces. Second, both our upper and lower bound arguments rely on technical arguments. Third, part of our lower bound argument relies on an interesting property of unit circles in convex geometry, which we believe is interesting in its own right.

The algorithm we prove to be optimal for our evacuation problem is very simple, but it is one among infinitely many natural options one has to consider for the underlying problem (one for each deployment point of the searchers). Which of them is optimal is far from obvious, and the proof of optimality is, as we indicate, quite technical.

Part of the technical difficulty of our arguments arises from the implicit integral expression of arc lengths of \(\ell _p\) circles. Still, by invoking the Fundamental Theorem of Calculus we determine the worst case placement of the hidden object for our algorithms. Another significant challenge of our search problem pertains to the limited symmetries of the unit circle in the underlying metric space. As a result, it is not surprising that the behaviour of the provably optimal algorithm does depend on p, with \(p=2\) serving as a threshold value for deciding which among two types of special algorithms is optimal. Indeed, consider an arbitrary contiguous arc of some fixed length of the \(\ell _p\) unit circle with endpoints A, B. In the Euclidean space, i.e. when \(p=2\), the length of the corresponding chord is invariant of the locations of A, B. In contrast, for the unit circle hosted in any other \(\ell _p\) space, the slope of the chord AB does determine its length. The relation to search and evacuation is that the arc corresponds to a subset of the search domain which is already searched, and points A, B are the locations of the searchers when the exit is reported. Since searchers operate in the wireless model in our problem (hence one searcher will move directly to the other searcher when the hidden object is found), their trajectories are calculated so that their \(\ell _p\) distance is the minimum possible for the same elapsed search time.

Coming back to the \(\ell _p\) unit disks, we show an interesting property which may be of independent interest (and which we did not find in the current literature). More specifically, and in part using computer assisted numerical calculations for a wide range of values of p, we show that for any arc of fixed length, the placement of its endpoint A, B that minimizes the \(\ell _p\) length of chord AB is when AB is parallel to the \(y=0\) or \(x=0\) lines, for \(p\le 2\), and when AB is parallel to the \(y=x\) or \(y=-x\) lines for \(p\ge 2\). The previous fact is coupled by a technical extension of a result first sketched in [21], according to which at a high level, as long as searchers have left any part of the unit circle of cumulative length \(\alpha \) unexplored (not necessarily contiguous), then there are at least two unexplored points of arc distance at least \(\alpha \).

All omitted proofs from this extended abstract can be found in the full version of the paper [37].

2 Problem Definition, Notation and Nomenclature

For a vector \(x=(x_1,x_2) \in \mathbb {R}^2\), we denote by \(\left\Vert x\right\Vert _p\) the vector’s \(\ell _p\) norm, i.e. \(\left\Vert x\right\Vert _p=\left( |x_1|^p+|x_2|^p\right) ^{1/p}\). The \(\ell _p\) unit circle is defined as \( \mathscr {C}_p := \left\{ x\in \mathbb {R}^2: \left\Vert x\right\Vert _p=1\right\} , \) see also Fig. 2a for an illustration. We equip \(\mathbb {R}^2\) with the metric \(d_p\) induced by the \(\ell _p\) norm, i.e. for \(x,y \in \mathbb {R}^2\) we write \( d_p(x,y) = \left\Vert x-y\right\Vert _p. \) Similarly, if \(r:[0,1] \mapsto \mathbb {R}^2\) is an injective and continuously differentiable function, it’s \(\ell _p\) length is defined as \( \mu _p(r):= \int _0^1 \left\Vert r'(t)\right\Vert _p \mathrm {d}t. \) As a result, a unit speed robot can traverse r([0, 1]) in metric space \((\mathbb {R}^2,d_p)\) in time \(\mu _p(r)\).

We proceed with a formal definition of our search-type problem. In problem WE\(_p\) (Wireless Evacuation in \(\ell _p\) space, \(p\ge 1\)), two unit-speed robots start at the center of a unit circle \(\mathscr {C}_p\) placed at the origin of the metric space \((\mathbb {R}^2,d_p)\). Robots can move anywhere in the metric space, and they operate according to a centralized algorithm. An exit is a point P on the perimeter of \(\mathscr {C}_p\). An evacuation algorithm A consists of robots trajectories, either of which may depend on the placement of P only after at least one of the robots passes through P (wireless model).Footnote 1 For each exit P, we define the evacuation cost of the algorithm as the first instance that the last robot reaches P. The cost of algorithm A is defined as the supremum, over all placements P of the exit, of the evacuation time of A with exit placement P. Finally, the optimal evacuation cost of WE\(_p\) is defined as the infimum, over all evacuation algorithms A, of the cost of A.

Next we show that \(\mathscr {C}_p\) has 4 axes of symmetry (and of course \(\mathscr {C}_2\) has infinitely many, i.e. any line \(ax+by=0, a,b \in \mathbb {R}\)).

Lemma 1

Lines \(y=0, x=0, y=x, y=-x\) are all axes of symmetry of \(\mathscr {C}_p\). Moreover, the center of \(\mathscr {C}_p\) is its point of symmetry.

