P. L'Ecuyer, `Efficient and Portable Combined Random Number Generators ', Communications of the ACM, Vol.31, No.6, pp.742--774,  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....the sum of the marked tiles equalled z of the sum of all the tiles. A boundary drawing algorithm was then applied to draw around the marked area to give the z confidence boundary. 6.Simulation Study To validate the method and the algorithm a Monte Carlo simulation was performed. TheL Ecuyer [30] pseudo random number generator with a period in excess of 2 10 18 , incorporating a Bays Durham shuffle with added safeguards, was used. Samples with 2 n , n=0. 10 cases were simulated. Each sample size was simulated with a different set of frequencies of disease in the population. Samples ....

P. L'Ecuyer, `Efficient and Portable Combined Random Number Generators ', Communications of the ACM, Vol.31, No.6, pp.742--774,

....but are known to have some major defects. The main problem is that the least significant bits of the numbers produced are correlated, and a scatter plot of ordered tuples (x i ; x i 1 , of random floating point numbers plotted in the unit hypercube shows regular lattice structure [2, 27, 32, 33, 7]. This problem becomes worse in higher dimensions, which may affect some high dimensional simulations. The problem of lattice structure can be quantified using the Spectral Test [1] Examples of LCGs with good parameters that perform well in the Spectral Test are given in Refs. 1, 40, 15, 41] ....

....However on modern processors addition is not markedly slower than XOR, so there is little reason to use these generators. 3. 4 Combined Generators Combining two different generators has been shown (both theoretically and empirically) to produce an improved quality generator in many circumstances [2, 32, 6, 42]. Based on an algorithm introduced by Wichmann and Hill [43] L Ecuyer [32] has shown how to additively combine two different 32 bit LCGs to produce a generator that passes all known statistical tests and has a long period of around 10 18 , thus overcoming the major drawbacks of standard 32 bit ....

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P. L'Ecuyer, Efficient and portable combined random number generators, Comm. ACM 31:6, 742 (1988). 22

....bigger, no random number generator is able to generate traffic according to the Markov chain. Therefore, we used an iterative method. Suppose for example that a = 10 and that the Markov chain is in the first state, then a number between 0 and 1 is generated with a reliable random number generator [13]. If the next number is less than 1=a = 1=10, the next state is not 2 but between 3 and n. The procedure continues until the random number is more than 1=a or until the last state is reached. To resume, the probability of being in the next step in the state i knowing that we are in the first state ....

P. L'Ecuyer, "Efficient and Portable Combined Random Number Generators," Communications of the ACM, vol. 31, pp. 742--774, June 1988.

....order statistics maximum and minimum of the quantity it was entrusted to monitor. Each run of MAGELLAN was executed by seeding the random number generators with different seeds. MAGELLAN uses the portable variant of MLCG (multiple linear congruent generators implemented in libg [Lea92] L E88] MLCG has a period of about 2:30584 Theta 10 18 . All our experiments were verified to be no longer than this period and thus, were safe from any effects of recycling. MAGELLAN maintains multiple random number generators for each random variable in the simulation model. This is to prevent any ....

P. L'Ecuyer. Efficient and Portable Combined Random Number Generators. Communications of the ACM, 31(6):742--749, June 1988.

..... The following facts hold for DTES processes of the forms (13) and (14) Background DTES processes are stationary Markovian, the former with a stationary transition structure and the latter with a time dependent transition structure. Their marginal distributions are all uniform on ZM (see e.g. [14] and a continuous state case in [8] Their step transition probabilities, denoted h (kjj) PfC n = kjC n = jg and h Gamma (kjj) PfC Gamma n = kjC Gamma n = jg, respectively, have the representations as given in the following lemma. Lemma 1. For k; j 2 ZM and = 0; 1; ....

L'Ecuyer, P., "Efficient and Portable Combined Random Number Generators", CACM 31, 742-749, 1988. 20

....random variables. Data presented in this paper are summarized from 8 runs for each experiment. Each run of MAGELLAN was executed by seeding the random number generators with different seeds. MAGELLAN uses a portable version of MLCG (multiple linear congruent generators implemented in libg [12] [13]. This MLCG has a period of about 2:30584 Theta 10 18 . All our experiments were verified to be no longer than this period and thus, were safe from any recycling effects. MAGELLAN maintains multiple random number generators for each random variable in the simulation model. This is to prevent ....

P. L'Ecuyer. Efficient and Portable Combined Random Number Generators. Communications of the ACM, 31(6):742--749, June 1988.

