[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-56166-en":3,"doc-seo-56166-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},56166,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","On the Evaluation of Powers and Related Problems (Preliminary Version)","Evaluation of powers via addition chains for non-negative integer matrices is studied through the function L(p,q,N), which measures the worst-case minimum number of vector-addition steps needed to compute all p rows of a p-by-q matrix with entries from {0,1,…,N}. The work motivates applications to optimal multiplication of multivariate sparse polynomials and to simulating linear fixed-point transformations using only additions. It develops a constructive graph-theoretic framework for Lupanov-style bounds and presents Theorem 1 under structural size conditions.","ON THE EVALUATION OF POWERS AND RELATED PROBLEMS  \n(Preliminary Version)  \nby  \nNicholas PippengerMathematical Sciences DepartmentIBM Thomas J.Watson Research CenterYorktown Heights,N.Y.10598  \n# Introduction\n\n   \nLet M be a p-by-q matrix of non-negative inte-gers.We shall consider the problem of obtaining thep rows of M by computations in which  \n1.the zero vector(0,...,0)and the qunit vectors(1,.…,0),……,(0,.,1)are availableat no cost;  \n2.the sum of two(not necessarily dstinct)previously computed vectors is available at a cost of  \nSuch a computation is called an addition chain for M.Let (M)  denote the minimum possible number of stepsin an addition chain for M.Let L(p,q,N)denotethe maximum of (M)over al¹p-by-q matrices Mwhose entries are drawn from {0,1,…,N}.  \n   \nLet us mention two applications of this problem.  \nseek to compute the p monomials  \nstarting from the q indeterminates x₁,x₂,…x,and using the minimum possible number of two-operandmultiplications.  \nThis problem has obvious relevance for evaluatingsparse multivariate polynomials.  Another applicationemerges upon \"taking the logarithm of the problem\".  \nProblem 2:We are given a matrix M as before.We  \n   \nM1×1+M12×2+…+M1q×qM21×1+M22×2 +…+M2q*qMp1×1+Mp2×2 +…+MPq*q,  \nstarting with the q indeterminates X1,X2,…×q'and using the minimum possible number of two-operandadditions.  \nThis problem arises in signal-processing applica-  \ntions:we are called upon to perform a linear trans-  \nformation whose coefficients are fixed-point numbers  \n(which can be regarded as integers);if we have noMULTIPLY instruction in our machine,we must simulatescalar multiplication by addition.(This problemwould be more realistic if we allowed negative coeffi-cients,subtractions,and short shifts.These changeswould not affect our analysis or results in any sig-nificant way.)  \nMany special cases of this problem have been con-sidered by various authors.The first of these was thecase p=1,q=1 proposed in 1937 by Scholz [1],whoobserved that  \nlog N≤L(1,1,N)≤2 log N.  \n(In this paper,log denotes the binary logarithm,whileln denotes the natural logarithm.)Brauer [2]im-proved the upper bound to  \nL(1,1,N)≤1og N  \nand thus obtained an asymptotic formula for L(1,1,N).Erdös[3]worked out the next term in the expansion:  \nIn 1956,Lupanov [4]considered the case N=1.(Lupanov's problem was not formally equivalent to thiscase,but his ideas require only trivial modificationsfor application in the present context.)He showedthat if  \n=2°()and q=2°(p)  \n(1)  \n(this condition says that the matrix is not \"exponen-tially lopsided\"),then  \nFurthermore,he obtained  \n(2)  \nand thus an asymptotic formula for L(p,q,1),on the additional condition that  \n(3)  \ntend to 0 or ∞.Nechiporuk [5]weakened this condi-tion to require only that(3)tend to a member of aninfinite,but nowhere dense,set of rational numbers.His solution involves a \"nonconstructive\"averaging ar-gument.Below,we shall give a constructive solutionthat requires no condition on(3)and establishes(2)on the assumption of(1)alone.  \nIn 1963,Bel1man [6]posed the case p=1,andStraus [7]showed that for any fixed q,  \nL(1,q,N)～log N.  \nFinally,in 1969 Knuth[8]posed the case q=1,andYao [9]showed that for any fixed p,  \nL(p,1,N)～log N.  \nThis result actually holds for p=o(1og log N).Below,we shall show that for p=n°(1),  \n# Lupanov's Problem\n\n   \nWe shall develop our first theorem in a graph-theoretic setting introduced by Lupanov.Let r be anacyclic directed graph with p distinguished verticesand q distinguished verticesn₁…,npWe shall say that r realizes M if the number offor 1≤1≤pdfrected paths from n1 to⁵18 M1and 1≤jsq.Let '(M)denote the minimum possiblenumber of edges in a graph that realizes M.LetL'(p,q,N)denote the maximum of 8'(M)over allp-by-q matríces whose entries are drawn from  \n{0,1,...N}.  \nTheorem 1:If p=2°(9)and q=2°(P),  \n   \nsece DPpyq,1)-1(9per crea12e the branLeposeof M,then reverse the direction of all edges),wemay assume p≥q.The upper bound in the case","cbCaifCh5His88ZW","https://ap.wps.com/l/cbCaifCh5His88ZW","pdf",473583,1,6,"English","en",105,"# Introduction\n# Lupanov's Problem\n## Graph-theoretic model and realization\n## Transformation of realizing graphs","[{\"question\":\"What is an addition chain for a non-negative integer p-by-q matrix?\",\"answer\":\"An addition chain is a computation model where the zero vector and the q qunit vectors are free, and each additional step computes the sum of two previously computed vectors at a cost of one step.\"},{\"question\":\"How are L(p,q,N) and L′(p,q,N) defined in the document?\",\"answer\":\"L(p,q,N) is the maximum, over all p-by-q matrices with entries in {0,1,…,N}, of the minimum number of steps needed in an addition chain. L′(p,q,N) is defined similarly but uses a graph realization concept and measures the maximum over such matrices of the minimum number of edges in realizing acyclic directed graphs.\"},{\"question\":\"What is Lupanov’s problem approach used in this paper?\",\"answer\":\"The paper presents a graph-theoretic setting where an acyclic directed graph with p inputs and q distinguished vertices realizes a matrix by counting directed paths. It then constructs a constructive solution that establishes upper bounds like those associated with Lupanov under conditions comparing the growth rates of p and q.\"}]",1783717325,15,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"on-the-evaluation-of-powers-and-related-problems-preliminary-version","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/on-the-evaluation-of-powers-and-related-problems-preliminary-version/56166/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-10",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is an addition chain for a non-negative integer p-by-q matrix?","Question",{"text":75,"@type":76},"An addition chain is a computation model where the zero vector and the q qunit vectors are free, and each additional step computes the sum of two previously computed vectors at a cost of one step.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are L(p,q,N) and L′(p,q,N) defined in the document?",{"text":80,"@type":76},"L(p,q,N) is the maximum, over all p-by-q matrices with entries in {0,1,…,N}, of the minimum number of steps needed in an addition chain. L′(p,q,N) is defined similarly but uses a graph realization concept and measures the maximum over such matrices of the minimum number of edges in realizing acyclic directed graphs.",{"name":82,"@type":73,"acceptedAnswer":83},"What is Lupanov’s problem approach used in this paper?",{"text":84,"@type":76},"The paper presents a graph-theoretic setting where an acyclic directed graph with p inputs and q distinguished vertices realizes a matrix by counting directed paths. It then constructs a constructive solution that establishes upper bounds like those associated with Lupanov under conditions comparing the growth rates of p and q.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]