A nearly optimal time-space lower bound for directed st-connectivity on the NNJAG model

Author(s):  
Jeff Edmonds ◽  
Chung-Keung Poon
Keyword(s):  
1998 ◽  
Vol 5 (10) ◽  
Author(s):  
Jakob Pagter ◽  
Theis Rauhe

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.


2001 ◽  
Vol 8 (2) ◽  
Author(s):  
Rasmus Pagh ◽  
Jakob Pagter

<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>


Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T17-T40 ◽  
Author(s):  
Zhiming Ren ◽  
Yang Liu

Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbing boundary condition was applied to remove artificial boundary reflections. We compared our optimal SFD with the conventional, TE-based time-space-domain, and LS-based SFD schemes. Dispersion analysis and numerical simulation results suggested that the new SFD schemes had a smaller numerical dispersion than the other three schemes when the same operator lengths were adopted. In addition, our LS-based time-space-domain SFD can obtain the same modeling accuracy with shorter spatial operator lengths. We also derived the stability condition of our schemes. The experiment results revealed that our new LS-based SFD schemes needed a slightly stricter stability condition.


1994 ◽  
Vol 23 (5) ◽  
pp. 966-975 ◽  
Author(s):  
Andrew Chi-Chih Yao
Keyword(s):  

2000 ◽  
Vol 7 (11) ◽  
Author(s):  
Jakob Pagter

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).


2020 ◽  
Vol 34 ◽  
pp. 151-170
Author(s):  
Costel Cioancă

"Semiotics of The Right To Dream: Mythical Dimensions of Time From The Romanian Fantastic Fairy Tale A fundamental concept of human existence as a species, Time has always been a defining landmark of the depth of thought of homo sapiens. With religious or scientific character, the ensemble of myths, beliefs, ideas, ideologies, representations and significances given to this concept led, …in time, to the birth of a rich, distinct and complex symbolic imaginary. Both a physical continuum (space-time in which biological, chemical, physical and mechanical processes occur that cause changes in Nature) and a philosophical one (events are perceived and cognitively systematized by man from the past to present towards future), Time it has always aroused peoples interest. We have deities of Time (Cronus, Zurvan, Maku). We have, also, the characteristic concepts that mark the fictional-mythical transfiguration and the triumph of the irreducible search for the truth of meaning. Such as the promise of a (possible) paradisiacal land of eternity, such as the Aion concept of the ancient Greeks (in the sense of cyclical time/eternity), or the existence of specialized divine beings (Moirs of ancient Greece, Roman Parce or Scandinavian Norns) who measure the profane time (past, present, future), and relates it to the celestial, relativistic, perpetual-eternal time. From the area of pure philosophy who approache the subject, inevitably passing through the field of quantum physics that tries to define as precisely as possible the notion of Time (definition, dimensions, units of measurement etc.), we have approaches to this concept at the level of music, literature, art. An true illud tempus, moving the content from metaphysics to myth, and viceversa, there are the many reflections of famous people about the concept of time. Approaching topics about the existence of ,,fashionable” references (billionaires, famous or just controversial politicians, footballers, actors, etc.), we have a post-modern mythological imaginary offered daily by Time Magazine, The New York Times, The Sunday Times etc. As well, being an important landmark in the editing policy, we have a font agreed by more and more magazines, periodicals, publishing houses - Times New Roman. The social life of the traditional Romanian communities, who generates and consumes fantastic fairy tales, tried to reconcile the human activities with the constant phenomena of the environment (terrestrial, cosmic). Starting from certain constants that counted human activity with the cosmic and terrestrial rhythms of Nature (day-night succession; the succession of seasons; the rhythmicity of some manifestations of the vegetal and animal kingdom), the calendars had appeared lunar, solar, solar-lunar, popular, Christian, civil). Their existence and use made that the passage of time to be more easily perceived and memorized. The calendar practices and habits, performed in a predetermined time and in a certain way (= ritual), did nothing but mark in the traditional symbolic thinking the specificity of that human time, to perform in that tradition, in Cosmic Time, trans-human time. This study deal with the valorizations and symbolism given to this concept by the popular imagination from Romanian fantastic fairy tale. The collections of fairy tales offered me some major directions that defines Time, sometimes the traditional imagination being a subtle game of physical constants and mythical-epic variations. Thus, I discovered metaphysical dimensions of time, the reason for linking Time, an optimal time of action, but also the exercise of distance (Time-Space) to be traveled by the hero or realms of eternity, where Time does not even exist as an abstraction… Everything followed, naturally, by a series of conclusions. Keywords: imaginary, phenomenology, hermeneutics, Romanian fairytale, Time "


2016 ◽  
Vol 24 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Hongyong Yan ◽  
Lei Yang ◽  
Xiang-Yang Li ◽  
Hong Liu

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.


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