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VoS: Variate Ordering Strategies for Skyline Query Optimization

Abhinav Gorantla, Pratanu Mandal, K. Selçuk Candan, Maria Luisa Sapino

arXiv:2608.26464v1cs.DB

TL;DR

Skyline optimization has focused on tuple-pair dominance checks even though each check may stop before comparing every preference attribute. This paper develops correlation-informed variate-ordering strategies and finds consistent time gains for min-correlation-first and incremental maximum correlation last, with especially significant gains for SaLSa and D&C.

  • Problem

    The paper addresses the mismatch between tuple-pair dominance-check counts and execution cost because dominance checks may evaluate different numbers of preference attributes.

  • Method

    It represents ordering as tuple order plus variate order and uses pairwise attribute correlations to construct several strategies for reducing per-attribute checks.

  • Results

    Min-correlation-first and incremental maximum correlation last provide the most consistent time gains, often close to the best gains possible.

  • Takeaways & Limitations

    Min-correlation-first provides significant execution-time gains, especially for SaLSa and D&C algorithms and larger numbers of variates.

Abstract

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Efficiency of skyline algorithms is highly influenced by the underlying data characteristics. Traditionally, optimization efforts have focused on minimizing the total number of tuple-pair dominance checks to improve query performance. However, in practice, a dominance check between two tuples does not necessarily require evaluating dominance relationships for each and every preference attribute of the data and this creates a disconnect between dominance checks optimization and query execution performance. In this paper, we argue that skyline algorithms need to optimize total per-attribute dominance checks, along with per-tuple dominance checks and that, for both of these goals, the ordering of the attributes (or variates) can have a substantial impact on the efficiency of skyline computation. Based on this premise, we present several strategies for identifying an effective variate order to minimize redundant attribute comparisons. Extensive experiments on both synthetic and real-world datasets, and on both scalar and SIMD architectures, confirm the effectiveness of the proposed approach in reducing computational overhead and improving skyline query performance.

I. INTRODUCTION

Skyline queries return Pareto-optimal tuples for multi-criteria decisions, but traditional tuple-pair-check optimization can miss execution costs. The paper argues that per-attribute checks and variate ordering better expose and reduce this cost.

  • I. INTRODUCTION: Skyline queries extract tuples that are not dominated by any other tuple under per-attribute preference criteria.A tuple must be at least as good on every preference attribute and strictly better on at least one.
  • I. INTRODUCTION: Hyperparameter selection can use skylines to present alternative configurations minimizing MSE_Loss, Huber_Loss, and Model_Size.The running example treats the skyline as a set of configurations for the researcher to consider.
  • I. INTRODUCTION: Tuple-pair dominance checks do not necessarily correlate with execution time because checks may stop before evaluating every preference attribute.Figure 1 contrasts tuple-pair checks with per-attribute checks as execution-time proxies.
  • I. INTRODUCTION: Per-attribute dominance checks correlate with skyline query performance more closely than the total number of tuple-pair checks.The paper identifies this mismatch as a disconnect between conventional dominance-check optimization and query execution performance.
  • I. INTRODUCTION: Changing the variate order can shorten a dominance check, as the example completes after two attributes instead of three.The default order compares MSE_Loss, Huber_Loss, and Model_Size, whereas Model_Size first enables earlier completion in the example.
  • I. INTRODUCTION: The paper proposes using attribute correlations to identify variate orders that reduce redundant per-attribute comparisons.Its premise is that the data distribution, represented by the attribute correlation matrix, supplies signals for selecting an effective order.

C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks

The paper connects variate ordering to both per-attribute and tuple-pair comparison costs. It argues that correlation-informed orders can improve pruning and filtering, while effects depend on the skyline algorithm and ordering-sensitive preprocessing.

