A union-find data structure maintains a collection of disjoint sets under makeset, union and find operations. Kaplan, Shafrir and Tarjan [SODA 2002] designed data structures for an extension of the unionfind problem in which elements of the sets maintained may be deleted. The cost of a delete

At SODA 2009, Demaine et al. presented a novel connection between binary search trees (BSTs) and subsets of points on the plane. This connection was independently discovered by Derryberry et al. As part of their results, Demaine et al. considered GreedyFuture, an offline BST algorithm

with bijective analysis [SODA 2007] to list update. Using this framework, we prove that Move-to-Front (MTF) is the unique optimal algorithm for list update. This addresses the open question of defining an appropriate model for capturing locality of reference in the context of list update [Hester and Hirschberg

; this addresses the open question of defining an appropriate model for capturing locality of reference in the context of list update [Hester and Hirschberg ACM Comp. Surv. 1985]. We then apply this model in conjunction with a recent technique for comparing online algorithms, namely bijective analysis [SODA 2007

-time preprocessing scheme for O(1)-RMQs by reducing this problem to the computation of lowest common ancestors in trees (Harel and Tarjan, SICOMP 13(2): 1984). The drawback of this and most later schemes is their huge space consumption of O(n log n) bits, which is not optimal. We make the assumption that the input

the question posed by Cabello and Mohar in 2005, and by Kawarabayashi and Reed (STOC’07), respectively. Note that the case k = 0 is the planarity case. Thus our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan (J. ACM 1974), which determines if a given graph is planar