scholarly journals Complexity of Nondeterministic Functions

1994 ◽  
Vol 1 (2) ◽  
Author(s):  
Alexander E. Andreev
Keyword(s):  
Upper Bound ◽  
Randomized Algorithms ◽  
Nonempty Intersection ◽  
Linear Size ◽  
Worst Case ◽  
K Value ◽  
Hitting Set ◽  
Set Size

The complexity of a nondeterministic function is the minimum possible complexity of its determinisation. The entropy of a nondeterministic function, F, is minus the logarithm of the ratio between the number of determinisations of F and the number of all deterministic functions.<br /> <br />We obtain an upper bound on the complexity of a nondeterministic function with restricted entropy for the worst case.<br /> <br /> These bounds have strong applications in the problem of algorithm derandomization. A lot of randomized algorithms can be converted to deterministic ones if we have an effective hitting set with certain parameters (a set is hitting for a set system if it has a nonempty intersection with any set from the system).<br /> <br />Linial, Luby, Saks and Zuckerman (1993) constructed the best effective hitting set for the system of k-value, n-dimensional rectangles. The set size is polynomial in k log n / epsilon.<br /> <br />Our bounds of nondeterministic functions complexity offer a possibility to construct an effective hitting set for this system with almost linear size in k log n / epsilon.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

NEW WORST-CASE UPPER BOUND FOR COUNTING EXACT SATISFIABILITY

2014 ◽  
Vol 25 (06) ◽  
pp. 667-678 ◽  
Author(s):  
JUNPING ZHOU ◽  
WEIHUA SU ◽  
JIANAN WANG
Keyword(s):  
Upper Bound ◽  
Worst Case

The counting exact satisfiablity problem (#XSAT) is a problem that computes the number of truth assignments satisfying only one literal in each clause. This paper presents an algorithm that solves the #XSAT problem in O(1.1995n), which is faster than the best algorithm running in O(1.2190n), where n denotes the number of variables. To increase the efficiency of the algorithm, a new principle, called common literals principle, is addressed to simplify formulae. This allows us to further eliminate literals. In addition, we firstly apply the resolution principles into solving #XSAT problem, and therefore it improves the efficiency of the algorithm further.

scholarly journals The Complexity of Identifying Large Equivalence Classes

1998 ◽  
Vol 5 (34) ◽  
Author(s):  
Peter G. Binderup ◽  
Gudmund Skovbjerg Frandsen ◽  
Peter Bro Miltersen ◽  
Sven Skyum
Keyword(s):  
Upper Bound ◽  
Equivalence Classes ◽  
Worst Case ◽  
Equivalence Tests

<p>We prove that at least (3k−4) / k(2k−3) n(n-1)/2 − O(k) equivalence tests and no<br />more than 2/k n(n-1)/2 + O(n)<br />equivalence tests are needed in the worst case to identify the equivalence classes with at least k members in set of n elements. The upper bound is an improvement by a factor 2 compared to known results. For k = 3 we give tighter bounds. Finally, for k > n/2 we prove that it is necessary and it suffices to make 2n − k − 1 equivalence tests which generalizes a known result for k = [(n+1)/2] .</p>

scholarly journals On Compact Encoding of Pagenumber $k$ Graphs

2008 ◽  
Vol Vol. 10 no. 3 ◽  
Author(s):  
Cyril Gavoille ◽  
Nicolas Hanusse
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Time ◽  
Worst Case ◽  
Encoding Scheme ◽  
Information Theoretic ◽  
Auxiliary Table ◽  
Minimum Number ◽  
International Audience

International audience In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

Risk assessment of chronic dietary exposure to the conjugated mycotoxin deoxynivalenol-3-β-glucoside in the Dutch population

2015 ◽  
Vol 8 (5) ◽  
pp. 561-572 ◽  
Author(s):  
E.M. Janssen ◽  
R.C. Sprong ◽  
P.W. Wester ◽  
M. De Boevre ◽  
M.J.B. Mengelers
Keyword(s):  
Risk Assessment ◽  
Body Weight ◽  
The Netherlands ◽  
Food Consumption ◽  
Upper Bound ◽  
Chronic Exposure ◽  
Dietary Exposure ◽  
Food Products ◽  
Dutch Population ◽  
Worst Case

In this study, a risk assessment of dietary exposure to the conjugated mycotoxin deoxynivalenol-3-β-glucoside (DON-3G) in the Dutch population was conducted. Data on DON-3G levels in food products available in the Netherlands are scarce. Therefore, data on co-occurring levels of DON-3G and deoxynivalenol (DON), its parent compound, were used to estimate the DON-3G/DON ratio for several food product categories. This resulted in a DON-3G/DON ratio of 0.2 (90% confidence interval (CI): 0.04-0.9) in grains & grain-milling products, 0.3 (90% CI: 0.03-2.8) in grain-based products and 0.8 (90% CI: 0.4-1.8) in beer. These ratios were applied to the Dutch monitoring data of DON to estimate the DON-3G concentrations in food products available in the Netherlands. DON and DON-3G concentrations were combined with food consumption data of two Dutch National Food Consumption Surveys to assess chronic exposure in young children (2-6 years), children (7-16 years) and adults (17-69 years) using the Monte Carlo Risk Assessment program. The chronic exposure levels of DON, DON-3G and the sum of both compounds (DON+DON-3G) were compared to the tolerable daily intake (TDI) of 1 μg/kg body weight/day which is based on the most critical effect of DON, namely decreased body weight gain. The assumption was made that DON-3G is deconjugated and then fully absorbed as DON in the gastro-intestinal tract. Exposure (P97.5) of the population aged 7-16 years and 17-69 years to DON or DON-3G separately, did not exceed the TDI. However, exposure to upper bound levels of DON+DON-3G (i.e. worst-case scenario) in the same age categories (P97.5) exceeded the TDI with a maximum factor of 1.3. Exposure (P97.5) of the 2-6 year-olds to DON was close to the TDI. Within this group, exposure (P97.5) to upper bound levels of DON+DON-3G exceeded the TDI with not more than a factor 2.

scholarly journals On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold

2015 ◽  
Vol 58 (2) ◽  
pp. 306-316 ◽  
Author(s):  
Kaveh Khoshkhah ◽  
Manouchehr Zaker
Keyword(s):  
Polynomial Time ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Planar Graphs ◽  
Worst Case ◽  
Time Solution

AbstractLet G be a graph and let τ be an assignment of nonnegative integer thresholds to the vertices of G. A subset of vertices, D, is said to be a τ-dynamicmonopoly if V(G) can be partitioned into subsets D0 , D1, …, Dk such that D0 = D and for any i ∊ {0, . . . , k−1}, each vertex v in Di+1 has at least τ(v) neighbors in D0∪··· ∪Di. Denote the size of smallest τ-dynamicmonopoly by dynτ(G) and the average of thresholds in τ by τ. We show that the values of dynτ(G) over all assignments τ with the same average threshold is a continuous set of integers. For any positive number t, denote the maximum dynτ(G) taken over all threshold assignments τ with τ ≤ t, by Ldynt(G). In fact, Ldynt(G) shows the worst-case value of a dynamicmonopoly when the average threshold is a given number t. We investigate under what conditions on t, there exists an upper bound for Ldynt(G) of the form c|G|, where c < 1. Next, we show that Ldynt(G) is coNP-hard for planar graphs but has polynomial-time solution for forests.

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