Treasure hunt is the task of finding an inert target by a mobile agent in an unknown environment. We consider treasure hunt in geometric terrains with obstacles. Both the terrain and the obstacles are modeled as polygons and both the agent and the treasure are modeled as points. The agent navigates in the terrain, avoiding obstacles, and finds the treasure when there is a segment of length at most 1 between them, unobstructed by the boundary of the terrain or by the obstacles. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the amount of information that the agent needs a priori in order to find the treasure at cost @math , where @math is the length of the shortest path in the terrain from the initial position of the agent to the treasure, avoiding obstacles. Following the paradigm of algorithms with advice, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the whole environment: the terrain, the position of the treasure and the initial position of the agent. Information complexity of treasure hunt is the minimum length of the advice string (up to multiplicative constants) that enables the agent to find the treasure at cost @math . We first consider treasure hunt in regular terrains which are defined as convex polygons with convex @math -fat obstacles, for some constant @math . A polygon is @math -fat if the ratio of the radius of the smallest disc containing it to the radius of the largest disc contained in it is at most @math . For the class of regular terrains, we establish the exact information complexity of treasure hunt. We then show that information complexity of treasure hunt for the class of arbitrary terrains (even for non-convex polygons without obstacles, and even for those with only horizontal or vertical sides) is exponentially larger than for regular terrains.
Algorithms with advice. The paradigm of algorithms with advice was developed mostly for tasks in graphs. Providing arbitrary types of knowledge that can be used to increase efficiency of solutions to network problems has been proposed in @cite_38 @cite_2 @cite_20 @cite_4 @cite_1 @cite_18 @cite_19 @cite_29 @cite_10 @cite_24 @cite_35 @cite_13 @cite_32 @cite_8 @cite_5 @cite_11 . This approach was referred to as algorithms with advice . The advice is given either to the nodes of the network or to mobile agents performing some task in it. In the first case, instead of advice, the term informative labeling schemes is sometimes used if different nodes can get different information.
@cite_13 @cite_38 @cite_18 @cite_35 @cite_4 @cite_8 @cite_29 @cite_1 @cite_32 @cite_24 @cite_19 @cite_2 @cite_5 @cite_10 @cite_20 @cite_11
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Prior knowledge, in the form of simple advice rules, can greatly speed up convergence in learning algorithms. Online learning methods predict the label of the current point and then receive the correct label (and learn from that information). The goal of this work is to update the hypothesis taking into account not just the label feedback, but also the prior knowledge, in the form of soft polyhedral advice, so as to make increasingly accurate predictions on subsequent examples. Advice helps speed up and bias learning so that generalization can be obtained with less data. Our passive-aggressive approach updates the hypothesis using a hybrid loss that takes into account the margins of both the hypothesis and the advice on the current point. Encouraging computational results and loss bounds are provided. Graphical models (e.g., Bayesian and constraint networks, influence diagrams, and Markov decision processes) have become a central paradigm for knowledge representation and reasoning in both artificial intelligence and computer science in general. These models are used to perform many reasoning tasks, such as scheduling, planning and learning, diagnosis and prediction, design, hardware and software verification, and bioinformatics. These problems can be stated as the formal tasks of constraint satisfaction and satisfiability, combinatorial optimization, and probabilistic inference. It is well known that the tasks are computationally hard, but research during the past three decades has yielded a variety of principles and techniques that significantly advanced the state of the art. In this book we provide comprehensive coverage of the primary exact algorithms for reasoning with such models. The main feature exploited by the algorithms is the model's graph. We present inference-based, message-passing schemes (e.g., variable-elimination) and search-based, conditioning schemes (e.g., cycle-cutset conditioning and AND OR search). Each class possesses distinguished characteristics and in particular has different time vs. space behavior. We emphasize the dependence of both schemes on few graph parameters such as the treewidth, cycle-cutset, and (the pseudo-tree) height. We believe the principles outlined here would serve well in moving forward to approximation and anytime-based schemes. The target audience of this book is researchers and students in the artificial intelligence and machine learning area, and beyond. Table of Contents: Preface Introduction What are Graphical Models Inference: Bucket Elimination for Deterministic Networks Inference: Bucket Elimination for Probabilistic Networks Tree-Clustering Schemes AND OR Search Spaces and Algorithms for Graphical Models Combining Search and Inference: Trading Space for Time Conclusion Bibliography Author's Biography We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice is a function, defined by the online algorithm, of the whole request sequence. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b=0 corresponds to the classical online model, and b=@?log|A|@?, where A is the algorithm's action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones. In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1@?b@?@Q(logn), where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio @W(log(n) b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n) b). For the k-server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k^O^(^1^ ^b^) for any choice of @Q(1)@?b@?logk. We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m,t)-advising scheme for a distributed problem P is a way, for every possible input I of P, to provide an "advice" (i.e., a bit string) about I to each node so that: (1) the maximum size of the advices is at most m bits, and (2) the problem P can be solved distributively in at most t rounds using the advices as inputs. In case of MST, the output returned by each node of a weighted graph G is the edge leading to its parent in some rooted MST T of G. Clearly, there is a trivial (log n,0)-advising scheme for MST (each node is given the local port number of the edge leading to the root of some MST T), and it is known that any (0,t)-advising scheme satisfies t ≥ Ω (√n). Our main result is the construction of an (O(1),O(log n))-advising scheme for MST. That is, by only giving a constant number of bits of advice to each node, one can decrease exponentially the distributed computation time of MST in arbitrary graph, compared to algorithms dealing with the problem in absence of any a priori information. We also consider the average size of the advices. On the one hand, we show that any (m,0)-advising scheme for MST gives advices of average size Ω(log n). On the other hand we design an (m,1)-advising scheme for MST with advices of constant average size, that is one round is enough to decrease the average size of the advices from log(n) to constant.