....3 shows the times for qrs and qhull. 5.2 Truncated Cubes As well as qhull, we tested the following insertion orders for cdd: minindex, maxindex, maxcutoff, and random. To simulate random insertion order, we permuted the input 200 times using the Combined Random Number Generator of L Ecuyer [14, 15] and reported the average time. In our data files, the cube constraints come last, so the results of Section 4 tell us to expect exponential performance for maxindex. The scaled order scales the cube constraints to be lexicographically larger than the stacked polytope constraints, then uses ....

L'Ecuyer. Efficient and portable combined random number generators. Communications of the ACM, 31, 1988.

....0.89 0.57 0.65 0.54 0.70 0.62 0.73 here 2 31 Gamma 1 1682459953 0.81 0.83 0.64 0.59 0.71 0.74 0.54 here 2 31 Gamma 1 1512629553 0.87 0.71 0.53 0.66 0.61 0.64 0.70 [32] 2 31 Gamma 1 630360016 0.82 0.43 0.78 0.80 0.57 0.68 0.72 [31] 2 31 Gamma 1 16807 0.34 0.44 0.58 0.74 0.65 0.57 0. 61 [24] 2 31 Gamma 1 39373 0.79 0.75 0.79 0.76 0.75 0.78 0.56 [26] 2 31 Gamma 1 41358 0.83 0.70 0.70 0.73 0.67 0.69 0.60 [16] 2 31 Gamma 1 742938285 0.87 0.86 0.86 0.83 0.83 0.62 0.74 [2] 2 31 Gamma 1 1327217885 0.86 0.59 0.73 0.57 0.69 0.69 0.57 [27] 2 31 Gamma 1 46344 0.93 0.08 0.44 0.53 ....

P. L'Ecuyer. Efficient and portable combined random number generators. Communications of the ACM, 31:742--749,774, 1988.

....rate) Thus an ARQ scheme could be implemented on the probe, although we chose not to use it. 1 .1 . 2 The uniform binary source To generate our random binary source, we used a very simple algorithm based on the combination of two Multiplicative Linear Congruential Generators (MLCGs) as shown in [3]. The uniform random source generator is given in Appendix 1. 1 .1 .3 The AWGN channel 1 .1 .3 .1 Definition The AWGN channel is defined by its noise power single sided spectral density N 0 . Its distribution is then: p(y) # 1 2## 2 # exp## 1 2# 2 y 2 # with # 2 defined by: # 2 # ....

P. l'Ecuyer, "Efficient and portable combined random number generator," Simulation Modelling and Statistical Computing, Commun. of the ACM, number 6, June 1988.

....x i = x i ; x i s Gamma1 ) i 0, produced by LCGs with maximal period. More precisely, the spectral test gives the maximal distance d s between adjacent parallel hyperplanes, the maximum being taken over all families of parallel hyperplanes that cover all overlapping tuples x i (see [20, 21, 22]) For example consider one of the top five generators of an exhaustive analysis of multiplicative congruential random number generators made by Fishman[16] FISH9 = LCG(2 32 ; 3934873077; 0; 1) The maximal period subsequence of FISH9 with k = 23 and i = 0 is produced by the generator FISH9 23 ....

P. L'Ecuyer. Efficient and portable combined random number generators. Communic. of the ACM, 31(6):742--774, 1988. Parallel Numerics 97/Zakopane,Poland/September 5-7, REFERENCES PACT

.... [10] for which the internal state is usually a single integer or floating point number) the phenomenally long period Fibonacci generators [9] for which the internal state is usually a modest array of integer or floating point components, plus a carry or borrow bit) or combination generators [4, 11] (for which the internal state is usually a small array of integers) Since no one algorithm is best for all applications, implementations are required to document the algorithm used, so that the user can judge its suitability for the application at hand. 1.3 Complex Arithmetic Three library ....

P. L'Ecuyer. Efficient and Portable Combined Random Number Generators. CACM 31(6):742--749,774, June 1988.

....[7] using the Fortran 90 random number generator intrinsic function as a basis, and implemented a parallel version in High Performance Fortran for both 32 and 64 bit computers. 2 Background One of the most commonly used random number generators is the Linear Congruential Generator (LCG) [1, 2, 3, 5, 10]. These generators produce their sequence of numbers by using equations of the form: S i = AS i Gamma1 C mod m; 1) where S i is the i th random number generated, S 0 being the initial number, or seed (which is selected by the user) A is some multiplier; m is a modulus and C is some ....

