Showing papers by "Gerth Stølting Brodal published in 2020"

A Simple Greedy Algorithm for Dynamic Graph Orientation

Abstract: Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in $$\mathcal {O}\left( \log n\right) $$ flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case $$\mathcal {O}\left( 1\right) $$ and $$\mathcal {O}\left( \sqrt{\log n}\right) $$ flips nearly matching out-degree bounds to their respective amortized solutions.

Soft Sequence Heaps.

Abstract: Chazelle [JACM00] introduced the soft heap as a building block for efficient minimum spanning tree algorithms, and recently Kaplan et al. [SOSA2019] showed how soft heaps can be applied to achieve simpler algorithms for various selection problems. A soft heap trades-off accuracy for efficiency, by allowing $\epsilon N$ of the items in a heap to be corrupted after a total of $N$ insertions, where a corrupted item is an item with artificially increased key and $0 < \epsilon \leq 1/2$ is a fixed error parameter. Chazelle's soft heaps are based on binomial trees and support insertions in amortized $O(\lg(1/\epsilon))$ time and extract-min operations in amortized $O(1)$ time. In this paper we explore the design space of soft heaps. The main contribution of this paper is an alternative soft heap implementation based on merging sorted sequences, with time bounds matching those of Chazelle's soft heaps. We also discuss a variation of the soft heap by Kaplan et al. [SICOMP2013], where we avoid performing insertions lazily. It is based on ternary trees instead of binary trees and matches the time bounds of Kaplan et al., i.e. amortized $O(1)$ insertions and amortized $O(\lg(1/\epsilon))$ extract-min. Both our data structures only introduce corruptions after extract-min operations which return the set of items corrupted by the operation.

Soft Sequence Heaps.

Abstract: Chazelle [JACM00] introduced the soft heap as a building block for efficient minimum spanning tree algorithms, and recently Kaplan et al. [SOSA2019] showed how soft heaps can be applied to achieve simpler algorithms for various selection problems. A soft heap trades-off accuracy for efficiency, by allowing $\epsilon N$ of the items in a heap to be corrupted after a total of $N$ insertions, where a corrupted item is an item with artificially increased key and $0 < \epsilon \leq 1/2$ is a fixed error parameter. Chazelle's soft heaps are based on binomial trees and support insertions in amortized $O(\lg(1/\epsilon))$ time and extract-min operations in amortized $O(1)$ time. In this paper we explore the design space of soft heaps. The main contribution of this paper is an alternative soft heap implementation based on merging sorted sequences, with time bounds matching those of Chazelle's soft heaps. We also discuss a variation of the soft heap by Kaplan et al. [SICOMP2013], where we avoid performing insertions lazily. It is based on ternary trees instead of binary trees and matches the time bounds of Kaplan et al., i.e. amortized $O(1)$ insertions and amortized $O(\lg(1/\epsilon))$ extract-min. Both our data structures only introduce corruptions after extract-min operations which return the set of items corrupted by the operation.