1 Introduction
We study the problem of delivering a message in minimum time from a source to a destination using autonomous mobile robots with different maximum speeds. We extend the work on this communication problem studied previously on graphs [1, 3, 8, 9]. In our setting, the robots are initially distributed in arbitrary locations in the plane and the locations of the source and destination are known by all. The robots may move with their own (maximum) speed. Robots cooperate by exchanging information (the message) using facetoface (F2F) communication. We study message transmission and allow messages to be replicated (as opposed to package delivery). The goal is to give an algorithm which minimizes the time required to deliver the message from the source to the destination through a series of F2F message transfers. In this paper we study how to complete this task efficiently and propose various centralized offline and distributed online algorithms which take into account the knowledge that the robots have about their speeds and initial locations.
1.1 Model, Notation and Terminology
The setup of our pony express problem will be in the Euclidean plane and points will be identified with their cartesian coordinates. We use capital letters to denote points and lowercase letters with subscripts to denote their components (e.g. point ). For any points , denotes the Euclidean distance between and , denotes the angle formed by in this order, and denotes the triangle formed by . Finally, denotes a circle centered at with radius .
Assume that robots are placed at arbitrary positions in the Euclidean plane. The respective speeds of the robots are . The movement of a robot is determined by a welldefined trajectory. A robot trajectory is a continuous function , with the location of the robot at time , such that , for all , where is the speed of the robot. A robot can move with its own constant speed and during the traversal of its trajectory it may stop and/or change direction instantaneously and at any time. Robot communication is F2F in that two robots can communicate (instantaneously) with each other only when they are colocated.
Algorithms describe the trajectories robots will follow and we will take into account the time it takes the algorithm to conclude the delivery task from the start, obtaining the message at a given source , and eventually delivering it to a given destination . In general, we are interested in offline and online algorithms. In the offline setting, the locations and speeds of all robots are known in advance and are available to a central authority that assigns trajectories to the robots. In the online setting, the robots know only their own initial position and speed, along with the positions of and . To measure the performance of our online algorithms, we consider their competitive ratio defined as follows. Let be the optimal delivery time for an instance of a given problem and be the time needed by some online algorithm for the same instance. The competitive ratio of is Our goal is to find online algorithms that minimize this competitive ratio.
1.2 Related work
Communicating mobile robots or agents have been used to address problems such as search, exploration, broadcasting and convergecasting, patrolling, connectivity maintenance, and area coverage (see [11]). For example, [6] addresses the problem of how well a group of collaborating robots with limited communication range is able to monitor a given geographical space. To this end, they study broadcasting and coverage resilience, which refers to the minimum number of robots whose removal may disconnect the network and result in uncovered areas, respectively. Similarly, rendezvous is a relevant communication paradigm and in [13, 17] the authors investigate rendezvous in a continuous domain under the presence of spies. A related study on message transmission in a synchronized multirobot system may be found in [6]. Another application is patrolling whereby mobile robots are required to keep perpetually moving along a specified domain so as to minimize the time a point in the domain is left unvisited by an agent, e.g., see [16] for a related survey.
Data delivery and convergecast with energy exchange under a centralized scheduler were studied in [15]. A restricted version concerns mobile agents of limited energy that are placed on a straight line and which need to collectively deliver a single piece of data from a given source point to a given target point on the line can be found in [10]. In [12] it is shown that deciding whether the agents can deliver the data is (weakly) NPcomplete. Additional research under various conditions and topological assumptions can be found in [4] which studies the gametheoretic task of selecting mobile agents to deliver multiple items on a network and optimizing or approximating the total energy consumption over all selected agents, in [2, 5, 7] which study data delivery and combine energy and time efficiency, and in [18, 19] which are concerned with collaborative exploration in various topologies.
Our problem was previously studied on graphs in [1, 3, 8, 9]. In particular it is shown in [8] that the problem can be solved with agents on an node, edge weighted graph in time . We use this algorithm in the development of our approximation algorithm.
Our current work is related to the Pony Express communication problem proposed in [14]. In that paper, the authors provide both optimal offline and online algorithms for the anycast and broadcast problems in the case where the underlying domain was a continuous line segment. To our knowledge, the planar case studied in our paper has not been considered previously.
1.3 Outline and results of the paper
In Section 2 we propose an optimal offline algorithm for two robots. For ease of exposition, we first consider the case when the slower robot starts at the source and then the general case of arbitrary starting positions. In Section 3 we study the offline multirobot case. We propose an algorithm which approximates the optimal delivery time to within a factor of . Section 4 is dedicated to online algorithms. For two robots we give an algorithm with competitive ratio of and show that for robots, this same algorithm has a competitive ratio of at most . We also analyze lower bounds for this specific algorithm showing these bounds are tight. In Section 5 we prove lower bounds on the competitive ratio of arbitrary online algorithms. We discuss two approaches, one where the position of a robot is unknown and the other where the speed of a robot is unknown. These different approaches provide lower bounds of 1.0391 and 1.0405, respectively. We conclude in Section 6 by discussing possibilities for additional research in this area.
2 Optimal Offline Algorithm for Two Robots
In this section, we will consider two robots and which can move with respective constant speeds and () and design optimal offline algorithms with respect to the F2F communication model (observe that by scaling the distances, setting the speed of the slow robot to be yields no loss of generality.) Let and be the starting positions of robots and , respectively. There are three cases to consider:

