A Class of Optimal Structures for Node Computations in Message Passing Algorithms
Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages <inline-formula> <tex-math notation="LaTeX">\mathbf {x} = (x_{1}, x_{2}, \ldots, x_{n}) </tex-math></inline-formula> and <inline-formula>...
Saved in:
| Published in: | IEEE transactions on information theory Vol. 68; no. 1; pp. 93 - 104 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
IEEE
01.01.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects: | |
| ISSN: | 0018-9448, 1557-9654 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages <inline-formula> <tex-math notation="LaTeX">\mathbf {x} = (x_{1}, x_{2}, \ldots, x_{n}) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathbf {y} = (y_{1}, y_{2}, \ldots, y_{n}) </tex-math></inline-formula>, respectively. In this paper, we investigate a class of structures that can be adopted by the node for computing <inline-formula> <tex-math notation="LaTeX">\mathbf {y} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula>, where each <inline-formula> <tex-math notation="LaTeX">y_{j}, j = 1, 2, \ldots, n </tex-math></inline-formula> is computed via a binary tree with leaves <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula> excluding <inline-formula> <tex-math notation="LaTeX">x_{j} </tex-math></inline-formula>. We make three main contributions regarding this class of structures. First, we prove that the minimum complexity of such a structure is <inline-formula> <tex-math notation="LaTeX">3n - 6 </tex-math></inline-formula>, and if a structure has such complexity, its minimum latency is <inline-formula> <tex-math notation="LaTeX">\delta + \lceil \log (n-2^{\delta }) \rceil </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\delta = \lfloor \log (n/2) \rfloor </tex-math></inline-formula>, where the logarithm always takes base two. Second, we prove that the minimum latency of such a structure is <inline-formula> <tex-math notation="LaTeX">\lceil \log (n-1) \rceil </tex-math></inline-formula>, and if a structure has such latency, its minimum complexity is <inline-formula> <tex-math notation="LaTeX">n \log (n-1) </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">n-1 </tex-math></inline-formula> is a power of two. Third, given <inline-formula> <tex-math notation="LaTeX">(n, \tau) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\tau \geq \lceil \log (n-1) \rceil </tex-math></inline-formula>, we propose a construction for a structure which we conjecture to have the minimum complexity among structures with latencies at most <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>. Our construction method runs in <inline-formula> <tex-math notation="LaTeX">O(n^{3} \log ^{2}(n)) </tex-math></inline-formula> time, and the obtained structure has complexity at most (generally much smaller than) <inline-formula> <tex-math notation="LaTeX">n \lceil \log (n) \rceil - 2 </tex-math></inline-formula>. |
|---|---|
| AbstractList | Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages [Formula Omitted] and [Formula Omitted], respectively. In this paper, we investigate a class of structures that can be adopted by the node for computing [Formula Omitted] from [Formula Omitted], where each [Formula Omitted] is computed via a binary tree with leaves [Formula Omitted] excluding [Formula Omitted]. We make three main contributions regarding this class of structures. First, we prove that the minimum complexity of such a structure is [Formula Omitted], and if a structure has such complexity, its minimum latency is [Formula Omitted] with [Formula Omitted], where the logarithm always takes base two. Second, we prove that the minimum latency of such a structure is [Formula Omitted], and if a structure has such latency, its minimum complexity is [Formula Omitted] when [Formula Omitted] is a power of two. Third, given [Formula Omitted] with [Formula Omitted], we propose a construction for a structure which we conjecture to have the minimum complexity among structures with latencies at most [Formula Omitted]. Our construction method runs in [Formula Omitted] time, and the obtained structure has complexity at most (generally much smaller than) [Formula Omitted]. Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages <inline-formula> <tex-math notation="LaTeX">\mathbf {x} = (x_{1}, x_{2}, \ldots, x_{n}) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathbf {y} = (y_{1}, y_{2}, \ldots, y_{n}) </tex-math></inline-formula>, respectively. In this paper, we investigate a class of structures that can be adopted by the node for computing <inline-formula> <tex-math notation="LaTeX">\mathbf {y} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula>, where each <inline-formula> <tex-math notation="LaTeX">y_{j}, j = 1, 2, \ldots, n </tex-math></inline-formula> is computed via a binary tree with leaves <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula> excluding <inline-formula> <tex-math notation="LaTeX">x_{j} </tex-math></inline-formula>. We make three main contributions regarding this class of structures. First, we prove that the minimum complexity of such a structure is <inline-formula> <tex-math notation="LaTeX">3n - 6 </tex-math></inline-formula>, and if a structure has such complexity, its minimum latency is <inline-formula> <tex-math notation="LaTeX">\delta + \lceil \log (n-2^{\delta }) \rceil </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\delta = \lfloor \log (n/2) \rfloor </tex-math></inline-formula>, where the logarithm always takes base two. Second, we prove that the minimum latency of such a structure is <inline-formula> <tex-math notation="LaTeX">\lceil \log (n-1) \rceil </tex-math></inline-formula>, and if a structure has such latency, its minimum complexity is <inline-formula> <tex-math notation="LaTeX">n \log (n-1) </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">n-1 </tex-math></inline-formula> is a power of two. Third, given <inline-formula> <tex-math notation="LaTeX">(n, \tau) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\tau \geq \lceil \log (n-1) \rceil </tex-math></inline-formula>, we propose a construction for a structure which we conjecture to have the minimum complexity among structures with latencies at most <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>. Our construction method runs in <inline-formula> <tex-math notation="LaTeX">O(n^{3} \log ^{2}(n)) </tex-math></inline-formula> time, and the obtained structure has complexity at most (generally much smaller than) <inline-formula> <tex-math notation="LaTeX">n \lceil \log (n) \rceil - 2 </tex-math></inline-formula>. |
| Author | He, Xuan Zhou, Liang Cai, Kui |
| Author_xml | – sequence: 1 givenname: Xuan orcidid: 0000-0002-8934-1981 surname: He fullname: He, Xuan email: xhe@swjtu.edu.cn organization: School of Information Science and Technology, Southwest Jiaotong University, Chengdu, China – sequence: 2 givenname: Kui orcidid: 0000-0003-2059-0071 surname: Cai fullname: Cai, Kui email: cai_kui@sutd.edu.sg organization: Science, Mathematics and Technology (SMT) Cluster, Singapore University of Technology and Design, Singapore – sequence: 3 givenname: Liang orcidid: 0000-0002-9453-3734 surname: Zhou fullname: Zhou, Liang email: lzhou@uestc.edu.cn organization: National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, China |
| BookMark | eNo9kE1LwzAYgIMouE3vgpeA5858pzmO4sdgOsV5DrFNZkfX1CQ9-O_N2PAUXnie9yXPFJz3vrcA3GA0xxip-81yMyeI4DnFWClOzsAEcy4LJTg7BxOEcFkoxspLMI1xl0fGMZmA9wWsOhMj9A6uh9TuTQc_UhjrNAYbofMBvvrGwsrvhzGZ1Po-wraHLzZGs7XwLbttv4WLbutDm7738QpcONNFe316Z-Dz8WFTPRer9dOyWqyKmjCWipqXTSMZ5UIQ-cXLWtSS8bqmRijsmFTKUSYwkdTIshGOIGFKlnmLuTPW0Rm4O-4dgv8ZbUx658fQ55OaZI8JiRHKFDpSdfAxBuv0EPInw6_GSB_C6RxOH8LpU7is3B6V1lr7jysuEZWU_gGJAGoE |
| CODEN | IETTAW |
| Cites_doi | 10.1109/SiPS.2015.7345024 10.1109/LCOMM.2019.2937112 10.1109/ACSSC.2015.7421419 10.1109/18.910577 10.1109/IEEESTD.2003.94282 10.1109/ICC.2016.7510906 10.1109/TIT.1962.1057683 10.1109/TCOMM.2005.852852 10.1109/GLOCOM.2008.ECP.214 10.1109/TCSII.2014.2362663 10.1109/ACCESS.2018.2797694 10.1109/TCSI.2008.924892 10.1109/ISIT.2015.7282490 10.1109/ICSPCS.2018.8631719 10.1587/transcom.2018TTI0001 10.1109/GLOBECOM38437.2019.9013335 10.1109/GLOCOM.2001.965575 10.1109/JSAC.2016.2603708 10.1109/TCOMM.2019.2944159 10.1109/TVLSI.2017.2766925 |
