15448623
Computing Roots of Graphs Is Hard
Abstract The square of an undirected graph G is the graph G2 on the same vertex set such that there is an edge between two vertices in G2 if and only if they are at distance at most 2 in G. The kth power of a graph is defined analogously. It has been conjectured that the problem of computing any square root of a square graph, or even that of deciding whether a graph is a square, is NP-hard. We settle this conjecture in the affirmative.
3122021
A Visibility-Based Pursuit-Evasion Problem
This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simply-connected free spaces, it is shown that the minimum number of pursuers required is Θ(lg n). For multiply-connected free spaces, the bound is pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a finite graph that is constructed on the basis of critical information changes. It has been implemented and computed examples are shown.
15385590
Dynamic itemset counting and implication rules for market basket data
We consider the problem of analyzing market-basket data and present several important contributions. First, we present a new algorithm for finding large itemsets which uses fewer passes over the data than classic algorithms, and yet uses fewer candidate itemsets than methods based on sampling. We investigate the idea of item reordering, which can improve the low-level efficiency of the algorithm. Second, we present a new way of generating “implication rules,” which are normalized based on both the antecedent and the consequent and are truly implications (not simply a measure of co-occurrence), and we show how they produce more intuitive results than other methods. Finally, we show how different characteristics of real data, as opposed by synthetic data, can dramatically affect the performance of the system and the form of the results.
1578969
Similarity Search in High Dimensions via Hashing
The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search/index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the \curse of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and/or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50).
10260140
Approximating Capacitated Routing and Delivery Problems
We provide approximation algorithms for some capacitated vehicle routing and delivery problems. These problems can all be viewed as instances of the following k-delivery TSP: given n source points and n sink points in a metric space, with exactly one item at each source, find a minimum length tour by a vehicle of finite capacity k to pick up and deliver exactly one item to each sink. The only known approximation algorithm for this family of problems is the 2.5-approximation algorithm of Anily and Hassin [ Networks, 22 (1992), pp. 419--433] for the special case k=1. For this case, we use matroid intersection to obtain a 2-approximation algorithm. Based on this algorithm and some additional lower bound arguments, we devise a factor-approximation for k-delivery TSP with arbitrary finite k. We also present a 2-approximation algorithm for the case $k = \infty$. 
We then initiate the study of dynamic variants of k-delivery TSP that model problems in industrial robotics and other applications. Specifically, we consider the situation where a robot arm (with finite or infinite capacity) must collect n point-objects moving in the Euclidean plane, and deliver them to the origin. The point-objects are moving in the plane with known, identical velocities---they might, for instance, be on a moving conveyor belt. We derive several useful structural properties that lead to constant-factor approximations for problems of this type that are relevant to the robotics application. Along the way, we show that maximum latency TSP is implicit in the dynamic problems, and that the natural "farthest neighbor" heuristic produces a good approximation for several notions of latency.
31439485
Sampling from a moving window over streaming data
We introduce the problem of sampling from a moving window of recent items from a data stream and develop two algorithms for this problem. The first algorithm, "chain-sample", extends reservoir sampling to deal with the expiration of data elements from the sample. The expected memory usage of our algorithm is O(k) when maintaining a sample of size k over a window of the n most recent elements from the data stream, and with high probability the algorithm requires no more than O(k log n) memory.When the number of elements in the window is variable, as is the case when the size of the window is defined as a time duration rather than as a fixed number of data elements, the sampling problem becomes harder. Our second algorithm, "priority-sample", works even when the number of elements in the window can vary dynamically over time. With high probability, the "priority-sample" algorithm uses no more than O(k log n) memory.
1647246
A Random Sampling Scheme for Path Planning
Several randomized path planners have been proposed dur ing the last few years. Their attractiveness stems from their applicability to virtually any type of robots, and their empir ically observed success. In this article, we attempt to present a unifying view of these planners and to theoretically explain their success. First, we introduce a general planning scheme that consists of randomly sampling the robot 's configuration space. We then describe two previously developed planners as instances of planners based on this scheme, but applying very different sampling strategies. These planners are probabilis tically complete: if a path exists, they will find one with high probability, if we let them run long enough. Next, for one of the planners, we analyze the relation between the probability of failure and the running time. Under assumptions characteriz ing the "goodness" of the robot's free space, we show that the