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A geometric construction of a superconcentrator of depth 2

1984
*
Theoretical Computer Science
*

We

doi:10.1016/0304-3975(84)90032-x
fatcat:pjz5bpippvgqllxv3o2uhrn75u
*construct*an N-*superconcentrator**of**depth**2*, with 3 N"' +0( N """) edges, by essentially duplicating the lines vs. points incidence graph*of**a*projective plane. ... n lines in the space which interseu both*A*and 8. When*A*= B, this result reduces to*a*theorem*of*I?!_ Bruijn and Erdijs (1948).*Geometric*consrruction qf superconcentraror*of**depth**2*219 4. ...*A*characterization*of**depth*-*2**superconcentrators*Let G=( V, E) be*a*digraph*of**depth**2*, as in (1) . For X c I, Y c 0, define Proof. ...##
###
Expanders, sorting in rounds and superconcentrators of limited depth

1985
*
Proceedings of the seventeenth annual ACM symposium on Theory of computing - STOC '85
*

Using our graphs we can also

doi:10.1145/22145.22156
dblp:conf/stoc/Alon85
fatcat:pksf5i2bt5debldvwk2kfk57gm
*construct*efficient n-*superconcentrators**of*limited*depth*. ... For example, we*construct*an n*superconcentrator**of**depth*3 with O(n4j3) edges; better than the previous known results. ... Mcshularn [*2*/l]*constructed*explicitly an n-s.c.*of**depth*4.*Superconcentrators**of*limit&*depth*.*2*and size O(# ). The rcsulls*of*S&an, Duguid and LcCorre ...##
###
Self -Routing Superconcentrators

1996
*
Journal of computer and system sciences (Print)
*

If the best

doi:10.1006/jcss.1996.0005
fatcat:d26ddwr6dngepb7gple3et6ib4
*constructions*known (see Ajtai, Komlo s, and Szemere di [3, 4] ) are used for the sorting networks, we obtain*a**construction*for self-routing*superconcentrators**of*size O(n log n),*depth*O ... Pippenger [17] showed (by*a*direct randomized*construction*, without using concentrators as*a*building block) the existence*of*n-*superconcentrators*with size O(n),*depth*O(log n), and valence O(1). ...##
###
Notes on the Complexity of Switching Networks
[chapter]

2001
*
Network Theory and Applications
*

All linear

doi:10.1007/978-1-4613-0281-0_14
fatcat:xh3nzinvdrb27njxqmeas5z5je
*superconcentrators**constructed*above have logarithmic*depths*. Wigderson and Zuckerman (1999, [88] )*constructed**a*linear-sized*superconcentrator*with sub-logarithmic*depth*: . ... To mention*a*few, for example, Noga Alon (1986, [7] ) used*geometric*expanders (expanders*constructed*from finite geometry) to deduce*a*certain strengthening*of**a*theorem*of*de Bruijn and Erdős on the ... In the case*of*connectors with*a*given*depth*, it is clear that , de Bruijn, Erdős and Spencer (1974, [28] ), while solving*a*problem*of*van Lint (1973, [86] ), used*a*probabilistic argument to show ...##
###
Author index volume 32 (1984)

1984
*
Theoretical Computer Science
*

Weyhrauch,

doi:10.1016/0304-3975(84)90052-5
fatcat:g2qcqovkqjdxvddkl26beynyhi
*A*decidable fragment*of*predicate calculus Connections in acyclic hypergraphs McAloon, K., Petri nets and large finite sets Meshulam, R.,*A**geometric**construction**of**a**superconcentrator**of**depth*... Maier (l,*2*) 121-156 (1,2) 6l-76 (3 ) 279-295 (1,2) 77-86 (I,*2*) 121-156 (3) 227-247 (3) 309-3 I9 (1.2) 47-60 (l,*2*)*2*% 46 (I,*2*) 201-213 (3) 321-330 ... 339-340 (3) 331-337 (1.2) 87-120 (I,*2*) 185-199 0304-3';'75/84/$3.00*a*1984, Elst:vier Science Publishers B.V. (North-Holland) ...##
###
Page 2946 of Mathematical Reviews Vol. , Issue 83h
[page]

1983
*
Mathematical Reviews
*

“We

*construct**a*family*of*explicit graphs {G,} for n=m’, m= 1,2,---, and our main result is the following Theorem*2*: For n=m?, m=1,2,---, G, is an (n,5,d 9) expander, where d)= (*2*—V3)/4.” ...*A*family*of*linear concentrators [*superconcentrators*]*of*density k is*a*set*of*(n,m,k+o0(1)) concentrators [(n,k+0(1)) s.c.’s] for 1<m<n<oo [for 1<n<oo]. “G.*A*. ...##
###
On the complexity of bilinear forms

1995
*
Proceedings of the twenty-seventh annual ACM symposium on Theory of computing - STOC '95
*

We also thank Michael Ben-Or for helpful comments on an earlier version

doi:10.1145/225058.225290
dblp:conf/stoc/NisanW95
fatcat:vrnb35wfs5bypihomndgxiegea
*of*this paper. ... The proof uses small,*depth**2**superconcentrators*to*construct*matrices M in which all minors have high rank. ... n)*2*). • T S(l M ) = Ω(n*2*/(log n)*2*) Proof: For any m set t = 4 log m, n = mt, and G(V, E)*a**depth**2*m-*superconcentrator*. ...##
###
Page 2848 of Mathematical Reviews Vol. , Issue 86g
[page]

1986
*
Mathematical Reviews
*

*a*

*superconcentrator*

*of*

*depth*

*2*. ... Sci. 32 (1984), no. 1-

*2*, 215-219. Author summary: “We

*construct*an N-

*superconcentrator*

*of*

*depth*

*2*, with 3N°3/

*2*+ O(N'!7/! ...

