No O(N) queries for checking if N intervals cover everything or for piercing N pairs of intervals. An O(N log N)-steps algorithm for piercing

TL;DR

This paper proves that certain interval coverage and piercing problems cannot be solved with linear queries and introduces an efficient O(N log N) algorithm for the piercing problem.

Contribution

It establishes lower bounds on query complexity for coverage and piercing problems and provides an O(N log N) algorithm for the piercing problem.

Findings

Linear query complexity is insufficient for coverage and piercing problems.

No Helly property exists for these interval problems.

An O(N log N) algorithm efficiently finds piercing points if they exist.

Abstract

The complexity of two related geometrical (indeed, combinatorial) problems is considered, measured by the number of queries needed to determine the solution. It is proved that one cannot check in a linear in N number of queries whether N intervals cover a whole interval, or whether for N pairs of intervals on two lines there is a pair of points intersecting each of these pairs of intervals ("piercing all pairs of intervals"). The proofs are related to examples which show that there is no "Helly property" here - the whole set of N may cover the whole interval (resp. may have no pair of points piercing all pairs of intervals) while any proper subset does not. Also, for the piercing problem we outline an algorithm, taking O(N log N) steps, to check whether there is a pair of points piercing all pairs of intervals and if there is, to find it.