[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81998-en":3,"doc-seo-81998-105":30,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81998,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Gap-Majority Lemmas in Communication Complexity","Information-theoretically optimal gap-majority lemmas are proved for the two-player randomized communication model. For a base function f:X→{±1}, the n-fold composition GapMAJ◦f^n distinguishes whether the sum f(X1)+…+f(Xn) is at least 0.01√n or at most −0.01√n. If computing f with success probability 0.501 needs I bits, then computing GapMAJ◦f^n with success probability 0.99 requires n·(I−O(1)) bits, achieving optimal linear information scaling and a constant–constant error tradeoff.","arXiv :2607 .07396v 1 [ cs .CC] 8 Jul 2026  \nGap-Majority Lemmas in Communication Complexity  \nPachara Sawettamalya∗ Huacheng Yu†  \nJuly 9, 2026  \nAbstract  \nWe prove an information-theoretically optimal gap-majority lemma in the two-player randomized communication model. For a base function f : X → {±1}, its n-fold gap-majority composition, denoted GapMAJ ◦ fn , takes n inputs (X1 ,..., Xn) and distinguishes whether f +n(X1 ,..., Xn ) := f(X1 )+ ... +f(Xn) is at least 0.01 √n or at most −0 .01 √n. We show that if computing f with success probability 0.501 requires I bits of information, then computing GapMAJ◦fn with success probability 0 .99 requires n·(I −O(1)) bits of information. This result is asymptotically optimal in two aspects: it achieves the correct linear scaling of information cost and the correct constant-constant tradeoff between error rates. This makes GapMAJ, to our knowledge, only the third explicit outer gadget that admits a strong composition theorem in the two-player communication setting, following the identity and XOR gadgets.  \nFrom an application side, our gap-majority lemma can be viewed as a generic amplification tool that lifts the hardness of deciding f into the hardness of approximating f +n. Using this framework, we give a new proof to the communication lower bound of Gap-Hamming and derive a tight streaming lower bound of triangle counting, demonstrating the versatility of the gapmajority lemma.  \n∗ Department of Computer Science, Princeton University. Supported by NSF CAREER award CCF-233994 . [ps3122@princeton.edu](ps3122@princeton.edu)  \n†Department of Computer Science, Princeton University. Supported by NSF CAREER award CCF-233994 . [yuhch123@gmail.com](yuhch123@gmail.com)  \n1 Introduction  \nGiven a base function f, consider the task of computing its n-fold product function fn(x1 ,..., xn) :=(f(x1 ) ,..., f (xn)) . A natural approach is to apply an optimal algorithm for f independently to each input, using n times the cost of computing a single instance. But is this optimal?  \nThis problem is known as the direct-sum question, and it is among the most central in computational complexity theory. It has been studied extensively across a variety of computational models such as circuit complexity [Yao82 ; Lev87 ; Imp95 ; IW97], communication complexity [FKNN95 ; BJKS04 ; BBCR10 ; JPY12 ; BRWY13 ; MS25], information complexity [BBCR10 ; BR11 ; Bra15], and query complexity [JKS10 ; Dru12 ; BK18 ; BB19 ; BB25 ; BGG+25], to name just a few.  \nIn parallel, another line of research investigates the complexity of composition functions g ◦ fn. For an arbitrary “outer gadget” g, the optimal bounds (i.e. proportional to the complexity of f and g) are known under certain complexity measures such as deterministic query complexity [Sav02 ; Tal13 ; Mon14] and quantum query complexity [LMR+11 ; Rei11] . However, for general complexity measures, a starting point is to understand the complexity of g ◦ fn when g is an explicit gadget. That is, to study how the cost of computing g ◦ fn scales with the cost of computing f and with n, for a specific g. Naively, one could still compute f on each input and then apply g to the outputs. In the spirit of the direct-sum question, we may ask whether this simple strategy is optimal.  \nQuestion 1.1 (Composition Theorem) . For a prescribed n-ary gadget g, how does the cost of computing g ◦ fn scale with the cost of computing f, and with n?  \nBesides the identity gadget for which Question 1.1 becomes the direct-sum problem, perhaps the most extensively studied gadget is the XOR function, which computes the parity of its n-bit input. Results of this type, often called the XOR lemma, have also been well-established in various computational models, e.g. [IW97 ; BBCR10 ; BKLS20 ; Yu22 ; IR24a; IR24b; SY25] . Beyond the XOR lemma, however, our understanding of Question 1.1 is relatively limited and is mostly contained to query complexity with respect to the OR, GapOR, and MAJ gadget","cbCaioKqmEORtDzI","https://ap.wps.com/l/cbCaioKqmEORtDzI","pdf",687374,6,1,27,"English","en",105,"# Abstract\n# Introduction\n## Direct-sum and composition questions\n## Gap-majority lemma question\n# Main Results","[{\"question\":\"What does the gap-majority gadget GapMAJ◦f^n aim to decide?\",\"answer\":\"It distinguishes whether f(X1)+…+f(Xn) is at least 0.01√n or at most −0.01√n, with high success probability in the randomized communication model.\"},{\"question\":\"How does the paper relate the information cost of computing f to computing GapMAJ◦f^n?\",\"answer\":\"If computing f with success probability 0.501 requires I bits, then computing GapMAJ◦f^n with success probability 0.99 requires n·(I−O(1)) bits.\"},{\"question\":\"Why is the result considered optimal?\",\"answer\":\"It matches the correct linear scaling of information cost and the correct constant–constant tradeoff between error rates in two aspects of optimality.\"}]",1784177486,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"gap-majority-lemmas-in-communication-complexity","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/gap-majority-lemmas-in-communication-complexity/81998/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does the gap-majority gadget GapMAJ◦f^n aim to decide?","Question",{"text":76,"@type":77},"It distinguishes whether f(X1)+…+f(Xn) is at least 0.01√n or at most −0.01√n, with high success probability in the randomized communication model.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the paper relate the information cost of computing f to computing GapMAJ◦f^n?",{"text":81,"@type":77},"If computing f with success probability 0.501 requires I bits, then computing GapMAJ◦f^n with success probability 0.99 requires n·(I−O(1)) bits.",{"name":83,"@type":74,"acceptedAnswer":84},"Why is the result considered optimal?",{"text":85,"@type":77},"It matches the correct linear scaling of information cost and the correct constant–constant tradeoff between error rates in two aspects of optimality.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]