{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T06:41:11Z","timestamp":1698043271485},"reference-count":8,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2007,3,21]],"date-time":"2007-03-21T00:00:00Z","timestamp":1174435200000},"content-version":"vor","delay-in-days":6653,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems &amp;amp; Computers in Japan"],"published-print":{"date-parts":[[1989,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the usual <jats:italic>P<\/jats:italic>\u2010ary number digit for the addition and subtraction of two numbers (operands), 0, 1, 2, \u2026, <jats:italic>P<\/jats:italic> \u2010 1 are employed. In the minimum redundant <jats:italic>P<\/jats:italic>\u2010ary representation, 1 is further used, and the addition or subtraction can be performed in a constant time independent of the number of operand digits (<jats:italic>P<\/jats:italic>\u03f5{2, 3, 4, \u2026}). This paper extends this property. First, a high\u2010speed addition\u2010subtraction algorithm is discussed. It can execute the addition and subtraction of <jats:italic>L<\/jats:italic>(\u03f5 {2, 3, \u2026, <jats:italic>P<\/jats:italic>}) operands in a constant time independent of <jats:italic>n<\/jats:italic> and <jats:italic>L.<\/jats:italic> The following property is shown as the result. Using the redundant <jats:italic>P<\/jats:italic>\u2010ary representation (called <jats:italic>K<\/jats:italic>\u2010redundant <jats:italic>P<\/jats:italic>\u2010ary representation), where <jats:italic>K<\/jats:italic>,\u2026, 2, 1, 0, 1, 2, \u2026, <jats:italic>P<\/jats:italic>\u2010<jats:italic>l<\/jats:italic> can be used as the digit, the minimum necessary value for <jats:italic>K<\/jats:italic> is <jats:italic>L<\/jats:italic> \u2010 1, and the desired algorithm can be found if <jats:italic>L<\/jats:italic>\u20101<jats:italic>K<\/jats:italic>(\u03f5<jats:italic>P<\/jats:italic>\u20141). The presented algorithm can easily be extended to an algorithm which can execute the addition and subtraction of <jats:italic>L<\/jats:italic> operands in a constant time independent of <jats:italic>n<\/jats:italic>, where <jats:italic>L<\/jats:italic> is an arbitrary number such that <jats:italic>P<\/jats:italic>+I<jats:italic>L.<\/jats:italic> It is shown that if <jats:italic>f(x)<\/jats:italic> = [log<jats:italic>px<\/jats:italic>]+1 ([<jats:italic>y<\/jats:italic>] is the minimum integer not less than <jats:italic>y<\/jats:italic>), the computation time <jats:italic>T<\/jats:italic> is <jats:italic>O<\/jats:italic>(loga), where \u03b1 is the minimum number of operations for the function <jats:italic>f<\/jats:italic>( ) satisfying <jats:italic>f<\/jats:italic>(<jats:italic>f<\/jats:italic>(<jats:italic>f<\/jats:italic>(<jats:italic>F<\/jats:italic>))))<jats:italic>P<\/jats:italic>; \u03b1 increases stepwise with <jats:italic>L<\/jats:italic>. When <jats:italic>P<\/jats:italic> is increased, <jats:italic>T<\/jats:italic> decreases, and the range of <jats:italic>L<\/jats:italic>, for which <jats:italic>T<\/jats:italic> is constant, increases exponentially. Using the proposed algorithm, one can expect that a multi\u2010input high\u2010speed adder\u2010subtractor can be constructed. Various kinds of high\u2010speed data processings can be realized, including the multiplication, where the shifted partial products are added.<\/jats:p>","DOI":"10.1002\/scj.4690200406","type":"journal-article","created":{"date-parts":[[2007,7,7]],"date-time":"2007-07-07T17:37:28Z","timestamp":1183829848000},"page":"59-70","source":"Crossref","is-referenced-by-count":0,"title":["High\u2010speed multioperand addition and subtraction using a p\u2010ary representation with necessary and minimum redundancy"],"prefix":"10.1002","volume":"20","author":[{"given":"Takeomi","family":"Tamesada","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2007,3,21]]},"reference":[{"issue":"10","key":"e_1_2_1_2_2","first-page":"1853","article-title":"High\u2010speed addition and subtraction using minimum redundant P\u2010ary representation","volume":"70","author":"Tamesada K.","year":"1987","journal-title":"Trans. (D), I.E.C.E., Japan"},{"issue":"12","key":"e_1_2_1_3_2","first-page":"2800","article-title":"A high\u2010speed subtraction using minimum redundant P\u2010ary representation","volume":"70","author":"Tamesada T.","year":"1987","journal-title":"Trans. (D), I.E.C.E., Japan"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1109\/TEC.1961.5219227"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1109\/T-C.1973.223670"},{"key":"e_1_2_1_6_2","volume-title":"Computer Arithmetic\/Principles, Architecture, and Design","author":"Hwang K.","year":"1979"},{"key":"e_1_2_1_7_2","volume-title":"Computer Arithmetic\/Logic and Design","author":"Spaniol O.","year":"1981"},{"issue":"6","key":"e_1_2_1_8_2","first-page":"683","article-title":"VLSI\u2010oriented high\u2010speed multiplier using redundant binary addition tree","volume":"66","author":"Takagi Yasuura","year":"1983","journal-title":"Trans. (D), I.E.C.E., Japan"},{"issue":"4","key":"e_1_2_1_9_2","first-page":"450","article-title":"VLSI\u2010oriented high\u2010speed divider using redundant binary representation","volume":"67","author":"Takagi Yasuura","year":"1984","journal-title":"Trans. (D), I.E.C.E., Japan"}],"container-title":["Systems and Computers in Japan"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fscj.4690200406","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/scj.4690200406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T13:17:20Z","timestamp":1697980640000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/scj.4690200406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,1]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1989,1]]}},"alternative-id":["10.1002\/scj.4690200406"],"URL":"https:\/\/doi.org\/10.1002\/scj.4690200406","archive":["Portico"],"relation":{},"ISSN":["0882-1666","1520-684X"],"issn-type":[{"value":"0882-1666","type":"print"},{"value":"1520-684X","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,1]]}}}