## Abstract

A general theory is developed for constructing the shallowest possible circuits and the shortest possible formulas for the carry-save addition of n numbers using any given basic addition unit. More precisely, it is shown that if BA is a basic addition unit with occurrence matrix N, then the shortest multiple carry-save addition formulas that could be obtained by composing BA units are of size n^{1/p+o(1)}, where p is the unique real number for which the L_{p} norm of the matrix N equals 1. An analogous result connects the delay matrix M of the basic addition unit BA and the minimal q such that multiple carry-save addition circuits of depth (q + o(1)) log n could be constructed by combining BA units. On the basis of these optimal constructions of multiple carry-save adders, the shallowest known multiplication circuits are constructed.

Original language | English |
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Pages (from-to) | 642-650 |

Number of pages | 9 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

Volume | 2 |

State | Published - 1990 |

Externally published | Yes |

Event | Proceedings of the 31st Annual Symposium on Foundations of Computer Science - St. Louis, MO, USA Duration: 22 Oct 1990 → 24 Oct 1990 |