Y. Ofman. On the algorithmic complexity of discrete functions. Cybernetics and Control Theory, 7(7):589--591, 1963.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....polylogarithmic. 3.4 Roots. The standard proof of Prop. 3 involves a clever addition circuit that computes the number of 1s in the input with logarithmic depth. This construction was discovered by several authors independently in the early 60s, among them Wallace [69] and, a la russe, Ofman [55]. Today, complicated constructions have reduced the constants in the complexity bound far below those of Prop. 3, see the references in [10] Hastad s bound (1) for the depth of the parity function is from [33] and improves work of Ajtai [1] and Furst, Saxe, and Sipser [26] Constant depth ....

Yu. Ofman. On the algorithmic complexity of discrete functions. Dokl. Akad. Nauk SSSR, 145:48--51,

....segmented scan, look ahead, pipeline interleaving, and cyclic reduction, found an ubiquitous role in both theoretical and experimental computing. In 1963, the fast computation technique for linear recurrences was used by Ofman to design a fast parallel circuit for the addition of binary numbers [22]. Since then, a great variety of this speed up has been employed to design a variety of arithmetic computation structures, including carry lookahead, carry save, and Wallace tree multipliers. Ercegovac and Lang recently surveyed this line of research [7] In 1967, a study of linear recurrence in ....

Y. Ofman, "On the algorithmic complexity of discrete functions," Soviet Physics Doklady, vol. 7, no. 7, pp. 589--591, 1963.

....communication operations. Binary means binary is used. prefix scan prefix operation 1 implemented prefix scan program, takes a vector x and associative operator, and maps from 0 , x , 2 , 0 , x 0 x , 0 1 # x 2 , x x 2 x N 1 With O(N) processes, prefix scan executed O(lg time [81]. The data parallel expression prefix scan is fairly straightforward (Algo rithm 5.2) Even naive translation this node code a workstation cluster using message passing communication is noticeably more involved (Algorithm 5.3) Algorithm 5.3 fact, simplified from what actually implemented. An ....

Yu. Ofman. On algorithmic complexity discrete functions. Soviet Physics Doklady, 7(7):589--591, January

....p (2.57) A proof that (2. 57) has a unique solution can be found in [Ada94] We define the function ps : PowerList:Y:n Gamma PowerList:Y:n to realize a well known algorithm for computing the prefix sum due to Ladner and Fischer [LF80] This algorithm has roots in an algorithm presented by Ofman [Ofm63] and later implemented on a perfect shuffle network by Stone [Sto71] Misra [Mis94] derived the algorithm for PowerLists; we show a slightly different derivation below: u . v = f define u . v : ps: p . q) g ps: p . q) f defining equation for ps (2.57) g 0 ps: p . q) Phi p . q = f ....

Yu. Ofman. On the algorithmic complexity of discrete function. Soviet Physics Doklady, 7(7):289--591, 1963.

....i j0 k Gamma1 i j0 k Gamma1 i Figure 2: Quantum circuit for the approximate preparation of j 0:x j : x 0 i. respectively) where x; y 2 f0; 1g n and additions and subtractions are performed as integers modulo 2 n . By appealing to classical results about the complexity of arithmetic [30], one can construct quantum circuits of size O(n) and depth O(log n) for these operations (using an ancilla of size O(n) It is straightforward to show that applying a subtraction to the state j x ij y i results in the state j x y ij y i. Also, the state j 0 i can be obtained from j0 n i ....

Y. Ofman. On the algorithmic complexity of discrete functions. Cybernetics and Control Theory, 7(7):589--591, 1963.

....His present affiliation is the Department of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. PATERSON et al. OPTIMAL CARRY SAVE NETWORKS 175 number of bit operations required, or equivalently the circuit size. A different approach was pursued by Avizienis [2] Dadda [7] 8] Ofman [20], Wallace [35] and others. They investigated the depth, rather than the size of multiplication circuits. The main result proved by the above authors in the early 1960 s was that, using a process called Carry Save Addition, n numbers (of linear length) could be added in depth O(log n) As a ....