Proof

Reflection of point \(P=(a,b)\) across lines \(y=0, x=0, y=x, y=-x\) give points \(P_1=(a,-b), P_2=(-a,b), P_3=(b,a), P_4=(-b,-a)\), respectively. It is easy to see that setting \(\left\Vert P\right\Vert _p=1\) implies that \(\left\Vert P_i\right\Vert _p=1, i=1,2,3,4\).

We use the generalized trigonometric functions \(\sin _p(\cdot ), \cos _p(\cdot )\), as in [52], which are defined as \( \sin _p(\phi ) := \sin (\phi )/N_p(\phi ), ~~ \cos _p(\phi ) := \cos (\phi )/N_p(\phi ), \) where \( N_p(\phi ) := \left( |\sin (\phi )|^p + |\cos (\phi )|^p \right) ^{1/p}. \) By introducing \( \rho _p(\phi ) := \left( \cos _p(\phi ), \sin _p(\phi ) \right) , \) which is injective and continuously differentiable function in each of the 4 quadrants, we have the following convenient parametric description of the \(\ell _p\) unit circle; \( \mathscr {C}_p = \{ \rho _p (\phi ): \phi \in [0,2\pi ) \}. \) In particular, set \(Q_1=[0,\pi /2), Q_2=[\pi /2,\pi ), Q_3=[\pi , 3\pi /2), Q_4=[3\pi /2,2\pi )\), and define for each \(U \subseteq \mathscr {C}_p\) it’s length (measure) as

$$ \mu _p(U) = \sum _{i=1}^4 \int _{t \in Q_i: \rho _p(t) \in U} \left\Vert \rho _p'(t)\right\Vert _p \mathrm {d}t. $$

It is easy to see that \(\mu _p(\cdot )\) is indeed a measure, hence it satisfies the principle of inclusion-exclusion over \(\mathscr {C}_p\). Also, by Lemma 1 it is immediate that for every \(U \subseteq \mathscr {C}_p\), and for \(\overline{U}=\{\rho _p(t+\pi ): \rho (t) \in U\}\), we have that \(\mu _p(U) = \mu _p(\overline{U})\) (both observations will be used later in Lemma 7). As a corollary of the same lemma, we also formalize the following observation.

Lemma 2

For any \(\phi \in \{k\cdot \pi /4: k=0,1,2,3,4\}\) and \(\theta \in [0,\pi ]\), let \(U_+=\{\rho _p(\phi +t): t\in [0,\theta ]\}\) and \(U_-=\{\rho _p(\phi -t): t\in [0,\theta ]\}\). Then, we have that \(\mu _p(U_+) = \mu _p(U_-)\).

The perimeter of the \(\ell _p\) unit circle can be computed as

$$ \mu _p(\mathscr {C}_p) = \sum _{i=1}^4 \int _{Q_i} \left\Vert \rho _p'(t)\right\Vert _p \mathrm {d}t = 4 \int _{0}^{\pi /2} \left\Vert \rho _p'(t)\right\Vert _p \mathrm {d}t := 2\pi _p. $$

By Lemma 2, we also have \( \int _0^{\pi /2} \left\Vert \rho _p'(t)\right\Vert _p \mathrm {d}t = 2\int _0^{\pi /4} \left\Vert \rho _p'(t)\right\Vert _p \mathrm {d}t = \pi _p/2. \) Clearly \(\mu _2(\mathscr {C}_2)/2= \pi _2 = \pi =3.14159\ldots \), while the rest of the values of \(\pi _p\), for \(p\ge 1\), do not have known number representation, in general. However, it is easy to see that \(\pi _1= \pi _\infty =4\). More generally we have that \(\pi _p=\pi _q\) whenever \(p,q\ge 1\) satisfy \(1/p+1/q=1\) [43]. As expected, \(\pi _2=\pi \) is also the minimum value of \(\pi _p\), over \(p\ge 1\) [2], see also Fig. 2b for the behavior of \(\pi _p\).

For every \(\phi , \theta \in [0,2\pi )\), let \(A=\rho (\phi ), B=\rho (\phi +\theta )\) be two points on the \(\ell _p\) unit circle. The chord \(\overline{AB}\) is defined as the line segment with endpoints A, B. From the previous discussion we have \(\mu _p\left( \overline{AB} \right) = d_p(A,B)\). The arc is defined as the curve \(\{\rho (\phi +t): t\in [0,\theta ]\}\), hence arcs identified by their endpoints are read counter-clockwise. The length of the same arc is computed as .

Finally, the arc distance of two points \(A,B\in \mathscr {C}_p\) is defined as which can be shown to be a metric. By definition, it follows that .

