scholarly journals An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture

2021 ◽  
Author(s):  
Yuansi Chen
Keyword(s):  
Lower Bound ◽  
Thin Shell ◽  
Mixing Time ◽  
Lipschitz Functions ◽  
Concentration Inequalities ◽  
Mcmc Sampling ◽  
Sampling Algorithms ◽  
Time Bounds

AbstractWe prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency $$d^{-o_d(1)}$$ d - o d ( 1 ) . When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency $$d^{-1/4}$$ d - 1 / 4 . Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain’s slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concave measures and better mixing time bounds for MCMC sampling algorithms on log-concave measures.

PERMUTATION ROUTING AND SORTING ON THE RECONFIGURABLE MESH

1998 ◽  
Vol 09 (02) ◽  
pp. 199-211
Author(s):  
SANGUTHEVAR RAJASEKARAN ◽  
THEODORE MCKENDALL
Keyword(s):  
Lower Bound ◽  
Randomized Algorithms ◽  
Linear Array ◽  
Order Term ◽  
Routing Algorithm ◽  
Sorting Algorithm ◽  
Routing Problem ◽  
Reconfigurable Mesh ◽  
Permutation Routing ◽  
Time Bounds

In this paper we demonstrate the power of reconfiguration by presenting efficient randomized algorithms for both packet routing and sorting on a reconfigurable mesh connected computer. The run times of these algorithms are better than the best achievable time bounds on a conventional mesh. Many variations of the reconfigurable mesh can be found in the literature. We define yet another variation which we call as Mr. We also make use of the standard PARBUS model. We show that permutation routing problem can be solved on a linear array Mr of size n in [Formula: see text] steps, whereas n-1 is the best possible run time without reconfiguration. A trivial lower bound for routing on Mr will be [Formula: see text]. On the PARBUS linear array, n is a lower bound and hence any standard n-step routing algorithm will be optimal. We also show that permutation routing on an n×n reconfigurable mesh Mr can be done in time n+o(n) using a randomized algorithm or in time 1.25n+o(n) deterministically. In contrast, 2n-2 is the diameter of a conventional mesh and hence routing and sorting will need at least 2n-2 steps on a conventional mesh. A lower bound of [Formula: see text] is in effect for routing on the 2D mesh Mr as well. On the other hand, n is a lower bound for routing on the PARBUS and our algorithms have the same time bounds on the PARBUS as well. Thus our randomized routing algorithm is optimal upto a lower order term. In addition we show that the problem of sorting can be solved in randomized time n+o(n) on Mr as well as on PARBUS. Clearly, this sorting algorithm will be optimal on the PARBUS model. The time bounds of our randomized algorithms hold with high probability.

scholarly journals Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

2018 ◽  
Vol 28 (3) ◽  
pp. 365-387
Author(s):  
S. CANNON ◽  
D. A. LEVIN ◽  
A. STAUFFER
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Mixing Time ◽  
Perpendicular Bisector ◽  
Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

The Move-to-Front Rule: A Case Study for two Perfect Sampling Algorithms

1998 ◽  
Vol 12 (3) ◽  
pp. 283-302 ◽  
Author(s):  
James Allen Fill
Keyword(s):  
Finite Population ◽  
Mixing Time ◽  
Test Case ◽  
Perfect Sampling ◽  
Mcmc Algorithms ◽  
Time Result ◽  
Elementary Problem ◽  
Sampling Algorithms ◽  
Coupling Algorithm

The elementary problem of exhaustively sampling a finite population without replacement is used as a nonreversible test case for comparing two recently proposed MCMC algorithms for perfect sampling, one based on backward coupling and the other on strong stationary duality. The backward coupling algorithm runs faster in this case, but the duality-based algorithm is unbiased for user impatience. An interesting by-product of the analysis is a new and simple stochastic interpretation of a mixing-time result for the move-to-front rule.

Mixing time bounds for graphlet random walks

2019 ◽  
Vol 152 ◽  
pp. 105851 ◽  
Author(s):  
Matteo Agostini ◽  
Marco Bressan ◽  
Shahrzad Haddadan
Keyword(s):  
Random Walks ◽  
Mixing Time ◽  
Time Bounds

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

scholarly journals Mixing Time Bounds via the Spectral Profile

2006 ◽  
Vol 11 (0) ◽  
pp. 1-26 ◽  
Author(s):  
Sharad Goel ◽  
Ravi Montenegro ◽  
Prasad Tetali
Keyword(s):  
Mixing Time ◽  
Spectral Profile ◽  
Time Bounds

VLBI Observations of Supernova 1987A

1988 ◽  
Vol 129 ◽  
pp. 185-186
Author(s):  
I. Shapiro ◽  
N. Bartel ◽  
R. Preston ◽  
D. Jones ◽  
A. Kemball ◽  
...  
Keyword(s):  
Lower Bound ◽  
Thin Shell ◽  
Outer Diameter ◽  
Supernova 1987A ◽  
Vlbi Observations

VLBI observations at 2.3 GHz of SN1987A on 28 February 1987 yielded no fringes, implying, for an optically thin shell, a lower bound on the (outer) diameter of 1.9 mas. From the comparison of the VLBI and optical results, we infer that the radiosphere of SN1987A was either about equal to, or larger than, the photosphere of the supernova five days after the explosion.

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