L. Stockmeyer: On Approximation Algorithms for #P. SIAM Journal on Computing 14 (1985), 849--861.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....information about the structure of the prime and are hence not suited unless one is willing to give away this information. Examples of such tests are the Miller Rabin test [26, 29] or the one based on Pocklington s theorem. A test that does not reveal such information is the one due to Lehmann [25] which we describe in the next subsection. 4.1 Lehmann s Primality Test Lehmann s primality test is variation of the Solovay Strassen [31] primality test and based on the following theorem [24] Theorem 3. An odd integer n 1 is prime if and only if 8a 2 Z 1 (mod n) and 9a 2 Z ....

D. J. Lehmann. On primality tests. SIAM Journal of Computing, 11(2):374--375, May 1982.

....or using n O(1) memory cells. Thus, even one extra time step can be more useful than increasing the number of processors or shared memory cells by a polynomial factor or using a more powerful write resolution rule. An early example of this approach was applied to the following problem [Sni85] SEARCH AN ORDERED LIST Given x 1 ; xn ; y 2 f1; rg such that x 1 x 2 : xn , determine that either y x 1 or xn y or find the index i such that x i y x i 1 . Using (p 1) ary search, a CREW PRAM with p processors can solve this problem in time O(log n= log(p ....

....obtained on an EREW PRAM, even for the restricted version of the problem in which x 1 ; xn ; y 2 f0; 1g, by bounding the number of variables affecting processors and memory cells as a function of time. This lower bound remains valid even if concurrent writes are allowed. EXERCISE 21.34 [Sni85] Prove that an EREWPRAM with p processors requires Omega Gammaqui n Gamma log p) steps to SEARCH AN ORDERED LIST of length n. EXERCISE 21.35 [Sni85] Prove that an EREW PRAM can SEARCH AN ORDERED LIST of length n in O(logn Gamma log p) steps using p processors, provided p copies of y are ....

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M. Snir. On parallel searching. SIAM Journal on Computing, 14(3):688--708, August 1985.

....CRCW and CREW PRAMs. They proved that the OR of n bits, which can be easily computed in constant time on a CRCW PRAM, requires Omega# log n) time on any CREW PRAM. This result was improved by Kuty lowski [Kut91] and Dietzfelbinger et al. DKR90] who determined the exact complexity of OR. Snir [Sni85] proved that the problem of searching a sorted list is more difficult on the EREW PRAM than on the CREW PRAM. Gafni, Naor, and Ragde [GNR89] extended 3 Beame and Sinha s research supported by NSF DARPA under grant CCR 8907960 and NSF under grant CCR 8858799. Fich s research supported by the ....

M. Snir. On parallel searching. SIAM Journal on Computing, 14(3):688--708, 1985.

....accepts when x 62 L. The important connection is now that if there is a proof system for NP with a polynomial time verifier which uses logarithmic randomness and has perfect completeness with k amortized free bits, then for any ffl 0 we get an in approximability result of n 1 k 1 Gammaffl [26]. Bellare, Goldreich and Sudan [6] proved that in fact we essentially have an equivalence, in that if it is NP hard to approximate Max Clique within a factor n 1 k 1 then there is also a proof system with essentially k amortized bits. In this paper, for any ffi 0, we find a proof system which ....

....verifier only reads either unknown or completely known bits. Let us state formally the connection between amortized free bits and inapproximability of clique. The basic construction is from [14] although it appeared earlier in a primitive form in [10] The first result essentially appears in [26] but the stated version below which is most suitable for our purposes appears in [6] Theorem 2.4 [6] Suppose NP admits a PCP which uses logarithmic randomness and f amortized free bit. Then, unless NP=ZPP, it follows for any ffl 0 that Max Clique cannot be approximated within n 1= 1 f ffl) ....

D. Zuckerman On unapproximable versions of NP-complete problems. SIAM Journal on Computing. Vol 25, 1996, pp 1293-1304.

....p the number of processors in a synchronous processor group calling a library procedure. The primary procedures in this part of PAD include: ffl Binary and parallel searching in ordered arrays, the latter running in time O(log n= log p) The parallel algorithm is folklore, but see for instance [11]. The expected speed up of the parallel algorithm is low, proportional only to log p. See Section 5. ffl Merging of ordered arrays. An implementation of the CREW algorithm in [4] is described, running in O(log n log m n m p ) time. The algorithm is work optimal. See Section 10. ffl Merge ....

