On the complexity of constructing optimal ordered binary decision diagrams
Identifieur interne : 001647 ( Istex/Corpus ); précédent : 001646; suivant : 001648On the complexity of constructing optimal ordered binary decision diagrams
Auteurs : Christoph Meinel ; Anna SlobodováSource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 1994.
Abstract
Abstract: In the following we prove the widely assumed conjecture that the construction of an optimal OBDD-representation of a Boolean function f is intractable if f is given in terms of DNFs, CNFs, logical circuits, or Ω-BDDs. The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance. Reflecting these theoretical results and the requirements in the area of practical applications (above all in CAD of ciruits), the problem of construction of the minimal OBDD for a given variable ordering becomes of growing importance. We show that, starting with a free BDD (FBDD) P of a Boolean function f on X, for each ordering π of X, the minimal π OBDD- representation Q of f can be constructed output-efficiently, i.e. in time that is polynomial in the sizes of input and output. This result can be considerably improved in the case of starting with an OBDD instead of an FBDD. The time requirement is then smaller than the product of input length and the square of output length (worst case), and almost the product of input and output length in average.
Url:
DOI: 10.1007/3-540-58338-6_98
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The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance. Reflecting these theoretical results and the requirements in the area of practical applications (above all in CAD of ciruits), the problem of construction of the minimal OBDD for a given variable ordering becomes of growing importance. We show that, starting with a free BDD (FBDD) P of a Boolean function f on X, for each ordering π of X, the minimal π OBDD- representation Q of f can be constructed output-efficiently, i.e. in time that is polynomial in the sizes of input and output. This result can be considerably improved in the case of starting with an OBDD instead of an FBDD. 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EURO-DAC'93</title></host><title>Dynamic Variable Reoredering for BDD Minimization</title></json:item><json:item><author><json:item><name>M,R Garey</name></json:item><json:item><name>D,S Johnson</name></json:item></author><host><author></author><title>Computers and Intractability. W. H. Freeman and Company</title><publicationDate>1979</publicationDate></host><publicationDate>1979</publicationDate></json:item><json:item><host><author><json:item><name>J Gergov</name></json:item><json:item><name>Ch </name></json:item></author><title>Meinel: Efficient Analysis and Manipulation of OBDDs Can Be Extended to FBDDs, to appeare in IEEE Trans. on Computers</title></host></json:item><json:item><author><json:item><name>J Gergov</name></json:item><json:item><name> Ch</name></json:item><json:item><name> Meinel</name></json:item></author><host><pages><last>320</last><first>310</first></pages><author></author><title>Proc. WG'92</title><publicationDate>1993</publicationDate></host><title>Analysis and Manipulation of Boolean Functions in Terms of Decision Graphs</title><publicationDate>1993</publicationDate></json:item><json:item><author><json:item><name>N Ishiura</name></json:item><json:item><name>H Sawada</name></json:item><json:item><name>S </name></json:item></author><host><pages><last>475</last><first>472</first></pages><author></author><title>Proc. IEEE International Conference on Computer Aided Design</title><publicationDate>1991</publicationDate></host><title>Yajima: Minimization of Binary Decision Diagrams Based on Exchanges of Variables</title><publicationDate>1991</publicationDate></json:item><json:item><author><json:item><name> Ch</name></json:item></author><host><author></author><title>Modified Branching Programs and Their Computational Power</title><publicationDate>1989</publicationDate></host><publicationDate>1989</publicationDate></json:item><json:item><author><json:item><name> Ch</name></json:item></author><host><pages><last>170</last><first>149</first></pages><issue>46</issue><author></author><title>EATCS-Bulletin</title><publicationDate>1992</publicationDate></host><title>Branching Programs -An Efficient Data Structure for Computer- Aided Circuit Design</title><publicationDate>1992</publicationDate></json:item><json:item><author><json:item><name>R Rudeu</name></json:item></author><host><pages><last>47</last><first>42</first></pages><author></author><title>Proc. 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The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance. Reflecting these theoretical results and the requirements in the area of practical applications (above all in CAD of ciruits), the problem of construction of the minimal OBDD for a given variable ordering becomes of growing importance. We show that, starting with a free BDD (FBDD) P of a Boolean function f on X, for each ordering π of X, the minimal π OBDD- representation Q of f can be constructed output-efficiently, i.e. in time that is polynomial in the sizes of input and output. This result can be considerably improved in the case of starting with an OBDD instead of an FBDD. 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Type="Secondary">Software Engineering</BookSubject><BookSubject Code="I14037" Priority="6" Type="Secondary">Programming Languages, Compilers, Interpreters</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup></BookInfo><BookHeader><EditorGroup><Editor><EditorName DisplayOrder="Western"><GivenName>Igor</GivenName><FamilyName>Prívara</FamilyName></EditorName></Editor><Editor><EditorName DisplayOrder="Western"><GivenName>Branislav</GivenName><FamilyName>Rovan</FamilyName></EditorName></Editor><Editor><EditorName DisplayOrder="Western"><GivenName>Peter</GivenName><FamilyName>Ruzička</FamilyName></EditorName></Editor></EditorGroup></BookHeader><Chapter ID="Chap44" Language="En"><ChapterInfo ChapterType="ReviewPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ChapterID>44</ChapterID><ChapterDOI>10.1007/3-540-58338-6_98</ChapterDOI><ChapterSequenceNumber>44</ChapterSequenceNumber><ChapterTitle Language="En">On the complexity of constructing optimal ordered binary decision diagrams</ChapterTitle><ChapterCategory>Contributions</ChapterCategory><ChapterFirstPage>515</ChapterFirstPage><ChapterLastPage>524</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1994</CopyrightYear></ChapterCopyright><ChapterHistory><OnlineDate><Year>2005</Year><Month>6</Month><Day>4</Day></OnlineDate></ChapterHistory><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>3540583386</BookID><BookTitle>Mathematical Foundations of Computer Science 1994</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Christoph</GivenName><FamilyName>Meinel</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Anna</GivenName><FamilyName>Slobodová</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>FB IV-Informatik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><Postcode>D-54286</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>In the following we prove the widely assumed conjecture that the construction of an optimal OBDD-representation of a Boolean function f is intractable if f is given in terms of DNFs, CNFs, logical circuits, or Ω-BDDs. The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance.