## A PROBABILISTIC STUDY ON COMBINATORIAL EXPANDERS AND HASHING ∗

### BibTeX

@MISC{Bradford_aprobabilistic,

author = {Phillip G. Bradford and Michael and N. Katehakis},

title = {A PROBABILISTIC STUDY ON COMBINATORIAL EXPANDERS AND HASHING ∗},

year = {}

}

### OpenURL

### Abstract

Abstract. This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber–Galil expander. Our results give an expansion factor of 3 for subgraphs of these three-regular graphs 16 with (p − 1) 2 inputs for p prime. The proofs are not based on eigenvalue methods or higher algebra. The same methods show the expected number of probes for unsuccessful search in double hashing is 1 bounded by, where α is the load factor. This assumes a double hashing scheme in which two 1−α hash functions are randomly and independently chosen from a specified uniform distribution. The result is valid regardless of the distribution of the inputs. This is analogous to Carter and Wegman’s result for hashing with chaining. This paper concludes by elaborating on how any sufficiently sized subset of inputs in any distribution expands in the subgraph of the Gabber–Galil graph expander of 1 focus. This is related to any key distribution having expected probes for unsuccessful search 1−α for double hashing given the initial random, independent, and uniform choice of two universal hash functions.

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Citation Context ...iven a key x, determining the (i + 1)st hash location using double hashing is done by h(i, x)=(h1(x)+ih2(x)) mod m. Double hashing is a classical data structure, and discussions of it can be found in =-=[11, 24, 17]-=-, for example. Inserting the element x into the table T is done by first searching for x in T. If T does not contain x, then x can be inserted into T. Likewise, to delete x from T, then it must be det... |

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Citation Context ...83s84 PHILLIP G. BRADFORD AND MICHAEL N. KATEHAKIS Finally, Alon [2] showed that a graph is an expanding graph iff its largest and secondlargest eigenvalues are well separated. See also, for example, =-=[5, 4, 26, 10, 18, 29]-=- for varying depths of coverage of eigenvalue methods for graph expansion. The eigenvalue methods have been central in much research on graph expanders. Eigenvalue methods do not give the best possibl... |

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Citation Context ...o show that randomly independently and uniformly selecting two double hash functions from a strongly universal set gives a double hashing result analogous to the classical result of Carter and Wegman =-=[8]-=- for hashing with chaining. Section 4 completes the expander result, showing the subgraphs expand by 3 16 by enunciating the trade-off of local and global expansion. Finally, in section 5 we give our ... |

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Citation Context ...l Gabber–Galil expansion was shown to be (2 − √ 3)/4 orabout0.067. Suppose double hashing is based on randomly, independently, and uniformly choosing two hash functions h1 and h2 from a universal set =-=[11]-=-. Then this paper shows the expected number of probes for unsuccessful search in double hashing is bounded by 1 1−α , where α is the load factor. This holds regardless of the distribution of the input... |

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Citation Context ...gers Business School— Newark and New Brunswick, Rutgers University, 180 University Ave., Newark, NJ 07102 (mnk@andromeda.rutgers.edu). 83s84 PHILLIP G. BRADFORD AND MICHAEL N. KATEHAKIS Finally, Alon =-=[2]-=- showed that a graph is an expanding graph iff its largest and secondlargest eigenvalues are well separated. See also, for example, [5, 4, 26, 10, 18, 29] for varying depths of coverage of eigenvalue ... |

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Citation Context ...ular expanding graphs. This algorithm is complex and takes O(n3 log 3 n) time to construct an expander. The expansion factor of these expanders is unknown but positive. Lubotzky, Phillips, and Sarnak =-=[19]-=- and independently Margulis [22] gave the best possible expanders using the eigenvalue methods [2, 18, 19, 29]. Kahale [16] gave the best expansion constant to date for Ramanujan and related graphs. R... |

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Citation Context ...83s84 PHILLIP G. BRADFORD AND MICHAEL N. KATEHAKIS Finally, Alon [2] showed that a graph is an expanding graph iff its largest and secondlargest eigenvalues are well separated. See also, for example, =-=[5, 4, 26, 10, 18, 29]-=- for varying depths of coverage of eigenvalue methods for graph expansion. The eigenvalue methods have been central in much research on graph expanders. Eigenvalue methods do not give the best possibl... |

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Citation Context ...] gave the best possible expanders using the eigenvalue methods [2, 18, 19, 29]. Kahale [16] gave the best expansion constant to date for Ramanujan and related graphs. Reingold, Vadhan, and Wigderson =-=[28]-=- give very important combinatorial constructions of constant degree expanders based on their new “zig-zag” graph product. By showing how the zig-zag product maintains the eigenvalue bounds (then break... |

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Citation Context ... 2.2 defines local and global expansion. Subsection 2.3 explains the relation of double hashing to the expander graph representation. Next, subsection 2.4 focuses on the results of Chor and Goldreich =-=[9]-=- showing randomly choosing such functions and computing their values gives pairwise independent and uniformly distributed values. Finally, subsection 2.5 bounds functions that are necessary for our fi... |

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Citation Context ... of eigenvalue methods for graph expansion. The eigenvalue methods have been central in much research on graph expanders. Eigenvalue methods do not give the best possible expanding graph coefficients =-=[33]-=-. For example, probabilistic methods show the existence of expanders that have better expansion than is possible to show by the separation of the largest and second-largest eigenvalues. Pinsker [27] f... |

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Citation Context ...xpanders recursively starting from a small expander. Further, Meshulam and Wigderson [25] give group theoretic techniques whose expansion they show depends on universal hash functions. Capalbo et al. =-=[7]-=- give constant degree d lossless expanders. These expand by (1 − ɛ)d, for ɛ>0, which is just about as much as possible. 3 We demonstrate expansion of 16 =0.1875 for three of the five permutations that... |