| ContentType | Journal Article |
| Copyright | Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2022 |
| Copyright_xml | – notice: Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2022 |
| DBID | 97E RIA RIE AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| DOI | 10.1109/TIT.2021.3119952 |
| DatabaseName | IEEE Xplore (IEEE) IEEE All-Society Periodicals Package (ASPP) 1998–Present IEEE Electronic Library (IEL) CrossRef Computer and Information Systems Abstracts Electronics & Communications Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional |
| DatabaseTitle | CrossRef Technology Research Database Computer and Information Systems Abstracts – Academic Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Technology Research Database |
| Database_xml | – sequence: 1 dbid: RIE name: IEEE Electronic Library (IEL) url: https://ieeexplore.ieee.org/ sourceTypes: Publisher |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Computer Science |
| EISSN | 1557-9654 |
| EndPage | 104 |
| ExternalDocumentID | 10_1109_TIT_2021_3119952 9570373 |
| Genre | orig-research |
| GrantInformation_xml | – fundername: National Natural Science Foundation of China grantid: 62101462 funderid: 10.13039/501100001809 – fundername: Singapore Ministry of Education Academic Research Fund Tier 2 grantid: MOE2019-T2-2-123 funderid: 10.13039/501100001459 – fundername: RIE2020 Advanced Manufacturing and Engineering (AME) Programmatic grantid: A18A6b0057 |
| GroupedDBID | -~X .DC 0R~ 29I 3EH 4.4 5GY 5VS 6IK 97E AAJGR AARMG AASAJ AAWTH ABAZT ABFSI ABQJQ ABVLG ACGFO ACGFS ACGOD ACIWK AENEX AETEA AETIX AGQYO AGSQL AHBIQ AI. AIBXA AKJIK AKQYR ALLEH ALMA_UNASSIGNED_HOLDINGS ASUFR ATWAV BEFXN BFFAM BGNUA BKEBE BPEOZ CS3 DU5 E.L EBS EJD F5P HZ~ H~9 IAAWW IBMZZ ICLAB IDIHD IFIPE IFJZH IPLJI JAVBF LAI M43 MS~ O9- OCL P2P PQQKQ RIA RIE RNS RXW TAE TN5 VH1 VJK AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| ID | FETCH-LOGICAL-c244t-c58dd74356627b58c6c745cc3a691f4799f3461273a78d6f206a84435e15faef3 |
| IEDL.DBID | RIE |
| ISICitedReferencesCount | 0 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000732981200010&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0018-9448 |
| IngestDate | Sun Jun 29 15:37:38 EDT 2025 Sat Nov 29 03:31:46 EST 2025 Wed Aug 27 05:07:52 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Language | English |
| License | https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html https://doi.org/10.15223/policy-029 https://doi.org/10.15223/policy-037 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c244t-c58dd74356627b58c6c745cc3a691f4799f3461273a78d6f206a84435e15faef3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0003-2059-0071 0000-0002-8934-1981 0000-0002-9453-3734 |
| PQID | 2612467100 |
| PQPubID | 36024 |
| PageCount | 12 |
| ParticipantIDs | proquest_journals_2612467100 ieee_primary_9570373 crossref_primary_10_1109_TIT_2021_3119952 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-Jan. 2022-1-00 20220101 |
| PublicationDateYYYYMMDD | 2022-01-01 |
| PublicationDate_xml | – month: 01 year: 2022 text: 2022-Jan. |
| PublicationDecade | 2020 |
| PublicationPlace | New York |
| PublicationPlace_xml | – name: New York |
| PublicationTitle | IEEE transactions on information theory |
| PublicationTitleAbbrev | TIT |
| PublicationYear | 2022 |
| Publisher | IEEE The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Publisher_xml | – name: IEEE – name: The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| References | ref13 ref12 ref15 ref14 ref20 ref11 ref10 ref21 ref2 ref1 Cormen (ref22) 2009 ref17 ref16 He (ref5) 2019 ref19 ref18 ref8 ref7 ref9 ref4 ref3 ref6 |
