Deterministic Rendezvous Algorithms

Pełc, Andrzej

doi:10.1007/978-3-030-11072-7_17

Cited by 17 publications

(25 citation statements)

References 37 publications

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“…Rendezvous problems are far from being limited to the infinite line or ring. Problems may be stated for agents moving on the plane, on graphs, on a torus, on networks and so on, see for instance [8,2,34]. These problems are different from the ones considered in this paper.…”

Section: Introductionmentioning

confidence: 96%

Rendezvous search with markers that can be dropped at chosen times

Leone

Alpern

2018

Naval Research Logistics

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The Rendezvous search problem asks how two noncomunnicating players, who move at unit‐speed after a random placement in the search region (in this article, the line), can minimize their expected meeting time. Baston and Gal () solved a version where each player leaves a marker at his starting point, for example parachutes left after jumping into the search region. Here we consider that one or both players have markers which they can drop at chosen times. When the players are placed facing random directions at a distance D on the line the optimal expected meeting times R are known to be as follows: With no markers, R = 13D/8; with one each dropped at the start, R = 3D/2. Here we show that when only one player has a marker, but it can be dropped at any time, we still have R = 3D/2, obtained by dropping the marker at time D/4. Having both players drop markers at chosen times does not further reduce R. We adopt a new algorithmic approach which first fixes the dropping times and then reduces the resulting problem to a finite one. We also consider evanescent markers, which are detectable for a specified time T after being dropped, modeling pheromone scent markers used by some species in mate search or stain dropped by sailors. From a less theoretical and more practical point of view, we can see our problem as that faced by two hikers who get separated when they are walking not too far from a coast. A reasonable Rendezvous strategy is for each of them to head in the general direction of the coastline. On reaching the sea, they can calculate (based on their speed and time taken to reach to coast) the maximum distance along the coastline where the other one arrives. Then they can use our solution, taking that distance as our parameter D. Of course if it turns out that they reached the sea at points closer than D, their Rendezvous time will only be shorter. So our solution can be interpreted as a worst case expected meeting time. The marker might be a note scribbled on the sand, or a pile of rocks. Perhaps before getting separated they would have decided which roles (I or II) each would take up, being careful and cautious hikers. So in particular the hiker taking the role of the marker placer would put down a marker of some sort when he had walked along the coast one quarter the maximum distance to his partner. If the other finds it, he would know for sure the direction of the hiker who dropped it. Another thing to take away from this article is our result that says “two markers are not better than one.” This observation has some significance in the mate search context, where males and females use pheromone deposits to help find each other. In most species, it is only the females who can make such deposits, and the males can detect it. Giving both species the depositing facility would be more costly in terms of energy and, given our negative result, might not significantly reduce Rendezvous times. So in a speculative sense, our result has some explanatory value in this context.

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Section: Introductionmentioning

confidence: 96%

Rendezvous search with markers that can be dropped at chosen times

Leone

Alpern

2018

Naval Research Logistics

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“…also [1,5]. Deterministic rendezvous in networks is surveyed in [23,24]. Several authors considered geometric settings (rendezvous in an interval of the real line, e.g., [5,6], or in the plane, e.g., [3,8,11]).…”

Section: Related Workmentioning

confidence: 99%

Deterministic Rendezvous in Infinite Trees

Bhagat¹,

Pełc²

2022

Preprint

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The rendezvous task calls for two mobile agents, starting from different nodes of a network modeled as a graph to meet at the same node. Agents have different labels which are integers from a set {1, . . . , L}. They wake up at possibly different times and move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous algorithms. The time of such an algorithm is the number of rounds since the wakeup of the earlier agent till the meeting. In most of the literature concerning rendezvous in graphs, the graph is finite and the time of rendezvous depends on its size. This approach is impractical for very large graphs and impossible for infinite graphs. For such graphs it is natural to design rendezvous algorithms whose time depends on the initial distance D between the agents. In this paper we adopt this approach and consider rendezvous in infinite trees. All our algorithms work in finite trees as well. Our main goal is to study the impact of orientation of a tree on the time of rendezvous.We first design a rendezvous algorithm working for unoriented regular trees, whose time is in O(z(D) log L), where z(D) is the size of the ball of radius D, i.e, the number of nodes at distance at most D from a given node. The algorithm works for arbitrary delay between waking times of agents and does not require any initial information about parameters L or D. Its disadvantage is its complexity: z(D) is exponential in D for any degree d > 2 of the tree. We prove that this high complexity is inevitable: Ω(z(D)) turns out to be a lower bound on rendezvous time in unoriented regular trees, even for simultaneous start and even when agents know L and D. Then we turn attention to oriented trees. While for arbitrary delay between waking times of agents the lower bound Ω(z(D)) still holds, for simultaneous start the time of rendezvous can be dramatically shortened. We show that if agents know either a polynomial upper bound on L or a linear upper bound on D, then rendezvous can be accomplished in oriented trees in time O(D log L), which is optimal. If no such extra knowledge is available, a significant speedup is still possible: in this case we design an algorithm working in time O(D 2 + log 2 L).

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“…The study of algorithms for mobile robots is extensive, as evidenced by a recent survey [13]. The Gathering problem has been investigated thoroughly under a wide variety of model assumptions, as summarized in [3,9,12] for continuous models and in [8,18] for discrete models. Of particular interest to our current work are discrete models where the robots are located in a network, have some amount of visibility beyond its own position [1,2,7,11,15], and where faults may occur [6,16,17].…”

Section: Related Workmentioning

confidence: 99%

Fast Byzantine Gathering with Visibility in Graphs

Miller¹,

Saha²

2020

Preprint

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We consider the gathering task by a team of m synchronous mobile robots in a graph of n nodes. Each robot has an identifier (ID) and runs its own deterministic algorithm, i.e., there is no centralized coordinator. We consider a particularly challenging scenario: there are f Byzantine robots in the team that can behave arbitrarily, and even have the ability to change their IDs to any value at any time. There is no way to distinguish these robots from non-faulty robots, other than perhaps observing strange or unexpected behaviour. The goal of the gathering task is to eventually have all non-faulty robots located at the same node in the same round. It is known that no algorithm can solve this task unless there at least f + 1 non-faulty robots in the team. In this paper, we design an algorithm that runs in polynomial time with respect to n and m that matches this bound, i.e., it works in a team that has exactly f + 1 non-faulty robots. In our model, we have equipped the robots with sensors that enable each robot to see the subgraph (including robots) within some distance H of its current node. We prove that the gathering task is solvable if this visibility range H is at least the radius of the graph, and not solvable if H is any fixed constant.

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Deterministic Rendezvous Algorithms

Cited by 17 publications

References 37 publications

Rendezvous search with markers that can be dropped at chosen times

Rendezvous search with markers that can be dropped at chosen times

Deterministic Rendezvous in Infinite Trees

Fast Byzantine Gathering with Visibility in Graphs

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