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.
The search problem has been extensively studied, e.g. see the survey by Benkoski @cite_2 ; deterministic algorithms for optimal linear search @cite_4 ; incorporating a when a robot changes direction during the search @cite_13 ; when bounds on the distance to the target are known in advance @cite_11 ; and for moving targets or more general linear cost functions @cite_15 . Other approaches include optimal randomized algorithms for the related @cite_0 , and stochastic and game theoretic investigations @cite_6 @cite_17 .
@cite_4 @cite_17 @cite_6 @cite_0 @cite_2 @cite_15 @cite_13 @cite_11
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We consider the problem of searching for an object on a line at an unknown distance OPT from the original position of the searcher, in the presence of a cost of d for each time the searcher changes direction. This is a generalization of the well-studied linear-search problem. We describe a strategy that is guaranteed to find the object at a cost of at most 9 ċ OPT + 2d, which has the optimal competitive ratio 9 with respect to OPT plus the minimum corresponding additive term. Our argument for upper and lower bound uses an infinite linear program, which we solve by experimental solution of an infinite series of approximating finite linear programs, estimating the limits, and solving the resulting recurrences for an explicit proof of optimality. We feel that this technique is interesting in its own right and should help solve other searching problems. In particular, we consider the star search or cowpath problem with turn cost, where the hidden object is placed on one of m rays emanating from the original position of the searcher. For this problem we give a tight bound of (1 + 2mm (m - 1)m-1)OPT + m((m (m - 1))m-1 - 1)d. We also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line. Searching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as the @math -lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem, and we give the first randomized algorithms in this paper. We show that our algorithm is optimal for two paths ( @math ), and give evidence that it is indeed optimal for larger values of @math . The randomized algorithms give expected performance that is almost twice as good as is possible with a deterministic algorithm. The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes. Consider a setting where robots must visit sites represented as nodes in a graph, but each robot may fail when traversing an edge. The goal is to find a set of paths for a team of robots which maximizes the expected number of nodes collectively visited, while guaranteeing that the paths satisfy a notion of “independence” formalized by a matroid (e.g. limits on team size, number of visits to regions), and that the probabilities that each robot survives to its destination are above a given threshold. We call this problem the Matroid Team Surviving Orienteers (MTSO) problem, which has broad applications such as environmental monitoring in risky regions and search and rescue in dangerous conditions. We present the MTSO formally and detail numerous examples of matroids in a path planning context. We then propose an approximate greedy algorithm for selecting a feasible set of paths and prove that the value of the output is within a factor p s p s + λ of the optimum, where p s is the per-robot survival probability threshold and 1 λ ≤ 1 is the approximation factor of an oracle routine for the well known orienteering problem. We demonstrate the efficiency of our approach by applying it to a scenario where a team of robots must gather information while avoiding pirates in the Coral Triangle. Let us consider an ordered vector A[1 : n]. If the cost of testing each position is similar, then the standard binary search is the best strategy to search the vector. This is true in both average and worst case. However, if the costs are non-uniform, then the best strategy is not necessarily the standard binary search. The best algorithm to construct a strategy that minimizes the expected search cost runs in O(n3) time and requires O(n2) space. The same complexities hold for the best algorithm to construct a strategy that minimizes the worst case search cost. Here, we show how to efficiently construct search strategies that are at most at a constant factor from the optimal one. These constructions take linear time and use only linear space. For the problem of minimizing the expected search cost, we present an algorithm that requires O(n) space and gives a (2 + ∈ + O(1))-approximated solution in O(n) time, for any fixed value of ∈ > 0. On the other hand, for the problem of minimizing the worst case search cost, we describe an algorithm that requires O(n) space and gives a (2 + ∈ + O(1))- approximated solution in O(n) time, for any fixed value of ∈ > 0. These two problems arise when processing a query in a distributed text database indexed by a suffix array. We consider the problem of searching on m current rays for a target of unknown location. If no upper bound on the distance to the target is known in advance, then the optimal competitive ratio is 1+2mm (m−1)m−1. We show that even if an upper bound of D on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least 1+2mm (m−1)m−1−O(1 log2D). This is again optimal – but in a stricter sense. In particular, this result implies the same lower bound for a robot searching for a target on infinite rays and finding it at a distance of D. To show that our lower bound is, indeed, optimal we construct a search strategy that achieves this ratio. Our strategy does not need to know an upper bound on the distance to the target in advance; it achieves a competitive ratio of 1+2mm (m−1)m−1−O(1 log2D) if the target is found at distance D. Finally, we also present a linear time algorithm to compute the strategy that allows the robot to search the farthest for a given competitive ratio C. The linear search problem concerns a search made in the real line for a point selected according to a given probability distribution. The search begins at zero and is made by continuous motion with constant speed along the line, first in one direction and then the other. The problem is to search in such a manner that the expected time required for finding the point according to the chosen plan of search is a minimum. This plan of search is usually conceived of as having a first step, a second,etc., and in that case, this author has previously shown a necessary and sufficient condition on the probability distribution for the existence of a search plan which minimizes the expected searching time. In this paper, we define a notion of search in which there is no first step, but the steps are instead numbered from negative to positive infinity. These new rules change the problem, and under them, there is always a minimizing search procedure. In those cases which satisfy the earlier criterion, the solutions obtained are essentially the same as those obtained previously. 