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An exact, cache-localized algorithm for the sub-quadratic convolution of hypercubes
[article]

2016
*
arXiv
*
pre-print

Fast multidimensional convolution can be performed naively in quadratic time and can often be performed more efficiently via the Fourier transform; however, when the dimensionality is

arXiv:1608.00206v1
fatcat:bhqmio2ubvesxflce5ds4dj6le
*large*, these algorithms ... Embeddings in*hypercubes*can be paired with sub-quadratic*hypercube*convolution method to construct sub-quadratic algorithms for variants of vector convolution. ... This means that max-convolution*on**hypercubes*containing*integers*of bounded dynamic range can be performed exactly by using a value of p*large*enough that the relative error 1 − N −1 p drops low enough ...##
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Page 3948 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

An efficient

*multiple*-copy embedding of the cube-connected-cycles network within the*hypercube*is presented. ... In this paper a new technique is presented for mapping computa- tions into*hypercubes*. The method used speeds up the transfer of a*large*amount of data. ...##
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Fast, deterministic routing, on hypercubes, using small buffers

1990
*
IEEE transactions on computers
*

We identify nodes with

doi:10.1109/12.61048
fatcat:2joyondogbfydm3b3qd2maiw5i
*integers*by using the standard numbering scheme for a*hypercube*. ...*On*the other hand, it is very hard to build*hypercubes*of very*large*numbers of nodes, which indicates that we may never reach the point where the constants involved become unimportant. ...##
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Liminf Sets in Simultaneous Diophantine Approximation

2016
*
Journal de Théorie des Nombres de Bordeaux
*

*On*the logarithmic density of a subset of

*integers*. ...

*One*key-step in the proof of Theorem 1.1 is to approximate an infinite set of positive

*integers*by arbitrarily

*large*subsets, the size of a subset being measured by its exponent of convergence. ... only by

*integer*vectors (r, q) such that q is a

*multiple*of the

*integer*p provided it is

*large*enough. ...

##
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Liminf Sets in Simultaneous Diophantine Approximation
[article]

2013
*
arXiv
*
pre-print

Let Q be an infinite set of positive

arXiv:1308.3806v1
fatcat:zhdjhbmw3rgyxat4wtpakrppiu
*integers*. ... of µ k*on*each of these*hypercubes*being uniform. ...*One*key-step in the proof of Theorem 1 is to approach an infinite set of positive*integers*by arbitrarily*large*subsets, the size of a subset being measured by its exponent of convergence. ...##
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Vertex Ramsey Problems in the Hypercube

2012
*
SIAM Journal on Discrete Mathematics
*

If we 2-color the vertices of a

doi:10.1137/110832239
fatcat:n5iobe67yracpovkmrle2pjju4
*large**hypercube*what monochromatic substructures are we guaranteed to find? ... A key tool is a lemma which reduces questions concerning monochromatic configurations in the*hypercube*to questions about monochromatic translates of sets of*integers*. ... The*integer*c L (s, t) produced by this "tower of Ramsey numbers" is obviously rather*large*if s is*large*. ...##
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Vertex Ramsey problems in the hypercube
[article]

2012
*
arXiv
*
pre-print

If we 2-color the vertices of a

arXiv:1211.0168v1
fatcat:tybb3ztlmnfozihdg2gedifon4
*large**hypercube*what monochromatic substructures are we guaranteed to find? ... A key tool is a lemma which reduces questions concerning monochromatic configurations in the*hypercube*to questions about monochromatic translates of sets of*integers*. ... The*integer*c L (s, t) produced by this "tower of Ramsey numbers" is obviously rather*large*if s is*large*. ...##
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Decomposing the vertex set of a hypercube into isomorphic subgraphs
[article]

2016
*
arXiv
*
pre-print

Let G be an induced subgraph of the

arXiv:1611.02021v1
fatcat:prcn5wvm5jhmrhnarn7o3jm3sa
*hypercube*Q_k for some k. We show that if |G| is a power of 2 then, for sufficiciently*large*n, the vertex set of Q_n can be partitioned into induced copies of G. ... In fact, we prove a stronger statement: if X is a subset of {0,1}^k for some k and if |X| is a power of 2, then, for sufficiently*large*n, {0,1}^n can be partitioned into isometric copies of X. ... Let F and S be as above and let r be a positive*integer*. We say that F contains an r-partition of S if there exists a weight function*on*F for which every element of S has*multiplicity*r. ...##
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MIMD-Factorisation on hypercubes
[chapter]

