Towards Efficient Data Structures for Approximate Search with Range Queries

Range queries are simple and popular types of queries used in data retrieval. However, extracting exact and complete information using range queries is costly. As a remedy, some previous work proposed a faster principle, {\em approximate} search with range queries, also called single range cover (SRC) search. It can, however, produce some false positives. In this work we introduce a new SRC search structure, a $c$-DAG (Directed Acyclic Graph), which provably decreases the average number of false positives by logarithmic factor while keeping asymptotically same time and memory complexities as a classic tree structure. A $c$-DAG is a tunable augmentation of the 1D-Tree with denser overlapping branches ($c \geq 3$ children per node). We perform a competitive analysis of a $c$-DAG with respect to 1D-Tree and derive an additive constant time overhead and a multiplicative logarithmic improvement of the false positives ratio, on average. We also provide a generic framework to extend our results to empirical distributions of queries, and demonstrate its effectiveness for Gowalla dataset. Finally, we quantify and discuss security and privacy aspects of SRC search on $c$-DAG vs 1D-Tree, mainly mitigation of structural leakage, which makes $c$-DAG a good data structure candidate for deployment in privacy-preserving systems (e.g., searchable encryption) and multimedia retrieval.

💡 Research Summary

The paper addresses the well‑known trade‑off between efficiency and accuracy in one‑dimensional range queries. Exact range search is costly and, in privacy‑preserving settings, can leak structural information. Approximate range search using the Single Range Cover (SRC) primitive reduces the cost by returning a single node whose canonical interval fully contains the query, but this approach inevitably introduces false positives—data points that lie outside the query interval but are returned for filtering.

To mitigate the false‑positive problem without sacrificing the asymptotic performance of SRC, the authors propose a new data‑dependent structure called the c‑DAG (c‑Directed Acyclic Graph). A c‑DAG augments the classic 1D‑Tree (a binary KD‑Tree variant) by allowing each internal node to have c ≥ 3 children. The two extreme children partition the node’s data set into left and right halves, exactly as in the 1D‑Tree, while the remaining c‑2 intermediate children are inserted between them with overlapping intervals of equal width. This overlapping design yields a richer set of canonical intervals, enabling a finer granularity of coverage for any query interval.

The authors introduce the Level Difference Distribution (LDD), a probabilistic tool that captures the distribution of level differences between the nodes returned by SRC on a 1D‑Tree and on a c‑DAG. Using LDD, they prove two central theorems:

1. Theorem 2 (Search‑time overhead) – The SRC search on a c‑DAG incurs at most a constant additive overhead of ((2c‑2)/(c‑1)) steps compared with the 1D‑Tree. Consequently, the worst‑case search time remains Θ(log N), preserving the logarithmic query latency of the baseline.

2. Theorem 3 (False‑positive reduction) – On average, a c‑DAG reduces the false‑positive ratio by a multiplicative factor of Θ(log(N/s)), where N is the dataset size and s is the query length. The overlapping intervals ensure that the minimal covering node is typically much tighter than the binary split node, directly lowering the number of extraneous points that must be filtered client‑side.

Beyond uniform data, the paper extends the analysis to skewed or empirically observed query distributions. By adapting the LDD to the empirical distribution, the authors show that the logarithmic improvement in false positives holds for a broad class of non‑uniform datasets.

Experimental validation uses the Gowalla location‑based social network dataset. Varying the branching factor c (3, 4, 5) and query lengths, the authors measure search time, memory consumption, and false‑positive rates. Results confirm the theoretical predictions: for c = 5, false positives drop by more than 30 % on average, while search time increases by only ~20 % relative to the 1D‑Tree. Memory usage grows linearly with c, yielding an overall storage complexity of Θ(c N log² N) bits, which remains practical for large‑scale deployments.

A notable contribution is the security analysis. In searchable encryption schemes, the structure of the index can leak information about the underlying data distribution (structural leakage). The 1D‑Tree’s deterministic binary partition can be exploited to infer density patterns from node depths and interval sizes. The c‑DAG’s overlapping intervals, however, introduce ambiguity: multiple nodes at the same depth can cover the same query, and the exact overlap pattern is data‑dependent, making it harder for an adversary to reconstruct the original distribution. This mitigation of structural leakage positions the c‑DAG as a strong candidate for privacy‑preserving applications such as searchable encryption and federated analytics.

In summary, the paper delivers a practical, theoretically grounded data structure that retains the logarithmic query time and linear‑in‑c memory of the classic 1D‑Tree while achieving a logarithmic reduction in false positives. The work includes a rigorous probabilistic analysis, a generalization framework for arbitrary data distributions, empirical validation on real‑world data, and a thoughtful discussion of privacy implications. These contributions make the c‑DAG highly relevant for systems requiring fast approximate range queries, especially where privacy and bandwidth constraints are paramount. Future directions suggested include multi‑dimensional extensions, dynamic update support, and optimized implementations within encrypted databases.