: the fast robot can get to before the slow robot. In this case, it is clear that in the optimal solution, the fast robot should move to , acquire the message, and carry it to .

: the slow robot can deliver the message to before the fast robot can even reach . In this case, the optimal solution is also clear. The slow robot should move to , acquire the message, and carry it to .

In all other cases, the slow robot can get to before the fast robot, but the fast robot can get to the destination faster. The optimal solution, then, must involve a handover between the robots at some point in the plane.
For the first two cases, the optimal solution is trivial. The third case, however, is not as we must find the point at which the robots meet. First, we characterize the optimal meeting point for Case 3 through a series of lemmas.
Lemma 1.
For Case , there exists an optimal solution such that if is the handover, then .
For the sake of contradiction, suppose . First, it is obvious that should move directly toward and then directly toward and, similarly, should move directly toward . Any other path could only increase the time to deliver the message. Then, since , either or must arrive at before the other. Thus, there must be a time where one robot is waiting at for the other to arrive. Let be the time that the first robot arrives to and be the position of the other robot at time . We claim that an equal or better solution than waiting at would be for the first robot to move along until it meets the other robot at some point . Then the faster of the two robots carries the message from to (Figure 1).
If arrives at first, then if it waits at for to arrive, the total delivery time is , but since is clearly larger (in perimeter) than , then
Thus, meeting at results in a quicker delivery, a contradiction to the assumption that is optimal. If arrives at first, then if it waits at for the total delivery time is , but
Again, meeting at results in an equal or quicker delivery, and so by contradiction, there must exist an optimal solution where . ∎ Intuitively, Lemma 1 says that robots must move at their maximum speeds directly towards the location they will acquire the message and then directly toward the location they handover or deliver the message. This restricts the set of feasible meeting points to the set of points in the plane such that both robots, moving in one direction at their maximum speeds, meet at the same time. For the case where the slow robot starts at the source (), this is directly related to an ancient theorem by the Greek philosopher Apollonius, which states “the trajectory traced by a point which moves in such a way that its Euclidean distance from a given point is a constant multiple of its Euclidean distance from another point is a circle” [20]. As a consequence, if the robots start at positions , respectively, then the locus of points at which the two robots may travel directly towards and meet at the same time is the circle of Apollonius (see Figure 2).
This circle, then is the locus of all possible handover points between the two robots. The precise statement in the context of mobile robots is stated in Lemma 2.
Lemma 2.
Two robots and with speeds and () are initially placed at points and , respectively. The locus of points such that robots and are equal time away from points and , respectively, i.e., , forms a circle with center and radius so that
(1) 
The proof follows easily by using the representation of the points in cartesian coordinates and solving the equation . ∎
The following definition of the Apollonius Circle will be used throughout this paper.
Definition 1 (Apollonius Circle).
The circle with center and radius given by Equations (1) is called the Apollonius circle between robots and when their respective starting positions are .
For instances of the problem where and whose optimal solutions do not involve either robot delivering the message by itself, the previous discussion results in the following lemma whose proof follows directly from Lemmas 1 and 2.
Lemma 3.
The optimal meeting point is the point on the Apollonius circle between robots and which minimizes the total delivery time . ∎
2.1 Optimal algorithm when a robot starts at the source
First we give an algorithm in the restricted case where one robot starts at the source where the message is located ().
