Partial match queries in relaxed K-dt trees

The study of partial match queries on random hierarchical multidimensional data structures dates back to Ph. Flajolet and C. Puech’s 1986 seminal paper on partial match retrieval. It was not until recently that fixed (as opposed to random) partial match queries were studied for random relaxed K-d trees, random standard K-d trees, and random 2-dimensional quad trees. Based on those results it seemed natural to classify the general form of the cost of fixed partial match queries into two families: that of either random hierarchical structures or perfectly balanced structures, as conjectured by Duch, Lau and Martínez (On the Cost of Fixed Partial Queries in K-d trees Algorithmica, 75(4):684–723, 2016). Here we show that the conjecture just mentioned does not hold by introducing relaxed K-dt trees and providing the average-case analysis for random partial match queries as well as some advances on the average-case analysis for fixed partial match queries on them. In fact this cost –for fixed partial match queries– does not follow the conjectured forms.Peer ReviewedPostprint (author's final draft

Partial match queries in quad-K-d trees

Quad-K-d trees [Bereckzy et al., 2014] are a generalization of several well-known hierarchical Kdimensional data structures. They were introduced to provide a unified framework for the analysis of associative queries and to investigate the trade-offs between the cost of different operations and the memory needs (each node of a quad-K-d tree has arity 2 m for some m, 1 ≤ m ≤ K). Indeed, we consider here partial match – one of the fundamental associative queries – for several families of quad-K-d trees including, among others, relaxed K-d trees and quadtrees. In particular, we prove that the expected cost of a random partial match Pˆn that has s out of K specified coordinates in a random quad-K-d tree of size n is Pˆn ∼ β · n α where α and β are constants given in terms of K and s as well as additional parameters that characterize the specific family of quad-K-d trees under consideration. Additionally, we derive a precise asymptotic estimate for the main order term of Pn,q – the expected cost of a fixed partial match in a random quad-K-d tree of size n. The techniques and procedures used to derive the mentioned costs extend those already successfully applied to derive analogous results in quadtrees and relaxed K-d trees; our results show that the previous results are just particular cases, and states the validity of the conjecture made in [Duch et al., 2016] to a wider variety of multidimensional data structures.This work has been supported by funds from the MOTION Project (Project PID2020-112581GB-C21) of the Spanish Ministery of Science and Innovation MCIN/AEI/10.13039/501100011033.Peer ReviewedPostprint (published version

On the cost of fixed partial match queries in K-d trees

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-015-0097-4Partial 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,q Pn,q=¿·(¿i:qi is specifiedqi(1-qi))a/2·na+l.o.t. (l.o.t. lower order terms, throughout this work) in many multidimensional data structures, which differ only in the exponent a and the constant ¿, 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.Peer ReviewedPostprint (author's final draft

Fixed partial match queries in quadtrees

Several recent papers in the literature have addressed the analysis of the cost P_{n,q} of partial match search for a given fixed query q - that has s out of K specified coordinates - in different multidimensional data structures. Indeed, detailed asymptotic estimates for the main term in the expected cost P_{n,q} = E {P_{n,q}} in standard and relaxed K-d trees are known (for any dimension K and any number s of specified coordinates), as well as stronger distributional results on P_{n,q} for standard 2-d trees and 2-dimensional quadtrees. In this work we derive a precise asymptotic estimate for the main order term of P_{n,q} in quadtrees, for any values of K and s, 0 infty exists, where alpha is the exponent of n in the expected cost of a random partial match query with s specified coordinates in a random K-dimensional quadtree.Peer ReviewedPostprint (published version

Joc d’estructures de dades i algorismes

L'activitat consisteix en la implementació d'un jugador per a un joc d'ordinador. L'objectiu és que els estudiants hi apliquin els algorismes i estructures de dades explicats en el curs. Un joc consisteix en un tauler on es mouen agents controlats pels jugadors. Segons les seves accions, cada jugador rep una puntuació, que en finalitzar la partida en determina la classificació. En el joc no hi ha interacció humana: els programes dels estudiants estan escrits abans de cada partida. Per programar un jugador, els estudiants disposen d'informació completa sobre l'estat del tauler i de tots els jugadors. La documentació del joc explica la interfície que el programa de l'estudiant o estudianta ha d'usar per comunicar-se amb el programa principal. Els estudiants disposen d'un servidor web al qual envien els seus jugadors. A més, se'ls proporciona el codi font del joc per poder-ne desenvolupar localment la implementació. L'activitat consta de dues fases. A la primera, els estudiants han de vèncer un jugador de prova, el "beneit", implementat pel professorat i que segueix una estratègia simple. A la segona fase, els estudiants que han superat la primera participen en una eliminatòria per determinar el millor jugador del quadrimestre.Peer Reviewe

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 player's weights) and there is a critical distance ß as well as a link cost a. 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 ß ) 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 ß>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 ß introduced in Bilò 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=max{1,W/a}; for star celebrity games PoS=1 and PoA=O(min{n/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft

Dynamic pipelining of multidimensional range queries

The problem of evaluating orthogonal range queries efficiently has been studied widely in the data structures community. It has been common wisdom for several years that for queries containing more than 20% of the elements of the dataset a linear scanning of the data was the most efficient solution. In recent experimental works using modern hardware –with main memory and parallelism– the conclusion is that linear scan is preferable for almost every query configuration (even containing a 1% of the data). In this work we propose an alternative approach to evaluate multidimensional range queries based on the dynamic pipeline paradigm –using main memory and concurrency. Our aim is to prove that under this framework, it is possible to beat the performance of linear scanning by the one of hierarchical multidimensional data structures –such as kd trees, quad trees, Rtrees or similar.Peer ReviewedPostprint (published version

On the existence of Nash equilibria in strategic search games

We consider a general multi-agent framework in which a set of n agents are roaming a network where m valuable and sharable goods (resources, services, information ....) are hidden in m different vertices of the network. We analyze several strategic situations that arise in this setting by means of game theory. To do so, we introduce a class of strategic games that we call strategic search games. In those games agents have to select a simple path in the network that starts from a predetermined set of initial vertices. Depending on how the value of the retrieved goods is splitted among the agents, we consider two game types: finders-share in which the agents that find a good split among them the corresponding benefit and firsts-share in which only the agents that first find a good share the corresponding benefit. We show that finders-share games always have pure Nash equilibria (pne ). For obtaining this result, we introduce the notion of Nash-preserving reduction between strategic games. We show that finders-share games are Nash-reducible to single-source network congestion games. This is done through a series of Nash-preserving reductions. For firsts-share games we show the existence of games with and without pne. Furthermore, we identify some graph families in which the firsts-share game has always a pne that is computable in polynomial time.Peer ReviewedPostprint (author’s final draft

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.Preprin

Rank selection in multidimensional data

Suppose we have a set of K-dimensional records stored in a general purpose spatial index like a K-d tree. The index efficiently supports insertions, ordinary exact searches, orthogonal range searches, nearest neighbor searches, etc. Here we consider whether we can also efficiently support search by rank, that is, to locate the i-th smallest element along the j-th coordinate. We answer this question in the affirmative by developing a simple algorithm with expected cost O(na(1/K) log n), where n is the size of the K-d tree and a(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 efficiently than with our algorithm, those solutions will rely on ad-hoc data structures or superlinear space. Our solution adds to an existing data structure (K-d trees) the capability of search by rank with very little overhead. The simplicity of the algorithm makes it easy to implement, practical and very flexible; however, its correctness and efficiency are far from self-evident. Furthermore, it can be easily adapted to other spatial indexes as well.Postprint (published version