Ofman Y., On the algorithmic complexity of discrete functions. Doklady Akademii Nauk SSSR, 145 pp. 48-51 (in Russian). English translation in Sov. Phys. Doklady, Vol. 7 (1963) pp. 589-591.

.... will reside in a[n Gamma 1] in parallel for each processor i sum[i] a[ n=p)i] for j from 1 to n=p sum[i] sum[i] a[ n=p)i j] result reduce(sum) 4 7 1 z processor 0 0 5 2 z processor 1 6 4 8 z processor 2 1 9 5 z processor 3 ] Processor Sums = [12 7 18 15] Total Sum = 52 Figure 3: The reduce operation with more elements than processors. We assume that n=p is an integer. on Phi being associative. The operator, however, does not need to be commutative since the order of the operands is maintained. On an EREW P RAM, each level of the tree can be ....

.... Gamma 1] Set left child a[i 2 d 1 Gamma 1] t a[i 2 d 1 Gamma 1] Set right child Step Array in Memory 0 [ 3 1 7 0 4 1 6 3 ] up 1 [ 3 4 7 7 4 5 6 9 ] 2 [ 3 4 7 11 4 5 6 14 ] 3 [ 3 4 7 11 4 5 6 25 ] clear 4 [ 3 4 7 11 4 5 6 0 ] down 5 [ 3 4 7 0 4 5 6 11 ] 6 [ 3 0 7 4 4 11 6 16 ] 7 [ 0 3 4 11 11 15 16 22 ] (b) Executing a prescan on a P RAM. Figure 4: A parallel prescan on a tree using integer addition as the associative operator Phi. v R[v] L[v] 0 A B Figure 5: Illustration for Theorem 1. Theorem 1 After a complete down sweep, each vertex of the tree contains the sum of all the leaf values ....

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Yu. Ofman. On the Algorithmic Complexity of Discrete Functions. Soviet Physics Doklady, 7(7):589--591, January 1963.

....3.3 discusses the tradeoff, and Section 4.3 gives an example of how they are used. The uses of both unsegmented and segmented scans are extensive. A list of some of them is given below. 1. To add arbitrary precision numbers. These are numbers that cannot be represented in a single machine word [17]. 2. To evaluate polynomials [19] 3. To solve recurrences such as x i = a i x i Gamma1 b i x i Gamma2 or x i = a i b i =x i Gamma1 (Livermore loops 6, 11, 19 and 23) 12] 4. To solve tridiagonal linear systems (Livermore Loop 5) 20] 5. To pack an array so that marked elements are deleted ....

Yu. Ofman. On the Algorithmic Complexity of Discrete Functions. Soviet Physics Doklady, 7(7):589--591, January 1963.

....p is a function of the problem size N , and with resource constraints where p is independent of N . A comprehensive survey of parallel computing using the prefix problem is given in [14] Special forms of prefix circuits have been previously known to Ofman as early as 1963, whose carry circuit[18] is a form of the carry circuits later discussed by Ladner and Fischer [13] Muraoka[16] showed that this simplest linear recurrence can be computed in log N with N=2 processors, and used the name tree height reduction for this technique. Kogge and Stone s recursive doubling[12] for first order ....

Y. Ofman, "On the algorithmic complexity of discrete functions", Cybernetics and control theory, Soviet Physics Doklady, Vol. 7(7), pp. 589-591, January 1963.

....to perform many of the same functions as on lists as recursive doubling. Section 5 presents a linearspace, conservative tree contraction algorithm based on the ideas of Miller and Reif [22] Section 6 presents treefix computations, which are generalizations of the parallel prefix computation [3, 7, 23] to trees. We show that treefix computations can be performed using the tree contraction algorithm of Section 5. Section 7 gives short, efficient, parallel algorithms for tree and graph problems, most of which are based on treefix computations. Section 8 discusses the relationship between the DRAM ....