Let be the source, be the starting position of the fast robot which we assume to be on the axis, and the destination. Without loss of generality, we assume (if , the instance can be reflected about the axis and solved equivalently, since is on the axis). By Lemma 3, our goal is to find the point on the robots’ Apollonius circle which minimizes the delivery time (see Figure 3).
Consider the following offline algorithm for computing the optimal delivery time.
Theorem 2.1.
First, note that Case 1 (from the three cases at the beginning of the section) is not considered since the slow robot, is assumed to start at the source. Case 2 is obviously handled by line 1 in the algorithm. Case 3 is divided into two subcases based on whether or not the condition in line 4 is satisfied. First, we consider the case where it is not. Let be the angle . Observe that if is tangent to the Apollonius circle (Figure 4), then . Clearly for any smaller value for , intersects the Apollonius circle at two points (and for any larger value, does not intersect the circle).
Then, let and (Figure 5 left). By the law of sines Thus and is just the associated point on the Apollonius circle.
Observe is the intersection point closest to . Since the condition in line 4 is satisfied, can move directly toward and, without veering from a direct path, meet at , acquire the message, and continue towards to deliver the message. We know, since the first case was not satisfied, that can reach the destination before can, so this is clearly the optimal trajectory.
Now, suppose the condition in line 4 is not satisfied. Consider the ellipse with foci and whose semimajor axis has length for some time . Then, by a defining property of an ellipse, the sum of the distances from each foci to a point on the ellipse is equal to a constant value . Consequently, a robot starting at with speed takes exactly time to travel to a point on the ellipse and then to . Observe that if the ellipse and Apollonius circle intersect, then the two robots can meet at one of the intersection points and, by the previous statement, the fast robot can deliver the message in time . If they intersect at two points, though, then any point on the Apollonius circle between these two intersections would yield a better solution. The solution, then, is to find the minimal which causes the Apollonius circle and the ellipse to intersect at exactly one point . Thus must be normal to both the Apollonius circle and the ellipse. That , therefore, must bisect follows from a wellknown property of the ellipse, namely that a normal line through a point on an ellipse bisects the angle it forms with the ellipse’s foci.
Next, we show the algorithm can be implemented to run using a constant number of operations (including trigonometric functions). The only lines in the algorithm that are not clearly computable with a constant number of operations are lines 8 and 10. To show that can be computed in constant time, we provide a formulation which can be given as input to Equation Solving tools (e.g., Mathematica) to find a closedform solution ^{1}^{1}1 Link to Mathematica solution for Theorem 2.1 . Let and be the point given by rotating around (into the positive halfplane, Figure 5 right). Observe that if bisects , then is collinear with , or: where . ∎
2.2 Optimal algorithm in the general case
In this subsection we consider the more general case where the slow robot does not start at the source. Let the starting positions of source and destination be and and let the robots and start from arbitrary points and in the plane, respectively. Again, we are interested in finding the point for the third case (from the cases at the beginning of the section), since optimal solutions for the first two cases are trivial to find. As depicted in Figure 6,
robot follows a trajectory which first visits a point at distance from its starting position, then continues along a straightline trajectory to meet robot at a suitable point , and finally delivers the message to the destination . The main steps of the algorithm are as follows.

If , then reaches before and should complete the delivery on its own.

Otherwise, if , then can deliver the message on its own before can even reach the destination.

