scholarly journals Kolmogorov Basic Graphs and Their Application in Network Complexity Analysis

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1604
Author(s):  
Amirmohammad Farzaneh ◽  
Justin P. Coon ◽  
Mihai-Alin Badiu
Keyword(s):  
Upper Bound ◽  
Kolmogorov Complexity ◽  
Complexity Analysis ◽  
Networks And Graphs ◽  
Simple Graphs ◽  
Graph Compression ◽  
Graphs And Networks

Throughout the years, measuring the complexity of networks and graphs has been of great interest to scientists. The Kolmogorov complexity is known as one of the most important tools to measure the complexity of an object. We formalized a method to calculate an upper bound for the Kolmogorov complexity of graphs and networks. Firstly, the most simple graphs possible, those with O(1) Kolmogorov complexity, were identified. These graphs were then used to develop a method to estimate the complexity of a given graph. The proposed method utilizes the simple structures within a graph to capture its non-randomness. This method is able to capture features that make a network closer to the more non-random end of the spectrum. The resulting algorithm takes a graph as an input and outputs an upper bound to its Kolmogorov complexity. This could be applicable in, for example evaluating the performances of graph compression methods.

scholarly journals Cost Recurrences for DML Programs

2001 ◽  
Vol 8 (25) ◽  
Author(s):  
Bernd Grobauer
Keyword(s):  
Upper Bound ◽  
Complexity Analysis ◽  
Intermediate Step ◽  
Dependent Types ◽  
Conservative Extension ◽  
Running Time ◽  
Size Information ◽  
First Order

A cost recurrence describes an upper bound for the running time of a program in terms of the size of its input. Finding cost recurrences is a frequent intermediate step in complexity analysis, and this step requires an abstraction from data to data size. In this article, we use information contained in dependent types to achieve such an abstraction: Dependent ML (DML), a conservative extension of ML, provides dependent types that can be used to associate data with size information, thus describing a possible abstraction. We systematically extract cost recurrences from first-order DML programs, guiding the abstraction from data to data size with information contained in DML type derivations.

scholarly journals Low Complexity, Pairwise Layered Tabu Search for Large Scale MIMO Detection

2021 ◽  
Author(s):  
SOURAV CHAKRABORTY ◽  
Nirmalendu Bikas Sinha ◽  
Monojit Mitra
Keyword(s):  
Tabu Search ◽  
Upper Bound ◽  
Large Scale ◽  
Complexity Analysis ◽  
Mimo System ◽  
Multiple Input Multiple Output ◽  
Low Complexity ◽  
Detection Algorithm ◽  
Detection Performance ◽  
Mimo Detection

Abstract This paper presents a low complexity pairwise layered tabu search (PLTS) based detection algorithm for a large-scale multiple-input multiple-output (MIMO) system. The proposed algorithm can compute two layers simultaneously and reduce the effective number of tabu searches. A metric update strategy is developed to reuse the computations from past visited layers. Also, a precomputation technique is adapted to reduce the redundancy in computation within tabu search iterations. Complexity analysis shows that the upper bound of initialization complexity in the proposed algorithm reduces from O(Nt4) to O(Nt3). The detection performance of the proposed detector is almost the same as the conventional complex version of LTS for 64QAM and 16QAM modulations. However, the proposed detector outperforms the conventional system for 4QAM modulation, especially in 16x16 and 8x8 MIMO. Simulation results show that the per cent of complexity reduction in the proposed method is approximately 75% for 64x64, 64QAM and 85% for 64x64 16QAM systems to achieve a BER of 10-3. Moreover, we have proposed a layer-dependent iteration number that can further reduce the upper bound of complexity with minor degradation in detection performance.

scholarly journals The Use of Sholl and Kolmogorov Complexity Analysis in Researching on the Sustainable Development of Creative Economies in the Development Region of Bucharest‒Ilfov, Romania

2019 ◽  
Vol 11 (22) ◽  
pp. 6195
Author(s):  
Karina Andreea Gruia ◽  
Razvan-Cătălin Dobrea ◽  
Cezar-Petre Simion ◽  
Cristina Dima ◽  
Alexandra Grecu ◽  
...  
Keyword(s):  
Sustainable Development ◽  
Kolmogorov Complexity ◽  
Regional Economy ◽  
Complexity Analysis ◽  
Spatial Dynamics ◽  
Spatial Behaviour ◽  
The Sustainable Development ◽  
Development Region ◽  
The Government

Nowadays, creative economies stand as a relevant indicator of the sustainable development of local and regional ones. The study aims to highlight the spatial behaviour of creative economies in the Bucharest‒Ilfov Development Region, the most dynamic and complex regional economy in Romania. In order to assess the spatial dynamics of creative economies in the region, an economic database was created, at the level of the territorial administrative unit, for the two economic indicators considered important for the study, number of employees and turnover, under the auspices of the Classification of National Economy Activities (NACE). The establishment of creative economies was made following the Government Decision no. 859 of 2014, with 66 codes for this sector. Annual cartographic models were developed for each indicator in QGIS (a free and open–source cross–platform desktop geographic information system application that supports viewing, editing, and analysis of geospatial data), for the period 2000–2016. For a relevant analysis of spatial behaviour, we used Sholl and Kolmogorov complexity, which highlighted specific patterns of spatial dynamics that help us to understand the role of creative economies in the sustainable development of regional economies. The results highlighted the role of accessibility corridors in the development of the regional economy.

scholarly journals COMPLEXITY OF COMPLEXITY AND STRINGS WITH MAXIMAL PLAIN AND PREFIX KOLMOGOROV COMPLEXITY

2014 ◽  
Vol 79 (2) ◽  
pp. 620-632 ◽  
Author(s):  
B. BAUWENS ◽  
A. SHEN
Keyword(s):  
Upper Bound ◽  
Kolmogorov Complexity ◽  
Short Proof ◽  
Simple Game ◽  
Tight Bound ◽  
Image Position ◽  
Conditional Complexity

