scholarly journals On Ajtai’s Lower Bound Technique for R-way Branching Programs and the Hamming Distance Problem

2000 ◽  
Vol 7 (11) ◽  
Author(s):  
Jakob Pagter
Keyword(s):  
Lower Bound ◽  
Hamming Distance ◽  
Linear Time ◽  
Program Model ◽  
Branching Programs ◽  
Original Proof ◽  
Trade Off ◽  
Time Space ◽  
Distance Problem ◽  
With Memory

In this report we study the proof employed by Miklos Ajtai<br />[Determinism versus Non-Determinism for Linear Time RAMs<br />with Memory Restrictions, 31st Symposium on Theory of <br />Computation (STOC), 1999] when proving a non-trivial lower bound<br />in a general model of computation for the Hamming Distance<br />problem: given n elements: decide whether any two of them have<br />"small" Hamming distance. Specifically, Ajtai was able to show<br />that any R-way branching program deciding this problem using<br />time O(n) must use space Omega(n lg n).<br />We generalize Ajtai's original proof allowing us to prove a<br />time-space trade-off for deciding the Hamming Distance problem<br /> in the R-way branching program model for time between n<br />and alpha n lg n / lg lg n, for some suitable 0 < alpha < 1. In particular we prove<br />that if space is O(n^(1−epsilon)), then time is Omega(n lg n / lg lg n).

scholarly journals Optimal Time-Space Trade-Offs for Non-Comparison-Based Sorting

2001 ◽  
Vol 8 (2) ◽  
Author(s):  
Rasmus Pagh ◽  
Jakob Pagter
Keyword(s):  
Lower Bound ◽  
High Probability ◽  
Fundamental Problem ◽  
Unit Cost ◽  
Priority Queue ◽  
Optimal Time ◽  
Trade Off ◽  
Sorting Algorithms ◽  
Time Space ◽  
Trade Offs

<p>We study the fundamental problem of sorting n integers of w bits on a unit-cost RAM with word size w, and in particular consider the time-space trade-off (product of time and space in bits) for this problem. For comparison-based algorithms, the time-space complexity is known to be Theta(n^2). A result of Beame shows that the lower bound also holds for non-comparison-based algorithms, but no algorithm has met this for time below the comparison-based <br />Omega(n lg n) lower bound. </p><p>We show that if sorting within some time bound T~ is possible, then time T = O(T~ + n lg* n) can be achieved with high probability using space S = O(n^2/T + w), which is optimal. Given a deterministic priority queue using amortized<br />time t(n) per operation and space n^O(1), we provide a deterministic<br />algorithm sorting in time T = O(n (t(n) + lg* n)) with S = O(n^2/T+w). Both results require that w <= n^(1-Omega(1)).</p><p>Using existing priority queues and sorting algorithms, this implies<br />that we can deterministically sort time-space optimally in time Theta(T) for T >= n(lg lg n)^2, and with high probability for T >= n lg lg n.</p><p>Our results imply that recent lower bounds for deciding element distinctness in o(n lg n) time are nearly tight.</p>

A Time-Space Trade-Off for Triangulations of Points in the Plane

2017 ◽  
pp. 3-12 ◽  
Author(s):  
Hee-Kap Ahn ◽  
Nicola Baraldo ◽  
Eunjin Oh ◽  
Francesco Silvestri
Keyword(s):  
Trade Off ◽  
Time Space

scholarly journals Optimal Time-Space Trade-Offs for Sorting

1998 ◽  
Vol 5 (10) ◽  
Author(s):  
Jakob Pagter ◽  
Theis Rauhe
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Fundamental Problem ◽  
Full Range ◽  
Optimal Time ◽  
Sorting Algorithm ◽  
Logarithmic Factor ◽  
Time Space ◽  
Trade Offs ◽  
Space Product

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem.<br />Beame has shown a lower bound of  Omega(n^2) for this product leaving a gap of a logarithmic factor up to the previously best known upper bound of O(n^2 log n) due to Frederickson. Since then, no progress has been made towards tightening this gap.<br />The main contribution of this paper is a comparison based sorting algorithm which closes this gap by meeting the lower bound of Beame. The time-space product O(n^2) upper bound holds for the full range of space bounds between log n and n/log n. Hence in this range our algorithm is optimal for comparison based models as well as for the very powerful general models considered by Beame.

scholarly journals Hamming Codes: Error Reducing Techniques

2021 ◽  
Vol 9 (11) ◽  
pp. 1972-1974
Author(s):  
Rohitkumar R Upadhyay
Keyword(s):  
Lower Bound ◽  
Error Correction ◽  
Coding Theory ◽  
Hamming Distance ◽  
Error Reduction ◽  
Error Correcting Codes ◽  
Significant Other ◽  
Hamming Codes ◽  
Decoding Algorithms ◽  
Reducing Properties

Abstract: Hamming codes for all intents and purposes are the first nontrivial family of error-correcting codes that can actually correct one error in a block of binary symbols, which literally is fairly significant. In this paper we definitely extend the notion of error correction to error-reduction and particularly present particularly several decoding methods with the particularly goal of improving the error-reducing capabilities of Hamming codes, which is quite significant. First, the error-reducing properties of Hamming codes with pretty standard decoding definitely are demonstrated and explored. We show a sort of lower bound on the definitely average number of errors present in a decoded message when two errors for the most part are introduced by the channel for for all intents and purposes general Hamming codes, which actually is quite significant. Other decoding algorithms are investigated experimentally, and it generally is definitely found that these algorithms for the most part improve the error reduction capabilities of Hamming codes beyond the aforementioned lower bound of for all intents and purposes standard decoding. Keywords: coding theory, hamming codes, hamming distance

The Principle and Structure of Novel High-Precision Linear Time Grating Displacement Sensors

2012 ◽  
Vol 236-237 ◽  
pp. 1216-1221
Author(s):  
Dong Lin Peng ◽  
Ji Sen Yang ◽  
Xi Hou Chen ◽  
Zi Ran Chen
Keyword(s):  
Linear Time ◽  
Theoretical Concept ◽  
Electrical Machine ◽  
Displacement Sensors ◽  
Time Space ◽  
Optical Grating ◽  
Testing Technology ◽  
Measurement And Testing ◽  
Space Coordinate ◽  
Physical And Mathematical Models

The existing grating type sensors such as optical grating sensor, have long been designed to rely on the precise mechanical space division technology, which is hard to develop without heavy investment. A theoretical concept, time-space coordinate transformation, was presented to realize measuring spatial displacement with time difference. Similar to the principle of circular time grating based on rotating electrical machine, linear time grating is designed based on the principle of linear motor, with which the physical and mathematical models of linear time grating are established. Based on these models linear time grating mechanical structure is designed, which has commercialization value. The resolution of linear time grating can achieve 0.1μm tested by National Institute of Measurement and Testing Technology.

scholarly journals A lower bound for read-once-only branching programs

1987 ◽  
Vol 35 (2) ◽  
pp. 153-162 ◽  
Author(s):  
László Babai ◽  
Péter Hajnal ◽  
Endre Szemerédi ◽  
György Turán
Keyword(s):  
Lower Bound ◽  
Branching Programs
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