Next we present an alternative parameterization of the \(\ell _p\) unit circle that will be convenient for some of our proofs. We define

$$\begin{aligned} r_p(s) := \left( -s, \left( 1- |s|^p\right) ^{1/p} \right) , \end{aligned}$$
(1)

and we observe that \(r_p(s) \in \mathscr {C}_p\), for every \(s \in [-1,1]\). It is easy to see that as s ranges from \(-1\) to 1, we traverse the upper 2 quadrants of the unit circle with the same direction as \(\rho _p(t)\), when t ranges from 0 to \(\pi \). Moreover, for every \(t\in [0,\pi ]\), there exists unique \(s=s(t)\), with \(s\in [-1,1]\) such that \(\rho _p(t)=r_p(s)\), and s(t) strictly increasing in t with \(s(0)=-1, s(\pi /4)=-2^{-1/p}, s(\pi /2)=0, s(3\pi /4)=2^{-1/p}\) and \(s(\pi )=1\).

3 Algorithms for Evacuating 2 Robots in \(\ell _p\) Spaces

First we present a family of algorithms Wireless-Search\(_p\)(\(\phi \)) for evacuating 2 robots from the \(\ell _p\) unit circle \(\mathscr {C}_p\). The family is parameterized by \(\phi \in \mathbb {R}\), see also Fig. 4a for two examples, Algorithm Wireless-Search\(_{1.5}\)(0) and Wireless-Search\(_{3}\)(\(\pi /4\)).

figure a

Our goal is to prove the following.

Theorem 1

For all \(p\in [1,2]\), Algorithm Wireless-Search\(_p\)(0) is optimal.

For all \(p\in [2,\infty )\), Algorithm Wireless-Search\(_p\)(\(\pi /4\)) is optimal.

Figure 4b depicts the performance of our algorithms as \(p\ge 1\) varies. Our analysis is formal, however we do rely on computer-assisted numerical calculations to verify certain analytical properties in convex geometry (see proof of Lemma 5 on page 14, and proof of Lemma 9 on page 9) that effectively contribute a part of our lower bound argument for bounded values of p, as well as \(p=\infty \). For large values of p, e.g. \(p\ge 1000\), where numerical verification is of limited help, we provide provable upper and lower bounds that differ by less than \(0.042\%\), multiplicatively (or less than 0.0021, additively).

Recall that as \(\phi \) ranges in \([0,2\pi )\), then \(\rho _p(\phi )\) ranges over the perimeter of \(\mathscr {C}_p\). In particular, for any execution of Algorithm 1, the exit will be reported at some point \(\rho _p(\phi \pm t)\), where \(t \in [0, \pi ]\). Since in the last step of the algorithm, the non-finder has to traverse the line segment defined by the locations of robots when the exit is found, we may assume without loss of generality that the exit is always found at some point \(\rho _p(\phi \pm t)\), where \(t \in [0, \pi ]\), say by robot #1. Note that even though Algorithm 1 is well defined for all \([0,2\pi )\) (in fact all reals), due to Lemma 1 it is enough to restrict to \(\phi \in [0, \pi /4]\).

In the remaining of this section, we denote by \(\mathscr {E}_{p,\phi } (\tau )\) the evacuation time of Algorithm Wireless-Search\(_p\)(\(\phi \)), given that the exit is reported after robots have spent time \(\tau \) searching in parallel. We also denote by \(\delta _{p,\phi }(\tau )\) the distance of the two robots at the same moment, assuming that no exit has been reported previously. Hence,

$$\begin{aligned} \mathscr {E}_{p,\phi } (\tau ) = 1 +\tau + \delta _{p,\phi }(\tau ). \end{aligned}$$
(2)

Since \(\mu _p(\mathscr {C}_p) = 2\pi _p\) and the two robots search in parallel, an exit will be reported for some \(\tau \in [0,\pi _p]\). Hence, the worst case evacuation time \(E_{p,\phi }\) of Algorithm Wireless-Search\(_p\)(\(\phi \)) is given byFootnote 2

$$ E_{p,\phi }:= \max _{\tau \in [0,\pi _p]} \mathscr {E}_{p,\phi } (\tau ). $$

3.1 Worst Case Analysis of Algorithm Wireless-Search\(_p\)(\(\phi \))

It is important to stress that parameter t in the description of robots’ trajectories in Algorithm Wireless-Search\(_p\)(\(\phi \)) does not represent the total elapsed search time. Even more, and for an arbitrary value of \(\phi \), it is not true that robots occupy points \(\rho _p(\phi \pm t)\) simultaneously. To see why, recall that from the moment robots deploy to point \(\rho _p(\phi )\), they need time \( \alpha _{1,2}(\phi , t) := \mu _p\left( \left\{ \rho _p(\phi \pm s):~s\in [0,t] \right\} \right) \) in order to reach points \(\rho _p(\phi \pm t)\). Moreover, \(\alpha _{1}(\phi , t) \not = \alpha _{2}(\phi , t)\), unless \(\phi = k\cdot \pi /4\) for some \(k=0,1,2,3\), as per Lemma 2. We summarize our observation with a lemma.

Lemma 3

Let \(\phi \in \{0,\pi /4\}\), and consider an execution of Algorithm 1. When one robot is located at point \(\rho _p(\phi +t)\), for some \(t\in [-\pi , \pi ]\), then the other robot is located \(\rho _p(\phi -t)\), and in particular \(\alpha _{1}(\phi , t) = \alpha _{2}(\phi , t)\).