  • C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks: Variate ordering does not directly change tuple-pair comparison counts unless the order is incorporated into tuple enumeration.The paper highlights this distinction when comparing variate ordering with tuple-ordering strategies.
  • C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks: Ordering-sensitive preprocessing can improve filtering efficiency and outperform lexicographic ordering or conventional pre-sorting methods.The paper specifically identifies lexicographic sort, sort-first skylines, SFS, and SaLSa as relevant comparisons.
  • C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks: Appropriate variate ordering can improve skyline efficiency by prioritizing conflicting preference attributes.The paper states that this can allow dominance decisions without comparing all variates.
  • C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks: Correlation-based strategies can approximate effective variate orders while reducing tuple-pair dominance checks and per-attribute comparisons.The proposed strategies use correlations present in the data as signals for ordering preference attributes.
  • C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks: The evaluation covers synthetic and real-world datasets across varying sizes and distributions on scalar and SIMD architectures.The experiments analyze the impact of variate ordering across skyline algorithms and hardware settings.
  • C. Argument #3: Variate Ordering and the Number of Tuple-Pair Dominance Checks: Skyline computation uses preference criteria that specify minimization or maximization for each preference attribute.The paper defines tuple dominance using weak dominance on every preference attribute and strict dominance on at least one.

B. Problem Formulation

The problem formulation represents an ordering strategy by tuple order and variate order, then seeks the strategy minimizing total per-variate dominance checks. The analysis explains how early stopping makes variate order consequential.

  • B. Problem Formulation: An ordering strategy O = ⟨Torder,Vorder⟩ specifies tuple-read order and within-tuple variate-evaluation order.Torder controls how tuples are read, while Vorder controls the order of variate comparisons.
  • B. Problem Formulation: For strategy O and algorithm A, CO,A(D,Θ) is the sequence of tuple-pair dominance checks used to compute the skyline.The sequence is defined for dataset D under preference criteria Θ.
  • B. Problem Formulation: γ(O,A,D,Θ) denotes the total number of per-variate dominance checks required by algorithm A under ordering strategy O.Each tuple-pair check contributes the number of per-variate comparisons needed to evaluate it.
  • B. Problem Formulation: The optimization goal is to identify O that minimizes γ(O,A,D,Θ) for a given dataset and skyline algorithm.The target is the number of per-variate checks during skyline evaluation.
  • 1) Skyline Algorithms that Do Not Modify the Data:: For algorithms such as BNL that preserve tuple order, variate order can still affect v-checks through early stopping.Tuple processing follows dataset storage order, but the order of preference attributes remains influential.
  • 1) Skyline Algorithms that Do Not Modify the Data:: Confirming dominance requires all preference attributes to satisfy weak dominance and at least one to satisfy strict dominance.These are the two conditions used in the tuple-dominance definition.
  • 1) Skyline Algorithms that Do Not Modify the Data:: Disproving weak dominance can stop at the first preference attribute where the candidate fails.The strict-dominance condition cannot be disproven without comparing all preference attributes, so the check terminates when weak dominance fails.

2) Pre-Sorting based Skyline Algorithms:

Pre-sorting skyline algorithms order tuples using a monotonic function, and variate ordering can influence both tuple order and the number of dominance checks. The paper frames the goal as approximating the variate permutation that minimizes required v-checks.

  • Pre-sorting algorithms such as SFS and SaLSa sort data using a monotonic function before skyline computation.
  • Lexicographic sorting depends on the variate order Vorder, which determines the resulting tuple order Torder.
  • For pre-sorting algorithms, the total number of v-checks is influenced by both tuple order and variate order.
  • Partitioning-based algorithms initially read tuples in dataset order, but partitioning depends on variate order and ultimately affects tuple order and v-checks.
  • The paper defines an optimal permutation π∗ and seeks an approximate ordering that minimizes the v-checks required to compute a skyline.

C. Number of Per-Attribute Comparisons

Attribute correlations affect how many per-variate comparisons are redundant during tuple-dominance checks. The examples motivate ordering weakly correlated or negatively correlated variates earlier to reduce unnecessary evaluations.

  • With perfectly correlated X and Y and anti-correlated Z, order ⟨X,Y,Z⟩ makes Y redundant but still requires checking Z to disprove dominance.
  • Order ⟨Z,X,Y⟩ makes the final Y comparison unnecessary because Z and X determine the outcome while X and Y are perfectly correlated.
  • Pairwise attribute correlations play a vital role in optimizing per-variate dominance checks.
  • Minimum absolute correlation first places variates with low absolute correlation to the others early in the order.
  • The rationale is that early comparisons on statistically independent variates are less likely to be redundant when early stopping occurs.
  • Max-correlation-first increases redundant work significantly and performs close to the worst possible variate order in the reported experiments.
  • Min-correlation-first instead prioritizes variates most anti-correlated with the remaining preference variates.

