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A geometric construction of a superconcentrator of depth 2

Roy Meshulam
1984 Theoretical Computer Science  
We 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.  ... 
doi:10.1016/0304-3975(84)90032-x fatcat:pjz5bpippvgqllxv3o2uhrn75u

Expanders, sorting in rounds and superconcentrators of limited depth

N Alon
1985 Proceedings of the seventeenth annual ACM symposium on Theory of computing - STOC '85  
Using our graphs we can also 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  ... 
doi:10.1145/22145.22156 dblp:conf/stoc/Alon85 fatcat:pksf5i2bt5debldvwk2kfk57gm

Self -Routing Superconcentrators

Nicholas Pippenger
1996 Journal of computer and system sciences (Print)  
If the best 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).  ... 
doi:10.1006/jcss.1996.0005 fatcat:d26ddwr6dngepb7gple3et6ib4

Notes on the Complexity of Switching Networks [chapter]

Hung Quang Ngo, Ding-Zhu Du
2001 Network Theory and Applications  
All linear 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  ... 
doi:10.1007/978-1-4613-0281-0_14 fatcat:xh3nzinvdrb27njxqmeas5z5je

Author index volume 32 (1984)

1984 Theoretical Computer Science  
Weyhrauch, 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)  ... 
doi:10.1016/0304-3975(84)90052-5 fatcat:g2qcqovkqjdxvddkl26beynyhi

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

Noam Nisan, Avi Wigderson
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 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.  ... 
doi:10.1145/225058.225290 dblp:conf/stoc/NisanW95 fatcat:vrnb35wfs5bypihomndgxiegea

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/!  ... 

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

Manindra Agrawal, Thomas Thierauf, Christopher Umans, Marc Herbstritt
2013 Dagstuhl Reports  
This seminar aims to capitalize on recent progress and bring together researchers who are using 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.  ... 
doi:10.4230/dagrep.2.10.60 dblp:journals/dagstuhl-reports/AgrawalTU12 fatcat:dg7ithf6xfgadkzwkxjtjhy7ge

Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory

Noga Alon
1986 Combinatorica  
The expansion properties 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.  ... 
doi:10.1007/bf02579382 fatcat:yamx2oirgvfujmx3gp3ohja3qm

Computationally Data-Independent Memory Hard Functions

Mohammad Hassan Ameri, Jeremiah Blocki, Samson Zhou, Michael Wagner
2020 Innovations in Theoretical Computer Science  
For any > 0 we show how to 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  ... 
doi:10.4230/lipics.itcs.2020.36 dblp:conf/innovations/AmeriBZ20 fatcat:sxg6xwp3dzemjflj6h77niupym

Computationally Data-Independent Memory Hard Functions [article]

Mohammad Hassan Ameri, Jeremiah Blocki, Samson Zhou
2019 arXiv   pre-print
We then ask whether it is possible to circumvent known upper bound for iMHFs and build 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.  ... 
arXiv:1911.06790v1 fatcat:w2h7v5abxjawphfauxwfw6tozm

Parameters of selective martite ores disintegration in structured ore bodies deposits by borehole hydraulic monitors

Mykola Stupnik, Viktor Tarasiutin, Pavlo Fedorenko, V. Bondarenko, I. Kovalevska, R. Lysenko, O. Malova, F. Cawood, M. Hardygora
2018 E3S Web of Conferences  
and obtaining 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  ... 
doi:10.1051/e3sconf/20186000032 fatcat:am7xpwootjg7xh35fcnko2dglm

Invariant and geometric aspects of algebraic complexity theory I

Jacques Morgensern
1991 Journal of symbolic computation  
The search for interesting lower bounds is 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.  ... 
doi:10.1016/s0747-7171(08)80115-0 fatcat:mmwad3gp5rhrbgo23jbjwmfixa

Complexity Lower Bounds using Linear Algebra

Satyanarayana V. Lokam
2007 Foundations and Trends® in Theoretical Computer Science  
Acknowledgments I am grateful to the following people for their careful reading 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).  ... 
doi:10.1561/0400000011 fatcat:jso5gaqnhnbitierw4ytzhixna
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