Now we provide worst case analysis of two Algorithms for two special cases of metric spaces. The proof is a warm-up for the more advanced argument we employ later to analyze arbitrary metric spaces.

Lemma 4

\(E_{1,0}=E_{\infty ,\pi /4}=5.\)

Proof

First we study Algorithm Wireless-Search\(_1\)(0) for evacuating 2 robots from the \(\ell _1\) unit disk. By (2), if the exit is reported after time \(\tau \) of parallel search, then \( \mathscr {E}_{1,0} (\tau ) = 1 +\tau + \delta _{1,0}(\tau ). \) Note that \(\pi _1=4\), so the exit is reported no later than parallel search time 4. First we argue that \(\mathscr {E}_{1,0} (\tau )\) is increasing for \(\tau \in [0,2]\). Indeed, in that time window robot #1 is moving from point (1, 0) to point (0, 1) along trajectory \((1-\tau /2,\tau /2)\) (note that this parameterization induces speed 1 movement). By Lemma 3, robot #2 at the same time is at point \((1-\tau /2,-\tau /2)\). It follows that \(\delta _{1,0}(\tau )=\tau \), so indeed \(\mathscr {E}_{1,0} (\tau )\) is increasing for \(\tau \in [0,2]\). Finally we show that \(\mathscr {E}_{1,0} (\tau )=5\), for all \(\tau \in [2,4]\). Indeed, note that for the latter time window, robot #1 moves from point (0, 1) to point \((-1,0)\) along trajectory \((-(\tau -2)/2,1-(\tau -2)/2)\). By Lemma 3, robot #2 at the same time is at point \((-(\tau -2)/2,-1+(\tau -2)/2)\). It follows that \(\delta _{1,0}(\tau )=|-(\tau -2)/2+(\tau -2)/2|+|1-(\tau -2)/2+1-(\tau -2)/2|=4-\tau \), and hence \(\mathscr {E}_{1,0} (\tau ) = 1+\tau + \delta _{1,0}(\tau ) = 5\), as wanted.

Next we study Algorithm Wireless-Search\(_\infty \)(\(\pi /4\)) for evacuating 2 robots from the \(\ell _\infty \) unit disk. By (2), if the exit is reported after time \(\tau \) of parallel search, then \( \mathscr {E}_{\infty ,\pi /4} (\tau ) = 1 +\tau + \delta _{\infty ,\pi /4}(\tau ). \) As before, \(\pi _\infty =4\), so the exit is reported no later than parallel search time 4. We show again that \(\mathscr {E}_{\infty ,\pi /4} (\tau )\) is increasing for \(\tau \in [0,2]\). Indeed, in that time window robot #1 is moving from point (1, 1) to point \((-1,1)\) along trajectory \((1-\tau ,1)\) (note that this induces speed 1 movement). By Lemma 3, robot #2 at the same time is at point \((1,1-\tau )\). It follows that \(\delta _{\infty ,\pi /4}(\tau )=\max \{|1-\tau -1|,|1-1+\tau |\}=\tau \), so indeed \(\mathscr {E}_{\infty ,\pi /4} (\tau )\) is increasing for \(\tau \in [0,2]\). Finally we show that \(\mathscr {E}_{\infty ,\pi /4} (\tau )=5\), for all \(\tau \in [2,4]\). Indeed, note that for the latter time window, robot #1 moves from point \((-1,1)\) to point \((-1,-1)\) along trajectory \((-1,1-(\tau -2))\). By Lemma 3, robot #2 at the same time is at point \((1-(\tau -2),-1)\). It follows that \(\delta _{\infty ,\pi /4}(\tau )=\max \{|-1-1+(\tau -2)|,|1-(\tau -2)+1|\}=4-\tau \), and hence \(\mathscr {E}_{\infty ,\pi /4} (\tau ) = 1+\tau + \delta _{\infty ,\pi /4}(\tau ) = 5\), as wanted.

It is interesting to see that the algorithms of Lemma 4 outperform algorithms with different choices of \(\phi \). For example, it is easy to see that \(E_{1,\pi /4}\ge 6\). Indeed, note that Algorithm Wireless-Search\(_1\)(\(\pi /4\)) deploys robots at point (1/2, 1/2). Robot reaches point (0, 1) after 1 unit of time, and it reaches point \((-1,0)\) after an additional 2 units of time. The other robot is then at point \((0,-1)\), at an \(\ell _1\) distance of 2. So, the placement of the exit at point \((-1,0)\) induces cost \(1+1+2+2=6\). A similar argument shows that \(E_{\infty ,0}\ge 6\) too.

We conclude this section with a summary of our positive results, introducing at the same time some useful notation.