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Citation Context ...83s84 PHILLIP G. BRADFORD AND MICHAEL N. KATEHAKIS Finally, Alon [2] showed that a graph is an expanding graph iff its largest and secondlargest eigenvalues are well separated. See also, for example, =-=[5, 4, 26, 10, 18, 29]-=- for varying depths of coverage of eigenvalue methods for graph expansion. The eigenvalue methods have been central in much research on graph expanders. Eigenvalue methods do not give the best possibl... |

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Citation Context |

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Citation Context ...r of these expanders is unknown but positive. Lubotzky, Phillips, and Sarnak [19] and independently Margulis [22] gave the best possible expanders using the eigenvalue methods [2, 18, 19, 29]. Kahale =-=[16]-=- gave the best expansion constant to date for Ramanujan and related graphs. Reingold, Vadhan, and Wigderson [28] give very important combinatorial constructions of constant degree expanders based on t... |

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Citation Context ...is proving they expand. In fact, the decision problem of determining expansion is co-NP-complete [6]. A series of classic papers firmly established that certain graph families expand. First, Margulis =-=[21]-=- showed that expanders exist, without giving bounds on their expansion. However, he did show how to construct them explicitly. Next, Gabber and Galil [12] gave an explicit expander construction with b... |

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Citation Context ...r essentially equivalent definitions for graph expanders. The “hard part” in designing graph expanders is proving they expand. In fact, the decision problem of determining expansion is co-NP-complete =-=[6]-=-. A series of classic papers firmly established that certain graph families expand. First, Margulis [21] showed that expanders exist, without giving bounds on their expansion. However, he did show how... |

26 | On universal classes of extremely random constant-time hash functions
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(Show Context)
Citation Context ...l [30] and Siegel [31] answer this, giving clog n-independent functions that are computable in constant time for a standard word model random access machine. Their results are quite general; see also =-=[32]-=-. Next, we focus on another answer to Carter and Wegman’s question using the standard set H of strongly universal hash functions, see Definition 6, as they are represented in the G3 graph. Although th... |

25 |
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Citation Context ...ion of the largest and second-largest eigenvalues. Pinsker [27] first showed the existence of expanders using probabilistic methods. There are some other constructions of expanders. According to Alon =-=[3]-=-, the (eigenvalue-based) construction of Jimbo and Maruoka [15] “only uses elementary but rather complicated tools from linear algebra.” Ajtai [1] also gives an algorithm using linear algebra for cons... |

17 |
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Citation Context ...e other constructions of expanders. According to Alon [3], the (eigenvalue-based) construction of Jimbo and Maruoka [15] “only uses elementary but rather complicated tools from linear algebra.” Ajtai =-=[1]-=- also gives an algorithm using linear algebra for constructing three-regular expanding graphs. This algorithm is complex and takes O(n3 log 3 n) time to construct an expander. The expansion factor of ... |

15 |
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(Show Context)
Citation Context ...iven a key x, determining the (i + 1)st hash location using double hashing is done by h(i, x)=(h1(x)+ih2(x)) mod m. Double hashing is a classical data structure, and discussions of it can be found in =-=[11, 24, 17]-=-, for example. Inserting the element x into the table T is done by first searching for x in T. If T does not contain x, then x can be inserted into T. Likewise, to delete x from T, then it must be det... |

11 |
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Citation Context ...s using double hashing take asymptotically the same number of probes as idealized uniform hashing does for any fixed load factor α less than about 0.319. For any fixed α<1, see Lueker and Molodowitch =-=[20]-=-. However, as pointed out in Schmidt and Siegel [30], these last results assume ideal randomized functions, whereas [30] utilizes more realistic k-wise independent and uniform functions (where k = clo... |

10 |
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(Show Context)
Citation Context ...). See Mehlhorn [24, 23] for lower bounds on the sizes of universal hash sets. As future research they suggest extending such an analysis to double or open hashing. Schmidt and Siegel [30] and Siegel =-=[31]-=- answer this, giving clog n-independent functions that are computable in constant time for a standard word model random access machine. Their results are quite general; see also [32]. Next, we focus o... |

6 |
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(Show Context)
Citation Context ... open addressed double hashing on a table T[0,...,m− 1]. Of course, T[j] corresponds to Ij. Double hashing approximates uniform open address hashing [26, 11, 24]. More precisely, Guibas and Szemerédi =-=[14]-=- showed unsuccessful searches using double hashing take asymptotically the same number of probes as idealized uniform hashing does for any fixed load factor α less than about 0.319. For any fixed α<1,... |

6 |
Expanders from symmetric codes
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(Show Context)
Citation Context ...By showing how the zig-zag product maintains the eigenvalue bounds (then breaks them), they show how to construct expanders recursively starting from a small expander. Further, Meshulam and Wigderson =-=[25]-=- give group theoretic techniques whose expansion they show depends on universal hash functions. Capalbo et al. [7] give constant degree d lossless expanders. These expand by (1 − ɛ)d, for ɛ>0, which i... |

3 | Double hashing is computable and randomizable with universal hash functions, submitted
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(Show Context)
Citation Context ...number of probes as idealized uniform hashing does for any fixed load factor α less than about 0.319. For any fixed α<1, see Lueker and Molodowitch [20]. However, as pointed out in Schmidt and Siegel =-=[30]-=-, these last results assume ideal randomized functions, whereas [30] utilizes more realistic k-wise independent and uniform functions (where k = clog n for a suitable constant c). Theorem 1 (see [20] ... |