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TraHist: Fast and Accurate Cardinality Estimation for Trajectory Data (Extended Version)

Yujeong Jo, Incheol Baek, Changbeom Shim, Yon Dohn Chung

arXiv:2610.22310v1cs.DB

TL;DR

Trajectory cardinality estimation is difficult because records from the same moving object are dependent, while conventional histograms partition records as independent points. TraHist approximates density-aligned partitioning with adaptive rectilinear buckets. The paper reports higher accuracy than existing methods while maintaining fast query performance and competitive construction cost.

  • Problem

    Point-centric histogram partitioning does not account for dependencies among samples from the same trajectory, leading to fragmentation and less representative buckets.

    Source: §1 · pp. 1–2

  • Method

    TraHist approximates density-aligned partitioning by constructing density-adaptive rectilinear buckets for trajectory data.

    Source: §1 · p. 2

  • Results

    Experiments on large-scale real-world trajectory datasets show significantly higher estimation accuracy than existing methods, with fast query performance and competitive construction cost.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Trajectory-aware, density-adaptive partitions provide a histogram-based approach that addresses fragmentation while retaining efficient estimation.

    Source: §1 · p. 2

    Directly optimizing the density-aligned partitioning objective is computationally impractical because it entails non-convex combinatorial optimization over continuous boundaries.

    Source: §4.1 · p. 6

Abstract

from arXiv · show

Cardinality estimation is fundamental for query optimization in database systems. Histogram-based estimators are widely used for their simplicity and efficiency but rely on point-centric partitioning that assumes record independence. This assumption breaks in trajectory data, where records from the same moving object show strong spatiotemporal dependencies. Conventional axis-aligned histograms therefore fragment trajectories across partition boundaries, leading to overestimation through duplicate counting. This error reflects a structural limitation of partition boundaries that are misaligned with trajectory-induced density patterns. To this end, we propose TraHist, a histogram-based estimator approximating density-aligned partitioning via efficient grid-based propagation. TraHist builds density-adaptive rectilinear buckets to preserve trajectory continuity and local density coherence while keeping histogram efficiency. Experiments on large-scale real-world datasets show that TraHist reduces estimation error by over 20x relative to the strongest competing estimator in query regimes where trajectory fragmentation is a dominant source of estimation error, while maintaining sub-millisecond query latency and competitive construction cost.

1 Introduction

Trajectory data has strong dependencies among records from the same moving object, which point-centric histograms do not capture. TraHist instead adapts histogram partitions to trajectory-induced density patterns while retaining efficient histogram-based estimation.

1 Introduction

  • Trajectory records from the same object are spatiotemporally dependent, and axis-aligned buckets can fragment them, causing duplicate counting even when bucket summaries are accurate.The resulting overestimation reflects partition boundaries that break object-level continuity, not simply coarse partitions or inaccurate bucket summaries.Source: §1 · pp. 1–2
  • Prior estimators refine distributions, model within-bucket data, or predict cardinality directly, but do not capture inter-record dependencies from trajectory continuity.Model-based learned estimators typically require training and inference, unlike lightweight, training-free histograms.Source: §1 · p. 2
  • TraHist approximates density-aligned partitioning with density-adaptive rectilinear buckets that adapt to trajectory traversal patterns and preserve local density coherence.Its design aims to reduce trajectory fragmentation while retaining efficient histogram-based estimation.Source: §1 · p. 2
  • Experiments on large-scale real-world trajectory datasets show higher estimation accuracy than existing methods, with fast query performance and competitive construction cost.The contribution statement reports these outcomes without specifying numerical values here.Source: §1 · p. 3

2 Background

The background defines trajectory range queries over trajectory polylines and targets the number of distinct trajectories they return. It frames histogram estimation as summarizing data in buckets under a bucket-count budget.

