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Top-k Stabbing Interval Queries

Waseem Akram, Sanjeev Saxena

TL;DR

This paper presents a linear space solution with O(\log n + k) query time for a weighted variant of the interval stabbing problem and presents another trade-off for the problem.

Abstract

We investigate a weighted variant of the interval stabbing problem, where the goal is to design an efficient data structure for a given set $\mathcal{I}$ of weighted intervals such that, for a query point $q$ and an integer $k>0$, we can report the $k$ intervals with largest weights among those stabbed by $q$. In this paper, we present a linear space solution with $O(\log n + k)$ query time. Moreover, we also present another trade-off for the problem.

Top-k Stabbing Interval Queries

TL;DR

This paper presents a linear space solution with O(\log n + k) query time for a weighted variant of the interval stabbing problem and presents another trade-off for the problem.

Abstract

We investigate a weighted variant of the interval stabbing problem, where the goal is to design an efficient data structure for a given set of weighted intervals such that, for a query point and an integer , we can report the intervals with largest weights among those stabbed by . In this paper, we present a linear space solution with query time. Moreover, we also present another trade-off for the problem.

Paper Structure

This paper contains 5 sections, 1 theorem, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal Solution
  4. Segment Tree-Based Solution
  5. Conclusion

Key Result

Theorem 1

Given a set $\mathcal{I}$ of weighted intervals on the real line, we can represent $\mathcal{I}$ into a data structure that supports the top-$k$ interval stabbing queries in $O(\log n + k\log\log n)$ time. The data structure can be built in $O(n\log n)$ space and time.

Figures (1)

  • Figure 1: An interval $[s_i, e_i]$ with weight $w_i$ (dashed and blue) is interpreted as a horizontal line segment $[s_i, e_i]\times w_i$ in the plane (solid and red).

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3