Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems

In this paper I present several novel, efficient, algorithmic techniques for solving some multidimensional geometric data management and analysis problems. The techniques are based on several data structures from computational geometry (e.g. segment tree and range tree) and on the well-known sweep-line method.

💡 Research Summary

The paper presents a collection of novel algorithmic techniques for managing and analyzing multidimensional geometric data, built upon classic computational‑geometry structures such as segment trees, range trees, and the sweep‑line paradigm. Three main problem families are addressed.

1. Largest Empty Circle / Fixed‑Aspect‑Ratio Hyper‑Rectangle – Given n points inside a bounding circle (or a d‑dimensional box), the goal is to find the largest empty circle (or hyper‑rectangle with prescribed aspect ratios) that fits entirely within the container and contains no input points. The authors binary‑search the candidate radius (or the base side length) and, for each candidate, shrink the container accordingly. They then collect all x‑coordinates of circle intersections (or rectangle edges) to form O(n²) vertical slabs. Inside each slab the problem reduces to a one‑dimensional interval‑cover test: project all circles (or rectangles) onto the slab’s mid‑line, obtain at most n intervals, sort their endpoints, and scan to see whether the interval representing the shrunken container is fully covered. The naïve implementation yields O(n³ log n) time, but by maintaining the ordering of curves in a balanced tree across consecutive slabs, the authors improve this to O(M·(n+log n)), where M is the number of distinct x‑coordinates (often O(n) for axis‑aligned objects). The same slab‑based approach is also applied to compute the union area of circles or polygons, with similar complexity improvements.

2. Containment Queries for Circles and Hyper‑Rectangles – For a set of circles that are either disjoint or nested, the paper shows how to decide for each circle whether it is contained in another. Each circle is split into left and right half‑circles; a vertical sweep line processes the right halves from left to right (and symmetrically the left halves). The active half‑circles intersecting the sweep line are stored as y‑intervals in an AVL tree. When a new half‑circle appears, only its neighboring intervals need to be examined, allowing O(log n) insertion, deletion, and containment checks, leading to an overall O(n log n) algorithm. For d‑dimensional axis‑aligned hyper‑rectangles (allowing partial overlap), the authors extend the idea: they sweep along the d‑th dimension while maintaining a (2d‑2)‑dimensional range tree. Each rectangle’s left endpoint inserts a point with weight equal to one of its coordinates; the right endpoint resets the weight to –∞. A query for the maximum weight inside a specific orthogonal range tells whether the current rectangle is fully covered by any previously active rectangle. The resulting time bound is O(n·log^{2d‑2} n), with an alternative O(n·log^{2d‑1} n) solution based on a 2d‑dimensional point representation and orthogonal range counting.

3. Maximum‑Weight Subsequence Accepted by a NFA – The third contribution tackles a weighted longest‑subsequence problem constrained by a nondeterministic finite automaton (NFA) whose transitions are defined by d‑dimensional interval constraints on the differences between successive points. For each state j a d‑dimensional range tree is kept, initially filled with –∞. While scanning the input points in order, the algorithm computes for every state the best subsequence ending at the current point by querying, for each incoming transition (q_{j′}→q_j), the maximum weight among points that satisfy the transition’s interval constraints. This query is a d‑dimensional orthogonal range‑maximum query, answered in O(log^{d} n) time per transition. After the DP value W_max(i,j) is obtained, the point is inserted into the range tree of state j with that weight, making it available for future queries. The total running time is O(n·(m+|E|)·log^{d} n). When d=1 the structure collapses to a segment tree and the algorithm matches the classic O(n log n) solution for the longest increasing subsequence; with two states and alternating interval constraints it yields the longest alternating subsequence in the same bound.

Overall, the paper demonstrates a unifying methodology: decompose complex multidimensional geometric problems into a series of one‑dimensional interval or orthogonal range queries, and then exploit dynamic balanced trees (segment trees, range trees, AVL trees) to maintain the necessary information while sweeping through the data. This approach yields provably optimal or near‑optimal asymptotic complexities for a variety of tasks that are central to computational geometry and multidimensional data analysis, and it opens avenues for practical implementations on large‑scale distributed systems where identifiers are embedded in high‑dimensional metric spaces.