2.1 Trajectory Data and Query

  • A trajectory is an ordered sequence of spatial samples from one object, represented as a polyline connecting consecutive samples.A trajectory range query returns trajectories whose polylines intersect the spatial range; its cardinality counts distinct returned trajectories.Source: §2.1 · p. 3
  • The target is trajectory-level cardinality, but point-based summaries can miss intersections between sparse samples and can count one trajectory multiple times across buckets.The paper focuses on spatial range predicates, with timestamps determining sample order rather than appearing in the predicate.Source: §2.1 · p. 3

2.2 Cardinality Estimation Problem

  • Histograms partition a domain into disjoint buckets, store representative statistics, and seek to minimize estimation error under a bucket-count constraint.The bucket-count constraint corresponds to a memory budget; estimation compares estimated and true query cardinalities using an error metric.Source: §2.2 · pp. 3–4
  • Conventional axis-aligned histograms can fragment trajectories for duplicate counting and combine regions with substantially different densities, weakening bucket-level summaries.Figure 2 illustrates both limitations, including density differences within a Porto bucket.Source: Figure 2 · p. 4

3 Motivations

Trajectory histograms face two linked challenges: partition boundaries fragment object-level continuity and can mix regions with different densities. Addressing both matters because fragmented trajectories can be counted repeatedly, while heterogeneous buckets weaken local summaries.

3 Motivations

  • Trajectory workloads require histograms to account jointly for object-level continuity and density heterogeneity, which conventional point-centric partitions cannot address simultaneously.These are distinct structural challenges: trajectory fragmentation causes duplicate counting, while biased, spatially correlated density patterns undermine locally coherent partitions.Source: §3 · p. 4

3.1 Fragmented Trajectories

  • Axis-aligned boundaries can fragment a trajectory polyline, with expected grid crossings proportional to d/h(|cosθ| + |sinθ|), minimized along an axis and maximized diagonally.The point-centric abstraction assigns dependent samples from one object to different buckets when its path crosses their boundaries.Source: §3.1 · pp. 4–5
  • A trajectory overlapping k query buckets may contribute up to k counts instead of one, even when each bucket accurately summarizes its local point statistics.The overcount arises from fragmented object-level aggregation rather than estimation noise within buckets.Source: §3.1 · p. 5
  • Grouping regions traversed by the same trajectory into one bucket can combine spatially heterogeneous areas and degrade bucket-level summaries.Continuity-preserving partitioning alone therefore does not resolve fragmentation without regard to local density coherence.Source: §3.1 · p. 5

3.2 Loss of Density Coherence in Buckets

  • Trajectory points form anisotropic density patterns concentrated along elongated paths, so density coherence is often stronger along movement directions than across them.The intra-bucket variance Var(B) quantifies density heterogeneity; keeping it low supports accurate bucket-level representations.Source: §3.2 · p. 5
  • Axis-aligned rectangles can mix regions with substantially different densities, inflating intra-bucket variance and degrading estimation accuracy.Refining bucket granularity or using more expressive bucket-level models helps only to a limited extent while boundaries remain misaligned with density structure.Source: §3.2 · pp. 5–6

4 Proposed Method: TraHist

TraHist approximates density-aligned partitioning with grid-based propagation, forming rectilinear buckets that reflect trajectory-induced density. It uses these buckets and compact statistics to estimate trajectory cardinality.

4.1 Density-aligned Partitioning

  • Density-aligned partitioning places boundaries along directions of minimal density change, but directly optimizing this ideal criterion is computationally impractical.The criterion aligns boundaries with local density geometry rather than fixed coordinate axes.Source: §4.1 · p. 6

4.2 Density-adaptive Approximation

  • TraHist approximates the ideal partitioning with a two-stage process: discretize density in a CountGrid, then aggregate grid cells into buckets.The output is a set of rectilinear buckets built through density-adaptive aggregation.Source: §4 · p. 6, §4.2 · p. 7
  • The CountGrid aggregates points by cell and applies log(1+C(x)) to stabilize local density variation.The transformed values serve as a discrete surrogate for the continuous density field.Source: §4.2 · p. 7
  • Propagation orders cells by cumulative density contrast, favoring paths of least aggregate contrast and boundaries at persistent density transitions.Neighboring cells are merged by default; a significant transition triggers a new bucket, with the adaptive threshold governed by β.Source: §4.2 · pp. 8–9

4.3 Cardinality Estimation

  • For query estimation, TraHist samples grid cells, aggregates bucket statistics, and applies a ratio-based correction for object-level dependencies.Grid-aligned sampling avoids segment–cell intersections and geometric overlap computation; query time is linear in selected cells.Source: §4.3 · pp. 9–10
  • TraHist also examines synopsis updates under trajectory insertions and deletions with a fixed partition, and its behavior under data shift.Source: §4.3 · p. 10

5 Analysis

TraHist’s construction and query costs scale with grid resolution and the number of query-covered cells, respectively. Its propagation has a local optimality property, and its contrast objective is related to a statistical surrogate for estimation error.