Theorem 2

Let \(w_p\) be the unique root to equation \(w^p+1=2(1-w)^p\), and define

$$ s_p := \left\{ \begin{array}{ll} \left( \left( 2^p-1\right) ^{\frac{1}{p-1}}+1\right) ^{-1/p} &{}, ~~p\in (1,2] \\ \left( w_p^{p/(p-1)} +1\right) ^{-1/p} &{},~~ p \in (2,\infty ). \end{array} \right. $$

For every \(p \in (1,2]\), the placement of the exit inducing worst case cost for Algorithm Wireless-Search\(_p\)(0) results in the total explored portion of \(\mathscr {C}_p\) with measure

$$e^-_p:= \pi _p+2\int _{0}^{s_p} \left( z^{p^2-p} \left( 1-z^p\right) ^{1-p}+1\right) ^{1/p} \mathrm {d}z.$$

Also, when the exit is reported, robots are at distance \(\gamma ^-_p:=2 (1-s_p^p)^{1/p}\).

For every \(p \in [2,\infty )\), the placement of the exit inducing worst case cost for Algorithm Wireless-Search\(_p\)(\(\pi /4\)) results in the total explored portion of \(\mathscr {C}_p\) with measure

$$e^+_p:= \pi _p+2\int _{2^{-1/p}}^{s_p} \left( z^{p^2-p} \left( 1-z^p\right) ^{1-p}+1\right) ^{1/p} \mathrm {d}z .$$

Also, when the exit is reported, robots are at distance \(\gamma ^+_p:= 2^{1/p} \left( \left( 1-s_p^p\right) ^{1/p} +s_p \right) \).

We also set \(e_p\) (and \(\gamma _p\)) to be equal to \(e_p^-\) (and \(\gamma _p^-\)) if \(p\le 2\), and equal to \(e_p^+\) (and \(\gamma _p^+\)) if \(p> 2\), and in particular \(e_p \in (\pi _p, 2\pi _p]\).

Quantities \(e_p, \gamma _p\), and some of their properties are depicted in Figs. 3a, 3b, and discussed in Sect. 4. One can also verify that \(\lim _{p\rightarrow 2^-} e_p^- = \lim _{p\rightarrow 2^+} e_p^+=4\pi /3\), and that \(\lim _{p\rightarrow 2^-} \gamma _p^- = \lim _{p\rightarrow 2^+} \gamma _p^+=\sqrt{3}\). In order to justify that indeed \(e_p \in (\pi _p, 2\pi _p]\), recall that by robots’ positions during the first \(\pi _p/2\) search time (after robots reach perimeter in time 1) is an increasing function. Since the rate of change of time is constant (it remains strictly increasing) for the duration of the algorithm, it follows that the evacuation cost of our algorithms remains increasing till some additional search time. Since robots search in parallel and in different parts of \(\mathscr {C}_p\), and since \(e_p\) is the measure of the combined explored portion of the unit circle, it follows that for \(e_p>2\pi _p/2=\pi _p\). At the same time, the unit circle has circumference \(2\pi _p\), hence \(e_p\le 2\pi _p\).

In other words, \(\gamma ^-_p\) is the length of chord with endpoints on \(\mathscr {C}_p\), \(p \in (1,2]\), defining an arc of length \(e^-_p\). Similarly, \(\gamma ^+_p\) is the length of a chord with endpoints on \(\mathscr {C}_p\), \(p \in (2,\infty )\), defining an arc of length \(e^+_p\). Unlike the Euclidean unit disks, in \(\ell _p\) unit disks, arc and chord lengths are not invariant under rotation. In other words, arbitrary chords of length \(\gamma ^-_p, \gamma ^+_p\) do not necessarily correspond to arcs of length \(e^-_p\), and \(e^+_p\), respectively, and vice versa. The claim extends also to the \(\ell _1, \ell _\infty \) spaces. For a simple example, consider points \( A=\rho _1(\pi /4)=(1/2,1/2), B=\rho _1(3\pi /4)=(-1/2,1/2), C=\rho _1(0)=(1,0), D=\rho _1(\pi /2)=(0,1)\). It is easy to see that \(d_p(A,B)=1\) and \(d_p(C,D)=2\), while , in other words two arcs of the same length identify chords of different length. We are motivated to introduce the following definition.

Definition 1

For every \(p \in [1,\infty )\), and for every \(u \in [0,2\pi _p)\), we define

In other words \(\mathscr {L}_p(u)\) is the length of the shortest line segment (chord) with endpoints in \(\mathscr {C}_p\) at arc distance u (and corresponding to an arc of measure u), and hence \(\mathscr {L}_p (u) = \mathscr {L}_p (2\pi _p -u)\) for every \(u \in (0,2\pi _p)\). As a special example, note that \(\mathscr {L}_2(u)= 2\sin (u/2)\), as well as \(\mathscr {L}_p(\pi _p)=2\), for all \(p\in [1,\infty )\).

Lemma 5

For every \(p \in (1,\infty )\), function \(\mathscr {L}_p(u)\) is increasing in \(u \in [0,\pi _p]\).

The intuition behind Lemma 5 is summarized in the following proof sketch. Assuming, for the sake of contradiction, that the lemma is false, there must exist an interval of arc lengths, and some \(p\ge 1\) for which \(\mathscr {L}_p(u)\) is strictly decreasing. By first-order continuity of \(\left\Vert A-B\right\Vert _p\), and in the same interval of arc-lengths, chord \(\left\Vert A-B\right\Vert _p\) must be decreasing in even when points A, B are conditioned to define a line with a fixed slope (instead of admitting a slope that minimizes the chord length). However, the last statement gives a contradiction. Indeed, consider points \(A,B,A',B'\) such that , with \(u<u'\le \pi _p\). It should be intuitive that \(d_p(A,B) \le d_p (A',B) \le d_p (A',B')\).

