TY - JOUR
T1 - Shallow circuits and concise formulae for multiple addition and multiplication
AU - Paterson, Michael
AU - Zwick, Uri
PY - 1993/9
Y1 - 1993/9
N2 - A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several different representation standards. The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {∧, ∨, ⊕}-circuits of depth 3.48 log2n and {∧, ∨, {bottom left crop}}-circuits of depth 4.95 log2n for the carry-save addition of n numbers of arbitrary length. As a consequence we get multiplication circuits of the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases respectively to 4.48 log2n and 5.95 log2n. We also get {∧, ⊕, {bottom left crop}}-formulae of size O (n3.13) and {∧, {bottom left crop}}-formulae of size O (n4.57) for all the output bits of a carry-save addition of n numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.
AB - A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several different representation standards. The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {∧, ∨, ⊕}-circuits of depth 3.48 log2n and {∧, ∨, {bottom left crop}}-circuits of depth 4.95 log2n for the carry-save addition of n numbers of arbitrary length. As a consequence we get multiplication circuits of the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases respectively to 4.48 log2n and 5.95 log2n. We also get {∧, ⊕, {bottom left crop}}-formulae of size O (n3.13) and {∧, {bottom left crop}}-formulae of size O (n4.57) for all the output bits of a carry-save addition of n numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.
KW - Multiplication
KW - Subject classifications: 68Q25, 06E30, 94C10
KW - carry-save addition
KW - circuits
KW - formulae
UR - http://www.scopus.com/inward/record.url?scp=0039033999&partnerID=8YFLogxK
U2 - 10.1007/BF01271371
DO - 10.1007/BF01271371
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AN - SCOPUS:0039033999
SN - 1016-3328
VL - 3
SP - 262
EP - 291
JO - Computational Complexity
JF - Computational Complexity
IS - 3
ER -