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

arXiv: https://arxiv.org/abs/2107.03277 · HTML: https://mlsanity.com/paper/2107.03277/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 46 references that are in the map.
- First posted on arXiv 7 Jul 2021.

## Builds on (most cited first, 4 of 4)

- [The optimality of syntactic dependency distances](https://mlsanity.com/paper/2007.15342/) (2020)
- [Minimum projective linearizations of trees in linear time](https://mlsanity.com/paper/2102.03277/) (2021)
- [Hubiness, length, crossings and their relationships in dependency trees](https://mlsanity.com/paper/1304.4086/) (2013)
- [Non-crossing dependencies: least effort, not grammar](https://mlsanity.com/paper/1411.2645/) (2014)

## Built on by (most cited first, 5 of 5)

- [The Linear Arrangement Library. A new tool for research on syntactic dependency structures](https://mlsanity.com/paper/2112.02512/) (2021)
- [The expected sum of edge lengths in planar linearizations of trees. Theory and applications](https://mlsanity.com/paper/2207.05564/) (2022)
- [Who is the root in a syntactic dependency structure?](https://mlsanity.com/paper/2501.15188/) (2025)
- [How to measure the optimality of word or gesture order with respect to the principle of swap distance minimization](https://mlsanity.com/paper/2604.01938/) (2026)
- [The Maximum Linear Arrangement Problem for trees under projectivity and planarity](https://mlsanity.com/paper/2206.06924/) (2022)

## Abstract (arXiv)

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.