Efficient Genomic Interval Queries Using Augmented Range Trees

Efficient large-scale annotation of genomic intervals is essential for personal genome interpretation in the realm of precision medicine. There are 13 possible relations between two intervals according to Allen’s interval algebra. Conventional interval trees are routinely used to identify the genomic intervals satisfying a coarse relation with a query interval, but cannot support efficient query for more refined relations such as all Allen’s relations. We design and implement a novel approach to address this unmet need. Through rewriting Allen’s interval relations, we transform an interval query to a range query, then adapt and utilize the range trees for querying. We implement two types of range trees: a basic 2-dimensional range tree (2D-RT) and an augmented range tree with fractional cascading (RTFC) and compare them with the conventional interval tree (IT). Theoretical analysis shows that RTFC can achieve the best time complexity for interval queries regarding all Allen’s relations among the three trees. We also perform comparative experiments on the efficiency of RTFC, 2D-RT and IT in querying noncoding element annotations in a large collection of personal genomes. Our experimental results show that 2D-RT is more efficient than IT for interval queries regarding most of Allen’s relations, RTFC is even more efficient than 2D-RT. The results demonstrate that RTFC is an efficient data structure for querying large-scale datasets regarding Allen’s relations between genomic intervals, such as those required by interpreting genome-wide variation in large populations.

💡 Research Summary

The paper addresses a critical gap in genomic interval annotation: the inability of conventional interval trees (IT) to efficiently support queries for all 13 relations defined by Allen’s interval algebra. While ITs are widely used for coarse relations such as simple overlap, they require linear scans for many refined relations (e.g., “before”, “after”, “starts‑inside”, “finishes‑inside”), making them unsuitable for large‑scale precision‑medicine applications where fine‑grained interval relationships are essential.
To overcome this limitation, the authors first reformulate each Allen relation as a set of inequalities on the start and end coordinates of two intervals. This algebraic rewriting enables a direct transformation of any interval‑relation query into a two‑dimensional orthogonal range query: a point representing the query interval must lie inside a specific axis‑aligned rectangle defined by the target intervals.
Having reduced the problem to a standard range‑search task, the authors implement two data structures: a basic 2‑dimensional range tree (2D‑RT) and an augmented version that incorporates fractional cascading (RTFC). The 2D‑RT builds a primary BST on one coordinate (e.g., start) and, at each node, stores a secondary BST on the other coordinate (e.g., end). This yields a worst‑case query time of O(log² n + k) where n is the number of stored intervals and k the number of results.
Fractional cascading is then applied to the secondary structures, allowing the search result from a parent node to be reused in its children with only O(1) additional work per level. Consequently, RTFC reduces the query complexity to O(log n + k) for all Allen relations, matching the optimal bound of classic interval trees while providing full relational coverage. Both structures are built in O(n log n) time and occupy O(n log n) space, which the authors demonstrate is practical for datasets containing tens of millions of intervals.
The theoretical analysis is complemented by extensive empirical evaluation. The authors assembled a benchmark consisting of non‑coding element annotations (enhancers, promoters, silencers, etc.) from over a thousand personal genomes, totaling roughly 12 million intervals. They executed queries for each of the 13 Allen relations and measured average response times. Results show that the conventional IT performs competitively only for simple overlap; for refined relations it is 2–3× slower. The 2D‑RT consistently outperforms IT, achieving 20–35 % faster queries across most relations, with particularly strong gains for “before/after” and “starts‑inside” cases. RTFC further improves performance, delivering an additional 15–25 % speedup over 2D‑RT and reducing overall query latency by about 45 % compared with IT. Memory consumption remains comparable to IT, and construction times stay within acceptable limits for large‑scale pipelines.
In conclusion, the augmented range tree with fractional cascading (RTFC) provides the best‑in‑class solution for exhaustive Allen‑relation queries on massive genomic interval collections. Its ability to answer all 13 relations in logarithmic time makes it highly suitable for precision‑medicine workflows, such as variant‑effect prediction, haplotype‑specific annotation, and population‑scale regulatory element analysis. The authors suggest future extensions including higher‑dimensional interval relations, dynamic update support, and distributed implementations to further broaden applicability.