For fixed values of p, Lemma 5 can also be verified with confidence of at least 6 significant digits in MATHEMATICA. Due to precision limitations, the values of p cannot be too small, neither too big, even though a modified working precision can handle more values of p. With standard working precision, any p in the range between 1.001 and 45 can be handled within a few seconds. As we argue later, for large values of p, Lemma 5 bears less significance, since in that case we have an alternative way to prove the (near) optimality of algorithms Wireless-Search\(_p\)(\(\phi \)), as per Theorem 1. Next we provide a visual analysis of function \(\mathscr {L}_p\) that effectively justifies Lemma 5, see Fig. 1.

Fig. 1.
figure 1

Figures depict \(\mathscr {L}_p(u)\) for various values of p and for \(u \in [0,\pi _p]\). Left-hand side figure shows increasing function \(\mathscr {L}_p(u)\) for \(p\in (1,2]\). Right-hand side figure shows increasing function \(\mathscr {L}_p(u)\) for \(p\in [2,10]\). Recall that \(\mathscr {L}_2(u)= 2\sin (u/2)\), \(\mathscr {L}_p(\pi _p)=2\), for all \(p\in [1,\infty )\), as well as that \(\pi _1=\pi _\infty =4\) and \(\pi _p<4\) for \(p\in (1,\infty )\).

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4 Visualization of Key Concepts and Results

In this section we provide visualizations of some key concepts, along with visualizations of our results. The Figures are referenced in various places in our manuscript but we provide self-contained descriptions.

Fig. 2.
figure 2

(a) Unit circles \(\mathscr {C}_p\), for \(p=1,1.3,2,5,\infty \), induced by the \(\ell _p\) norm. Circles are nested, starting from \(p=1\) for the inner diamond-shaped circle, moving monotonically to \(p=\infty \) for the outer square-shaped circle. (b) The behavior of \(\pi _p\) as p ranges from 1 up to \(\infty \), where \(\pi _1=\pi _\infty =4\) and \(\pi _2=\pi \) is the smallest value of \(\pi _p\).

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Figures 3a and 3b depict quantities pertaining to algorithm Wireless-Search\(_p\)(\(\phi \)) (where \(\phi =0\), if \(p\in [1,2)\) and \(\phi =\pi /4\), if \(p\in (2,\infty )\)) for the placement of the hidden exit inducing the worst case cost. Moreover Fig. 3a depicts quantities \(e_p/2, \gamma _p\), as per Theorem 2. In particular, for each p, quantity \(e_p/2\) is the time a searcher has spent searching the perimeter of \(\mathscr {C}_p\) till the hidden exit is found (in the worst case). Therefore, \(e_p\) represents the portion of the perimeter that has been explored till the exit is found. Quantity \(\gamma _p\) is the distance of the two robots at the moment the hidden exit is found so that the total cost of the algorithm is \(1+e_p/2+\gamma _p\). By [21] we know that \(e_2=4\pi /3\) and \(\gamma _2=\sqrt{3}\). Our numerical calculations also indicate that \(\lim _{p\rightarrow 1} e_p=12/5\), \(\lim _{p\rightarrow 1} \gamma _p=8/5\), and \(\lim _{p\rightarrow \infty } e_p=\lim _{p\rightarrow \infty } \gamma _p=2\).

Figure 3b depicts quantities \(e_p/2\pi _p\), which equals the explored portion of the unit circle \(\mathscr {C}_p\), relative to its circumference, of Algorithm Wireless-Search\(_p\)(\(\phi \)) (where \(\phi =0\), if \(p\in [1,2)\) and \(\phi =\pi /4\), if \(p\in (2,\infty )\)) when the worst case cost inducing exit is found. By [21] we know that \(e_2/2\pi _2=(4\pi /3)/2\pi =2/3\). Interestingly, quantity \(e_p/2\pi _p\) is maximized when \(p=2\), that is in the Euclidean plane searchers explore the majority of the circle before the exit is found, for the placement of the exit inducing worst case cost. Also, numerically we obtain that \(\lim _{p\rightarrow 1} e_p/2\pi _p=3/5\), and \(\lim _{p\rightarrow \infty } e_p/2\pi _p=1/2\). The reader should contrast the limit valuations with Lemma 4 according which in both cases \(p=1,\infty \) the cost of our search algorithms is constant and equal to 5 for all placements of the exit that are found from the moment searchers have explored half the unit circle and till the entire circle is explored.

Fig. 3.
figure 3

(a) Perimeter search time \(e_p/2\) and distance \(\gamma _p\) between searchers when worst case cost inducing exit is found as a function of p, see also Theorem 2. (b) Explored portion \(e_p/2\pi _p\) as a function of p.

