Optimal Size Integer Division Circuits

Division is a fundamental problem for arithmetic and algebraic computation. This paper describes Boolean circuits (of bounded fan-in) for integer division (finding reciprocals) that have size O(M(n)) and depth O(lognloglogn), where M(n) is the size complexity of O(logn) depth integer multiplication circuits. Currently, M(n) is known to be O(n logn log log n), but any improvement in this bound that preserves circuit depth will be reflected by a similar improvement in the size complexity of our division algorithm. Previously, no one has been able to derive a division circuit with size O(n logc n) for any c, and simultaneous depth less than &#x03A9(log2 n). The circuit families described in this paper are logspace uniform; that is, they can be constructed by a deterministic Turing machine in space O(log n). The results match the best-known depth bounds for logspace uniform circuits, and are optimal in size. The general method of high-order iterative formulas is of independent interest as a way of efficiently using parallel processors to solve algebraic problems. In particular, this algorithm implies that any rational function can be evaluated in these complexity bounds. As an introduction to high-order iterative methods a circuit is first presented for finding polynomial reciprocals (where the coefficients come from an arbitrary ring, and ring operations are unit cost in the circuit) in size O(PM(n)) and depth O(logn log logn), where PM(n) is the size complexity of optimal depth polynomial multiplication

Optimal size integer division circuits

Abstract. Division is a fundamental problem for arithmetic and algebraic computation. This paper describes Boolean circuits (of bounded fan-in) for integer division (finding reciprocals) that have size O(M (n)) and depth O(log n log log n), where M(n) is the size complexity of O(log n) depth integer multiplication circuits. Currently, M(n) isknowntobeO(nlog n log log n), but any improvement in this bound that preserves circuit depth will be reflected by a similar improvement in the size complexity of our division algorithm. Previously, no one has been able to derive a division circuit with size O(n log c n) for any c, and simultaneous depth less than Ω(log 2 n). The circuit families described in this paper are logspace uniform; that is, they can be constructed by a deterministic Turing machine in space O(log n). The results match the best-known depth bounds for logspace uniform circuits, and are optimal in size. The general method of high-order iterative formulas is of independent interest as a way of efficiently using parallel processors to solve algebraic problems. In particular, this algorithm implies that any rational function can be evaluated in these complexity bounds. As an introduction to high-order iterative methods a circuit is first presented for finding polynomial reciprocals (where the coefficients come from an arbitrary ring, and ring operations are unit cost in the circuit) in size O(PM(n)) and depth O(log n log log n), where PM(n) is the size complexity of optimal depth polynomial multiplication

OPTIMAL SIZE INTEGER DIVISION CIRCUITS 913

computation. In 3 (dealing with polynomial reciprocals) we use a circuit model with operations in an arbitrary ring as the basis. Optimal algorithms have been known for quite some time for addition and subtraction, and good algorithms exist for multiplication. Ifwe let SM(n) be the sequential time complexity of multiplication and M(n) be the size complexity of O(logn) depth multiplication using the circuit model, then the best known results are due to SchSnhage and Strassen [11] who give an algorithm based on discrete Fourier transforms with SM(n) O(nlognlog logn) and M(n) O(n lognlog log n). The problem of integer division was examined by Cook in his Ph.D. thesis [5], and it was shown by using second-order Newton approximations that the sequential time complexity of taking reciprocals is asymptotically the same as that of multiplication. Unfortunately, this method does not carry over to the circuit model for size O(M(n)) division circuits, we require depth t(log 2 n) from a direct translation of Cook’s method of Newton iteration. In addition, no one has been able to derive a new method for integer division with size O(M(n)) and depth less than ft(log 2 n