....or seed (which is selected by the user) A is some multiplier; m is a modulus and C is some constant. There are rules and tests that help choosing good values of A, C and m [1] for example to ensure that the parameters are chosen so they give full period. One of these tests is the spectral test [1, 3, 10], a popular test used to determine the randomness of the output of the generator for a particular A and m. This kind of generator has the advantage of being of uniform distribution when it is of full period. However, when the generator s output is grouped in pairs, that is, fS 1 ; S 2 g; fS 2 ; S ....

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Pierre L'Ecuyer, Efficient and Portable Combined Random Number Generators, Communications of the ACM 31, 742 (1988).

....discussed which are only valid for comparatively small multipliers a ( 11] 2] In [8] a FORTRAN implementation is suggested that applies the method of [11] twice. It is valid for all a that can be factorized into two integers smaller than 2 15 but has the disadvantage of being rather slow. In [6] and [9] computer searches were performed to find the best multipliers (with respect to the lattice structure from dimensions two to six) for which the method in [2] is applicable. Compared with the best multipliers of [3] not much is lost by this restriction as far as the lattice structure is ....

.... 164.27 193.79 99404.0 100015.8 950706376 suggested in [3] 233.14 188.81 100071.0 100195.2 630360016 suggested in [8] 148.34 202.38 99494.8 99529.0 397204094 used in SAS and IMSL 271.21 439.69 99894.6 100371.4 16807 the minimal standard of [9] 56799.39 48582.05 367131.6 206056.0 39373 found in [6] 24297.35 20789.49 214153.6 144163.4 48271 suggested in [9] 20076.95 16936.76 177831.2 131313.8 69621 suggested in [9] 13722.25 11733.92 131067.6 113625.0 The results of Table 1 support the considerations of above: The rejection method, using inversion to sample from the dominating density in ....

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L'Ecuyer, P. Efficient and portable combined random number generators. Communications of the ACM 31, 6 (June 1988), 742-749.

....which parameter to use. Prime modulus m Multiplier K P q i=1 a i P [1] 29 18 2 7 P [2] 541 331 3 14 P [3] 7919 3325 2 19 P [4] 104729 43955 2 25 P [5] 1299709 818870 2 31 P [6] 15485863 6507610 2 36 P [7] 179424673 104026441 3 42 P [8] 2038074743 1289818712 2 47 P [9] = 22801763489 9450445177 2 52 M = 2147483647 1257471787 2 48 Table 1. Some optimal multipliers Concerning efficiency algorithm C1 shows a very bad behavior. This parallelization may lead to a higher over all computational demand than the sequential version. This happens if a node spends much ....

.... In order to rate the quality of the calculated LCG parameters we performed a spectral test which gives the maximal distance between adjacent parallel hyperplanes, the maximum being taken over all families of parallel hyperplanes that cover all overlapping tuple generated in dimensions 2 s 8 (see [7, 9]) It should be noted that this test uses a different quality measure than discrepancy for judging the quality of LCGs especially we investigate the behavior of the LCG sequences for dimensions s 2. See the following normalized spectral test results (values near 1 imply a good lattice ....

P. L'Ecuyer. Efficient and portable combined random number generators. Communications of the ACM, 31:742--774, 1988.

....length of z will be the product of that of the basic generators. One can show that the statistical properties of z are no worse than those of x or y [11] In fact, one expects it would be much superior but little has yet been proven to date. Good combined generators have been developed by L Ecuyer [35], based on the addition of Linear Congruential sequences. 4 Testing Random Number Generators We have so far discussed the desired features of random number generators and described some of the popular generators used in Monte Carlo applications. Now the question arises: How good are the random ....

P. L'Ecuyer. Efficient and portable combined random number generators. Comm. of the ACM, 31:742--774, 1988.

....background of these conditions, see the excellent monograph of Niederreiter[17] and the discussion in Ripley[19] D5H 1 Rel 1.0 April 27, 1994 Random number generators for parallel processors PACT The LCGs of type 1. are easier to implement than it might appear at a first glance, see L Ecuyer[13] or Park and Miller[18] It is well known that the output (x n ) n0 of an LCG is a periodic sequence. Further, the s tuples (x n ; x n 1 ; x n s Gamma1 ) n = 0; 1; s = 2; 3; form a lattice in the s dimensional unit cube [0; 1[ s . The latter fact allows to rate LCGs by ....