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. Incremental search algorithms, such as Generalized Fringe-Retrieving A* and D* Lite, reuse search trees from previous searches to speed up the current search and thus often find cost-minimal paths for series of similar search problems faster than by solving each search problem from scratch. However, existing incremental search algorithms have limitations. For example, D* Lite is slow on moving target search problems, where both the start and goal states can change over time. In this paper, we therefore introduce Moving Target D* Lite, an extension of D* Lite that uses the principle behind Generalized Fringe-Retrieving A* to repeatedly calculate a cost-minimal path from the hunter to the target in environments whose blockages can change over time. We demonstrate experimentally that Moving Target D* Lite is four to five times faster than Generalized Adaptive A*, which so far was believed to be the fastest incremental search algorithm for solving moving target search problems in dynamic environments. In this paper, we study the problem of optimal trajectory generation for a team of heterogeneous robots moving in a plane and tracking a moving target by processing relative observations, i.e., distance and or bearing. Contrary to previous approaches, we explicitly consider limits on the robots' speed and impose constraints on the minimum distance at which the robots are allowed to approach the target. We first address the case of a single tracking sensor and seek the next sensing location in order to minimize the uncertainty about the target's position. We show that although the corresponding optimization problem involves a nonconvex objective function and a nonconvex constraint, its global optimal solution can be determined analytically. We then extend the approach to the case of multiple sensors and propose an iterative algorithm, i.e., the Gauss-Seidel relaxation (GSR), to determine the next best sensing location for each sensor. Extensive simulation results demonstrate that the GSR algorithm, whose computational complexity is linear in the number of sensors, achieves higher tracking accuracy than gradient descent methods and has performance that is indistinguishable from that of a grid-based exhaustive search, whose cost is exponential in the number of sensors. Finally, through experiments, we demonstrate that the proposed GSR algorithm is robust and applicable to real systems. We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL). This problem, previously studied in a paper of Alpern and Gal (1995), asks how two blind, speed one players placed a distance d apart on the line, can find each other in minimum expected time. The distance d is drawn from a known cumulative probability distribution G, and the players are faced in random directions. We show that the ARSPL is strategically equivalent to a new problem we call the double linear search problem (DLSP), where an object is placed equiprobably on one of two lines, and equiprobably at positions ±d. A searcher is placed at the origin of each of these lines. The two searchers move with a combined speed of one, to minimize the expected time before one of them finds the object. Using results from a concurrent paper of the first author and J. V. Howard (1998), we solve the DLSP (and hence the ARSPL) for the case where G is convex on its support, and show that the solution is that conjectured in a paper of Baston and Gal (1998). This paper studies a generalization of the classic binary search problem of locating a desired value within a sorted list. The classic problem can be viewed as determining the correct one-dimensional, binary-valued threshold function from a finite class of such functions based on queries taking the form of point samples of the function. The classic problem is also equivalent to a simple binary encoding of the threshold location. This paper extends binary search to learning more general binary-valued functions. Specifically, if the set of target functions and queries satisfy certain geometrical relationships, then an algorithm, based on selecting a query that is maximally discriminating at each step, will determine the correct function in a number of steps that is logarithmic in the number of functions under consideration. Examples of classes satisfying the geometrical relationships include linear separators in multiple dimensions. Extensions to handle noise are also discussed. Possible applications include machine learning, channel coding, and sequential experimental design. In this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object. The model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position. This model is meant to give a realistic cost measure for a robot moving in the plane. We also examine the effect of decreasing the amount of a priori information given to search problems. Problems of this type are very simple analogues of non-trivial problems on searching an unbounded region, processing digitized images, and robot navigation. We show that for some simple search problems, knowing the general direction of the goal is much more informative than knowing the distance to the goal. We investigate methods for planning in a Markov Decision Process where the cost function is chosen by an adversary after we fix our policy. As a running example, we consider a robot path planning problem where costs are influenced by sensors that an adversary places in the environment. We formulate the problem as a zero-sum matrix game where rows correspond to deterministic policies for the planning player and columns correspond to cost vectors the adversary can select. For a fixed cost vector, fast algorithms (such as value iteration) are available for solving MDPs. We develop efficient algorithms for matrix games where such best response oracles exist. We show that for our path planning problem these algorithms are at least an order of magnitude faster than direct solution of the linear programming formulation. We study a wide range of online covering and packing optimization problems. In an online covering problem, a linear cost function is