On showing lower bounds for external-memory computational geometry problems

1999 ◽  
pp. 139-159 ◽  
Author(s):  
Lars Arge ◽  
Peter Miltersen
Keyword(s):  
Computational Geometry ◽  
Lower Bounds ◽  
External Memory

IMPROVED POINTER MACHINE AND I/O LOWER BOUNDS FOR SIMPLEX RANGE REPORTING AND RELATED PROBLEMS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 233-251 ◽  
Author(s):  
PEYMAN AFSHANI
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Computational Geometry ◽  
Lower Bounds ◽  
Data Structures ◽  
Dimensional Space ◽  
Block Size ◽  
External Memory ◽  
Geometric Transformation ◽  
The Right

We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems. Simplex range reporting is the problem of storing n points in the d-dimensional space in a data structure such that the k points that lie inside a query simplex can be found efficiently. This is one of the fundamental and extensively studied problems in computational geometry. Currently, the best data structures for the problem achieve Q(n) + O(k) query time using [Formula: see text] space in which the [Formula: see text] notation either hides a polylogarithmic or an nε factor for any constant ε > 0, (depending on the data structure and Q(n)). The best lower bound on this problem is due to Chazelle and Rosenberg who showed any pointer machine data structure that can answer queries in O(nγ + k) time must use Ω(nd-ε-dγ) space. Observe that this bound is a polynomial factor away from the best known data structures. In this article, we improve the space lower bound to [Formula: see text]. Not only this bridges the gap from polynomial to sub-polynomial, it also offers a smooth trade-off curve. For instance, for polylogarithmic values of Q(n), our space lower bound almost equals Ω((n/Q(n))d); the latter is generally believed to be the “right” bound. By a simple geometric transformation, we also improve the best lower bounds for the halfspace range reporting problem. Furthermore, we study the external memory model and offer a new simple framework for proving lower bounds in this model. We show that answering simplex range reporting queries with Q(n)+O(k/B) I/Os requires [Formula: see text]) space or [Formula: see text] blocks, in which B is the block size.

Lower bounds for processing data with few random accesses to external memory

2009 ◽  
Vol 56 (3) ◽  
pp. 1-58 ◽  
Author(s):  
Martin Grohe ◽  
André Hernich ◽  
Nicole Schweikardt
Keyword(s):  
Lower Bounds ◽  
External Memory ◽  
Processing Data

scholarly journals A General Lower Bound on the I/O-Complexity of Comparison-based Algorithms

1992 ◽  
Vol 21 (407) ◽  
Author(s):  
Lars Arge ◽  
Mikael Knudsen ◽  
Kirsten Larsen
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
External Memory ◽  
Optimal Algorithms ◽  
Internal Memory

We show a relationship between the number of comparisons and the number of I/O operations needed to solve a given problem. We work in a model, where the permitted operations are l/O-operations and comparisons of two records in internal memory. An I/O- operation swaps <em>B</em> records between external memory and the internal memory (capable of holding <em>M</em> records). An algorithm for this model is called an I/O-algorithm. The main result of this paper is the following: Given an I/O-algorithm that solves an n-record problem P_n using I/O(bar{x}) I/O's on the input bar{x}, there exists an ordinary comparison algorithm that uses no more than <em>n</em> logB + I/O(bar{x}) € T_{merge}(M-B, B) comparisons on input bar{x}. T_{merge}(n, m) denotes the number of comparisons needed to merge two sorted lists of size n and m, respectively. We use the result to show lower bounds on the number of I/O-operations needed to solve the problems of sorting, removing duplicates from a multiset and determining the mode (the most frequently occurring element in a multiset). Aggarwal and Vitter have shown that the sorting bound is tight. We show the same result for the two other problems, by providing optimal algorithms.

scholarly journals Lower Bounds for External Memory Integer Sorting via Network Coding

2021 ◽  
pp. STOC19-87-STOC19-111
Author(s):  
Alireza Farhadi ◽  
MohammadTaghi Hajiaghayi ◽  
Kasper Green Larsen ◽  
Elaine Shi
Keyword(s):  
Network Coding ◽  
Lower Bounds ◽  
External Memory ◽  
Integer Sorting

TIGHT QUANTUM BOUNDS FOR COMPUTATIONAL GEOMETRY PROBLEMS

2009 ◽  
Vol 07 (05) ◽  
pp. 935-947
Author(s):  
NILTON VOLPATO ◽  
ARNALDO MOURA
Keyword(s):  
Computational Geometry ◽  
Lower Bounds ◽  
Open Problem ◽  
Arbitrary Order ◽  
Order Relation ◽  
Upper Bounds ◽  
General Algorithm ◽  
Asymptotically Optimal ◽  
Optimal Bounds

We present new quantum lower bounds and upper bounds for several computational geometry problems. The bounds presented here improve on currently known results in a number of ways. We give asymptotically optimal bounds for one of the problems considered, and we provide, up to logarithmic factors, optimal bounds for a number of other problems and, in particular, we settle an open problem of Bahadur et al. Some of these new bounds are obtained using a general algorithm for finding a minimum pair over a given arbitrary order relation.

scholarly journals Lower bounds for external memory integer sorting via network coding

2020 ◽  
Vol 63 (10) ◽  
pp. 97-105
Author(s):  
Alireza Farhadi ◽  
Mohammad Taghi Hajiaghayi ◽  
Kasper Green Larsen ◽  
Elaine Shi
Keyword(s):  
Network Coding ◽  
Lower Bounds ◽  
External Memory ◽  
Integer Sorting
Sign in / Sign up

Export Citation Format

Share Document