....all of the elements of a list and to accumulate values stored in each element of a list. More generally, contraction trees are useful for performing prefix computations in a conservative fashion. Let D be a domain with a binary associative operation Delta and an identity . A prefix computation [3, 7, 23] on a list with elements x 1 ; x 2 ; x n in D puts the value y i = x 1 Delta x 2 Delta Delta Delta x i in element i for each i = 1; 2; n. A prefix computation on a list can be performed by a conservative, twophase algorithm on the contraction tree. The leaves of the ....

Yu. Ofman. On the algorithmic complexity of discrete functions. Soviet Physics -- Doklady, 7(7):589--591, 1963. English translation.

....in two categories: with no resource constraints, where usually the number of processor p is a function of the problem size N , and with resource constraints where p is independent of N . Special forms of the prefix circuits had been previously known to Ofman as early as 1963, whose carry circuit[17] is a form of the carry circuits later discussed by Ladner and Fischer [12] Muraoka[15] showed that this simplest linear recurrence can be computed in log N with N=2 processors, and used the name treeheight reduction for this technique. Kogge and Stone s recursive doubling[11] for first order ....

Y. Ofman, "On the algorithmic complexity of discrete functions", Cybernetics and control theory, Soviet Physics Doklady, Vol. 7(7), pp. 589-591, January 1963.

....can be applied in order to obtain bounds on the complexity of the addition function. In this section there is also a theorem about the optimal size of the addition function stated (cf. 9] Moreover there can be found a table containing the depths and sizes of several known adders (cf. 1] [8], 10] 11] 4] 5] 12] Section 3 contains the design and analysis of the adder by Ladner and Fisher according to [5] In subsection 3.1 the general design of the adder is presented. The parallel algorithm solving the problem of adding two n bit binary numbers is developed step by step in ....

....subfunctions of f for x i = 0 and x i = 1 are different. Lamagna and Savage prove the following theorem in [6] as Wegener writes in [12] Theorem 2. 4 [6] Let f 2 Bn be non degenerated, then n Gamma 1 C(f) adder depth size Brent [1] log 2 n O( p log 2 n) O(nlog 2 n) Carry Look Ahead [8] O(log 2 n) O(n) Conditional Sum [10] 11] 2log 2 n 1 3nlog 2 n 10n Gamma 6 n = 2 k Krapchenko [4] dlog 2 ne 7 p 2dlog 2 ne 16 3n 6 Delta 2 dlog 2 ne Ladner Fisher [5] 2dlog 2 ne 2k 2 (8 6 Delta 2 Gammak )n 0 k dlog 2 ne School Method [12] 2n Gamma 1 5n Gamma 3 ....

Ofman, On the algorithmic complexity of discrete functions, Sov.Phys.Dokl. 7, pp. 589-591

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Y. Ofman. On the algorithmic complexity of discrete functions. Cybernetics and Control Theory, 7(7):589--591, 1963.

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Yu. Ofman. On the algorithmic complexity of discrete functions. Soviet Physics Doklady, 7(7):589--591, January 1963.

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Y. Ofman, On the algorithmic complexity of discrete functions, Soy. Phys. Dokl. ', 589-591 (1963).

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Y. Ofman: "On the Algorithmic Complexity of Discrete Functions", Soviet Physics Doklady, Vol. 7, No. 7, pp. 589-591, 1963.

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Y. Ofman; `On the Algorithmic Complexity of Discrete Functions' Soviet Physics -- Doklady, 7(7):589-591, Jan 63.

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Yu. P. Ofman. On the algorithmic complexity of discrete functions. Soviet Physics Doklady, 7:589--591, 1963.

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Yu. P. Ofman. On the algorithmic complexity of discrete functions. Soviet Physics Doklady, 7:589--591, 1963.

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Yu Ofman. On the algorithmic complexity of discrete functions. Cybernetics and Control Theory, Sov. Phys Dokl., 7(7):589--591, January 1963.

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