Otherwise reaches in time and, at the same time, robot goes to a specially selected point which lies on the circle centered at with radius .
Observe that by Lemma 1, , , and must be collinear. We can then generalize the result of section 2.1 using the following lemma. Recall that the center of similitude (also known as homothetic center) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another (see [21][Section 1.1.2]).
Lemma 4.
Let be the center of the Apollonius circle between and when is at and is at . Then, is the center of similitude of the circles and . Consider any point in the circumference of . Let be the angle , then is the center of the Apollonius circle of .
Refer to Figure 7. Consider any point at the circumference of the circle . Let be the center of the Apollonius circle between and when is at and is at . Therefore, and . Observe that the ratio is always constant.
Therefore, defines a circle where is the center of similitude. The lemma follows since the triangles and are similar. ∎
Consider two points and as described in Lemma 4, for some . We can now use Theorem 2.1 to characterize the solution. However, this approach does not lead to a closedform solution. Instead, in the following lemma, we present another approach using optimization which does.
Let . Then the optimal trajectory is obtained by a point which minimizes the objective function
(2) 
subject to the condition
(3) 
Recall points are collinear, the handover point point must lie at the intersection of two circles as depicted in Figure 8.
This can be expressed by the fact that satisfies the two equations
(4)  
(5) 
In turn, this gives a system of quadratic equations parametrized with respect to time . We can rewrite Equation (4) as and subtracting both sides of the last Equation from Equation (5) we derive the Equation
By collecting similar terms, using Equation (5), and simplifying we derive the following Equation
This is exactly Equation (3). ∎
The resulting optimization problem has two unknowns in the objective function (2) and must satisfy the condition of Equation (3). It can be used to substitute variables and express the final optimization function described in Formula (2) using only a single variable, say , which can then be minimized using standard analytical methods. This is easily seen since Equation (3) is of degree in the variable (as well as in the variable , for that matter). Therefore a closed form expression of the variable in terms of the variable and the known parameters is easily derived.
There are two symmetries in Equation (3) which simplify the objective function and make the calculation of the solution easier. They are easily revealed with simple geometric transformations.
For the first symmetry, consider a rotation of the axis and a translation of the entire configuration of points so that and lie on the horizontal axis, i.e., and . Then Equation (3) is transformed to the equation
(6) 
The resulting symmetry is along the horizontal axis in Equation (6). Namely, if is a solution so is . If we consider Equation (6) in the unknown we see that it is of degree , but which is also a quadratic in . Therefore can be easily expressed as a function of using the formula for the roots of the quadratic equation. The second symmetry is obtained in a similar manner. If is a solution so is . One considers a rotation of the axis and a translation of the entire configuration of points so that and lie on the vertical axis, i.e., and . Details can be completed as above. To sum up we have the following Algorithm 2 which determines the handover point which yields the optimal trajectory.
Algorithm 2 returns the optimal delivery time for two robots and with speeds and , respectively, and can be implemented using a constant number of operations (including trigonometric functions). The proof follows from the previous discussion. Indeed, without loss of generality we may consider only the case where the slow robot reaches first. As depicted in Figure 8 there are two competing trajectories. Given that the slow robot can arrive first at , either the robots follow the algorithm and the slow robot meets the faster robot at the meeting point to handover the message to which then delivers it to or the faster robot gets the message at and delivers it to without cooperating with the other robot.
In the former case the delivery time will be while in the latter case the delivery time will be . However this is easy to prove since the point must lie inside the triangle as depicted in Figure 8, i.e., . All other lines in the algorithm clearly require a constant number of operations. By the previous discussion, a closed form solution for the optimization required in line 6 exists ^{2}^{2}2 Link to Mathematica solution for Theorem 2.2 . ∎
3 Offline Approximation for Multiple Robots
In principle, the equations derived in the previous section can be generalized to solve the problem optimally for robots. Unfortunately, we are not able to solve the resulting set of equations. We do not speculate on the complexity of the general problem here. Instead, in this section we provide a approximation algorithm, The robots know the location of the source and destination but also all robots know the initial locations and speeds of all other robots. The basic idea of our proof is contained in the following observation depicted in Figure 9. Suppose that during the execution of an optimal “Euclidean” algorithm (i.e., optimal in the sense of the Euclidean distance) two robots placed at and , follow the straightline trajectories and , respectively, and meet at the point .
