Amalia Duch

Randomized K-Dimensional Binary Search Trees

We introduce randomized K-dimensional binary search trees (randomized Kd-trees), a variant of K-dimensional binary search trees that allows the efficient maintenance of multidimensional records for any sequence of insertions and deletions; and thus, is fully dynamic. We show that several types of associative queries are efficiently supported by randomized Kd-trees. In particular, a randomized Kd-tree with n records answers exact match queries in expected O(log n) time. Partial match queries are answered in expected O(n1-f(s/K)) time, when s out of K attributes are specified (with 0 < f(s/K) < 1 a real valued function of s/K). Nearest neighbor queries are answered on-line in expected O(log n) time. Our randomized algorithms guarantee that their expected time bounds hold irrespective of the order and number of insertions and deletions.

On the average performance of orthogonal range search in multidimensional data structures

In this work we present the average-case analysis of orthogonal range search for several multidimensional data structures. We first consider random relaxed K-d trees as a prototypical example. Later we extend these results to many different multidimensional data structures. We show that the performance of range searches is related to the performance of a variant of partial matches using a mixture of geometric and combinatorial arguments. This reduction simplifies the analysis and allows us to give exact upper and lower bounds for the performance of range searches (Theorems 3 and 4) and a useful characterization of the cost of range search as a sum of the costs of partial match-like operations (Theorem 5). Using these results, we can get very precise asymptotic estimates for the expected cost of range searches (Theorem 6).

Survey: Computational models for networks of tiny artifacts: A survey

We survey here some recent computational models for networks of tiny artifacts. In particular, we focus on networks consisting of artifacts with sensing capabilities. We first imagine the artifacts moving passively, that is, being mobile but unable to control their own movement. This leads us to the population protocol model of Angluin et al. (2004) [16]. We survey this model and some of its recent enhancements. In particular, we also present the mediated population protocol model in which the interaction links are capable of storing states and the passively mobile machines model in which the finite state nature of the agents is relaxed and the agents become multitape Turing machines that use a restricted space. We next survey the sensor field model, a general model capturing some identifying characteristics of many sensor networks settings. A sensor field is composed of kinds of devices that can communicate one to the other and also to the environment through input/output data streams. We, finally, present simulation results between sensor fields and population protocols and analyze the capability of their variants to decide graph properties.

Sensor Field: A Computational Model

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.

Quad-kd trees

We introduce the quad-kd tree: a general purpose and hierarchical data structure for the storage of multidimensional points. Quad-kd trees include point quad trees and kd trees as particular cases and therefore they could constitute a general framework for the study of fundamental properties of trees similar to them. Besides, quad-kd trees can be tuned by means of insertion heuristics and bucketing techniques to obtain trade-offs between their costs in time and space. We propose three such heuristics and we show analytically and experimentally their competitive performance. Our analytical results back the experimental outcomes and suggest that the quad-kd tree is a flexible data structure that can be tailored to the resource requirements of a given application.

Selection by rank in K-dimensional binary search trees

In this work we show how to augment general purpose multidimensional data structures, such as K-d trees, to efficiently support search by rank that is, to locate the i-th smallest element along the j-th coordinate, for given i and j and to find the rank of a given item along a given coordinate. To do so, we introduce two simple, practical and very flexible algorithms - Select-by-Rank and Find-Rank - with very little overhead. Both algorithms can be easily implemented and adapted to several spatial indexes, although their analysis is far from trivial. We are able to show that for random K-d trees of size n the expected number of nodes visited by Find-Rank is Pn,i=i¾?n1-1/K for i=on or i=n-on, and Pn,i=fKi/ni¾?nα+onα for i=xn+on with 0<x<1, where 1-1/Ki¾?α<1 depends on the dimension K and the variant of K-d tree under consideration. We also show that Select-by-Rank visits gKi/ni¾?nαlnn+Onα nodes on average, where i=xn+on is the given rank and the exponent α is as above. We give the explicit form of the functions fKx and gKx, both are bounded in [0, 1] and they depend on K, on the variant of K-d tree under consideration, and, eventually, on the specific coordinate j for which we execute our algorithms. As a byproduct of the analysis of our algorithms, but no less important, we give the average-case analysis of a partial match search in random K-d trees when the query is not random.

Celebrity games

We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the players weights) and there is a critical distance β as well as a link cost α. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than β) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if β 1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance β introduced in Bilo et al. 6. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA = PoS = m a x { 1 , W / α } ; for star celebrity games PoS = 1 and PoA = O ( m i n { n / β , W α } ) but if the Nash Equilibrium is a tree then the PoA is O ( 1 ) ; finally, when β = 1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for β = 2 .

Updating relaxed K -d trees

In this work we present an in-depth study of randomized relaxed <i>K</i>-d trees. It covers two fundamental aspects: the randomized algorithms that allow to preserve the random properties of relaxed <i>K</i>-d trees and the mathematical analysis of the expected performance of these algorithms. In particular, we describe randomized update algorithms for <i>K</i>-d trees based on the split and join algorithms of Duch et al. [1998]. We carry out an analysis of the expected cost of all these algorithms, using analytic combinatorics techniques. We show that the average cost of split and join is of the form ζ(<i>K</i>) ṡ <i>n</i>^φ(<i>K</i>) + <i>o</i>(<i>n</i>^φ(<i>K</i>)), with 1 ≤ φ(<i>K</i>) < 1.561552813, and we give explicit formulæ for both ζ(<i>K</i>) and φ(<i>K</i>). These results on the average performance of split and join imply that the expected cost of an insertion or a deletion is Θ(<i>n</i>^{φ(<i>K</i>)−1}) when <i>K</i> > 2 and Θ(log <i>n</i>) for <i>K</i> = 2.

Improving the performance of multidimensional search using fingers

We propose two variants of K-d trees where fingers are used to improve the performance of orthogonal range search and nearest neighbor queries when they exhibit locality of reference. The experiments show that the second alternative yields significant savings. Although it yields more modest improvements, the first variant does it with much less memory requirements and great simplicity, which makes it more attractive on practical grounds.

Fingered Multidimensional Search Trees

In this work, we propose two variants of K-d trees where fingers are used to improve the performance of orthogonal range search and nearest neighbor queries when they exhibit locality of reference. The experiments show that the second alternative yields significant savings. Although it yields more modest improvements, the first variant does it with much less memory requirements and great simplicity, which makes it more attractive on practical grounds.