5.1 Computational Complexity

  • Construction takes O(N_pts) time for CountGrid creation and O(N log N) time for bucket formation, using O(N) space.Here N is the number of grid cells and N_pts the number of trajectory points.Source: §5.1 · p. 10
  • Query estimation takes O(N_q) time for a query covering N_q grid cells, independent of bucket shape complexity.Source: §5.1 · p. 10
  • After construction, storage is O(N) for the LabelMap and bucket statistics, higher than traditional histograms but smaller than learned-estimator model footprints.Source: §5.1 · p. 10

5.2 Theoretical Properties

  • Propagation is locally optimal for its accumulated-contrast objective: reassigning a cell to a neighboring region cannot reduce E(Π).Each cell is assigned to the region that reaches it with minimum cumulative cost.Source: §5.2 · p. 11
  • The practical contrast cost serves as a tractable surrogate for the intra-bucket squared-error objective, with discrepancy controlled by local smoothness and bucket homogeneity.For 0 < α ≤ 1, Appendix D provides a formal bound on bucket deviation in terms of cumulative propagation cost.Source: §5.2 · p. 11

6 Experimental Study

The evaluation compares TraHist with histogram and spatial-object estimators on real-world trajectory range queries, measuring accuracy, trajectory continuity, and computational efficiency. TraHist’s strongest accuracy gains occur in low- and medium-selectivity regimes, while query latency remains sub-millisecond and construction costs are competitive.

6.1 Experimental Setup

  • The study evaluates 1,000 trajectory range queries per selectivity group using data-driven query centers, comparing estimators under matched memory and fixed bucket budgets.The evaluation covers Porto, T-Drive, and GeoLife; GeoLife results are presented separately in Appendix E.Source: §6.1 · pp. 12–13

6.2.1 TraHist achieves the highest accuracy for low and medium selectivity queries (RQ1).

  • On Porto, TraHist reduces median Q-error versus DEH by 3.5–5.4× at low and 1.1–3.7× at medium selectivity; on T-Drive, it improves over the strongest baseline by 4.7–11.0× and 10.7–16.2×, respectively.These results are under matched memory budgets.Source: Figure 7 · p. 14, §6.2.1 · p. 15, Table 2 · p. 15

6.2.2 TraHist reaches low estimation error under tight memory budgets (RQ1).

  • At 50 and 100 KB on Porto low-selectivity queries, TraHist has median Q-error of 1.4 at both budgets, versus DEH’s 5.2 and 4.2, respectively.At 500 KB, TraHist reaches approximately 1–3 median Q-error on Porto; no baseline reaches that range at low selectivity.Source: §6.2.2 · pp. 15–16

6.2.3 Accuracy differences diminish under high selectivity queries (RQ1).

  • Under high selectivity, TraHist’s median Q-error is 1.3–4.2× that of the most accurate estimator, while its gains reach 11.0× and 16.2× at low and medium selectivity.The relative ranking changes as fragmentation-induced overcounting becomes less pronounced relative to true cardinality.Source: §6.2.3 · p. 16, §6.2.2 · p. 16

6.3.1 TraHist maintains high accuracy under fixed bucket budgets (RQ1).

  • At 50 buckets, TraHist reduces median Q-error by over 20× versus LHist on T-Drive medium-selectivity queries and by 25–45× versus DEH and CASE+ across low and medium selectivity.Its accuracy advantage remains substantial across 50–500 buckets and is also observed on Porto.Source: Table 3 · p. 17, §6.3.1 · p. 17

6.3.2 Trajectory continuity is better preserved by TraHist under fixed bucket budgets (RQ2).

  • At 50 buckets on Porto, TraHist averages 1.3 fragments; on T-Drive, it stays near one fragment across budgets, lower than the competing partitioning strategies.Figure 9 reports average, median, and maximum fragmentation for both datasets as bucket budgets increase.Source: Figure 9 · p. 18, §6.3.2 · p. 18