2) Minimum Correlation First:

The paper develops correlation-based variate-ordering strategies that prioritize conflicting attributes globally, incrementally, or through pairwise sequence scores. These strategies aim to reduce redundant comparisons and improve tuple processing.

  • Correlation-based ordering assumes the optimal variate order can be approximated from data correlations, but two-attribute cases are outside the paper’s scope.
  • Min-correlation-first prioritizes preference variates that are most anti-correlated with the other preference variates.
  • Negative correlations can make early per-variate evaluations less redundant when dominance checks stop early.
  • For pre-sorting algorithms, placing negatively correlated variates first can expose Pareto trade-offs early and cause likely dominators to be visited earlier.
  • The incremental strategy selects at each step the variate with minimum correlation to the variates that remain unconsidered.
  • Unlike global strategies, the incremental approach ignores already considered variates when choosing the next one.
  • The pairwise strategy seeks a sequence whose early variates have lower pairwise correlations, so conflicting pairs may render later comparisons unnecessary.

D. Number of Tuple-Pair Comparisons

Variate ordering also affects tuple-pair comparisons when it determines tuple enumeration, especially for lexicographic sorting. The proposed L-VOS combines min-correlation-first ordering with lexicographic sorting and LIFO skyline scanning to improve filtering efficiency.

  • The paper examines whether variate ordering can affect tuple-pair dominance checks in addition to per-attribute checks.
  • Filtering power is the number of other tuples dominated by a tuple, and pre-sorting algorithms use it to prioritize tuples with stronger pruning potential.
  • Monotone tuple orders place tuples dominated by a given tuple after it, but pruning efficiency still varies with the number of candidate comparisons.
  • For a monotone order, scanning the current skyline list can stop when one skyline tuple dominating the current tuple is found.
  • The proposed combination uses lexicographic enumeration under min-correlation-first ordering and LIFO scanning of the current skyline list.
  • Reverse scanning often finds dominators sooner because recently inserted skyline tuples are closer in lexicographic order and more comparable.
  • L-VOS lexicographically sorts tuples according to min-correlation-first variate order and performs better than order-agnostic lexicographic sorting, SFS, and SaLSa in the reported experiments.

2) Variate Orders and D&C/BBS:

For D&C and BBS, variate order affects recursive partitioning, pruning, and SIMD early stopping. Min-correlation-first reduces tuple-pair checks for both algorithms, with larger execution-time gains for D&C.

  • D&C/BBS: D&C and BBS do not pre-sort tuples, but tuple consideration and pruning indirectly depend on variate order through data partitioning.
  • D&C: D&C recursively partitions by attributes, computes local skylines, and merges them; variate order determines split order and recursive-subproblem efficiency.The order also affects which local skyline points can be removed during merging.
  • Results: Min-correlation-first reduces tuple-pair dominance checks for both D&C and BBS, while D&C achieves higher execution-time gains.The result is reported for experiments in Section VII-B.
  • SIMD: Variate ordering can reduce average per-variate comparisons per dominance check, especially on scalar architectures that process one data element per operation.SIMD architectures process one operation across multiple data elements and therefore require separate consideration.
  • SIMD: When preference attributes exceed one SIMD vector, chunked processing makes early stopping through variate ordering useful for reducing per-chunk dominance checks.For 5 attributes on AVX-256, early stopping can reduce skyline computation costs by ∼50% by limiting the check to one SIMD vector comparison.

VI. COMPLEXITY ANALYSIS

VOS computes attribute correlations, derives a variate order, and applies skyline computation, with strategy-dependent ordering costs. Correlation computation dominates the stated preprocessing scale when m ≪ N, while lexicographic sorting adds O(NlogN) for L-VOS.

  • VOS pipeline: VOS computes correlations, identifies a variate order, rearranges variates, optionally lexicographically sorts data, and computes the skyline.The rearrangement is implicit during sorting and skyline execution rather than implemented as a separate step.
  • Complexity: Pairwise correlation computation costs O(Nm2), where N is dataset size and m is the number of preference attributes or SIMD chunks.
  • Complexity: Min-abs-correlation-first and min-correlation-first find orders in O(m), whereas inc-min-abs-correlation-first and incmax-correlation-last require O(m2).
  • Complexity: Min-pairwise-correlation has O(m!) ordering complexity, but the paper notes that m ≪ N and m! ≪ Nm2 in most cases.
  • Complexity: L-VOS adds lexicographic pre-sorting with O(NlogN) complexity before skyline computation.The final skyline step’s complexity depends on the selected skyline algorithm.
  • Experimental setup: Experiments vary dataset size, preference dimensionality, correlation structure, and real-world versus synthetic data, using repeated runs and reported averages.Synthetic data span correlation matrices with different ranks; real-world preference sets include three, five, and seven variates.