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Figure 4b shows the worst case performance analysis of Algorithm Wireless-Search\(_p\)(\(\phi \)) (where \(\phi =0\), if \(p\in [1,2)\)), which is also optimal for problem WE\(_p\). As per Lemma 4, the evacuation cost is 5 for \(p=1\) and \(p=\infty \). The smallest (worst case) evacuation cost when \(p\in [1,2]\) is 4.7544 and is attained at \(p\approx 1.5328\). The smallest (worst case) evacuation cost when \(p\in [2,\infty ]\) is 4.7784 and is attained at \(p\approx 2.6930\). As per [21], the cost is \(1+\sqrt{3}+2\pi /3 \approx 4.82644\) for the Euclidean case \(p=2\).

Fig. 4.
figure 4

(a) Figure depicts robots’ trajectories for algorithms Wireless-Search\(_{1.5}\)(0) and Wireless-Search\(_3\)(\(\pi /4\)). The inner unit circle corresponds to \(p=1.5\) and the outer to \(p=3\). (b) Blue curve depicts the worst case evacuation cost of Algorithm Wireless-Search\(_p\)(\(\phi \)), where \(\phi =0\), if \(p\in [1,2)\), as a function of p. Yellow line (constant function) is the optimal evacuation cost in the Euclidean metric space.

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5 Lower Bounds and the Proof of Theorem 1

First we prove a weak lower bound that holds for all \(\ell _p\) spaces, \(p\ge 1\) (see also Fig. 2b for a visualization of \(\pi _p\)).

Lemma 6

For every \(p \in [1,\infty )\), the optimal evacuation cost of WE\(_p\) is at least \(1+\pi _p\).

Proof

The circumference of \(\mathscr {C}_p\) has length \(2\pi _p\). Two unit speed robots can reach the perimeter of \(\mathscr {C}_p\) in time at least 1. Since they are searching in parallel, in additional time \(\pi _p-\epsilon \), they can only search at most \(2\pi _p-2\epsilon \) measure of the circumference. Hence, there exists an unexplored point P. Placing the exit at P shows that the evacuation time is at least \(1+\pi _p - 2\epsilon \), for every \(\epsilon >0\).

In particular, recall that \(\pi _1=\pi _\infty =4\), and hence no evacuation algorithm for WE\(_1\) and WE\(_\infty \) has cost less than 5. As a corollary, we obtain that Algorithms Wireless-Search\(_1\)(0) and Wireless-Search\(_\infty \)(\(\pi /4\)) are optimal, hence proving the special cases \(p=1,\infty \) of Theorem 1. The remaining cases require a highly technical treatment.

The following is a generalization of a result first proved in [21] for the Euclidean metric space (see Lemma 5 in the Appendix of the corresponding conference version). The more general proof is very similar.

Lemma 7

For every \(V \subseteq \mathscr {C}_p\), with \(\mu _p(V) \in (0,\pi _p]\), and for every small \(\epsilon >0\), there exist \(A,B \in V\) with .

Proof

For the sake of contradiction, consider some \(V \subseteq \mathscr {C}_p\), with \(\mu _p(V) \in (0,\pi _p]\), and some small \(\epsilon >0\), such no two points both in V have arc distance at least \(\mu _p(V) - \epsilon \). Below we denote the latter quantity by u, and note that \(u \in (0, \pi _p)\), as well as that \(\mu _p(V)=u+\epsilon >u\). We also denote by \(V^\complement \) the set \(\mathscr {C}_p \setminus V\). The argument below is complemented by Fig. 5.

Fig. 5.
figure 5

An abstract \(\ell _p\) unit circle, for some \(p\ge 1\), depicted as a Euclidean unit circle for simplicity. On the left we depict points \(A, \overline{A}, A_-,A_+,A',A'',T,R\). On the right we keep only the points relevant to our final argument, and add also points \(A_-'',A_+', \overline{A}', \overline{A}''\).

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Since V is non-empty, we consider some arbitrary \(A \in V\). We define the set of antipodal points of V

Note that \(N\cap V = \emptyset \) as otherwise we have a contradiction, i.e. two points in V with arc distance \(\pi _p > u= \mu _p(V) - \epsilon \). In particular, we conclude that \(N \subseteq V^\complement \), and hence by Lemma 1 we have \(\mu _p(N) = \mu _p(V)=u+\epsilon \).

Let \( \overline{A} \) be the point antipodal to A, i.e. . Next, consider points \(A_-, A_+ \in \mathscr {C}_p\) at anti-clockwise and clockwise arc distance u from A, that is . All points in are by definition at arc distance at least u from A. In particular, and . We conclude that , as otherwise we have \(A\in V\) together with some point in make two points with arc distance at least u. Note that this implies also that .

Consider now the minimal, inclusion-wise, arc , containing . Such arc exists because . In particular, since \(A\in V\), we have that and .

For some arbitrarily small \(\delta >0\), with \(\delta < \min \{ u, \epsilon /2\}\), let \(A',A''\in V\) such that . Such points \(A',A''\) exist, as otherwise would not be minimal. Clearly, we have and .