</Para><Para>Reflecting these theoretical results and the requirements in the area of practical applications (above all in CAD of ciruits), the problem of construction of the minimal OBDD for a given variable ordering becomes of growing importance. We show that, starting with a free BDD (FBDD) P of a Boolean function f on X, for each ordering π of X, the minimal π OBDD- representation Q of f can be constructed output-efficiently, i.e. in time that is polynomial in the sizes of input and output. This result can be considerably improved in the case of starting with an OBDD instead of an FBDD. The time requirement is then smaller than the product of input length and the square of output length (worst case), and almost the product of input and output length in average.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>On the complexity of constructing optimal ordered binary decision diagrams</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>On the complexity of constructing optimal ordered binary decision diagrams</title></titleInfo><name type="personal"><namePart type="given">Christoph</namePart><namePart type="family">Meinel</namePart><affiliation>FB IV-Informatik, Universität Trier, D-54286, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Anna</namePart><namePart type="family">Slobodová</namePart><affiliation>FB IV-Informatik, Universität Trier, D-54286, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference" displayLabel="ReviewPaper"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2005-06-04</dateIssued><copyrightDate encoding="w3cdtf">1994</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: In the following we prove the widely assumed conjecture that the construction of an optimal OBDD-representation of a Boolean function f is intractable if f is given in terms of DNFs, CNFs, logical circuits, or Ω-BDDs. The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance. Reflecting these theoretical results and the requirements in the area of practical applications (above all in CAD of ciruits), the problem of construction of the minimal OBDD for a given variable ordering becomes of growing importance. We show that, starting with a free BDD (FBDD) P of a Boolean function f on X, for each ordering π of X, the minimal π OBDD- representation Q of f can be constructed output-efficiently, i.e. in time that is polynomial in the sizes of input and output. This result can be considerably improved in the case of starting with an OBDD instead of an FBDD. The time requirement is then smaller than the product of input length and the square of output length (worst case), and almost the product of input and output length in average.</abstract><relatedItem type="host"><titleInfo><title>Mathematical Foundations of Computer Science 1994</title><subTitle>19th International Symposium, MFCS'94 Košice, Slovakia, August 22–26, 1994 Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Igor</namePart><namePart type="family">Prívara</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Branislav</namePart><namePart type="family">Rovan</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Peter</namePart><namePart type="family">Ruzička</namePart><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Proceedings"></genre><originInfo><copyrightDate encoding="w3cdtf">1994</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11645">Computer Science</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="I">Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I16013">Computation by Abstract Devices</topic><topic authority="SpringerSubjectCodes" authorityURI="I16021">Algorithm Analysis and Problem Complexity</topic><topic authority="SpringerSubjectCodes" authorityURI="I1603X">Logics and Meanings of Programs</topic><topic authority="SpringerSubjectCodes" authorityURI="I16048">Mathematical Logic and Formal Languages</topic><topic authority="SpringerSubjectCodes" authorityURI="I14029">Software Engineering</topic><topic authority="SpringerSubjectCodes" authorityURI="I14037">Programming Languages, Compilers, Interpreters</topic></subject><identifier type="DOI">10.1007/3-540-58338-6</identifier><identifier type="ISBN">978-3-540-58338-7</identifier><identifier type="eISBN">978-3-540-48663-3</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">42007</identifier><identifier type="BookID">3540583386</identifier><identifier type="BookChapterCount">54</identifier><identifier type="BookVolumeNumber">841</identifier><part><date>1994</date><detail type="volume"><number>841</number><caption>vol.</caption></detail><extent unit="pages"><start>515</start><end>524</end></extent></part><recordInfo><recordOrigin>Springer-Verlag, 1994</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Lecture Notes in Computer Science</title></titleInfo><name type="personal"><namePart type="given">G.</namePart><namePart type="family">Goos</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Hartmanis</namePart><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><copyrightDate encoding="w3cdtf">1994</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="SeriesID">558</identifier><recordInfo><recordOrigin>Springer-Verlag, 1994</recordOrigin></recordInfo></relatedItem><identifier type="istex">011F469CE5E2F317F2AB1B6F352D0C9B6F1461E1</identifier><identifier type="DOI">10.1007/3-540-58338-6_98</identifier><identifier type="ChapterID">44</identifier><identifier type="ChapterID">Chap44</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag, 1994</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Springer-Verlag, 1994</recordOrigin></recordInfo></mods></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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