Minimum projective linearizations of trees in linear time

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(n logdmax) 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.LAP is supported by Secretaria d’Universitats i Recerca de la Generalitat de Catalunya and the Social European Fund. RFC and LAP are supported by the grant TIN2017-89244-R from MINECO (Ministerio de Economía, Industria y Competitividad). RFC is also supported by the recognition 2017SGR-856 (MACDA) from AGAUR (Generalitat de Catalunya). JLE is funded by the grant PID2019- 109137GB-C22 from MINECO.Peer ReviewedPostprint (published version