text: We revisit the linear search problem where a robot, initially placed at the origin on an infinite line, tries to locate a stationary target placed at an unknown position on the line. Unlike previous studies, in which the robot travels along the line at a constant speed, we consider settings where the robot's speed can depend on the direction of travel along the line, or on the profile of the terrain, e.g. when the line is inclined, and the robot can accelerate. Our objective is to design search algorithms that achieve good competitive ratios for the time spent by the robot to complete its search versus the time spent by an omniscient robot that knows the location of the target. We consider several new robot mobility models in which the speed of the robot depends on the terrain.These include 1) different constant speeds for different directions, 2) speed with constant acceleration and or variability depending on whether a certain segment has already been searched, 3) speed dependent on the incline of the terrain. We provide both upper and lower bounds on the competitive ratios of search algorithms for these models, and in many cases, we derive optimal algorithms for the search time.
  @cite_18: The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems -- delay-tolerant networks, opportunistic-mobility networks, social networks -- obtaining closely related insights. Indeed, the concepts discovered in these investigations can be viewed as parts of the same conceptual universe; and the formal models proposed so far to express some specific concepts are components of a larger formal description of this universe. The main contribution of this paper is to integrate the vast collection of concepts, formalisms, and results found in the literature into a unified framework, which we call TVG (for time-varying graphs). Using this framework, it is possible to express directly in the same formalism not only the concepts common to all those different areas, but also those specific to each. Based on this definitional work, employing both existing results and original observations, we present a hierarchical classification of TVGs; each class corresponds to a significant property examined in the distributed computing literature. We then examine how TVGs can be used to study the evolution of network properties, and propose different techniques, depending on whether the indicators for these properties are a-temporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.
  @cite_14: 
  @cite_7: Graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers. In this annotated bibliography, we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field.
  @cite_8: 
  @cite_9: We consider the problem of searching on a line using n mobile robots, of which at most f are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a target placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target. If n ≥ 2f+2, there is a simple algorithm with competitive ratio 1. For f Our search algorithm is easily seen to be optimal for the case n=f+1. We also show that as @math tends to ∞ the competitive ratio of our algorithm for the case $n = 2f+1 approaches 3 and this is optimal. More precisely, we show that asymptotically, the competitive ratio of our proportional schedule algorithm A(2f+1,f) is at most 3 + 4ln n n , while any search algorithm has a lower bound 3 + 2ln n n on its competitive ratio.
  @cite_3: In this paper we investigate distributed computation in dynamic networks in which the network topology changes from round to round. We consider a worst-case model in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. The model captures mobile networks and wireless networks, in which mobility and interference render communication unpredictable. In contrast to much of the existing work on dynamic networks, we do not assume that the network eventually stops changing; we require correctness and termination even in networks that change continually. We introduce a stability property called T -interval connectivity (for T >= 1), which stipulates that for every T consecutive rounds there exists a stable connected spanning subgraph. For T = 1 this means that the graph is connected in every round, but changes arbitrarily between rounds. We show that in 1-interval connected graphs it is possible for nodes to determine the size of the network and compute any com- putable function of their initial inputs in O(n2) rounds using messages of size O(log n + d), where d is the size of the input to a single node. Further, if the graph is T-interval connected for T > 1, the computation can be sped up by a factor of T, and any function can be computed in O(n + n2 T) rounds using messages of size O(log n + d). We also give two lower bounds on the token dissemination problem, which requires the nodes to disseminate k pieces of information to all the nodes in the network. The T-interval connected dynamic graph model is a novel model, which we believe opens new avenues for research in the theory of distributed computing in wireless, mobile and dynamic networks.
  @cite_5: In this paper we consider the group search problem, or evacu- ation problem, in which k mobile entities ( ( M E )s) located on the line perform search for a specific destination. The ( M E )s are initially placed at the same origin on the line L and the target is located at an unknown distance d, either to the left or to the right from the origin. All ( M E )s must simultaneously occupy the destination, and the goal is to minimize the time necessary for this to happen. The problem with k = 1 is known as the cow-path problem, and the time required for this problem is known to be 9d − o(d) in the worst case (when the cow moves at unit speed); it is also known that this is the case for k ≥ 1 unit-speed ( M E )s. In this paper we present a clear argument for this claim by showing a rather counter-intuitive result. Namely, independent of the number of ( M E )s, group search cannot be performed faster than in time 9d − o(d). We also examine the case of k = 2 ( M E )s with different speeds, showing a surprising result that the bound of 9d can be achieved when one ( M E ) has unit speed, and the other ( M E ) moves with speed at least 1 3.
  @cite_16: 
  @cite_10: Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed @math . The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most @math and also proved that any algorithm has evacuation time at least @math . We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most @math and show that any algorithm has evacuation time at least @math . To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them.
summary: The search problem has also been studied in environments where search occurs in graphs (see, e.g. @cite_7 ) or along dynamically evolving links of networks @cite_18 @cite_3 . More recently, variants of search using collections of robots have been investigated. The robots can employ either communication (at any distance) or communication, where communication is only possible among co-located robots. For example, the problem of @cite_10 @cite_16 is essentially a search problem where search is completed only when the target is reached by the last robot. Linear group search in the face-to-face communication model has also been studied with robots that either operate at the same speed or with a pair of robots having distinct maximal speeds @cite_5 @cite_8 . Finally, a new direction of research seeks to analyze linear search with multiple robots where some fraction of the robots may exhibit either crash faults @cite_9 or @cite_14 .