Optimal Offline Dynamic $2,3$-Edge/Vertex Connectivity

Peng, Richard; Sandlund, Bryce; Sleator, Daniel D.

doi:10.48550/arxiv.1708.03812

Cited by 1 publication

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“…More generally, we also believe that separation between online and offline algorithms is a far more widespread phenomenon. The current best running times for offline and online algorithms differ in a multitude of problems related to dynamic graph data structures [11,31,17,21,6]. Formulating and investigating this separation is an intriguing task that is significantly beyond the scope of this paper.…”

Section: Discussionmentioning

confidence: 99%

Dynamic Optimality Refuted -- For Tournament Heaps

Munro,

Peng,

Wild

et al. 2019

Preprint

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We prove a separation between offline and online algorithms for finger-based tournament heaps undergoing key modifications. These heaps are implemented by binary trees with keys stored on leaves, and intermediate nodes tracking the min of their respective subtrees. They represent a natural starting point for studying self-adjusting heaps due to the need to access the root-to-leaf path upon modifications. We combine previous studies on the competitive ratios of unordered binary search trees by [Fredman WADS2011] and on order-by-next request by [Martínez-Roura TCS2000] and [Munro ESA2000] to show that for any number of fingers, tournament heaps cannot handle a sequence of modify-key operations with competitive ratio in o( √ log n).Critical to this analysis is the characterization of the modifications that a heap can undergo upon an access. There are exp(Θ(n log n)) valid heaps on n keys, but only exp(Θ(n)) binary search trees. We parameterize the modification power through the well-studied concept of fingers: additional pointers the data structure can manipulate arbitrarily. Here we demonstrate that fingers can be significantly more powerful than servers moving on a static tree by showing that access to k fingers allow an offline algorithm to handle any access sequence with amortized cost O(log k (n) + 2 lg * n ).

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Section: Discussionmentioning

confidence: 99%