## Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications (1993)

Venue: | Combinatorica |

Citations: | 89 - 26 self |

### BibTeX

@ARTICLE{Wigderson93expandersthat,

author = {Avi Wigderson and David Zuckerman},

title = {Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications},

journal = {Combinatorica},

year = {1993},

volume = {19},

pages = {245--251}

}

### Years of Citing Articles

### OpenURL

### Abstract

For every n and 0 ! ffi ! 1, we construct graphs on n nodes such that every two sets of size n ffi share an edge, having essentially optimal maximum degree n 1\Gammaffi+o(1) . Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n 1+1=k+o(1) comparisons. 2. A k round selection algorithm using n 1+1=(2 k \Gamma1)+o(1) comparisons. 3. A depth 2 superconcentrator of size n 1+o(1) . 4. A depth k wide-sense nonblocking generalized connector of size n 1+1=k+o(1) . All of these results improve on previous constructions by factors of n\Omega\Gamma37 , and are optimal to within factors of n o(1) . These results are based on an improvement to the extractor construction of Nisan & Zuckerman: our algorithm extracts an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits. 1

### Citations

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Citation Context ...phs on n nodes with maximum degree n 1\Gammaffi+o(1) . Remark: In fact, our n o(1) factors will be bounded by exp((log n) 2=3+o(1) ). Our result is obtained by improving the extractor construction of =-=[NZ]-=-. The motivation for extractors is that there are many fast and useful randomized algorithms. The extractor allows us to compute efficiently if the random source is defective, as long as we have a sma... |

111 | Sorting in c log n parallel steps - Ajtai, Komlós, et al. - 1983 |

106 | Simulating BPP Using a General Weak Random Source - Zuckerman - 1996 |

105 |
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Citation Context ...here is still a gap to close. Sorting and Selecting in Rounds Sorting and selecting in rounds has been an area of intensive study. This is the worst-case complexity in Valiant's comparison-tree model =-=[Val1]-=- using a constant number k of rounds. For sorting,\Omega\Gamma n 1+1=k (log n) 1=k ) comparisons are necessary [AA], and O(n 1+1=k log n) comparisons are sufficient [BT]. This last result, however, is... |

98 | Explicit construction of linear sized superconcentrators - GABBER, GALIL - 1981 |

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67 | A 1, isoperimetric inequalities for graphs and superconcentrators - Alon, Milman - 1985 |

60 | On extracting randomness from weak random sources
- Ta-Shma
- 1996
(Show Context)
Citation Context ... Subsequent Work Subsequent to this work, the n o(1) factors have been improved twice [SZ, TS96], by constructing stronger extractors and applying our methods. In the most recent improvement, Ta-Shma =-=[TS96]-=- obtained expanders where the n o(1) factors are exp((log log n) O(1) ). Hence all the applications have these new n o(1) factors and the depth of the linear-sized superconcentrator is (log log n) O(1... |

56 |
General weak random sources
- Zuckerman
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(Show Context)
Citation Context ...on't have any truly random bits, we can cycle through all possibilities -- see [NZ, Zuc2] for more details.) Our model for defective random source will essentially be the most general: Definition 1.3 =-=[Zuc1]-=- A distribution D on f0; 1g n is called a ffi-source if for all x 2 f0; 1g n , D(x)s2 \Gammaffin . Note that a particular type of ffi-source is the uniform distribution on a subset A ` f0; 1g n , jAjs... |

53 | Explicit construction of a concentrator - Margulis - 1973 |

51 | Security Preserving Amplification of Hardness - Goldreich, Impagliazzo, et al. |

49 | Eigenvalues and expansion of regular graphs
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(Show Context)
Citation Context ...s not tight. For a random d-regular graph, small sets S have roughly (d \Gamma 1)jSj neighbors, yet bounding the second eigenvalue can only be used to show the existence of roughly (d=2)jSj neighbors =-=[Kah]-=-. The situation gets much worse for larger degree and stronger expansion. A definition that captures such strong expansion is: Definition 1.1 [Pip3] An undirected graph is a-expanding if any two disjo... |

45 | Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6
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- 1986
(Show Context)
Citation Context ...ave been explicitly constructed (e.g. [GG]), these all have logarithmic depth. The best known explicit constructions for depth 2 is O(n 3=2 ) [Mes], and for depth 2k + 1 are of size O(n (k+3)=(k+2) ) =-=[Alo1]-=-. On the other hand, non-explicit constructions were known of size O(n log 2 n) for depth 2 [Pip2], and O(n(k; n)) for depth 2k, ks2, for an extremely slowly growing (k; n) (e.g. (2; n) = log n) [DDPW... |

