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Dependency trees have proven to be a very successful model to represent thesyntactic structure of sentences of human languages. In these structures,vertices are words and edges connect syntactically-dependent words. Thetendency of these dependencies to be short has been demonstrated using randombaselines for the sum of the lengths of the edges or its variants. A ubiquitousbaseline is the expected sum in projective orderings (wherein edges do notcross and the root word of the sentence is not covered by any edge), that canbe computed in time $O(n)$. Here we focus on a weaker formal constraint, namelyplanarity. In the theoretical domain, we present a characterization ofplanarity that, given a sentence, yields either the number of planarpermutations or an efficient algorithm to generate uniformly random planarpermutations of the words. We also show the relationship between the expectedsum in planar arrangements and the expected sum in projective arrangements. Inthe domain of applications, we derive a $O(n)$-time algorithm to calculate theexpected value of the sum of edge lengths. We also apply this research to aparallel corpus and find that the gap between actual dependency distance andthe random baseline reduces as the strength of the formal constraint ondependency structures increases, suggesting that formal constraints absorb partof the dependency distance minimization effect. Our research paves the way forreplicating past research on dependency distance minimization using randomplanar linearizations as random baseline.

The expected sum of edge lengths in planar linearizations of trees. Theory and applications