known in advance, but the linear constraints that define the feasible solution space are given one by one, in rounds. In an online packing problem, the profit function as well as the packing constraints are not known in advance. In each round additional information (i.e., a new variable) about the profit function and the constraints is revealed. An online algorithm needs to maintain a feasible solution in each round; in addition, the solutions generated over the different rounds need to satisfy a monotonicity property. We provide general deterministic primal-dual algorithms for online fractional covering and packing problems. We also provide deterministic algorithms for several integral online covering and packing problems. Our algorithms are designed via a novel online primal-dual technique and are evaluated via competitive analysis. We consider the problem of on-line searching on m rays. A point robot is assumed to stand at the origin of m concurrent rays one of which contains a goal g that the point robot has to find. Neither the ray containing g nor the distance to g are known to the robot. The only way the robot can detect g is by reaching its location. We use the competitive ratio as a measure of the performance of a search strategy, that is, the worst case ratio of the total distance DR traveled by the robot to find g to the distance D from the origin to g.We present a new proof of a tight lower bound of the competitive ratio for randomized strategies to search on m rays. Our proof allows us to obtain a lower bound on the optimal competitive ratio for a fixed m even if the distance of the goal to the origin is bounded from above.Finally, we show that the optimal competitive ratio converges to 1 +2(eα- 1) α2 m ∼ 1 + 2.1.544 m, for large m where α minimizes the function(ex- 1) x2. In the classical stochastic k-armed bandit problem, in each of a sequence of T rounds, a decision maker chooses one of k arms and incurs a cost chosen from an unknown distribution associated with that arm. The goal is to minimize regret, defined as the difference between the cost incurred by the algorithm and the optimal cost. In the linear optimization version of this problem (first considered by Auer [2002]), we view the arms as vectors in R, and require that the costs be linear functions of the chosen vector. As before, it is assumed that the cost functions are sampled independently from an unknown distribution. In this setting, the goal is to find algorithms whose running time and regret behave well as functions of the number of rounds T and the dimensionality n (rather than the number of arms, k, which may be exponential in n or even infinite). We give a nearly complete characterization of this problem in terms of both upper and lower bounds for the regret. In certain special cases (such as when the decision region is a polytope), the regret is polylog(T ). In general though, the optimal regret is Θ∗( √ T ) — our lower bounds rule out the possibility of obtaining polylog(T ) rates in general. We present two variants of an algorithm based on the idea of “upper confidence bounds.” The first, due to Auer [2002], but not fully analyzed, obtains regret whose dependence on n and T are both essentially optimal, but which may be computationally intractable when the decision set is a polytope. The second version can be efficiently implemented when the decision set is a polytope (given as an intersection of half-spaces), but gives up a factor of √ n in the regret bound. Our results also extend to the setting where the set of allowed decisions may change over time. ∗Department of Computer Science, University of Chicago, varsha@cs.uchicago.edu †Toyota Technological Institute at Chicago, hayest,sham @tti-c.org Abstract Real-time heuristic search methods interleave planning and plan executions and plan only in the part of the domain around the current state of the agents. So far, real-time heuristic search methods have mostly been applied to deterministic planning tasks. In this article, we argue that real-time heuristic search methods can efficiently solve nondeterministic planning tasks. We introduce Min-Max Learning Real-Time A∗ (Min-Max LRTA∗), a real-time heuristic search method that generalizes Korf's LRTA∗ to nondeterministic domains, and apply it to robot-navigation tasks in mazes, where the robots know the maze but do not know their initial position and orientation (pose). These planning tasks can be modeled as planning tasks in nondeterministic domains whose states are sets of poses. We show that Min-Max LRTA∗ solves the robot-navigation tasks fast, converges quickly, and requires only a small amount of memory. In this work, we propose a search-based planning method to compute dynamically feasible trajectories for a quadrotor flying in an obstacle-cluttered environment. Our approach searches for smooth, minimum-time trajectories by exploring the map using a set of short-duration motion primitives. The primitives are generated by solving an optimal control problem and induce a finite lattice discretization on the state space which can be explored using a graph-search algorithm. The proposed approach is able to generate resolution-complete (i.e., optimal in the discretized space), safe, dynamically feasibility trajectories efficiently by exploiting the explicit solution of a Linear Quadratic Minimum Time problem. It does not assume a hovering initial condition and, hence, is suitable for fast online re-planning while the robot is moving. Quadrotor navigation with online re-planning is demonstrated using the proposed approach in simulation and physical experiments and comparisons with trajectory generation based on state-of-art quadratic programming are presented. We consider the task of obtaining the maximum a posteriori estimate of discrete pairwise random fields with arbitrary unary potentials and semimetric pairwise potentials. For this problem, we propose an accurate hierarchical move making strategy where each move is computed efficiently by solving an st-MINCUT problem. Unlike previous move making approaches, e.g. the widely used a-expansion algorithm, our method obtains the guarantees of the standard linear programming (LP) relaxation for the important special case of metric labeling. Unlike the existing LP relaxation solvers, e.g. interior-point algorithms or tree-reweighted message passing, our method is significantly faster as it uses only the efficient st-MINCUT algorithm in its design. Using both synthetic and real data experiments, we show that our technique outperforms several commonly used algorithms.