Now we replace the Euclidean trajectories and with the rectilinear trajectories and , respectively. Elementary geometry implies that
(7) 
This observation leads to the following lemma.
Lemma 5.
Consider the pony express problem for robots, a source and a destination in the plane. Then , where are the delivery costs of the optimal trajectories of the pony express problem for delivering from a source to a destination measured in the rectilinear and Euclidean metrics, respectively.
Proof.
Consider an optimal Euclidean algorithm which ensures the delivery time is exactly , i.e., . Now use the idea discussed in Figure 9 to replace the Euclidean trajectory of algorithm with a rectilinear trajectory thus giving rise to a rectilinear algorithm . Note that in this rectilinear simulation of the optimal Euclidean solution, robots may not arrive at the endpoints at the same time. The robot that arrives first, should simply wait at the meeting point until the second robot arrives. The meeting time is thus determined by the last arrival of the two robots. By definition, the time it takes the rectilinear algorithm to deliver the message is at least , i.e., . From Inequality (7) we have that . Therefore we conclude that ∎∎
Consider robots in the plane with starting positions . Without loss of generality assume the slowest robot has speed . Further, let the source of a message be located at a point and the destination at a point and assume, without loss of generality, that the line segment is horizontal. Enclose the points and in a square with sides parallel to the axis, where is a positive real proportional to the diameter of the set . For arbitrarily small, partition the square with parallel vertical and horizontal lines with consecutive distances , respectively, so as to form a grid. Without loss of generality we may assume that and are vertices in this grid graph (This is easy to accomplish by choosing to be an integral fraction of the distance between and .) Clearly, this forms a grid graph with vertices so that are also vertices and edges. Now consider the following algorithm.
Theorem 3.1.
For any arbitrarily small, there exists an algorithm that finds trajectories for robots to deliver the message from the source to the destination in time whose delivery time is at most multiplied by the delivery time of the optimal Euclidean algorithm plus the additional additive overhead , where is the diameter of the point set .
Proof.
Let and run Algorithm 3. Let be the time the algorithm takes to deliver the message and let be the time for step 2 in the algorithm (the optimal delivery time for the given grid with starting positions ). Then, let and be the optimal delivery times for the rectilinear and Euclidean metrics respectively. First, observe
(8) 
The result will follow from Lemma 5 and the following claim:
Claim.
.
Proof.
(of Claim) Without loss of generality, assume the robots involved (in order) are robots through . Let be the handover points in the optimal rectilinear algorithm (where ). Let be the nearest point to on the grid. Let be the time it takes for the first robot to arrive at the source and, (for ) be the time that robot holds the message.
Consider the algorithm where robots emulate the rectilinear algorithm on the grid by meeting at instead of , for each handover. Note that robots may not arrive at the endpoints at the same time. However, the algorithm is offline the robot that arrives first, should simply wait at the meeting point until the second robot arrives.
The time each robot holds the message then is at most and the total time to deliver the message is
This proves the claim. ∎
4 Online Upper Bounds
In this section we discuss online algorithms. In Subsection 4.1 we give an online algorithm with competitive ratio for two robots with knowledge only of the source and destination . We show this bound is tight for the given algorithm. In Subsection 4.2 we show that the same algorithm when generalized to robots has competitive ratio at most 2. Further, we show that for any , the competitive ratio of our algorithm is at least .
4.1 Two Robot Algorithm with Competitive Ratio
Consider the following Algorithm 4 for multiple robots.
Observe that in this algorithm the robots act independently. In particular no attempt is made to coordinate the action of the robots and if two robots meet they ignore each other. This is not required in order to achieve the upper bounds below. For our lower bounds on this algorithm, we assume that the robots do not interact even if it may improve the time to complete the task.
Theorem 4.1.
For the case of two robots, Algorithm 4 delivers the message from the source to the destination in at most times the optimal offline time.
Given an arbitrary instance of the problem, let be the time taken by the optimal solution to deliver the message from to . Let be the robots involved in that optimal solution where . Observe that if only one robot is involved then our online solution is optimal. Let be the starting point of robot . Let be the point in the optimal solution where hands the message off to . Let be the time when this happens. Finally, let and .
We make the following observations:

