New Lower Bound Techniques for Dynamic Partial Sums and Related Problems

Husfeldt, Thore; Rauhe, Theis

doi:10.1137/s0097539701391592

Cited by 15 publications

(15 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The end of the 80s saw the publication of two landmark papers in the field: Ajtai's static lower bound for predecessor search [1], and the dynamic lower bounds of Fredman and Saks [26]. In the 20 years that have passed, cellprobe complexity has developed into a mature research direction, with a substantial bibliography: we are aware of [1,26,34,37,35,29,5,25,36,14,2,15,6,13,11,12,27,28,16,33,31,44,43,8,41,42,40,45,48].…”

Section: Introductionmentioning

confidence: 99%

Unifying the Landscape of Cell-Probe Lower Bounds

Ptraşcu¹

2011

SIAM J. Comput.

View full text Add to dashboard Cite

We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for:• high-dimensional problems, where the goal is to show large space lower bounds.• constant-dimensional geometric problems, where the goal is to bound the query time for space O(n · polylogn).• dynamic problems, where we are looking for a trade-off between query and update time. (In this case, our bounds are slightly weaker than the originals, losing a lg lg n factor.)Our reductions also imply the following new results:• an Ω(lg n/ lg lg n) bound for 4-dimensional range reporting, given space O(n · polylogn). This is quite timely, since a recent result [39] solved 3D reporting in O(lg 2 lg n) time, raising the prospect that higher dimensions could also be easy.• a tight space lower bound for the partial match problem, for constant query time.• the first lower bound for reachability oracles.In the process, we prove optimal randomized lower bounds for lopsided set disjointness. * The conference version of this paper appeared in FOCS'08 under the title (Data) Structures. † mip@alum.mit.edu. AT&T Labs. Parts of this work were done while the author was at MIT and IBM Almaden Research Center.

show abstract

Section: Introductionmentioning

confidence: 99%

Unifying the Landscape of Cell-Probe Lower Bounds

Ptraşcu¹

2011

SIAM J. Comput.

View full text Add to dashboard Cite

show abstract

“…As a consequence, our lower bounds hold even if the algorithm makes nondeterministic cell probes, or if an all-powerful prover reveals a minimal set of cells sufficient to show that a certain answer to a query is correct. One possible framework for nondeterministic computation in the context of online data structures is defined in [12]. Our bounds also hold in this framework.…”

Section: General Frameworkmentioning

confidence: 93%

“…Hon et al [11] noticed that the lower bound of Beame and Fich [1] for the predecessor problem must also hold for select, if δ = b. A stronger bound of Ω(lg n/ lg b), which applies for any δ ≥ 1 has been established by Husfeldt and Rauhe [12], by extending the chronogram technique of Fredman and Saks [7]. Combining the techniques of this paper with the proof from Section 5.2 of [15] (which gives a similar lower bound for update and sum) yields a tight lower bound of Ω(lg n/ lg(b/δ)) for update and select.…”

Section: Appendixmentioning

confidence: 97%

Lower bounds for dynamic connectivity

Pătraşcu¹,

Demaine²

2004

Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

We prove an Ω(lg n) cell-probe lower bound on maintaining connectivity in dynamic graphs, as well as a more general trade-off between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight for these restricted cases, proving optimality of these data structures (e.g., Sleator and Tarjan's dynamic trees). Our trade-off is known to be tight for trees, and the best two data structures for dynamic connectivity in general graphs are points on our trade-off curve. In this sense these two data structures are optimal, and this tightness serves as strong evidence that our lower bounds are the best possible. From a more theoretical perspective, our result is the first logarithmic cell-probe lower bound for any problem in the natural class of dynamic language membership problems, breaking the long standing record of Ω(lg n/ lg lg n). In this sense, our result is the first data-structure lower bound that is "truly" logarithmic, i.e., logarithmic in the problem size counted in bits. Obtaining such a bound is listed as one of three major challenges for future research by Miltersen [13] (the other two challenges remain unsolved). Our techniques form a general framework for proving cell-probe lower bounds on dynamic data structures. We show how our framework also applies to the partial-sums problem to obtain a nearly complete understanding of the problem in cell-probe and algebraic models, solving several previously posed open problems.

show abstract

“…Lower Bounds: The partial sum problem for threshold functions [21] captures the essence of the dynamic range α-majority problem: maintain n bits x 1 , ..., x n subject to updates and threshold queries. An update consists of flipping the bit at a specified index.…”

mentioning

confidence: 99%

“…The answer to query threshold(i) is "yes" if and only if i j=1 x j ≥ f (i), where f (i) is an integer function such that f (i) ∈ {0, ..., ⌈i/2⌉}. Husfeldt and Rauhe proved a lower bound [21] on the query time t q for a data structure that can answer threshold queries with update time t u .…”

mentioning

confidence: 99%

Dynamic range majority data structures

Elmasry

Munro

et al. 2016

Theoretical Computer Science

View full text Add to dashboard Cite

Given a set P of n coloured points on the real line, we study the problem of answering range α-majority (or "heavy hitter") queries on P. More specifically, for a query range Q, we want to return each colour that is assigned to more than an α-fraction of the points contained in Q. We present a new data structure for answering range α-majority queries on a dynamic set of points, where α ∈ (0, 1). Our data structure uses O(n) space, supports queries in O((lg n)/α) time, and updates in O((lg n)/α) amortized time. If the coordinates of the points are integers, then the query time can be improved to O(lg n/(α lg lg n)). For constant values of α, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in d-dimensions, for d ≥ 2, as well as dynamic arrays, in which each entry is a colour. IntroductionMany problems in computational geometry deal with point sets that have information encoded as colours assigned to the points. In this paper, we design dynamic data structures for the range α-majority problem, in which we want to report colours that appear frequently within an axis-aligned query rectangle. This problem is useful in database applications in which we would like to know typical attributes of the data points in a query range [23,24]. For the one-dimensional case, where the points represent time stamps, this problem has data mining applications for network traffic logs, similar to those of coloured range counting (cf. [17]).Formally, we are given a set, P, of n points, where each point p ∈ P is assigned a colour c from a set, C, of colours. We denote the colour of p as col(p) = c. We are also given a fixed parameter α ∈ (0, 1), that defines the threshold for determining whether a colour is to be considered frequent. Our goal is to design a dynamic range α-majority data structure that can perform the following operations:-Query(Q): We are given an axis-aligned hyperrectangle Q as a query. Let P(Q) be the set {p | p ∈ P ∩Q}, and P(Q, c) be the set {p | p ∈ P(Q), col(p) = c}. The answer to the query Q is the set of colours C ⋆ such that for each colour c ∈ C ⋆ , |P(Q, c)| > α|P(Q)|, and for all c ∈ C ⋆ , |P(Q, c)| ≤ α|P(Q)|. We refer to a colour c ∈ C ⋆ as an α-majority for Q, and this type of query as an α-majority query. When α = 1/2, the problem is to identify the majority colour in Q, if such a colour exists. -Insert(p, c): Insert a point p with colour c into P. -Delete(p): Remove the point p from P. Previous WorkStatic and Dynamic Range α-Majority: In all of the following results, unless mentioned otherwise, the threshold α ∈ (0, 1) is fixed at construction time, rather than specified for each query individually.

show abstract

New Lower Bound Techniques for Dynamic Partial Sums and Related Problems

Cited by 15 publications

References 30 publications

Unifying the Landscape of Cell-Probe Lower Bounds

Unifying the Landscape of Cell-Probe Lower Bounds

Lower bounds for dynamic connectivity

Dynamic range majority data structures

Contact Info

Product

Resources

About