M. Sipser, "Expanders, Randomness, or Time versus Space," Journal of Computer and System Sciences, 36: 379-383, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n , E is called an 3 (n; m; t; ffi; ffl) extractor. 2 As in [NZ] we observe that an extractor that adds t random bits yields a BPP simulation taking time 2 t poly(n) In the case when S is the class of ffi sources, it is often convenient to view the extractors graph theoretically, as in [Sip, San, CW]. Namely, construct a bipartite graph on f0; 1g n Theta f0; 1g m , where x 2 f0; 1g n is adjacent to z 2 f0; 1g m iff z = E(x; y) for some y. Then any set in f0; 1g n of size at least 2 ffin expands almost uniformly into f0; 1g m . In particular, it yields efficient constructions ....

....log n) O(log n) 2 n ; 2 p n ; n O(1) 2 n fl ) Just as extractors and dispersers for constant ffi had important applications [NZ, WZ] so too do our results for subconstant ffi. The first application is to a relationship between the RP = P question and time space tradeoffs. Sipser [Sip] showed that if certain expander graphs can be constructed efficiently, then for some ffl 0 and any time bound t(n) either RP = P or all unary languages in DTIME(t(n) are accepted infinitely often in SPACE(t(n) 1 Gammaffl ) If we had a polynomial time simulation of RP using a ffi source ....

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M. Sipser, "Expanders, Randomness, or Time versus Space," Journal of Computer and System Sciences, 36: 379-383, 1988.

....be used in practice. Linear congruential generators, although known not to be cryptographically secure (see e.g. FKL] P] continue to be used in practice. Because of these disadvantages, work has been done to construct good generators for more specific tasks. For example, Santha [Sa] and Sipser [Si] introduced the notion of a quasi perfect pseudo random generator. A quasi perfect prg can be used to decrease the probability of error of a BPP or RP algorithm from a constant 1=2 to an exponentially small amount, using only a constant factor more random bits. Until now, no prg was proven to ....

....congruential generators) still work well in practice. Sipser has shown that the existence of certain types of constructive expanders implies that of quasi perfect prg s; our results could be rephrased as explicit constructions of Sipser expanders (with somewhat weaker parameters than those used in [Si]) The first construction involves taking a random walk on a type of expander graph known to be constructible [GG] The proof, which was independently discovered by Cohen and Wigderson [CWi] uses techniques similar to those in [AKS] These quasi perfect prg s can be implemented simply and ....

M. Sipser, "Expanders, Randomness, or Time versus Space," Structure in Complexity, 1986.

....the advantage of using few random bits from the ffi source. Namely, if r truly random bits are used by an RP algorithm to achieve probability of success 1=2, then O(r log r) bits from a ffi source are used by our RP simulation. Hence this is also a quasi perfect pseudo random generator (see [San] [Sip]) Our BPP algorithm requires r O( log 2 ffi Gamma1 ) ffi) bits. Subsequent to this work, Noam Nisan and the author have extended and simplified the construction, building an extractor [NZ] although the construction there does not imply our result. However, the ideas there do help ....

....there does not imply our result. However, the ideas there do help simplify our original construction, so we present the simplified construction here. The simulations of RP and BPP are equivalent to the explicit construction of a certain type of expander graph, called a disperser (see [San] [Sip], CWi] It is easy to show that random 2 One of the bit fixing sources in [CWi] has a weaker entropy bound than that which we impose. Our model can be modified to generalize this source, too, but then our simulations would fail. graphs are dispersers, yet one cannot even use the eigenvalue ....

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M. Sipser, "Expanders, Randomness, or Time Versus Space," Journal of Computer and System Sciences, 36: 379-383, 1988.

....assigns probability at most 2 GammaR j to any string. It is minimally random in the sense that any weaker source is insufficient to do a black box polynomial time simulation of RP algorithms. 2 1 Introduction A disperser is a type of expander which was first introduced by Sipser in [Sip88]. Cohen and Wigderson [CW89] classified dispersers into two types: OR dispersers and MAJORITYdispersers. A bipartite multigraph G = V; W;E) with jV j = N and jW j = M is called an (N; M;T) OR disperser if any subset of V having at least T vertices has a neighbor set in W of size at least M=2; G ....

....probabilities are for a y picked uniformly at random from f0; 1g m where m = p(jxj) for some polynomial function p = p(ML ) We call W x L = fy 2 f0; 1g m jML (x; y) acceptsg the witness set of ML on input x, where m is the number of random bits used by ML on inputs of length jxj. Sipser [Sip88] defined the complexity class strong random polynomial time (Strong RP) to be the class of languages L for which there is an RP machine ML ( Delta; Delta) and a real number 0 j 1 such that on input a string x 2 L of (any) length n, ML uses q(n) random bits for some polynomial q( Delta) and ....

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M. Sipser, "Expanders, Randomness, or Time versus Space," Journal of Computer and System Sciences, 36: 379-383, 1988.

....absolute value. Ramanujan graphs have the property that their second eigenvalue is upper bounded by 2 p k Gamma 1. Furthermore, these graphs have been explicitly constructed in [7] In this work we also exploit properties of another class of bipartite graphs called dispersers. Definition2. [11] A bipartite graph G = U; V; E) is an (K; disperser if for each subset A U of size K there are at least (1 Gamma )jV j vertices of V that are adjacent to A. Sipser showed in [11] that such graphs exist. An almost optimal explicit construction of dispersers is reported in [12] Lemma 3. ....

....In this work we also exploit properties of another class of bipartite graphs called dispersers. Definition2. 11] A bipartite graph G = U; V; E) is an (K; disperser if for each subset A U of size K there are at least (1 Gamma )jV j vertices of V that are adjacent to A. Sipser showed in [11] that such graphs exist. An almost optimal explicit construction of dispersers is reported in [12] Lemma 3. 11] There exist a (K; ffl) disperser G(U; V; E) with jU j = N and jV j = M , such that each node v 2 U has degree maxf 2M K (1 ln 1=ffl) 2 ffl (1 ln N=Kg. The properties of ....

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M. Sipser, "Expanders, randomness, or time versus space". Journal of Computer and System Sciences, 36:379--383, 1988.

....k vertex i j X 2 A and vertex i j Y 2 B. Thus, there are at least k vertex disjoint paths from s to t. 4. A construction based on dispersers In this section we give a first and simple construction of converting families of graphs based on dispersers. An explicit construction of Definition 4.1. [21] A bipartite graph G = U; V; E) is an (K; disperser if for each subset A U of size K there are at least (1 Gamma )jV j vertices of V that are adjacent to A. Lemma 4.2. If Cw(l) is a (ffll 2; fl) disperser, with fl ffll 2 (1 ffl)l then Hw(l) is a w(l) converting graph. Proof. In ....

....paths in H l (A; B) from s to t. Thus, by Lemma 3.1 Hw(l) is a w(l) converting graph. Proof of Theorem 1.1. By Lemma 4.2 for the construction of a (1 ffl)l converting family, it is sufficient to have a (ffll 2; fl) dipserser, with fl ffll 2 (1 ffl)l . The probabilistic argument of [21] proves that such a disperser exists and it has degree less than 2(1 ffl) ffl Gamma ln 1 ffl ffl 1 Delta (existential result) In particular, for w(l) 3=2l the required disperser has degree less than 13. 5. An explicit construction based on expanding graphs In this section we give ....

M. Sipser. "Expanders, randomness, or time versus space". Journal of Computer and System Sciences, 36:379--383, 1988.

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