Figure 1 - available from: Theory of Computing Systems
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A snapshot of the searchers at time T. Here, each searcher R i is responsible for searching ray r i during time T (or right after time T, in the case i = 0). Moreover, each length corresponds to the time required to search a certain ray and return to the origin (and thus is equal to two times the length of some element in X)
Source publication
We study two optimization problems in a multiprocessor environment in the presence of set-up costs. The first problem involves multiple parallel searchers (e.g., robots) that must locate a target which lies in one of many concurrent rays, and at an unknown position from their common origin. Every time a searcher turns direction, it incurs a turn co...
Citations
... Moreover [24] gave optimal strategies for the generalized problem of star search, a result that was rediscovered later [10]. Some of the related work includes the study of randomization [29]; multi-searcher strategies [34]; multi-target searching [31,35]; searching with turn cost [2,5,21]; searching with an upper bound on the target distance [16,26]; fault-tolerant search [20,33] and searching with unreliable hints [4]; and the variant in which some probabilistic information on target placement is known [27,28]. This list is not exclusive; see also Chapter 8 in the book [1]. ...
... We are now ready to prove the main results of this section. Recall that for any two strategies X , Y , dr(X , Y ) is given by (2). Combining with (3), as well as with the fact that for Y ∈ S * , we have that D(Y , ) ≥ d * ( ), (from the definition of d * ), we obtain that ...
In search problems, a mobile searcher seeks to locate a target that hides in some unknown position of the environment. Such problems are typically considered to be of an on-line nature, in that the target’s position is unknown to the searcher, and the performance of a search strategy is usually analyzed by means of the standard framework of the competitive ratio, which compares the cost incurred by the searcher to an optimal strategy that knows the location of the target. However, one can argue that even for simple search problems, competitive analysis fails to distinguish between strategies which, intuitively, should have different performance in practice. Motivated by the above observation, in this work we introduce and study measures supplementary to competitive analysis in the context of search problems. In particular, we focus on the well-known problem of linear search, informally known as the cow-path problem, for which there is an infinite number of strategies that achieve an optimal competitive ratio equal to 9. We propose a measure that reflects the rate at which the line is being explored by the searcher, and which can be seen as an extension of the bijective ratio over an uncountable set of requests. Using this measure we show that a natural strategy that explores the line aggressively is optimal among all 9-competitive strategies. This provides, in particular, a strict separation from the competitively optimal doubling strategy, which is much more conservative in terms of exploration. We also provide evidence that this aggressiveness is requisite for optimality, by showing that any optimal strategy must mimic the aggressive strategy in its first few explorations.
... The exact and approximate competitive ratio of pathwise search has been studied in many settings, mostly assuming a star-like search domain. Examples include multi-Searcher strategies (López-Ortiz & Schuierer, 2004;Angelopoulos, , Arsénio, Dürr, & López-Ortiz, 2016), searching with turn cost (Demaine, Fekete, & Gal, 2006;Angelopoulos, Arsénio, & Dürr, 2017), searching with probabilistic information (Jaillet & Stafford, 1993), searching with upper/lower bounds on the distance of the Hider from the root (Hipke, Icking, Klein, & Langetepe, 1999;López-Ortiz & Schuierer, 2001;Bose, Carufel, & Durocher, 2015), and searching for multiple hiders (Angelopoulos, López-Ortiz, & Panagiotou, 2014;McGregor, Onak, & Panigrahy, 2009;Kirkpatrick, 2009). All these works assume that the search domain is either the unbounded line or the unbounded star. ...
We study the classic problem in which a Searcher must locate a hidden point, also called the Hider in a network, starting from a root point. The network may be either bounded or unbounded, thus generalizing well-known settings such as linear and star search. We distinguish between pathwise search, in which the Searcher follows a continuous unit-speed path until the Hider is reached, and expanding search, in which, at any point in time, the Searcher may restart from any previously reached point. The former has been the usual paradigm for studying search games, whereas the latter is a more recent paradigm that can model real-life settings such as hunting for a fugitive, demining a field, or search-and-rescue operations. We seek both deterministic and randomized search strategies that minimize the competitive ratio, namely the worst-case ratio of the Hider’s discovery time, divided by the length of the shortest path to it from the root. Concerning expanding search, we show that a simple search strategy that applies a “waterfilling” principle has optimal deterministic competitive ratio; in contrast, we show that the optimal randomized competitive ratio is attained by fairly complex strategies even in a very simple network of three arcs. Motivated by this observation, we present and analyze an expanding search strategy that is a 54-approximation of the randomized competitive ratio. Our approach is also applicable to pathwise search, for which we give a strategy that is a 5-approximation of the randomized competitive ratio, and which improves upon strategies derived from previous work.
... The exact and approximate competitive ratio of pathwise search has been studied in many settings, mostly assuming a star-like search domain. Examples include multi-Searcher strategies (López-Ortiz & Schuierer, 2004;Angelopoulos, , Arsénio, Dürr, & López-Ortiz, 2016), searching with turn cost (Demaine, Fekete, & Gal, 2006;Angelopoulos, Arsénio, & Dürr, 2017), searching with probabilistic information (Jaillet & Stafford, 1993), searching with upper/lower bounds on the distance of the Hider from the root (Hipke, Icking, Klein, & Langetepe, 1999;López-Ortiz & Schuierer, 2001;Bose, Carufel, & Durocher, 2015), and searching for multiple hiders (Angelopoulos, López-Ortiz, & Panagiotou, 2014;McGregor, Onak, & Panigrahy, 2009;Kirkpatrick, 2009). All these works assume that the search domain is either the unbounded line or the unbounded star. ...