6.4.1 TraHist enables fast query estimation with low and stable latency (RQ3).

  • TraHist maintains stable sub-millisecond average query latency and the lowest construction time under fixed memory budgets; at fixed bucket budgets, construction takes about 15–23 s on Porto and 6–7 s on T-Drive.Construction time on Porto rises from approximately 15 s at 50 buckets to about 23 s at 2,000 buckets; T-Drive remains nearly constant.Source: §6.4 · p. 18, Table 4 · p. 19, §6.4.1 · p. 19

6.5.1 Surrogate fidelity limits spatial-object adaptations under selective queries (RQ4).

  • Exact convex-hull counting reduces low-selectivity surrogate median Q-error relative to exact MBR counting by 1.5–2.0×, but errors remain substantial because enclosures include untraversed regions.Even when CDHistogram-MBR and DeepDB-MBR estimate MBR cardinality nearly exactly, their end-to-end errors remain close to the Exact-MBR reference; TraHist has 7.6–9.2× lower median Q-error at low and medium selectivity than the evaluated MBR estimators.Source: §6.5.1 · pp. 20–21

7 Related Work

Prior work spans histogram partitioning, moving-object histograms, learned cardinality estimation, and spatial cardinality estimation. TraHist differs by adapting trajectory-aware spatial partitions for range cardinality estimation rather than relying on fixed partitions or a single enclosing-object surrogate.

7 Related Work

  • Traditional multidimensional histograms use axis-aligned buckets, while workload-aware methods such as STGrid and STHoles refine buckets based on query feedback; LHist adds intra-bucket modeling without changing rectangular partitions.TraHist focuses on data-driven synopses independent of access patterns and changes the partitioning strategy.Source: §7 · p. 21
  • Moving-object histograms use velocity information for future-position queries, while DEH and CASE use connection metadata to correct overcounting on given partitions rather than optimizing partition structure.TraHist structurally reduces fragmentation by aligning boundaries with trajectory-induced density patterns.Source: §7 · pp. 21–22
  • Learned relational estimators model correlations or query workloads, but their original formulations estimate cardinalities over relational tuples rather than trajectory polylines.TraHist targets trajectory range cardinality estimation without training.Source: §7 · p. 22
  • Spatial estimators target points, rectangles, joins, or polygons; applying object-based methods to trajectories requires a surrogate such as an MBR or enclosing polygon, whose intersections can differ from the trajectory’s.TraHist estimates trajectory range cardinality without collapsing each trajectory into one enclosing object.Source: §7 · p. 22

B Topological Predicates Beyond Intersects

TraHist estimates Intersects cardinalities, while other topological predicates require information beyond its compact synopsis. Its statistics can be maintained under a fixed LabelMap, and incremental maintenance is faster than rebuilding in the reported settings.

B Topological Predicates Beyond Intersects

  • TraHist estimates Intersects cardinalities; Within, Crosses, and Touches select distinct subsets and require additional trajectory or segment information.The synopsis can estimate intersecting candidates for query plans, but estimating the reduction to those predicates’ final results requires additional statistics.Source: B Topological Predicates Beyond… · pp. 25–26

C Updating TraHist

  • The update strategies reuse both the LabelMap and statistics, reuse only the LabelMap while refreshing statistics, or rebuild the CountGrid, LabelMap, and statistics.The strategies are No refresh, Statistics only, and Full rebuild, respectively.Source: C Updating TraHist · p. 26

C.1 Exact Maintenance under a Fixed LabelMap

  • Under a fixed LabelMap, insertions and deletions update per-bucket point and distinct-cell totals to maintain S(k) exactly; deletion needs the prior trajectory record or auxiliary per-trajectory counts.Modifications are handled as deletion followed by insertion, while updates outside the fixed spatial domain require grid extension or reconstruction.Source: §C.1 · p. 26
  • Incremental maintenance takes 1.0–10.8 seconds and is 1.9–3.0× faster than full rebuild, with synopsis states of 53.8–64.7 KB.The comparison uses fixed spatial domains and previous trajectory records for deletions; Table C1 reports state and processing cost across five checkpoints.Source: §C.1 · p. 27, Table C1 · p. 27

C.2 Accuracy under Data Shift

Under complete trajectory replacement, updating per-bucket statistics generally preserves median accuracy near that of a full rebuild, while rebuilding gives the strongest tail accuracy. The comparison uses sequential replacement to shift the trajectory distribution.