| References_xml | – ident: ref17 doi: 10.1109/SiPS.2015.7345024 – volume-title: Introduction to Algorithms year: 2009 ident: ref22 – ident: ref6 doi: 10.1109/LCOMM.2019.2937112 – ident: ref15 doi: 10.1109/ACSSC.2015.7421419 – ident: ref2 doi: 10.1109/18.910577 – ident: ref8 doi: 10.1109/IEEESTD.2003.94282 – ident: ref16 doi: 10.1109/ICC.2016.7510906 – ident: ref1 doi: 10.1109/TIT.1962.1057683 – ident: ref3 doi: 10.1109/TCOMM.2005.852852 – ident: ref11 doi: 10.1109/GLOCOM.2008.ECP.214 – ident: ref10 doi: 10.1109/TCSII.2014.2362663 – ident: ref19 doi: 10.1109/ACCESS.2018.2797694 – ident: ref9 doi: 10.1109/TCSI.2008.924892 – ident: ref12 doi: 10.1109/ISIT.2015.7282490 – ident: ref20 doi: 10.1109/ICSPCS.2018.8631719 – ident: ref21 doi: 10.1587/transcom.2018TTI0001 – ident: ref4 doi: 10.1109/GLOBECOM38437.2019.9013335 – ident: ref7 doi: 10.1109/GLOCOM.2001.965575 – year: 2019 ident: ref5 article-title: Mutual information-maximizing quantized belief propagation decoding of regular LDPC codes publication-title: arXiv:1904.06666 – ident: ref13 doi: 10.1109/JSAC.2016.2603708 – ident: ref14 doi: 10.1109/TCOMM.2019.2944159 – ident: ref18 doi: 10.1109/TVLSI.2017.2766925 |
| SSID | ssj0014512 |
| Score | 2.3845465 |
| Snippet | Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages <inline-formula> <tex-math... Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages [Formula Omitted] and [Formula... |
| SourceID | proquest crossref ieee |
| SourceType | Aggregation Database Index Database Publisher |
| StartPage | 93 |
| SubjectTerms | Algorithms Binary structure Binary trees Complexity Complexity theory Decoding Directed graphs Electronic mail latency low-density parity-check (LDPC) code Message passing message passing algorithm Nodes Parity check codes |
| Title | A Class of Optimal Structures for Node Computations in Message Passing Algorithms |
| URI | https://ieeexplore.ieee.org/document/9570373 https://www.proquest.com/docview/2612467100 |
| Volume | 68 |
| WOSCitedRecordID | wos000732981200010&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVIEE databaseName: IEEE Electronic Library (IEL) customDbUrl: eissn: 1557-9654 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0014512 issn: 0018-9448 databaseCode: RIE dateStart: 19630101 isFulltext: true titleUrlDefault: https://ieeexplore.ieee.org/ providerName: IEEE |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LSwMxEB5q8aAHq1axWiUHL4Jr95FNNsciioLWihW8LZuXFrQrtvX3O8lui6IXbzkkwzKz80zmG4BjqTSXyvKAKVQ3SqUIhE5lILm2MS1kFiov6Rs-GGRPT2LYgNNlL4wxxj8-M2du6e_ydanmrlTWEw4uiicrsMI5q3q1ljcGNI0qZPAIFRhzjsWVZCh6o-sRJoJxhPmpa0iOf7ggP1PllyH23uWy9b_v2oSNOook_UrsW9Awk21oLSY0kFpht2H9G9xgG-77xM_AJKUld2gq3pDEg8ePnWPSTTB8JYNSG1LRqUp5ZDwht25MyrMhQzyLhEj_9bn8GM9e3qY78Hh5MTq_CuqRCoFCPz4LVJppjUFD6nDfZZoppjhNlUoKJiJLuRA2oRj08KTgmWY2DlmRUdxvotQWxia70JyUE7MHJEm1slksFUMPJ4UthKGcSlNYGzNcduBkweX8vULOyH3GEYocJZI7ieS1RDrQdlxd7qsZ2oHuQix5rVrT3GGeoXWPwnD_71MHsBa7HgVfJ-lCE_loDmFVfc7G048j_9d8AXCPwD0 |
| linkProvider | IEEE |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LSwMxEB60CurB-sT6zMGL4Np9ZDebYxHFYlsVK3hbNq9asF2x1d_vJLstil685ZAMy8zOM5lvAE6FVExIw7xEorpRKrjHVSw8wZQJaS5SXzpJd1ivlz4_8_sFOJ_3wmit3eMzfWGX7i5fFfLDlsqa3MJFsWgRluzkrKpba35nQOOgxAYPUIUx65hdSvq82W_3MRUMA8xQbUty-MMJuakqv0yx8y_X9f992QasV3EkaZWC34QFPd6C-mxGA6lUdgvWvgEObsNDi7gpmKQw5A6NxQhJPDoE2Q9MuwkGsKRXKE1KOmUxjwzHpGsHpQw0ucezSIi0XgfF-3D6MprswNP1Vf_yxquGKngSPfnUk3GqFIYNsUV-F3EqE8loLGWUJzwwlHFuIophD4tylqrEhH6SpxT36yA2uTbRLtTGxVjvAYliJU0aCpmgjxPc5FxTRoXOjQkTXDbgbMbl7K3EzshczuHzDCWSWYlklUQasG25Ot9XMbQBhzOxZJVyTTKLeob2PfD9_b9PncDKTb_byTrt3u0BrIa2Y8FVTQ6hhjzVR7AsP6fDyfux-4O-AKIww4Y |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Class+of+Optimal+Structures+for+Node+Computations+in+Message+Passing+Algorithms&rft.jtitle=IEEE+transactions+on+information+theory&rft.au=He%2C+Xuan&rft.au=Cai%2C+Kui&rft.au=Zhou%2C+Liang&rft.date=2022-01-01&rft.pub=The+Institute+of+Electrical+and+Electronics+Engineers%2C+Inc.+%28IEEE%29&rft.issn=0018-9448&rft.eissn=1557-9654&rft.volume=68&rft.issue=1&rft.spage=93&rft_id=info:doi/10.1109%2FTIT.2021.3119952&rft.externalDBID=NO_FULL_TEXT |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0018-9448&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0018-9448&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0018-9448&client=summon |