scholarly journals Computing the Atom Graph of a Graph and the Union Join Graph of a Hypergraph

Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 347
Author(s):  
Anne Berry ◽  
Geneviève Simonet
Keyword(s):  
Real Number ◽  
Time Complexity ◽  
Matrix Multiplication ◽  
Efficient Algorithms ◽  
Join Graph ◽  
Current Value ◽  
Minimal Separator

The atom graph of a graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(nωlogn,nm,n(n+m¯)) time, where n is the number of vertices of G, m is the number of its edges, m¯ is the number of edges of the complement of G, and ω, also denoted by α in the literature, is a real number, such that O(nω) is the best known time complexity for matrix multiplication, whose current value is 2,3728596. This time complexity is no more than the time complexity of computing the atoms in the general case. We extend our results to α-acyclic hypergraphs, which are hypergraphs having at least one join tree, a join tree of an hypergraph being defined by its hyperedges in the same way as an atom tree of a graph is defined by its atoms. We introduce the notion of union join graph, which is the union of all possible join trees; we apply our algorithms for atom graphs to efficiently compute union join graphs.

scholarly journals On a New Formula for Fibonacci’s Family m-step Numbers and Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 805 ◽  
Author(s):  
Monther Rashed Alfuraidan ◽  
Ibrahim Nabeel Joudah
Keyword(s):  
Number Theory ◽  
Time Complexity ◽  
Matrix Multiplication ◽  
Square Roots ◽  
Computational Number Theory ◽  
The Matrix ◽  
Theory Application ◽  
Constant Coefficients ◽  
New Formula

In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recurrence m-step relations with constant coefficients by using the last few terms of its corresponding Fibonacci’s family m-step sequence. As a computational number theory application, we develop a method to estimate the square roots.

scholarly journals Efficient Algorithms for Optimal 4-Bit Reversible Logic System Synthesis

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiqiang Li ◽  
Hanwu Chen ◽  
Guowu Yang ◽  
Wenjie Liu
Keyword(s):  
Time Complexity ◽  
Hash Table ◽  
Lossless Compression ◽  
Logic Circuits ◽  
Reversible Logic ◽  
Efficient Algorithms ◽  
Toffoli Gate ◽  
Reversible Circuits ◽  
Almost All ◽  
Logic System

Owing to the exponential nature of the memory and run-time complexity, many methods can only synthesize 3-bit reversible circuits and cannot synthesize 4-bit reversible circuits well. We mainly absorb the ideas of our 3-bit synthesis algorithms based on hash table and present the efficient algorithms which can construct almost all optimal 4-bit reversible logic circuits with many types of gates and at mini-length cost based on constructing the shortest coding and the specific topological compression; thus, the lossless compression ratio of the space ofn-bit circuits reaches near2×n!. This paper presents the first work to create all 3120218828 optimal 4-bit reversible circuits with up to 8 gates for the CNT (Controlled-NOT gate, NOT gate, and Toffoli gate) library, and it can quickly achieve 16 steps through specific cascading created circuits.

scholarly journals Text Indexing for Regular Expression Matching

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 133
Author(s):  
Daniel Gibney ◽  
Sharma V. Thankachan
Keyword(s):  
Time Complexity ◽  
Regular Expression ◽  
Fundamental Problem ◽  
Matrix Multiplication ◽  
Query Time ◽  
Algorithmic Problem ◽  
Time And Space ◽  
Matrix Vector Multiplication ◽  
Deterministic Solution ◽  
Strongly Sublinear

Finding substrings of a text T that match a regular expression p is a fundamental problem. Despite being the subject of extensive research, no solution with a time complexity significantly better than O(|T||p|) has been found. Backurs and Indyk in FOCS 2016 established conditional lower bounds for the algorithmic problem based on the Strong Exponential Time Hypothesis that helps explain this difficulty. A natural question is whether we can improve the time complexity for matching the regular expression by preprocessing the text T? We show that conditioned on the Online Matrix–Vector Multiplication (OMv) conjecture, even with arbitrary polynomial preprocessing time, a regular expression query on a text cannot be answered in strongly sublinear time, i.e., O(|T|1−ε) for any ε>0. Furthermore, if we extend the OMv conjecture to a plausible conjecture regarding Boolean matrix multiplication with polynomial preprocessing time, which we call Online Matrix–Matrix Multiplication (OMM), we can strengthen this hardness result to there being no solution with a query time that is O(|T|3/2−ε). These results hold for alphabet sizes three or greater. We then provide data structures that answer queries in O(|T||p|τ) time where τ∈[1,|T|] is fixed at construction. These include a solution that works for all regular expressions with Expτ·|T| preprocessing time and space. For patterns containing only ‘concatenation’ and ‘or’ operators (the same type used in the hardness result), we provide (1) a deterministic solution which requires Expτ·|T|log2|T| preprocessing time and space, and (2) when |p|≤|T|z for z=2o(log|T|), a randomized solution with amortized query time which answers queries correctly with high probability, requiring Expτ·|T|2Ωlog|T| preprocessing time and space.

scholarly journals Reduction of programs execution time for tasks related to sequences or matrices

2020 ◽  
Vol 75 ◽  
pp. 04019
Author(s):  
Oleksandr Mitsa ◽  
Yurii Horoshko ◽  
Serhii Vapnichnyi
Keyword(s):  
Computer Science ◽  
Execution Time ◽  
Time Complexity ◽  
Matrix Multiplication ◽  
Matrix Form ◽  
The Third

The article discusses three approaches to reducing runtime of the programs, which are solutions of Olympiad tasks on computer science, related to sequences or matrices. The first approach is based on the representation of some sequences in matrix form and then the program of calculating the members of the sequence will have asymptotics equal to the time complexity of the exponentiation algorithm and will be O(log (n)). The second approach is to upgrade the known code to obtain significant reduction of the program runtime. This approach is very important to know for scientists who write code for scientific researches and are faced with matrix multiplication operations. The third approach is based on reducing time complexity by search for regularities; the author's task is presented and this approach is used to solve it.