45 |
Explicit construction of concentrators from generalized N-guns
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(Show Context)
Citation Context ... upon ideas in [Zuc1, Zuc2]. Indeed, the explicit construction of expanders that beat the eigenvalue bound in a different scenario were first obtained in [Zuc1]. 1 Actually, using Tanner's inequality =-=[Tan]-=-, it suffices to have the degree slightly less: dasn=(1 + =d). 2 1.2 Applications Our graphs improve many explicit constructions. In all cases, our results improve upon previous constructions by facto... |

39 | Graph-theoretic properties in computational complexity - Valiant - 1976 |

37 |
Sorting and selecting in rounds
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(Show Context)
Citation Context ... to show the existence of roughly (d=2)jSj neighbors [Kah]. The situation gets much worse for larger degree and stronger expansion. A definition that captures such strong expansion is: Definition 1.1 =-=[Pip3]-=- An undirected graph is a-expanding if any two disjoint sets of vertices, each containing at least a vertices, are joined by an edge. Equivalently, every set with a vertices has more than n \Gamma a n... |

32 |
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Citation Context ...Alo1]. On the other hand, non-explicit constructions were known of size O(n log 2 n) for depth 2 [Pip2], and O(n(k; n)) for depth 2k, ks2, for an extremely slowly growing (k; n) (e.g. (2; n) = log n) =-=[DDPW]-=-. Here, we give an explicit construction for depth 2 of size O (n). This is our biggest improvement: a factor of O ( p n). We use this construction to give the first explicit construction of a linear-... |

26 |
Wide-sense nonblocking networks
- Feldman, Friedman, et al.
- 1988
(Show Context)
Citation Context ...ually built with expanders, with the exception of [Mor]. The best previous construction of depth 1 (n; n ffi ; \Omega\Gamma n ffi )) concentrators has size 3 O(n 1+minfffi=2;(1\Gammaffi)g ) (see e.g. =-=[FFP]-=-). Here we construct a generalization of these concentrators with size O (n). We use this generalized construction to give a construction of wide-sense nonblocking generalized connectors. To motivate ... |

22 | Superconcentrators
- Pippenger
- 1977
(Show Context)
Citation Context ...and depth k, then there are explicitly-constructible n-superconcentrators of linear size and depth k +O(log a). Proof: (Sketch) Use the recursive superconcentrator construction developed by Pippenger =-=[Pip1]-=-. After O(log a) levels, we need an n=a-superconcentrator. Assuming a = a(n) is a non-decreasing function of n, we use the n=a-superconcentrator of size at most n. 2 Theorem 4.8 For all n, there are e... |

18 |
Rearrangeable networks with limited depth
- PIPPENGER, YAO
- 1982
(Show Context)
Citation Context ....al. [FFP] gave non-explicit constructions for depth k wide-sense nonblocking generalized connectors of size O(n 1+1=k (log n) 1\Gamma1=k ), essentially matching the\Omega\Gamma n 1+1=k ) lower bound =-=[PY]-=-. They also gave explicit constructions for depth 2 of size O(n 5=3 ), for depth 3 of size O(n 11=7 ), and for depth k of size O(n 1+2=k ). Here we give an explicit construction for depth k of size O ... |

16 |
On the second eigenvalue and linear expansion of regular graphs
- Kahale
- 1992
(Show Context)
Citation Context ...s not tight. For a random d-regular graph, small sets S have roughly (d \Gamma 1)jSj neighbors, yet bounding the second eigenvalue can only be used to show the existence of roughly (d=2)jSj neighbors =-=[Kah]-=-. The situation gets much worse for larger degree and stronger expansion. A definition that captures such strong expansion is: Definition 1.1 [Pip3] An undirected graph is a-expanding if any two disjo... |

12 |
Superconcentrators of depth 2
- Pippenger
- 1982
(Show Context)
Citation Context ...icit constructions for depth 2 is O(n 3=2 ) [Mes], and for depth 2k + 1 are of size O(n (k+3)=(k+2) ) [Alo1]. On the other hand, non-explicit constructions were known of size O(n log 2 n) for depth 2 =-=[Pip2]-=-, and O(n(k; n)) for depth 2k, ks2, for an extremely slowly growing (k; n) (e.g. (2; n) = log n) [DDPW]. Here, we give an explicit construction for depth 2 of size O (n). This is our biggest improveme... |