P. L'Ecuyer. Efficient and portable combined random number generators. Comm. ACM, 31:742--749,774, 1988.

....a multiplicative congruential generator, such as the 4.3bsd rand( or rand48( One should be cautious with such generators, however. Frequently recommended 32 bit moduli and multipliers (e.g. the minimal standard generator [24] RNUN in the IMSL library) produce pinsets with noticeable stripes [22]. A combined scheme, based on two multiplicative generators, can be used to overcome this defect [22] 6 Conclusion We have described a method for computing rectilinear Steiner minimal trees on up to 23 pins on a contemporary workstation with a 500 megabyte disk. Our method is based on the ....

....with such generators, however. Frequently recommended 32 bit moduli and multipliers (e.g. the minimal standard generator [24] RNUN in the IMSL library) produce pinsets with noticeable stripes [22] A combined scheme, based on two multiplicative generators, can be used to overcome this defect [22]. 6 Conclusion We have described a method for computing rectilinear Steiner minimal trees on up to 23 pins on a contemporary workstation with a 500 megabyte disk. Our method is based on the Dreyfus Wagner dynamic program for solving Steiner problems on arbitrary graphs[12] To keep the ....

Pierre L'Ecuyer. Efficient and portable combined random number generators. Communications of the ACM, 31(6):742--774, June 1988.

....(N = 20; n = 50000; t = 9) Table 2: The selected generators. G1. LCG with m = 2 31 Gamma 1 and a = 16807. G2. LCG with m = 2 31 Gamma 1 and a = 630360016. G3. LCG with m = 2 31 Gamma 1 and a = 742938285. G4. CSD generator of Sherif and Dear (1990) G5. Combined generator in Fig. 3 of L Ecuyer (1988). G6. Combined Tausworthe generator G1 of Tezuka and L Ecuyer (1991) G7. Twisted GFSR with (r; s; p) 25; 7; 32) G8. Subtract with borrow generator with (b; r; s; L) 2 32 Gamma 5; 43; 22; 1) dom number generators. If the results agree, it certainly improves our confidence that this is ....

L'Ecuyer, P. 1988. Efficient and Portable Combined Random Number Generators. Communications of the ACM , 31, 6:742--749 and 774.

....are in Table VIII. We also indicate the period length, the type of implementation (FP for floating point and I for integer arithmetic) and the sum of the 10 7 numbers generated. In addition to the already mentioned CMRGs, we report the timings for a C version of the 32 bit combined LCG of L Ecuyer (1988) (comblec88a) the CMRG in Figure I of L Ecuyer (1996) combMRG96a) and one of the system s generators in UNIX (drand48) In all cases (except for drand48) each integer in the seed was 12345. It is a good idea to check that your implementations reproduce the same sums. For comblec88a and ....

L'Ecuyer, P. 1988. Efficient and portable combined random number generators.

....stretch a short (truly) random seed into a long sequence that is supposed to appear and behave like a true random sequence. The most commonly used in practice are still the linear congruential generators and its variants (matrix form, combinations, etc. Knuth (1981) Bratley et al. 1987) L Ecuyer (1988, 1989) These generators are quite efficient and show good statistical behavior when their parameters are well chosen. However, efficient algorithms have been designed to infer sequences produced by linear congruential generators, even when the multiplier, increment and modulus are unknown, by ....

....time. It took us 155 hours of CPU time to find two 128 bit special primes p and q, but for which 2 is a quadratic residue with respect to both (p Gamma 1) 2 and (q Gamma 1) 2. This experiment, like all others reported in this paper, were done on a MicroVax II, using our own software (based on L Ecuyer et al. 1988)) The truly random bits were replaced by the output of the fast generator proposed in L Ecuyer (1988) To generate random primes, we followed the approach discussed in Beauchemin et al. 1988) Since selecting from Sn is much faster than from S 0 n , it could be interesting to analyze the ....

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L'Ecuyer, P. (1988). Efficient and Portable Combined Random Number Generators. Communications of the ACM , 31, 6, 742--749 and 774.

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L'Ecuyer, P. (1988) "Efficient and Portable Combined Random Number Generators ", CACM 31, 742-749.

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Pierre L'Ecuyer, Efficient and portable combined random number generators, Communications of the ACM, vol. 31 (1988), no. 6, pp. 742-- 749, 774.

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P. L'Ecuyer. "Efficient and Portable Combined Random Number Generators". Communications of the ACM 31(6):742-749, 774; 1988.

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P. L'Ecuyer. Efficient and portable combined random number generators. Comm. ACM, 31:742--774, 1988.

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P. L'Ecuyer, "Efficient and portable combined random number generators", Communications of the ACM 31 (1988), 742.

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