Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Jeff Erickson, Danny Krizanc, Henk Meijer, Pat Morin, Mark Overmars, and Sue Whitesides, “Realizing Partitions Respecting Full and Partial Order Information”, Journal of Discrete Algorithms, volume 6, 2008, pages 51–58. Special issue of selected papers from the 16th Australasian Workshop on Combinatorial Algorithms, 2005.

Abstract:

For n ∈ ℕ, we consider the problem of partitioning the interval [0, n) into k subintervals of positive integer lengths ℓ₁, …, ℓ_k such that the lengths satisfy a set of simple constraints of the form ℓ_i ◇_i j ℓ_j where ◇_i j is one of <, >, or =. In the full information case, ◇_i j is given for all 1 ≤ i, j ≤ k. In the sequential information case, ◇_ij is given for all 1 < i < k and j = i ± 1. That is, only the relations between the lengths of consecutive intervals are specified. The cyclic information case is an extension of the sequential information case in which the relationship ◇_1k between ℓ₁ and ℓ_k is also given. We show that all three versions of the problem can be solved in time polynomial in k and log n.

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