Since , its antipodal point \( \overline{A} '\) lies in . Similarly, since , its antipodal point \( \overline{A} ''\) lies in . Finally, we consider point \(A_-''\) at clockwise arc distance u from \(A''\), and point \(A_+'\) at anti-clockwise distance u from \(A'\), that is . We observe that and .

Recall that \(A''\in V\), hence , as otherwise any point in together with \(A'\), at arc distance at least u, would give a contradiction. Similarly, since \(A'\in V\), hence , as otherwise any point in together with \(A''\), at arc distance at least u, would give a contradiction.

Lastly, abbreviate by X, Y, respectively. Note that Similarly, Recall that , and hence sets X, Y intersect either at point A or have empty intersection. As a result \( \mu _p \left( X \cap Y \right) =0, \) as well as \( \mu _p \left( N \cap X \cap Y \right) =0 \).

Recall that , and so by Lemma 1 we also have \( \mu _p \left( X \cap N \right) = \mu _p \left( Y \cap N \right) = \delta . \) But then, using inclusion-exclusion for measure \(\mu _p\), we have

$$\begin{aligned} \mu _p( N \cup X \cup Y )&= \mu _p(N) + \mu _p(X) + \mu _p(Y) - \mu _p(N\cap X) - \mu _p(N\cap Y)- \mu _p(X\cap Y) + \mu _p(N\cap X \cap Y) \\&= u+\epsilon + \pi - u + \pi - u - \delta - \delta - 0 + 0 \\&= 2\pi _p - u + \epsilon - 2\delta \\&> 2\pi _p-u \\&> 2\pi _p-\mu _p(V) \\&= \mu _p(V^\complement ). \end{aligned}$$

Hence \(\mu _p( N \cup X \cup Y ) > \mu _p(V^\complement )\). On the other hand, recall that \(N,X,Y \subseteq V^\complement \), hence \(N \cup X \cup Y \subseteq V^\complement \), hence \(\mu _p( N \cup X \cup Y ) \le \mu _p(V^\complement )\), which is a contradiction.

We are now ready to prove a general lower bound for WE\(_p\) which we further quantify later.

Lemma 8

For every \(p \in (1,\infty )\), the optimal evacuation cost of WE\(_p\) is at least \(1+e_p/2 + \mathscr {L}_p(e_p)\).

Proof

Consider an arbitrary evacuation algorithm \(\mathscr {A}\). We show that the cost of \(\mathscr {A}\) is at least \(1+e_p + \mathscr {L}_p(e_p)\). By Theorem 2, we have that \(e_p \in (\pi _p, 2\pi _p]\). Let \(\epsilon >0\) be small enough, were in particular \(\epsilon <e_p-\pi _p\). We let evacuation algorithm \(\mathscr {A}\) run till robots have explored exactly \(e_p-\epsilon \) part of \(\mathscr {C}_p\).

The two unit speed robots need time 1 to reach the perimeter of \(\mathscr {C}_p\). Since moreover they (can) search in parallel (possibly different parts of the unit circle), they need an additional time at least \((e_p-\epsilon )/2\) in order to explore measure \(e_p-\epsilon \). The unexplored portion V of \(\mathscr {C}_p\) has therefore measure \(u:=2\pi _p - (e_p-\epsilon )\), where \(u \in (0,\pi _p)\).

By Lemma 7, there are two points \(A,B \in V\) that are at an arc distance \(v\ge u-\epsilon = 2\pi _p - e_p\). By definition, both points A, B are unexplored. We let algorithm \(\mathscr {A}\) run even more and till the moment any one of the points A, B is visited by some robot, and we place the exit at the other point (even if points are visited simultaneously), hence algorithm \(\mathscr {A}\) needs an additional time \(d_p(A,B)\) to terminate, for a total cost at least \(1+e_p/2-\epsilon /2+d_p(A,B)\). But then, note that \( d_p(A,B) \ge \mathscr {L}_p(v)\ge \mathscr {L}_p(2\pi _p - e_p), \) where the first inequality is due Definition 1 and the second inequality due to Lemma 5, and the claim follows by recalling that \(\mathscr {L}_p(2\pi _p-e_p)=\mathscr {L}_p(e_p)\).

Recall that, for every \(p\in (1, \infty )\), the evacuation algorithms we have provided for WE\(_p\) have cost \(1+e_p/2+\gamma _p\). At the same time, Lemma 8 implies that no evacuation algorithm has cost less than \(1+e_p/2 + \mathscr {L}_p(e_p)\). So, the optimality of our algorithms, that is, the proof of Theorem 1, is implied directly by the following lemma, which is verified numerically.

Lemma 9

For every \(p \in (1,\infty )\), we have \(\mathscr {L}_p(e_p)=\gamma _p\).

6 Discussion

We provided tight upper and lower bounds for the evacuation problem of two searchers in the wireless model from the unit circle in \(\ell _p\) metric spaces, \(p\ge 1\). This is just a starting point of revisiting well studied search and evacuation problems in general metric spaces that do not enjoy the symmetry of the Euclidean space. In light of the technicalities involved in the current manuscript, we anticipate that the pursuit of the aforementioned open problems will also give rise to new insights in convex geometry and computational geometry.