Linear-time calculation of the expected sum of edge lengths in random projective linearizations of trees

TL;DR

This paper introduces a linear-time method to precisely compute the expected sum of dependency distances in random projective linearizations of trees, improving efficiency over previous Monte Carlo approaches.

Contribution

It provides exact formulas for expectation calculation and algorithms for identifying trees with maximum or minimum dependency distance sums.

Findings

Star trees maximize the expected sum

Efficient linear-time computation formulas are derived

Algorithms to find trees minimizing the sum are presented

Abstract

The syntactic structure of a sentence is often represented using syntactic dependency trees. The sum of the distances between syntactically related words has been in the limelight for the past decades. Research on dependency distances led to the formulation of the principle of dependency distance minimization whereby words in sentences are ordered so as to minimize that sum. Numerous random baselines have been defined to carry out related quantitative studies on languages. The simplest random baseline is the expected value of the sum in unconstrained random permutations of the words in the sentence, namely when all the shufflings of the words of a sentence are allowed and equally likely. Here we focus on a popular baseline: random projective permutations of the words of the sentence, that is, permutations where the syntactic dependency structure is projective, a formal constraint that sentences satisfy often in languages. Thus far, the expectation of the sum of dependency distances in random projective shufflings of a sentence has been estimated approximately with a Monte Carlo procedure whose cost is of the order of $Rn$, where $n$ is the number of words of the sentence and $R$ is the number of samples; it is well known that the larger $R$, the lower the error of the estimation but the larger the time cost. Here we present formulae to compute that expectation without error in time of the order of $n$. Furthermore, we show that star trees maximize it, and give an algorithm to retrieve the trees that minimize it.