[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85100-en":3,"doc-seo-85100-105":30,"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":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},85100,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Computing over Data Streams using Catalytic Space","A streaming model with catalytic memory is proposed, where an auxiliary workspace must be restored to its initial content after computation. Catalytic space enables dramatic space savings for exact data-stream algorithms. Exact frequency-moment computation for insertion-only streams admits four-pass solutions with O(k log m) clean space and (k+1) passes using k times less catalytic space. For small moments, F2 and F3 are computable exactly in two and three passes with O(log m) clean space, and F0 reduces to Fk, also in four passes. The approach extends to exact graph-stream induced subgraph counting, including triangle counting in three passes using O(log n) clean space, plus a single-pass limitation showing simulation without catalytic memory.","Computing over Data Streams using Catalytic Space  \nRipley Becker University of Nebraska–Lincoln  \nSourav Chakraborty Indian Statistical Institute, Kolkata  \nDebarshi Chanda  \nIndian Statistical Institute, Kolkata  \nA. Pavan Iowa State University  \narXiv :2607 .08559v 1 [ cs .DS] 9 Jul 2026  \nN. V. Vinodchandran  \nUniversity of Nebraska–Lincoln  \nAbstract  \nWe introduce a streaming model with catalytic memory, an auxiliary workspace that must be returned to its initial state at the end of the computation. We show that catalytic space yields dramatic space savings for data stream algorithms.  \nWe first study the exact computation of frequency moments in insertion-only data streams. For every k ≥ 1, we give an exact four-pass algorithm for computing Fk using O (k log m) clean space, where m is the stream length. We also present a (k + 1)-pass algorithm with the same clean-space complexity that uses a factor of k less catalytic space than the four-pass algorithm.  \nFor small moments, we obtain stronger results. In particular, we show that F2 and F3 can be computed exactly in two and three passes, respectively, using only O(log m) clean space. Additionally, we show that exact F0 computation reduces to computing Fk for a suitably chosen large value of k, resulting in an exact four-pass algorithm for F0 using only O(log m)  \nclean space. We further show how our frequency-moment algorithms can be used to exactly count induced occurrences of any fixed graph H in a graph stream, yielding a four-pass algorithm that uses OH(log n) clean space, where n is the number of vertices in the graph. Asa special case, we obtain an exact three-pass algorithm for triangle counting using O(log n) clean space.  \nAll of our algorithms are multi-pass. We complement these algorithmic results with a matching limitation showing that catalytic memory does not provide additional power in the single-pass setting. Specifically, we prove that every randomized or deterministic single-pass streaming algorithm using s bits of clean memory and catalytic space can be simulated in the standard streaming model, without catalytic memory, using O (s) space.  \n1 Introduction  \nWe introduce data streaming algorithms that have access to additional catalytic memory. In the catalytic computation model, an algorithm is given, besides its ordinary workspace, an auxiliary memory region called catalytic space. At the beginning of the computation, the catalytic space is initialized to arbitrary content, and this space can be used during the computation. However, it is required that the contents of the catalytic space must be restored to their initial contents at the end of the computation. Since its contents are available but cannot be consumed, this memory behaves like a catalyst. Since its introduction in [7], catalytic computation has emerged as a powerful model, with recent works showing that such reversible access to auxiliary memory can surprisingly enable efficient computation [1, 12, 11, 2, 8, 10, 9] . For example it is known that every language in nondeterministic logspace (NL) can be accepted by a deterministic logspace machine that has access to catalytic space. A recent survey by Mertz provides an introduction to the topic [16] .  \nThe main conceptual contribution of the present work is that such usable but non-consumable memory helps with data streaming computations too. In the standard streaming model, almost all non-trivial problems require randomness and approximation for space-efficient computation. For exact computation, the optimal deterministic/randomized algorithms essentially have to store the entire stream. We show that catalytic space can circumvent this barrier: problems that are intractable in the deterministic streaming setting indeed become feasible when catalytic memory is available.  \nIn this paper, we focus on the problem of computing frequency moments of an insertion-only data stream. For a data stream S = ⟨x1 , x2 , ··· , xm ⟩ of items xi ∈ [n] = ","cbCaiuY6N15AhzUi","https://ap.wps.com/l/cbCaiuY6N15AhzUi","pdf",379292,4,1,33,"English","en",105,"# Abstract\n# Introduction\n## Catalytic computation model\n## Streaming frequency moments and space lower bounds\n## Example: computing F2 in multiple passes","[{\"question\":\"What is catalytic memory in the proposed data streaming model?\",\"answer\":\"Catalytic memory is an auxiliary workspace initialized arbitrarily before processing the stream, usable during computation, and required to be restored to its original content at the end.\"},{\"question\":\"How can frequency moments Fk be computed for insertion-only streams?\",\"answer\":\"For every k ≥ 1, an exact four-pass algorithm computes Fk using O(k log m) clean space. A (k+1)-pass algorithm achieves the same clean-space complexity with k times less catalytic space.\"},{\"question\":\"What limitations are shown for single-pass streaming with catalytic space?\",\"answer\":\"Every randomized or deterministic single-pass algorithm using s bits of clean memory and catalytic space can be simulated in the standard streaming model without catalytic memory using O(s) space, so catalytic space does not add power in the single-pass setting.\"}]",1784201103,83,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"computing-over-data-streams-using-catalytic-space","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/computing-over-data-streams-using-catalytic-space/85100/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",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 catalytic memory in the proposed data streaming model?","Question",{"text":75,"@type":76},"Catalytic memory is an auxiliary workspace initialized arbitrarily before processing the stream, usable during computation, and required to be restored to its original content at the end.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How can frequency moments Fk be computed for insertion-only streams?",{"text":80,"@type":76},"For every k ≥ 1, an exact four-pass algorithm computes Fk using O(k log m) clean space. A (k+1)-pass algorithm achieves the same clean-space complexity with k times less catalytic space.",{"name":82,"@type":73,"acceptedAnswer":83},"What limitations are shown for single-pass streaming with catalytic space?",{"text":84,"@type":76},"Every randomized or deterministic single-pass algorithm using s bits of clean memory and catalytic space can be simulated in the standard streaming model without catalytic memory using O(s) space, so catalytic space does not add power in the single-pass setting.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"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":106,"slug":138},19,"General","general"]