Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1311.4394

Encoding Range Minimum Queries

Pooya Davoodi·Gonzalo Navarro·Rajeev Raman·S. Srinivasa Rao
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the problem of encoding range minimum queries (RMQs): given an array A[1..n] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ(i,j), which returns the index containing the smallest element in A[i..j], without access to the array A at query time. We give a data structure whose space usage is 2n + o(n) bits, which is asymptotically optimal for worst-case data, and answers RMQs in O(1) worst-case time. This matches the previous result of Fischer and Heun [SICOMP, 2011], but is obtained in a more natural way. Furthermore, our result can encode the RMQs of a random array A in 1.919n + o(n) bits in expectation, which is not known to hold for Fischer and Heun's result. We then generalize our result to the encoding range top-2 query (RT2Q) problem, which is like the encoding RMQ problem except that the query RT2Q(i,j) returns the indices of both the smallest and second-smallest elements of A[i..j]. We introduce a data structure using 3.272n+o(n) bits that answers RT2Qs in constant time, and also give lower bounds on the effective entropy} of RT2Q.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Pooya Davoodi, Gonzalo Navarro, Rajeev Raman, S. Srinivasa Rao. 2013-11-18. Encoding Range Minimum Queries. https://arxiv.org/abs/1311.4394

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↗