C.2 Accuracy under Data Shift

  • The controlled shift sequentially replaces groups ordered by their projection onto the first principal component, increasing replacement from 20% to 100%.No refresh and Statistics only share a fixed LabelMap; Full rebuild reconstructs the partition and may produce a different bucket count.Source: §C.2 · pp. 27–28
  • Across nine dataset-selectivity combinations, Statistics only has median Q-error ratios of 0.81–1.19 relative to Full rebuild; Full rebuild has the lowest 95th percentile in all nine.Statistics only lowers 95th percentile Q-error relative to No refresh in eight of nine combinations.Source: Table C2 · p. 28, §C.2 · p. 28
  • At complete replacement on Porto, Statistics only median Q-errors are 5.6, 2.3, and 3.2, versus 12.6, 5.7, and 5.3 for No refresh across low, medium, and high selectivity.The 95th percentile errors are lowest under Full rebuild for all three selectivities; No refresh yields 727, 452, and 501.Source: §C.2 · p. 28

D Derivations for Theoretical Properties

The derivations explain TraHist’s local optimality and justify its propagation contrast as a tractable surrogate for the ideal local objective. The approximation error is controlled by local density variation and cumulative propagation cost.

D Derivations for Theoretical Properties

  • At termination, propagation costs satisfy a neighbor inequality that establishes local optimality: reassigning a cell to a neighboring region cannot reduce E(Π).The stored cell cost is the accumulated contrast along the path assigning its label.Source: D Derivations for Theoretical Properties · pp. 28–29
  • TraHist uses the locally computable contrast D(x,n) instead of recomputing bucket means to calculate each cell’s marginal increase in the intra-bucket SSE objective.The contrast avoids recomputing bucket means at each propagation step.Source: D Derivations for Theoretical Properties · p. 29
  • The bound for D(x,n) relates local smoothing and intra-bucket non-uniformity to cumulative propagation cost, supporting its use as a surrogate for the ideal local objective.The approximation error is controlled by local density variation and cumulative propagation cost.Source: D Derivations for Theoretical Properties · p. 29

E Estimation Accuracy on the GeoLife Dataset

On GeoLife, TraHist maintains strong median accuracy across memory and bucket budgets, especially at low and medium selectivity, while CASE+ has stronger tail control. The appendix also examines robustness under trajectory subsampling and changing grid-cell sizes.

E Estimation Accuracy on the GeoLife Dataset

  • GeoLife covers walking, cycling, bus, and driving trajectories from 182 Beijing-area users, complementing the road-constrained Porto and T-Drive datasets.The authors attribute TraHist’s relative advantage over LHist in part to GeoLife’s heterogeneous trajectory distribution, which likely limits LHist’s intra-bucket regression models.Source: E Estimation Accuracy on the GeoLife… · pp. 29, 31
  • TraHist achieves 2×–9× lower median Q-error than LHist across GeoLife memory and bucket budgets, with median Q-error of 2.1–3.8 and within 1.4× of the lowest at every budget.Figures E1 and E2 report median and tail Q-error across budgets and selectivity regimes.Source: E Estimation Accuracy on the GeoLife… · pp. 29–30, Figure E1 · p. 30, Figure E2 · p. 30
  • CASE+ has the lowest 99th percentile and maximum Q-error at every evaluated budget, and the lowest 95th percentile at most budgets; TraHist’s accuracy gap narrows at high selectivity.TraHist’s accuracy advantage over LHist is more pronounced on GeoLife than on the taxi datasets.Source: E Estimation Accuracy on the GeoLife… · pp. 30–31

F Robustness and Scalability Analysis

  • The robustness analysis varies trajectory subsampling and CountGrid cell size, and examines accuracy and resource costs as cell size decreases.It also examines whether sparse storage can mitigate storage growth without changing the underlying partition.Source: F Robustness and Scalability Analysis · p. 31, Figure F1 · p. 31

F.1 Robustness to Trajectory Subsampling

TraHist remains accurate under substantial trajectory subsampling, but its accuracy depends on cell size relative to both retained sampling density and query size. Finer cells increase memory and do not consistently improve accuracy; tiling reduces CountGrid storage but barely changes TraHist memory.

