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Partial match queries constitute the most basic type of associative queries in multidimensional data structures such as K-d trees or quadtrees. Given a query q = (q0, . . . , qK−1) where s of the coordinates are specified and K − s are left unspecified (qi = * ), a partial match search returns the subset of data points x = (x0, . . . , xK−1) in the data structure that match the given query, that is, the data points such that xi = qi whenever qi = * . There exists a wealth of results about the cost of partial match searches in many different multidimensional data structures, but most of these results deal with random queries. Only recently a few papers have begun to investigate the cost of partial match queries with a fixed query q. This paper represents a new contribution in this direction, giving a detailed asymptotic estimate of the expected cost Pn,q for a given fixed query q. From previous results on the cost of partial matches with a fixed query and the ones presented here, a deeper understanding is emerging, uncovering the following functional shape for Pn,qin many multidimensional data structures, which differ only in the exponent α and the constant ν, both dependent on s and K, and, for some data structures, on the whole pattern of specified and unspecified coordinates in q as well. Although it is tempting to conjecture that this functional shape is "universal", we have shown experimentally that it seems not to be true for a variant of K-d trees called squarish K-d trees.

Abstract. Suppose we have a set of K-dimensional records stored in a general purpose spatial index like a K-d tree. The index eciently supports insertions, ordinary exact searches, orthogonal range searches, nearest neighbor searches, etc. Here we consider whether we can also eciently support search by rank, that is, to locate the i-th smallest element along the j-th coordinate. We answer this question in the armative by developing a simple algorithm with expected cost O(n (1=K) log n), where n is the size of the K-d tree and (1=K) < 1 for any K 2. The only requirement to support the search by rank is that each node in the K-d tree stores the size of the subtree rooted at that node (or some equivalent information). This is not too space demanding. Furthermore, it can be used to randomize the update algorithms to provide guarantees on the expected performance of the various operations on K-d trees. Although selection in multidimensional data can be solved more eciently than with our algorithm, those solutions will rely on ad-hoc data structures or superlinear space. Our solution adds to an existing data structure (Kd trees) the capability of search by rank with very little overhead, and it can be easily adapted to other spatial indexes as well. The simplicity of the algorithm makes it easy to implement, practical and very exible; however, its correctness and eciency are far from self-evident.

Abstract. We introduce a formal model of computation for networks of tiny artifacts, the static synchronous sensor field model (SSSF) which considers that the devices communicate through a fixed communication graph and interact with the environment through input/output data streams. We analyze the performance of SSSFs solving two sensing problems the Average Monitoring and the Alerting problems. For constant memory SSSFs we show that the set of recognized languages is contained in DSPACE(n+m) where n is the number of nodes of the communication graph and m its number of edges. Finally we explore the capabilities of SSSFs having sensing and additional non-sensing constant memory devices.