M. Snir. On parallel searching. SIAM Journal of Computing, 14:688--708, 1985.

....three most popular variants of the PRAM model and separations between variants of the CRCW, and between the CRCW and CREW models have already been shown, the big open problem is to determine the relative power of CREW and EREW PRAMs. 4. 4 Previous Attempts at Separating CREW and EREW PRAMs Snir [Sni85] proved that the problem of searching a sorted list is more difficult on the EREW PRAM than on the CREW PRAM. This result is unsatisfactory on two counts: 1) Separation is proved for a partial function, which may not say anything about their relative power for problems defined over complete ....

Marc Snir. On parallel searching. SIAM Journal on Computing, 14:688--708, 1985.

....Safe Primes 115 information about the structure of the prime and are hence not suited unless one is willing to expose this information. Examples of such tests are the MillerRabin test [30, 33] or the one based on Pocklington s theorem. A test that does not reveal such information is due to Lehmann [27] and described in the next subsection. 4.1 Lehmann s Primality Test Lehmann s test is variation of the Solovay Strassen [36] primality test and based on the following theorem [26] Theorem 3. An odd integer n 1 is prime if and only if 8a 2 Z n : a (n Gamma1) 2 j Sigma1 (mod n) and 9a 2 ....

....[36] primality test and based on the following theorem [26] Theorem 3. An odd integer n 1 is prime if and only if 8a 2 Z n : a (n Gamma1) 2 j Sigma1 (mod n) and 9a 2 Z n : a (n Gamma1) 2 j Gamma1 (mod n) This theorem suggest the following probabilistic primality test [27]: Choose k random bases a 1 ; a k 2 Z n , check whether a (n Gamma1) 2 i j Sigma1 (mod n) holds for all i s, and check whether a (n Gamma1) 2 i j Gamma1 (mod n) if true for at least one i 2 f1; kg. The probability that a non prime n passes this test is at most ....

D. J. Lehmann. On primality tests. SIAM Journal of Computing, 11(2):374--375, May 1982.

....shared memory, or perform any operation on the data stored in the local memory. The above definition still leaves some ambiguity regarding the behaviour of PRAM during the read and write operations. Thus a family of PRAM models each differ on their memory access characteristics have been defined [Sni85, KR90, J aJ92, Rei93] 1) Exclusive Read Exclusive Write, or EREW PRAM, does not allow concurrent access to the same memory location; 2) Concurrent Read Exclusive Access, or CREW PRAM, allows concurrent read from the same location, whereas exclusive write is required for concurrent writes; and ....

M. Snir. On Parallel Searching. SIAM Journal on Computing, 14(3):688--708, 1985.

....it is impossible to prove the no P Selective set is E P T hard unless the most fundamental problem in complexity theory can be solved. This indicates our result is almost optimal by the relativizable methods. We also study the symmetric difference between hard set with P selective sets. Yesha [29] first studied the symmetric difference between NP P tt hard sets and the sets in P. Schoning showed that no the symmetric difference between a E P T hard and a set in P is of exponential density. The symmetric difference of NP hard sets and P selective was investigated by Fu and Li [9] ....

Y.Yesha: On Certain Polynomial-time Truth-table Reducibilities of Complete Sets to Sparse Sets. SIAM Journal on Computing, 12(1983), 411-425.

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L. Stockmeyer: On Approximation Algorithms for #P. SIAM Journal on Computing 14 (1985), 849--861.

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C. Thomassen On the Complexity of Finding a Minimum Cycle Cover of a Graph. SIAM Journal of Computing, Volume 26, Number 3, pp 675-677, June 1997.

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D. Zuckerman: On Unapproximable Versions of NP-Complete Problems, SIAM Journal on Computing, 25(6) (1996), 1293--1304.

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, On the serial transitive closure problem, SIAM Journal on Computing, 24 (1995), pp. 109--122.

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