11 |
Parallel sorting
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(Show Context)
Citation Context ...iant's comparison-tree model [Val1] using a constant number k of rounds. For sorting,\Omega\Gamma n 1+1=k (log n) 1=k ) comparisons are necessary [AA], and O(n 1+1=k log n) comparisons are sufficient =-=[BT]-=-. This last result, however, is non-constructive. Pippenger [Pip3] showed a slightly worse non-explicit construction of O(n 1+1=k (log n) 2\Gamma2=k ), but his construction depends only on the existen... |

9 | Parallel selection - AZAR, PIPPENGER - 1990 |

9 |
A geometric construction of a superconcentrator of depth 2, Theoret
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- 1984
(Show Context)
Citation Context ...g expander graphs. While linear-sized superconcentrators have been explicitly constructed (e.g. [GG]), these all have logarithmic depth. The best known explicit constructions for depth 2 is O(n 3=2 ) =-=[Mes]-=-, and for depth 2k + 1 are of size O(n (k+3)=(k+2) ) [Alo1]. On the other hand, non-explicit constructions were known of size O(n log 2 n) for depth 2 [Pip2], and O(n(k; n)) for depth 2k, ks2, for an ... |

8 |
sorting in rounds and superconcentrators of limited depth
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- 1985
(Show Context)
Citation Context ...ave been explicitly constructed (e.g. [GG]), these all have logarithmic depth. The best known explicit constructions for depth 2 is O(n 3=2 ) [Mes], and for depth 2k + 1 are of size O(n (k+3)=(k+2) ) =-=[Alo1]-=-. On the other hand, non-explicit constructions were known of size O(n log 2 n) for depth 2 [Pip2], and O(n(k; n)) for depth 2k, ks2, for an extremely slowly growing (k; n) (e.g. (2; n) = log n) [DDPW... |

7 |
Almost sorting in one round
- Ajtai, Koml'os, et al.
- 1989
(Show Context)
Citation Context ... O (n 1+1=k ) and O (n 1+1=(2 k \Gamma1) ) comparisons, respectively. Proof: Use Lemma 3.1 with the graphs constructed in Theorem 1.2. 2 The following lemma about almost-sorting in 1 round appears in =-=[AKSS]-=-: Lemma 3.3 [AKSS] If G is an a-expanding graph, then after performing the comparisons according to G, all relations will be known except for O(an log n). This immediately gives: Theorem 3.4 There are... |

7 |
approximate sorting and searching in rounds
- Alon, Azar, et al.
- 1988
(Show Context)
Citation Context ...ensive study. This is the worst-case complexity in Valiant's comparison-tree model [Val1] using a constant number k of rounds. For sorting,\Omega\Gamma n 1+1=k (log n) 1=k ) comparisons are necessary =-=[AA]-=-, and O(n 1+1=k log n) comparisons are sufficient [BT]. This last result, however, is non-constructive. Pippenger [Pip3] showed a slightly worse non-explicit construction of O(n 1+1=k (log n) 2\Gamma2... |

7 | Security Preserving Amplification of Hardness", 31st FOCS - Goldreich, Impagliazzo, et al. - 1990 |

4 |
Explicit construction of natural bounded concentrators
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(Show Context)
Citation Context ...d superconcentrator with sublogarithmic depth (namely, depth (log n) 2=3+o(1) ). The main tool in most superconcentrator constructions is the concentrator, which is interesting in its own right (e.g. =-=[Mor]-=-). An (n; m; l)-concentrator is an (n; m)-network such that every set of at most l inputs is connected by vertex-disjoint paths to outputs. Concentrators of depth 1 are usually built with expanders, w... |

4 | Time Space Tradeoffs for Computing Functions, Using Connectivity Properties of Their Circuits - Tompa - 1980 |

3 |
Randomness-Optimal Oblivious Sampling," Random Structures & Algorithms
- Zuckerman
- 1997
(Show Context)
Citation Context ... a defective source until we get essentially all of its entropy out. It is made precise in lemmas 2.4 and 2.5. This idea was key for later results as well, e.g. the near-optimal samplers of Zuckerman =-=[Zuc3]-=-. The only tools we use are hash functions and k-wise independence. Our construction builds heavily on the one in [NZ], which in turn builds upon ideas in [Zuc1, Zuc2]. Indeed, the explicit constructi... |

1 |
General Weak Random Sources," 31st FOCS
- Zuckerman
- 1990
(Show Context)
Citation Context ...on't have any truly random bits, we can cycle through all possibilities -- see [NZ, Zuc2] for more details.) Our model for defective random source will essentially be the most general: Definition 1.3 =-=[Zuc1]-=- A distribution D on f0; 1g n is called a ffi -source if for all x 2 f0; 1g n , D(x)s2 \Gammaffin . Note that a particular type of ffi -source is the uniform distribution on a subset A ` f0; 1g n , jA... |