2) Skyline Algorithms:

Baseline skyline performance varies across algorithms and data conditions, while the experiments compare alternative variate orders against vanilla methods on synthetic and real data. BBS can be slow when outliers place the entire dataset in the skyline, and ordering computation is much cheaper than skyline execution.

  • Baseline performance: On the 100K-tuple, five-attribute baseline, BBS has the highest average execution time because outlier cases can place the entire dataset in the skyline.In those cases, BBS cannot exploit MBR-based pruning; its reported time excludes R-Tree construction.
  • Variants: The experiments compare proposed variate orders with empirically best, empirically worst, maximum-correlation-first, and vanilla skyline baselines.Vanilla algorithms are evaluated across all preference-variate permutations and reported with average performance.
  • Evaluation: Synthetic experiments report execution-time gains across alternative variate-ordering strategies, while real-data experiments provide the corresponding comparison for five attributes.The table captions identify the synthetic and real-world evaluation scopes.
  • Variants: Optimal variate-order computation is significantly cheaper than skyline query execution time in the reported comparison.
  • Evaluation: Evaluation measures tuple-pair dominance checks, per-variate dominance checks, and skyline computation time, reporting average percentage cost reductions against vanilla algorithms.Each experiment configuration is run 100 times with repeated data generation or augmentation.

B. Experiment Results

Experiments show that variate ordering can reduce per-attribute dominance checks and improve skyline execution time, with gains depending on the strategy, algorithm, and architecture. Minimum-correlation-first strategies are generally strongest, while NACF and XCF can hurt some algorithms.

  • Minimum-correlation-first (NCF) and incremental maximum correlation last (IXCL) provide the most consistent time gains, often close to the best possible gains.These results are reported for synthetic data relative to vanilla skyline algorithms and include comparisons with best- and worst-performing orders.
  • NACF produces negative gains for BNL, D&C, and BBS, whereas SFS and SaLSa benefit from its NCF-based lexicographical preprocessing.The effect therefore depends partly on whether the skyline algorithm uses correlation-sensitive presorting.
  • XCF performs poorly among the evaluated strategies and, except for SFS and SaLSa, yields negative gains close to the worst-case scenario.The experiments associate early placement of highly correlated variates with substantially more redundant work.
  • Performance gains remain consistent across data sizes, while variate ordering does not reduce t-checks for BNL and BBS.SFS, SaLSa, and D&C do see significant t-check reductions because their presorting or partitioning depends on variate order.
  • All algorithms except BNL show significant reductions in v-checks, and these reductions produce execution-time gains across algorithms.For BNL and D&C, time gains closely follow v-check reductions; other algorithms experience additional overheads.

2) Impact of the Number of Preference Attributes:

The benefits of variate ordering generally grow as the preference-attribute set expands, but they depend on correlation structure, algorithm, sorting choices, and hardware execution model. NCF-based ordering performs especially well in lexicographical sorting and remains effective under SIMD execution.

  • 2) Impact of the Number of Preference Attributes:: Time gains generally increase as the number of preference attributes grows, indicating more significant variate-pruning opportunities.The experiments vary the number of preference variates while keeping other default parameters constant.
  • Correlation-matrix rank affects gains: SFS and SaLSa perform best at low ranks, whereas D&C performs best at mid-range ranks.These results indicate sensitivity to the data’s correlation structure and to the skyline algorithm.
  • NCF variate ordering combined with LIFO skyline validation provides the best overall execution time among the evaluated lexicographical sorting variants.This comparison uses synthetic data with 100K tuples and five preference attributes.
  • Under SIMD execution, VOS supports chunk pruning rather than variate pruning and yields comparable pruning and execution-time gains for SFS, SaLSa, and D&C.The comparison covers scalar and SIMD hardware configurations.
  • The paper concludes that pairwise attribute correlations provide a useful signal for ordering preference attributes, with NCF especially beneficial for SaLSa, D&C, and larger variate sets.The conclusion frames per-attribute checks as a better proxy for skyline execution time than tuple-pair checks alone.
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