## Abstract

We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model. -It is well known that there are linear-space data structures for 2-D orthogonal range counting with worstcase optimal query time O(log n/ log log n). We give an O(nlog log n)-space adaptive data structure that improves the query time to O(log log n+ log k/ log log n), where k is the output count. When k = O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan et al., 2011]. -We give an O(nlog log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1+d)-factor approximation to the count in O(log log n) time for any fixed constant d > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem. -Last, we consider the 1-D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jørgensen and Larsen [2011] presented a linear-space adaptive data structure with query time O(log log n+ log k/ log log n). We give a new linear-space structure that improves the query time to O(1 + log k/ log log n), exactly matching the lower bound proved by Jørgensen and Larsen.

Original language | English (US) |
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Article number | 45 |

Journal | ACM Transactions on Algorithms |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - Sep 2016 |

Externally published | Yes |

## Keywords

- Adaptive
- Approximate
- Computational geometry
- Data structure
- Orthogonal
- Range counting
- Word RAM

## ASJC Scopus subject areas

- Mathematics (miscellaneous)