Efficient Optimally Lazy Algorithms for Minimal-Interval Semantics

TL;DR

This paper introduces efficient algorithms for minimal-interval semantics, enabling fast computation of query witnesses and proximity operators, with a focus on optimal laziness and broad applicability.

Contribution

It presents linear and logarithmic algorithms for conjunction, disjunction, and other operators, along with a formal notion of optimal laziness and its applicability to various domains.

Findings

Algorithms are linear in the number of intervals.

Algorithms for conjunction and disjunction are logarithmic in operands.

The framework applies to various domains with antichains of intervals.

Abstract

Minimal-interval semantics associates with each query over a document a set of intervals, called witnesses, that are incomparable with respect to inclusion (i.e., they form an antichain): witnesses define the minimal regions of the document satisfying the query. Minimal-interval semantics makes it easy to define and compute several sophisticated proximity operators, provides snippets for user presentation, and can be used to rank documents. In this paper we provide algorithms for computing conjunction and disjunction that are linear in the number of intervals and logarithmic in the number of operands; for additional operators, such as ordered conjunction and Brouwerian difference, we provide linear algorithms. In all cases, space is linear in the number of operands. More importantly, we define a formal notion of optimal laziness, and either prove it, or prove its impossibility, for each algorithm. We cast our results in a general framework of antichains of intervals on total orders, making our algorithms directly applicable to other domains.