We study the classic problem in which a Searcher must locate a hidden point, also called the Hider in a network, starting from a root point. The network may be either bounded or unbounded, thus generalizing well-known settings such as linear and star search. We distinguish between pathwise search, in which the Searcher follows a continuous unit-speed path until the Hider is reached, and expanding search, in which, at any point in time, the Searcher may restart from any previously reached point. The former has been the usual paradigm for studying search games, whereas the latter is a more recent paradigm that can model real-life settings such as hunting for a fugitive, demining a field, or search-and-rescue operations. We seek both deterministic and randomized search strategies that minimize the competitive ratio, namely the worst-case ratio of the Hider's discovery time, divided by the shortest path to it from the root. Concerning expanding search, we show that a simple search strategy that applies a "waterfilling" principle has optimal deterministic competitive ratio; in contrast, we show that the optimal randomized competitive ratio is attained by fairly complex strategies even in a very simple network of three arcs. Motivated by this observation, we present and analyze an expanding search strategy that is a 5/4 approximation of the randomized competitive ratio. Our approach is also applicable to pathwise search, for which we give a strategy that is a 5 approximation of the randomized competitive ratio, and which improves upon strategies derived from previous work.
... Optimal strategies for m-ray searching were given in [17,18,9]. Several variants of this problem have been studied, including: randomized algorithms [26]; multi-searcher algorithms [24]; searching with turn cost [16,4,6], searching with probabilistic information on targets [19,20]; searching with upper and/or lower bounds on the distance of the target from O [23,14]; searching with probabilistic location and/or fault-tolerance [5]; and the study of new cost formulations [7,3]. The partial information model was introduced independently in [22,25], in the unweighted setting (i.e., w i = 1, for all targets i). ...
Searching for a hidden target is an important algorithmic paradigm with numerous applications. We introduce and study the general setting in which a number of targets, each with a certain weight, are hidden in a star-like environment that consists of $m$ infinite, concurrent rays, with a common origin. A mobile searcher, initially located at the origin, explores this environment in order to locate a set of targets whose aggregate weight is at least a given value $W$. The cost of the search strategy is defined as the total distance traversed by the searcher, and its performance is evaluated by the worst-case ratio of the cost incurred by the searcher over the cost of an on optimal, offline strategy with (some) access to the instance. This setting is a broad generalization of well-studied problems in search theory; namely, it generalizes the setting in which only a single target is sought, as well as the case in which all targets have unit weights. We consider two models depending on the amount of information allowed to the offline algorithm. In the first model, which is the canonical model in search theory, the offline algorithm has complete information. Here, we propose and analyze a strategy that attains optimal performance, using a parameterized approach. In the second model, the offline algorithm has only partial information on the problem instance (i.e., the target locations). Here, we present a strategy of asymptotically optimal performance that is logarithmically related to $m$. This is in stark contrast to the full information model in which a linear dependency is unavoidable.
We study the online problem of evacuating k robots on m concurrent rays to a single unknown exit. All k robots start on the same point s, not necessarily on the junction j of the m rays, move at unit speed, and can communicate wirelessly. The goal is to minimize the competitive ratio, i.e., the ratio between the time it takes to evacuate all robots to the exit and the time it would take if the location of the exit was known in advance, in the worst-case instance.
When k=m, we show that a simple waiting strategy yields a competitive ratio of 4 and present a lower bound of 2+7/3≈3.52753 for all k=m≥3. For k=3 robots on m=3 rays, we give a class of parametrized algorithms with a nearly matching competitive ratio of 2+3≈3.73205.
We also present an algorithm for 1<k<m, achieving a competitive ratio of 1+2⋅m−1k⋅(1+km−1)1+m−1k, obtained by parameter optimization on a geometric search strategy. Interestingly, the robots can be initially oblivious to the value of m>2.
Lastly, by using a simple but fundamental argument, we show that for k<m robots, no algorithm can reach a competitive ratio better than 3+2⌊(m−1)/k⌋, for every k,m with k<m.
We study the online problem of evacuating k robots on m concurrent rays to a single unknown exit. All k robots start on the same point \(s\), not necessarily on the junction \(j\) of the m rays, move at unit speed, and can communicate wirelessly. The goal is to minimize the competitive ratio, i.e., the ratio between the time it takes to evacuate all robots to the exit and the time it would take if the location of the exit was known in advance, on a worst-case instance. When \(k=m\), we show that a simple waiting strategy yields a competitive ratio of 4 and present a lower bound of \(2+\sqrt{7/3} \approx 3.52753\) for all \(k=m\ge 3\). For \(k=3\) robots on \(m=3\) rays, we give a class of parametrized algorithms with a nearly matching competitive ratio of \(2+\sqrt{3} \approx 3.73205\). We also present an algorithm for \(1<k<m\), achieving a competitive ratio of \(1 + 2 \cdot \frac{m - 1}{k} \cdot \left( 1 + \frac{k}{m - 1} \right) ^{1 + \frac{m-1}{k}}\), obtained by parameter optimization on a geometric search strategy. Interestingly, the robots can be initially oblivious to the value of \(m > 2\). Lastly, by using a simple but fundamental argument, we show that for \(k<m\) robots, no algorithm can reach a competitive ratio better than \(3+2\left\lfloor (m-1)/k \right\rfloor \), for every k, m with \(k<m\).
