Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1407.5750

Fibonacci Heaps Revisited

Haim Kaplan·Robert E. Tarjan·Uri Zwick
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Fibonacci heap is a classic data structure that supports deletions in logarithmic amortized time and all other heap operations in O(1) amortized time. We explore the design space of this data structure. We propose a version with the following improvements over the original: (i) Each heap is represented by a single heap-ordered tree, instead of a set of trees. (ii) Each decrease-key operation does only one cut and a cascade of rank changes, instead of doing a cascade of cuts. (iii) The outcomes of all comparisons done by the algorithm are explicitly represented in the data structure, so none are wasted. We also give an example to show that without cascading cuts or rank changes, both the original data structure and the new version fail to have the desired efficiency, solving an open problem of Fredman. Finally, we illustrate the richness of the design space by proposing several alternative ways to do cascading rank changes, including a randomized strategy related to one previously proposed by Karger. We leave the analysis of these alternatives as intriguing open problems.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haim Kaplan, Robert E. Tarjan, Uri Zwick. 2014-07-22. Fibonacci Heaps Revisited. https://arxiv.org/abs/1407.5750

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