AbstractPéter Gács showed (Gács 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., $C\left( {C\left( x \right)|x} \right) \ge {\rm{log}}n - {\rm{log}}^{\left( 2 \right)} n - O\left( 1 \right)$ (Here ${\rm{log}}^{\left( 2 \right)} i = {\rm{loglog}}i$.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound ${\rm{log}}n - O\left( 1 \right)$ We also show the same bound for prefix-free complexity.Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that $|x| - C\left( x \right) \le c$ and $|x| + K\left( {|x|} \right) - K\left( x \right) \ge {\rm{log}}^{\left( 2 \right)} |x| - c{\rm{log}}^{\left( 3 \right)} |x|$ In fact, the results of Solovay and Gács are closely related. Using the result above, we provide a short proof for Solovay’s result. We also generalize it by showing that for some c and for all n there are strings x of length n with $n - C\left( x \right) \le c$ and $n + K\left( n \right) - K\left( x \right) \ge K\left( {K\left( n \right)|n} \right) - 3K\left( {K\left( {K\left( n \right)|n} \right)|n} \right) - c.$ We also prove a close upper bound $K\left( {K\left( n \right)|n} \right) + O\left( 1 \right)$Finally, we provide a direct game proof for Joseph Miller’s generalization (Miller 2006) of the same Solovay’s theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that$|x| + K\left( {|x|} \right) - K\left( x \right) \ge {\rm{log}}^{\left( 2 \right)} |x| - O\left( {{\rm{log}}^{\left( 3 \right)} |x|} \right).$

scholarly journals Upper bound for the energy of strongly connected digraphs

2011 ◽  
Vol 5 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Singaraj Ayyaswamy ◽  
Selvaraj Balachandran ◽  
Ivan Gutman
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Strongly Regular Graph ◽  
Simple Graphs ◽  
Strongly Connected Digraph ◽  
Complex Eigenvalues ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Strongly Regular ◽  
Appl Math

The energy of a digraph D is defined as E(D) = ?n,i=1 ?Re(zi)?, where z1, z2, ..., zn are the (possibly complex) eigenvalues of D . We show that if D is a strongly connected digraph on n vertices, a arcs, and c2 closed walks of length two, such that Re(z1) ? (a + c2)=(2n) ? 1 , then E(D) ? n(1 + ?n)=2. Equality holds if and only if D is a directed strongly regular graph with parameters (n, n+?n/2, 3n+2?n/8, n+2?n/8, n+2?n/8). This bound extends to digraphs an earlier result [J. H. Koolen, V. Moulton:, Maximal energy graphs. Adv. Appl. Math., 26 (2001), 47-52], obtained for simple graphs.

Algorithms for the Generalized NTRU Equations and their Storage Analysis

2020 ◽  
Vol 177 (2) ◽  
pp. 115-139
Author(s):  
Gook Hwa Cho ◽  
Seongan Lim ◽  
Hyang-Sook Lee
Keyword(s):  
Upper Bound ◽  
Complexity Analysis ◽  
Hierarchical Level ◽  
Storage Requirement ◽  
Mathematical Properties ◽  
Identity Based Encryption ◽  
Identity Based

In LATTE, a lattice based hierarchical identity-based encryption (HIBE) scheme, each hierarchical level user delegates a trapdoor basis to the next level by solving a generalized NTRU equation of level ℓ ≥ 3. For ℓ = 2, Howgrave-Graham, Pipher, Silverman, and Whyte presented an algorithm using resultant and Pornin and Prest presented an algorithm using a field norm with complexity analysis. Even though their ideas of solving NTRU equations can be conceptually extended for ℓ ≥ 3, no explicit algorithmic extensions with the storage analysis are known so far. In this paper, we interpret the generalized NTRU equation as the determinant of a matrix. By using the mathematical properties of the determinant, we show that how to construct algorithms for solving the generalized NTRU equation either using resultant or a field norm for any ℓ ≥ 3. We also obtain an upper bound of the size of solutions by using the properties of the determinant. From our analysis, the storage requirement of the algorithm using resultant is O(ℓ2n2 logB) and that of the algorithm using a field norm is O(ℓ2n logB), where B is an upper bound of the coefficients of the input polynomials of the generalized NTRU equations. We present examples of our algorithms for ℓ = 3 and the average storage requirements for ℓ = 3; 4.

scholarly journals Representing prefix and border tables: results on enumeration

2015 ◽  
Vol 27 (2) ◽  
pp. 257-276
Author(s):  
JULIEN CLÉMENT ◽  
LAURA GIAMBRUNO
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Complexity Analysis ◽  
Minimal Size ◽  
Golden Mean ◽  
The Real ◽  
Maximal Size ◽  
Linear Algorithms

For some text algorithms, the real measure for the complexity analysis is not the string itself but its structure stored in its prefix table or equivalently border table. In this paper, we define the combinatorial class of prefix lists, namely a sequence of integers together with their size, and an injection ψ from the class of prefix tables to the class of prefix lists. We call a valid prefix list the image by ψ of a prefix table. In particular, we describe algorithms converting a prefix/border table to a prefix list and inverse linear algorithms from computing from a prefix list L = ψ(P) two words respectively in a minimal size alphabet and on a maximal size alphabet with P as prefix table. We then give a new upper bound on the number of prefix tables for strings of length n (on any alphabet) which is of order (1 + ϕ)n (with $\varphi=\frac{1+\sqrt{5}}{2}$ the golden mean) and also present a corresponding lower bound.

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