Efficient Algorithms for Some Common Applications on GHCC

2005 ◽  
Vol 06 (04) ◽  
pp. 417-433
Author(s):  
Srabani Mukhopadhyaya ◽  
Bhabani P. Sinha
Keyword(s):  
Interconnection Network ◽  
Matrix Multiplication ◽  
Efficient Algorithms ◽  
Matrix Transpose ◽  
Generalized Hypercube

Generalized Hypercube-Connected-Cycles (GHCC), is a challenging interconnection network, proposed earlier in the literature. In this paper, we discuss how some important, useful algorithms, like, matrix transpose, matrix multiplication and sorting can efficiently be implemented on GHCC. Matrix transpose and matrix-by-matrix multiplication of matrices of order n × n, [Formula: see text], takes O(l) and [Formula: see text] time, respectively, on GHCC(l,m), with lml processors. Using the same number of processors, a list of ml numbers can be sorted in O(l2 log 3 m) time.

THE RECONFIGURABLE RING OF PROCESSORS: EFFICIENT ALGORITHMS VIA HYPERCUBE SIMULATION

1995 ◽  
Vol 05 (01) ◽  
pp. 37-48 ◽  
Author(s):  
ARNOLD L. ROSENBERG ◽  
VITTORIO SCARANO ◽  
RAMESH K. SITARAMAN
Keyword(s):  
Communication Protocol ◽  
Matrix Multiplication ◽  
Constant Factor ◽  
Efficient Algorithms ◽  
Parallel Computer ◽  
Computational Power ◽  
Processing Elements ◽  
Dynamically Reconfigurable ◽  
Small Constant ◽  
In Transit

We propose a design for, and investigate the computational power of a dynamically reconfigurable parallel computer that we call the Reconfigurable Ring of Processors ([Formula: see text], for short). The [Formula: see text] is a ring of identical processing elements (PEs) that are interconnected via a flexible multi-line reconfigurable bus, each of whose lines has one-packet width and can be configured, independently of the other lines, to establish an arbitrary PE-to-PE connection. A novel aspect of our design is a communication protocol we call COMET — for Cooperative MEssage Transmission — which allows PEs of an [Formula: see text] to exchange one-packet messages with latency that is logarithmic in the number of PEs the message passes over in transit. The main contribution of this paper is an algorithm that allows an N-PE, N-line [Formula: see text] to simulate an N-PE hypercube executing a normal algorithm, with slowdown less than 4 log log N, provided that the local state of a hypercube PE can be encoded and transmitted using a single packet. This simulation provides a rich class of efficient algorithms for the [Formula: see text], including algorithms for matrix multiplication, sorting, and the Fast Fourer Transform (often using fewer than N buslines). The resulting algorithms for the [Formula: see text] are often within a small constant factor of optimal.

scholarly journals EFFICIENT ALGORITHMS FOR HIGHLY COMPRESSED DATA: THE WORD PROBLEM IN HIGMAN'S GROUP IS IN P

2012 ◽  
Vol 22 (08) ◽  
pp. 1240008 ◽  
Author(s):  
VOLKER DIEKERT ◽  
JÜRN LAUN ◽  
ALEXANDER USHAKOV
Keyword(s):  
Word Problem ◽  
Time Complexity ◽  
Efficient Algorithms ◽  
Reduction Procedure ◽  
Algorithmic Algebra ◽  
Practical Algorithm ◽  
Power Circuits ◽  
The One ◽  
New Applications ◽  
Compressed Data

Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: (1) We define a modified reduction procedure on power circuits which runs in quadratic time, thereby improving the known cubic time complexity. The improvement is crucial for our other results. (2) We improve the complexity of the Word Problem for the Baumslag group to cubic time, thereby providing the first practical algorithm for that problem. (The algorithm has been implemented and is available in the CRAG library.) (3) The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.

Generalizations of Code Languages with Marginal Errors

2021 ◽  
pp. 1-21
Author(s):  
Sang-Ki Ko ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Time Complexity ◽  
Regular Language ◽  
Efficient Algorithms ◽  
Context Free ◽  
Free Language

The [Formula: see text]-prefix-free, [Formula: see text]-suffix-free and [Formula: see text]-infix-free languages generalize the prefix-free, suffix-free and infix-free languages by allowing marginal errors. For example, a string [Formula: see text] in a [Formula: see text]-prefix-free language [Formula: see text] can be a prefix of up to [Formula: see text] different strings in [Formula: see text]. We also define finitely prefix-free languages in which a string [Formula: see text] can be a prefix of finitely many strings. We present efficient algorithms that determine whether or not a given regular language is [Formula: see text]-prefix-free, [Formula: see text]-suffix-free or [Formula: see text]-infix-free, and analyze the time complexity of the algorithms. We establish undecidability results for deciding these properties for (linear) context-free languages.

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