1995
*
Lecture Notes in Computer Science
*

This paper describes the development and implementation of the MPQS factoring algorithm using

doi:10.1007/bfb0053454
fatcat:hvjzwjliknhddctlqc2munml3y
*multiple**hypercubes*customised to a MIMD parallel computer. ... The computationally most expensive steps ran*on*a Parsytec machine consisting of 1024 Inmos T805 microprocessors. General 100 decimal digit numbers can be factored in 1-2 days. ... MPQS*on*a*Hypercube*We assume familiarity with the '*Multiple*Polynomial Quadratic Sieve' (MPQS) algorithm [8, 11] and will sketch only the improved*hypercube*variation used by US. ...##
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Page 3202 of Mathematical Reviews Vol. , Issue 95f
[page]

1995
*
Mathematical Reviews
*

Our se- quence corresponds to a Hamiltonian path

*on*the n-dimensional*hypercube*C,,, where n is @(log N/loglog N). ... This yields a faster way of changing polynomials in the*multiple*polynomial quadratic sieve algorithm, since moving along the*hypercube*turns out to be very cheap.” ...##
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On the Density of Integer Points on the Generalised Markoff-Hurwitz and Dwork Hypersurfaces
[article]

2014
*
arXiv
*
pre-print

We use bounds of mixed character sums modulo a prime p to estimate the density of

arXiv:1404.5866v1
fatcat:gg3mngpzq5gubgl677hell44k4
*integer*points*on*the hypersurface f_1(x_1) + ... + f_n(x_n) =a x_1^k_1... x_n^k_n for some polynomials f_i ∈ Z[X], nonzero ...*integer*a and positive*integers*k_i i=1, ..., n. ... Let ν ≥ 1 be a fixed*integer*. Then for a sufficiently*large*positive*integer*T ≥ 3, for all but o(π(T )) primes p ≤ T and any*integers*u and h < p, we have the bound 1) . ...##
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Edge Sums Of De Bruijn Interconnection Networks

2003
*
International Journal of Computer Mathematics
*

This problem was introduced by Graham and Harary who presented the solution for

doi:10.1080/0020716031000087159a
fatcat:x7phmarbhfbzbah4o54rk4dlvm
*hypercubes*. ... An interconnection network is a highly symmetrical connected graph of order n nodes, size m edges, connectivity k and diameter d, where n and k are*large*but m and d are small. ... The*hypercube*Q n has for its nodes the set V n of all the binary sequences with n terms, two of which are adjacent whenever the sequences disagree in exactly*one*place. ...##
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On the Achromatic Number of Hypercubes

2000
*
Journal of combinatorial theory. Series B (Print)
*

For an m-dimensional

doi:10.1006/jctb.2000.1955
fatcat:fg6cticnzzeg3fjydmlnv3znfy
*hypercube*P m 2 we prove: There exist constants 0 < c 1 < c 2 , independent of m, such that c 1 (m2 m?1 ) 1=2 (P m 2 ) c 2 (m2 m?1 ) 1=2 : ... The achromatic number of a nite graph G, (G), is the maximum number of independent sets into which the vertex set may be partitioned, so that between any two parts there is at least*one*edge. ... The following theorem determines the achromatic number of a*hypercube*of a suciently*large*dimension up to a*multiplication*by a constant. Theorem 3. For a*hypercube*of dimension d, Q d : (2 ? ...##
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Decremental Scattering for Data Transport Between Host and Hypercube Nodes

1993
*
1993 International Conference on Parallel Processing - ICPP'93 Vol2
*

This strategy, based

doi:10.1109/icpp.1993.67
dblp:conf/icpp/SharmaW93
fatcat:3sg4knszcrfi5ivvxik24nfshi
*on*scatteeng*on**hypercubes*of decremental dimemiom, p e r f o r m much better than other sirraiegies known today. ... In this paper we study a new strategy for data: distribuUonfiom host to MIMD*hypercube*nodes. ... In this paper we derive a new strategy for host to*hypercube*node communication based*on**multiple*communications from host to decreasing dimension hypercubea. ...##
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Hybrid Parallel Bidirectional Sieve based on SMP Cluster
[article]

2012
*
arXiv
*
pre-print

The approach obtains huge speedup and efficiency

arXiv:1205.4883v1
fatcat:542sc5zinzgshmwfuxz26uls3u
*on*SMP Cluster. ... article, hybrid parallel bidirectional sieve method is implemented by SMP Cluster, the individual computational units joined together by the communication network, are usually shared-memory systems with*one*... As n gets*large*, an algorithm with exponential complexity quickly becomes impractical. ...
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