5 research outputs found
Distributional convergence for the number of symbol comparisons used by QuickSelect
When the search algorithm QuickSelect compares keys during its execution in
order to find a key of target rank, it must operate on the keys'
representations or internal structures, which were ignored by the previous
studies that quantified the execution cost for the algorithm in terms of the
number of required key comparisons. In this paper, we analyze running costs for
the algorithm that take into account not only the number of key comparisons but
also the cost of each key comparison. We suppose that keys are represented as
sequences of symbols generated by various probabilistic sources and that
QuickSelect operates on individual symbols in order to find the target key. We
identify limiting distributions for the costs and derive integral and series
expressions for the expectations of the limiting distributions. These
expressions are used to recapture previously obtained results on the number of
key comparisons required by the algorithm.Comment: The first paragraph in the proof of Theorem 3.1 has been corrected in
this revision, and references have been update
Sesquickselect: One and a half pivots for cache-efficient selection
Because of unmatched improvements in CPU performance, memory transfers have become a bottleneck of program execution. As discovered in recent years, this also affects sorting in internal memory. Since partitioning around several pivots reduces overall memory transfers, we have seen renewed interest in multiway Quicksort. Here, we analyze in how far multiway partitioning helps in Quickselect. We compute the expected number of comparisons and scanned elements (approximating memory transfers) for a generic class of (non-adaptive) multiway Quickselect and show that three or more pivots are not helpful, but two pivots are. Moreover, we consider "adaptive" variants which choose partitioning and pivot-selection methods in each recursive step from a finite set of alternatives depending on the current (relative) sought rank. We show that "Sesquickselect", a new Quickselect variant that uses either one or two pivots, makes better use of small samples w.r.t. memory transfers than other Quickselect variants
On a functional contraction method
In den letzten zwanzig Jahren hat sich die Kontraktionsmethode als ein wesentlicher Zugang zu Problemen der Konvergenz in Verteilung von Folgen von Zufallsvariablen, die additiven Rekurrenzen genĂŒgen, herausgestellt. Dabei beschrĂ€nkten sich ihre Anwendungen zunĂ€chst auf reellwertige Zufallsvariablen, in den letzten Jahren wurde die Methode allerdings auch fĂŒr komplexere Wertebereiche, wie etwa HilbertrĂ€ume entwickelt. Basierend auf der Klasse der Zolotarev-Metriken, die in den siebziger Jahren eingefĂŒhrt wurden, entwickeln wir die Methode im Rahmen von BanachrĂ€umen und prĂ€zisieren sie in den FĂ€llen von stetigen resp. cadlag Funktionen auf dem Einheitsintervall. Wir formulieren ausreichende Bedingungen an die unter Betrachtung stehende Folge und deren möglichen Grenzwert, welcher eine stochastische Fixpunktgleichung erfĂŒllt, die es erlauben, in Anwendungen funktionale GrenzwertsĂ€tze zu beweisen. Im Weiteren prĂ€sentieren wir als Anwendung zunĂ€chst einen neuen Beweis vom klassischen Invarianzprinzip nach Donsker, der auf additiven Rekursionen beruht. AuĂerdem wenden wir die Methode zur Analyse der KomplexitĂ€t von partiellen Suchproblemen in zweidimensionalen QuadrantenbĂ€umen und 2-d BĂ€umen an. Diese grundlegenden Datenstrukturen werden seit ihrer EinfĂŒhrung in den siebziger Jahren viel studiert. Unsere Ergebnisse liefern Antworten auf Fragen, die seit den Pionierarbeiten von Flajolet et al. in den achtziger und neunziger Jahren auf diesem Gebiet unbeantwortet blieben. Wir erwarten, dass die von uns entwickelte funktionale Kontraktionsmethode in den nĂ€chsten Jahren zur Lösung weiterer Fragen des asymptotischen Verhaltens von ZufallsgröĂen, die additive Rekursionen erfĂŒllen, beitragen wird.Within the last twenty years, the contraction method has turned out to be a fruitful approach to distributional convergence of sequences of random variables which obey additive recurrences. It was mainly invented for applications in the real-valued framework; however, in recent years, more complex state spaces such as Hilbert spaces have been under consideration. Based upon the family of Zolotarev metrics which were introduced in the late seventies, we develop the method in the context of Banach spaces and work it out in detail in the case of continuous resp. cadlag functions on the unit interval. We formulate sufficient conditions for both the sequence under consideration and its possible limit which satisfies a stochastic fixed-point equation, that allow to deduce functional limit theorems in applications. As a first application we present a new and considerably short proof of the classical invariance principle due to Donsker. It is based on a recursive decomposition. Moreover, we apply the method in the analysis of the complexity of partial match queries in two-dimensional search trees such as quadtrees and 2-d trees. These important data structures have been under heavy investigation since their invention in the seventies. Our results give answers to problems that have been left open in the pioneering work of Flajolet et al. in the eighties and nineties. We expect that the functional contraction method will significantly contribute to solutions for similar problems involving additive recursions in the following years
Exponential bounds for the running time of a selection algorithm
Hoareâs selection algorithm for finding the &h-largest element in a set of n elements is shown to use C comparisons where (i) E(P) < A,n â for some constant A,> 0 and all p> 1; (ii) P(C/n) u) < (i)â(â+â(â) â asu-m. Exact values for the âA p â and âo ( 1) â terms are given. 1