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 lan- guages. 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 per- mutations 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 is, the lower the error of the estimation but the larger the time cost. Here we pre- sent formulae to compute that expectation without error in time of the order of n. Furthermore, we show that star trees maximize it, and provide an algorithm to retrieve the trees that minimize it.

The Minimum Linear Arrangement problem (MLA) consists of finding a mapping π from vertices of a graph to distinct integers that minimizes ∑{u,v}∈E|π(u)−π(v)|. In that setting, vertices are often assumed to lie on a horizontal line and edges are drawn as semicircles above said line. For trees, various algorithms are available to solve the problem in polynomial time in n=|V|. There exist variants of the MLA in which the arrangements are constrained. Iordanskii, and later Hochberg and Stallmann (HS), put forward O(n)-time algorithms that solve the problem when arrangements are constrained to be planar (also known as one-page book embeddings). We also consider linear arrangements of rooted trees that are constrained to be projective (planar embeddings where the root is not covered by any edge). Gildea and Temperley (GT) sketched an algorithm for projective arrangements which they claimed runs in O(n) but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in O(nlog⁡dmax) where dmax is the maximum degree but did not provide sufficient detail. Here we correct an error in HS's algorithm for the planar case, show its relationship with the projective case, and derive simple algorithms for the projective and planar cases that run without a doubt in O(n) time.