Adaptive sampling strategies for quickselects

Martínez, Conrado; Panario, Daniel; Viola, Alfredo

doi:10.1145/1798596.1798606

Cited by 15 publications

(15 citation statements)

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“…Future research may focus on this scenario, trying to identify an optimal choice for the pivots. Related results are known for classic Quickselect [22,24] and Yaroslavskiy's algorithm in Quicksorting [39].…”

Section: Resultsmentioning

confidence: 91%

Analysis of Quickselect Under Yaroslavskiy’s Dual-Pivoting Algorithm

2014

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The algorithm is recursive, and at each level of recursion it uses a partitioning algorithm. Classic implementations of Quicksort use a variety of partitioning methods derived from fundamental versions invented by Hoare [13,14] and refined and popularized by Sedgewick; see for example [35].Very recently, a partitioning algorithm proposed by Yaroslavskiy created some sensation in the algorithms community. The excitement arises from various indications, theoretical and experimental, that Quicksort on average runs faster under Yaroslavskiy's dual pivoting (see [37]). Indeed, after extensive experimentation Oracle adopted Yaroslavskiy's dual-pivot Quicksort as the default sorting method for their Java 7 runtime environment, a software platform used on many computers worldwide. Dual PivotingAlgorithm 1 Dual-pivot Quickselect algorithm for finding the rth order statistic.Quickselect (A, left, right, r) // Assumes left ≤ r ≤ right. // Returns the element that would reside in A[r] after sorting A[left .. right]. 1 if right ≤ left2 return A[left] 3 else 4 (i p , i q ) := PartitionYaroslavskiy(A, left, right) 5 c := sgn(r − i p ) + sgn(r − i q ) // Here sgn denotes the signum function. 6 case distinction on the value of c 7 in case −2 do return Quickselect (A, left , i p − 1, r) 8 in case −1 do return A[i p ] 9 in case 0 do return Quickselect (A, i p + 1, i q − 1, r)

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Section: Resultsmentioning

confidence: 91%

Analysis of Quickselect Under Yaroslavskiy’s Dual-Pivoting Algorithm

2014

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“…larger of two sampled elements depending on whether α ≤ 1/2 or α > 1/2 holds. 5 This adaptive variant was considered in [29]; it is optimal w.r.t. comparisons for sample size 2 and beats YQS w.r.t.…”

Section: Ybb-select With Linear Ranksmentioning

confidence: 99%

“…Second, it is known that choosing the pivot adaptively, i.e., depending on the value of m (mimicking the asymptotically optimal Floyd-Rivest algorithm!) improves the average costs by a significant factor even for small sample sizes [29].…”

Section: Introductionmentioning

confidence: 99%

“…">Preliminaries3 Unfortunately, the grand-average analysis does not extend to adaptive methods like Sesquickselect. For the second part, we hence consider selecting the α-quantile in a large array for a fixed α ∈ (0, 1) (extending techniques from [29]). We give an elementary proof for the correctness of a resulting integral equation for the leading-term coefficient as a function of α (under reasonable assumptions fulfilled for our applications) that appears to be novel.The setting with two parameters makes computations appreciably more challenging.…”

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Sesquickselect: One and a half pivots for cache-efficient selection

Martínez

Nebel

Wild

2019

2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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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. After the Latin prefix sesqui-meaning "one and a half". Preliminaries3 Unfortunately, the grand-average analysis does not extend to adaptive methods like Sesquickselect. For the second part, we hence consider selecting the α-quantile in a large array for a fixed α ∈ (0, 1) (extending techniques from [29]). We give an elementary proof for the correctness of a resulting integral equation for the leading-term coefficient as a function of α (under reasonable assumptions fulfilled for our applications) that appears to be novel.The setting with two parameters makes computations appreciably more challenging. For Quickselect with YBB partitioning (without pivot sampling) and Sesquickselect with k = 2 we solve the integral equations analytically, and we obtain precise numerical solutions for more general cases. From these, we can derive promising candidates of cache-optimal Quickselect variants for all practical sample sizes.Outline. We give an overview of previous work in the remainder of this section. § 2 introduces notation and preliminaries. In § 3, we state a general distributional recurrence of costs. § 4 discusses the analysis for random ranks. We then switch to fixed ranks and derive the integral equation in § 5. We solve it for Quickselect with YBB-partitioning ( § 6) and for the novel "Sesquickselect" algorithm ( § 7). Our paper concludes with a discussion of our findings ( § 8).The appendix contains a comprehensive list of notation, as well as some technical proofs and details of the computations. Previous WorkThe first published analysis of (classic) Quickselect by Knuth served as one of two illustrating examples in an invited address at the IFIP Congress 1971, with the goal to advertise the emerging area of analysis of algorithms [24,25]. The expected number of comparisons in classic Quickselect is E[C n,m ] = 2 (n + 1)i . An asymptotic approximation for selecting the α-quantile, α ∈ (0, 1) fixed, follows with m = αn: E[C n,αn ] = 2(h(α)+1)·n−8 ln n±O(1), n → ∞, where h(x) = −x ln x−(1−x) ln(1−x); (here we set 0 ln 0 ...

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“…Martínez, Panario, and Viola [24] analyze the behavior of QUICKSELECT with small data sets and propose stopping QUICKSELECT's recursion early and using sorting as an alternative policy below a cutoff, essentially a simple multi-strategy QUICKSELECT. Same authors [25] propose several adaptive sampling strategies for QUICKSELECT that take into account the index searched.…”

Section: Related Workmentioning

confidence: 99%

Fast Deterministic Selection

Alexandrescu¹

2016

Preprint

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The Median of Medians (also known as BFPRT) algorithm, although a landmark theoretical achievement, is seldom used in practice because it and its variants are slower than simple approaches based on sampling. The main contribution of this paper is a fast linear-time deterministic selection algorithm QUICKSELECTADAPTIVE based on a refined definition of MEDIANOFMEDIANS. The algorithm's performance brings deterministic selection-along with its desirable properties of reproducible runs, predictable run times, and immunity to pathological inputs-in the range of practicality. We demonstrate results on independent and identically distributed random inputs and on normally-distributed inputs. Measurements show that QUICKSELECTADAPTIVE is faster than state-of-the-art baselines.

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Adaptive sampling strategies for quickselects

Cited by 15 publications

References 15 publications

Analysis of Quickselect Under Yaroslavskiy’s Dual-Pivoting Algorithm

Analysis of Quickselect Under Yaroslavskiy’s Dual-Pivoting Algorithm

Sesquickselect: One and a half pivots for cache-efficient selection

Fast Deterministic Selection

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