##
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Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)

2013
*
Dagstuhl Reports
*

This seminar aims to capitalize on recent progress and bring together researchers who are using

doi:10.4230/dagrep.2.10.60
dblp:journals/dagstuhl-reports/AgrawalTU12
fatcat:dg7ithf6xfgadkzwkxjtjhy7ge
*a*diverse array*of*algebraic and combinatorial methods in*a*variety*of*settings. ...*A*prominent example is Dinur's proof*of*the PCP Theorem via gap amplification which yielded short PCPs with only*a*polylogarithmic length blowup (which had been the focus*of*significant research effort ... For*depth**2*, our Ω(n(log n/ log log n)*2*) lower bound gives the largest known lower bound for computing any linear map. ...##
###
Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory

1986
*
Combinatorica
*

The expansion properties

doi:10.1007/bf02579382
fatcat:yamx2oirgvfujmx3gp3ohja3qm
*of*the graphs are proved using the eigenvalues*of*their adjacency matrices. ... These graphs enable us to improve previous results on*a*parallel sorting problem that arises in structural modeling, by describing an explicit algorithm to sort n elements in k time units using O(n ~k) ... As mentioned in [3] the*geometric*expanders are useful also in explicit*constructions**of*efficient*superconcentrators**of*limited*depth*.*2*. ...##
###
Computationally Data-Independent Memory Hard Functions

2020
*
Innovations in Theoretical Computer Science
*

For any > 0 we show how to

doi:10.4230/lipics.itcs.2020.36
dblp:conf/innovations/AmeriBZ20
fatcat:sxg6xwp3dzemjflj6h77niupym
*construct**a*k-restricted dynamic graph with k = Ω(N 1− ) that provably achieves maximum cumulative pebbling cost Ω(N*2*). ... By contrast, the dMHF scrypt achieves maximal CMC Ω(N*2*) -though the CMC*of*scrypt would be reduced to just O (N ) after*a*side-channel attack. ... We will*construct**a*dynamic pebbling strategy*A*that for all times t, maintains*a**depth*-reducing set S t such that*depth*(G t − S t ) ≤ d, where G t is the portion*of*G revealed after running*A*for time ...##
###
Computationally Data-Independent Memory Hard Functions
[article]

2019
*
arXiv
*
pre-print

We then ask whether it is possible to circumvent known upper bound for iMHFs and build

arXiv:1911.06790v1
fatcat:w2h7v5abxjawphfauxwfw6tozm
*a*ciMHF with CMC Ω(N^*2*). ... Memory hard functions (MHFs) are an important cryptographic primitive that are used to design egalitarian proofs*of*work and in the*construction**of*moderately expensive key-derivation functions resistant ... This work was done in part while Samson Zhou was*a*postdoctoral fellow at Indiana University. ...##
###
Parameters of selective martite ores disintegration in structured ore bodies deposits by borehole hydraulic monitors

2018
*
E3S Web of Conferences
*

and obtaining

doi:10.1051/e3sconf/20186000032
fatcat:am7xpwootjg7xh35fcnko2dglm
*a*new kind*of*product -martite*superconcentrate*. ... The scientific novelty*of*the research consists in determining*a*criterion*of*hydrodisintegration*of*martites, conditions*of*forming*a*required fractional composition*of*monitor disintegration products ... This work was conducted within the projects "Determination*of*regularities*of*the stress-strain state*of*rocks disturbed by workings with the purpose*of*developing resource-saving ore mining technologies ...##
###
Invariant and geometric aspects of algebraic complexity theory I

1991
*
Journal of symbolic computation
*

The search for interesting lower bounds is

doi:10.1016/s0747-7171(08)80115-0
fatcat:mmwad3gp5rhrbgo23jbjwmfixa
*a*good reason to study the complexity*of*computation*of*linear forms. ... The analysis*of*linear algorithms leads naturally to questions about computational networks with their combinatorial aspects and about special configurations*of*sets*of*points in projective spaces which ... These questions will be discussed in another paper, following suggestions*of**A*. Hirschowitz and B. Sturmfels, whom I thank for their help as well as L. Baratchart, N. White and the referees. ...##
###
Complexity Lower Bounds using Linear Algebra

2007
*
Foundations and Trends® in Theoretical Computer Science
*

Acknowledgments I am grateful to the following people for their careful reading

doi:10.1561/0400000011
fatcat:jso5gaqnhnbitierw4ytzhixna
*of*the early versions*of*this paper and valuable feedback: Amit Deshpande, Seny Kamara, Neeraj Kayal, Meena Mahajan, Jayalal ... We note that in the proof*of*the lower bound on w*2*(*A*) using the lower bound on*depth*-*2**superconcentrators*, only connectivity properties*of*the factorization*A*= BC are used; associate*a**depth*-*2*graph ... Since the lower bound*of*[82] is tight for*depth*-*2**superconcentrators*, this approach cannot yield better lower bounds on w*2*(*A*). ...
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