.

.

delivers the message in time . (Recall that is the time for to reach , , and .)

delivers the message in time which is less than .

and .

.
Let be the competitive ratio of our algorithm for our two robots. We have by observation 4. This is maximized when or when . In this case we have:
This is maximized when at which point ∎
Example 1.
Now we give a tight lower bound on the competitive ratio of Algorithm 4 for two robots. Consider the following example input. One robot is placed at the source which is the point and has speed . The destination is placed at the point . The second robot has speed and is placed at the point . The robots are initially placed at distance . In the optimal algorithm the robots meet in time at the point . The faster robot picks up the message at and delivers it to in additional time . Therefore the delivery time of the optimal algorithm is equal to . It follows that the competitive ratio satisfies
Remark 1.
Note that in Example 1 if we parametrize the speed of the slow robot to , and place the fast robot at position then similar calculations show that . Further, it is easy to see that the lower bound is maximized for .
4.2 Multi Robot Algorithm with Competitive Ratio
Theorem 4.2.
Algorithm 4 has competitive ratio at most for any robots.
To simplify notation, let the distance between the source and destination be . Observe that in the algorithm, every robot attempts to deliver the message entirely by itself. The robots do not cooperate at all. Clearly, then, if the robots can optimally deliver the message in , then Theorem 4.2 holds if and only if there exists at least one robot that can deliver the message by itself in time. In other words, we must show that there is a robot with speed and that starts a distance from the source such that ( such that ).
For the sake of contradiction, assume all robots have speed . Then we must show that they could not possibly deliver the message optimally in time or less. By restricting the robots speed as a function of their distance to the source, we allow the robots to choose everything else about their starting positions to minimize the optimal delivery time. Clearly, robots will most quickly deliver the message if they are positioned on the line from (robots are always moving directly toward the message or its destination).
Furthermore, the more robots in the system, the faster the message will be delivered (observe that given two participating robots, inserting an additional robot between them improves the delivery time). Therefore, we can assume there are an uncountably infinite number of robots (one at every point on the interval ) and the problem becomes continuous.
Consider a robot that starts at position on the line. By construction, its velocity is less than and after time its position is . Therefore, if the message is at position on the line segment at time
, its velocity at that moment is less than
since that is the upper bound on the speed of the fastest robot that could reach by time .It follows from the previous discussion that the speed of the message must satisfy the inequality: The resulting differential equation with unknown and the initial condition , yields
Finally, we can use this equation to show that the delivery time (when ) must be greater than . Observe , so the robots cannot deliver the message in time , a contradiction. Therefore, if the optimal delivery is , then there must exist a robot with speed which can deliver the message to the destination in at most time. ∎
Theorem 4.3.
Given robots, there is a robot deployment such that Algorithm 4 has competitive ratio at least .
Let be the source and the destination. Without loss of generality, we assume is a unit line where is at the origin and . Given the meeting points, we construct an instance with robots with competitive ratio . Let be the meeting point of robot and for . In our construction, robots are required to arrive at at the same time after reaching . Hence, we compute the speed from the meeting points as follows: Let be the time that robot arrives at point . Therefore, it arrives at after reaching at time . Further, at time robot has reached . Therefore, it arrives at after reaching at time . Setting both equal and factorizing we obtain the speed given , i.e. . Initially, we set which implies that is the fastest robot. Let be the initial position or robot on the line . Observe that all robots are in the interval . We claim that the competitive ratio of the setting is . Observe that arrives at at time . Therefore, in the optimal algorithm it takes an additional to reach meanwhile in Algorithm 4 robot takes to reach . Therefore, the competitive ratio is . Simplifying we obtain and the theorem follows. ∎
Remark 2.
Observe that for any by taking we have the competitive ratio of Algorithm 4 is at least .
5 Online Lower Bounds for Two Robots
In this section we prove two lower bounds on the competitive ratio for arbitrary online algorithms. Our lower bounds require only two mobile robots. In the first lower bound (Theorem 5) we assume that the speed of one of the robots is unknown and in the second (Theorem 5.1) we assume that the starting position of one of the robots is unknown. We provide both bounds (even though the second is slightly better) as the arguments are somewhat different and it seems plausible that an improved lower bound may come from combining the two approaches.
The lower bound for the competitive ratio when the fast robot does not know whether the speed of the slow robot is one or zero is at least . In the proof we consider two robots and where is placed at the source of the message and has speed either 0 or 1 as set by the adversary and has speed greater than 1. Without loss of generality assume that and are at distance one. By Lemma 2 the Apollonius circle is at distance
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