F.2 Sensitivity to Query Region Size

  • Queries nearest r_Q = 4 have lower median Q-error than those nearest r_Q = 1 at every evaluated cell size on T-Drive and GeoLife, and seven of eight on Porto.At the finest resolutions, median Q-error falls from 13.66 to 1.92 on Porto, 4.92 to 1.51 on T-Drive, and 4.14 to 2.16 on GeoLife.Source: Figure F2 · p. 32, §F.2 · p. 33, Table F1 · p. 33

F.1 Robustness to Trajectory Subsampling

  • At k = 64, only 4.62%, 2.74%, and 1.69% of points remain on Porto, T-Drive, and GeoLife, while accuracy stays low across most evaluated budgets.On Porto, fine cells can leave too few observations: at Δ = (25, 25), median Q-error reaches 40.91 and 61.30 under low and medium selectivity.Source: §F.1 · p. 32
  • At 50 KB and k = 64, TraHist's median Q-error is 18.1× and 16.6× lower than LHist and CASE+ on T-Drive, and 5.7× and 5.5× lower than CASE+ on GeoLife.These comparisons favor estimators built from all trajectory points; TraHist retains only a small fraction.Source: §F.1 · p. 32

F.3 Grid Cell Size Δ and Synopsis Scalability

  • As Δ decreases, GeoLife's grid grows from 506 to 1.83 million cells and TraHist memory from 5.0 KB to 30.2 MB, while construction time rises from 4.6 to 39.2 seconds.The cell count grows roughly 3,600×, making memory the primary scalability cost.Source: §F.3 · p. 34, Table F2 · p. 34
  • Finer cells do not consistently improve accuracy: they generally help low-selectivity Porto queries but increase medium- and high-selectivity error.On T-Drive, the finest setting yields median Q-error of 4.68 at low selectivity and 14.98 at high selectivity; no single evaluated Δ minimizes error across all selectivity groups.Source: §F.3 · pp. 34–35, Table F2 · p. 34
  • At the finest settings, tiling cuts CountGrid storage by 40.1–88.2%, but changes total TraHist memory by at most 6.4% and raises average query latency 2.3–3.0×.The LabelMap and per-bucket statistics, not the CountGrid, drive estimation and remain largely unaffected by tiling.Source: Table F2 · p. 34, §F.3 · p. 35

G.1 Contrast Balance 𝛼

The contrast balance α and threshold scaling β control density-adaptive partition granularity, but their accuracy effects depend on selectivity and dataset characteristics. The defaults, α = 0.5 and β = 1.0, offer practical trade-offs across the evaluated settings.

G.1 Contrast Balance 𝛼

  • As α increases, low- and medium-selectivity median Q-error generally rises, especially on GeoLife, where it grows from 3.8 to 14.7 at Δ = (3500, 3500).Larger α makes partitions finer; the default α = 0.5 stays within 1.8× of the configuration-wise optimum for low and medium selectivity.Source: §G.1 · pp. 35–37, Figure G2 · p. 36, Figure G3 · p. 37
  • At high selectivity, increasing α lowers GeoLife median Q-error from 4.1 to 2.1 at Δ = (2000, 2000), while T-Drive's response varies with cell size.Porto's error rises with α; the differing responses track query scale and dataset density heterogeneity.Source: §G.1 · pp. 36–37

G.2 Threshold Scaling 𝛽

  • Increasing β generally improves low- and medium-selectivity median Q-error by reducing fragmentation, including T-Drive's drop from 35.5 to 5.2 and GeoLife's from 23.1 to 3.0.The improvements are weaker or can reverse in Porto's densest partitioning regime.Source: §G.2 · pp. 37–38
  • At high selectivity, larger β raises GeoLife median Q-error from 2.4 to 5.2 at Δ = (3500, 3500), while Porto and T-Drive generally remain stable or improve.The default β = 1.0 stays within 2.2× of the low- and medium-selectivity configuration-wise optimum while avoiding pronounced high-selectivity degradation on GeoLife.Source: §G.2 · pp. 37–38, Figure G3 · p. 37

H Extended Results for the Main Evaluation

Extended-budget results preserve the main evaluation's accuracy pattern: TraHist leads in low-selectivity settings, while the strongest estimator varies by dataset and selectivity. DEH, CASE+, and TraHist retain sub-millisecond latency as budgets increase.

H Extended Results for the Main Evaluation

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