[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82360-en":3,"doc-seo-82360-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},82360,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Streaming with Catalytic Memory","Streaming with Catalytic Memory presents a catalytic streaming model combining catalytic and regular memory. It provides exact computation of frequency moments using logarithmic regular memory bits and polynomial catalytic memory bits, and extends to exact evaluation of arbitrary polynomials of item frequencies. Applications include exactly counting distinct elements, counting triangles (or small subgraphs) in streaming graphs, and identifying heavy hitters. The approach uses constant passes for moments and degree+1 passes for polynomial evaluation, with lower bounds and a two-pass design for the second frequency moment.","arXiv :2607 .09475v 1 [ cs .DS] 10 Jul 2026  \nStreaming with Catalytic Memory ∗ Tamara Kaplan, Nimrod Kaplan, Haim Kaplan  \nJuly 13, 2026  \nAbstract  \nWe introduce a streaming model that uses both catalytic and regular memory. In this model, we show how to exactly compute the frequency moments using a logarithmic number of bits of regular memory and a polynomial number of bits of catalytic memory. More generally, we show how to compute arbitrary polynomials of the item frequencies exactly within the same space bounds. As an application, we obtain catalytic streaming algorithms that exactly compute the number of distinct elements in a stream, count the number of triangles (or any other small subgraph) in a graph whose edges arrive in a stream, and identify heavy hitters.  \nOur algorithms for frequency moments perform a constant number of passes over the stream, and for polynomial evaluation, we require one more pass than the degree of the polynomial. By relating our catalytic streaming model to the catalytic communication model introduced in [PSW25], we show that catalytic memory is not useful for any one-pass streaming algorithm. For lower boundson multipass streaming algorithms, the impossibility results of [PSW25] are not strong enough. However, using a different technique, we show that under certain natural restrictions, no catalytic streaming algorithm can compute the second frequency moment in fewer than three passes.  \nThis definition of the restricted class of two-pass algorithms then guides us in the design of a two-pass algorithm for computing the second moment exactly that circumvents these restrictions and breaks the three-pass barrier.  \n1 Introduction  \nCatalytic computation was introduced by Buhrman, Cleve, Kouck´y, Loff, and Speelman [BCK+14] asa theoretical model for understanding the computational power of a “full hard drive” (i.e., memory which can be used but has to be restored at the end to its original content) . Since then, catalytic memory has attracted significant attention and has led to several interesting results in complexity theory. In particular, researchers have explored the power of non-determinism [BKLS18] and randomness [CLMP25] in this model, as well as non-uniform [GKM15,Pot17,CM22] and quantum [BFM+25] versions of it. Furthermore, interesting space-efficient algorithms using catalytic memory have been developed. Notable examples include the work of Henzinger, Pyne, and Ragavan [HPR26], who improved upon Cook and Mertz’s [CM24] results by solving the Tree Evaluation Problem using subpolynomial catalytic memory and logarithmic regular memory. Additionally, Chmel et al. [CDK+26] provided deterministic solutions for directed connectivity and sequence alignment.  \nA Catalytic Turing machine has a working tape as of a standard Turing machine and in addition it has a catalytic tape initialized arbitrarily that has to be restored to its original content by the end of the computation.1 The class Catalytic Logspace, denoted by CL, contains all problems computable by a catalytic Turing machine with a regular tape of size logarithmic in the input length and a polynomialsize catalytic tape. Surprisingly, Buhrman et al. [BCK+14] proved that TC 1 ⊆ CL,2 showing that  \n∗ An extended abstract of this paper appears in ESA 2026 .  \n1 The terminology comes from chemistry: a catalyst enables a reaction to proceed without being consumed or permanently altered.  \n2 TC1 denotes the class of decision problems solvable by a family of polynomial-size, logarithmic-depth Boolean circuits with unbounded fan-in AND, OR, and MAJORITY (aka threshold) gates.  \ncatalytic memory can be exploited to solve problems not known to lie in L.3  \nRecently, Pyne, Sheffield, and Wang [PSW25] introduced a catalytic communication model, in which Alice and Bob communicate by exchanging messages using a small clean memory and a large catalytic memory. They show that this model is substantially stronger than the standard one as it enables t","cbCaitKI7KgLKvGK","https://ap.wps.com/l/cbCaitKI7KgLKvGK","pdf",489963,5,1,21,"English","en",105,"# Abstract\n# Introduction\n## The Catalytic Streaming model","[{\"question\":\"What is the key idea of the catalytic streaming model in this paper?\",\"answer\":\"The model augments standard multi-pass streaming with a large catalytic memory that can be used during computation but must be restored to its original content by the end.\"},{\"question\":\"How are frequency moments computed exactly, and what are the memory and pass requirements?\",\"answer\":\"Frequency moments can be computed exactly with logarithmic regular memory bits and polynomial catalytic memory bits using a constant number of passes over the stream.\"},{\"question\":\"What does the paper conclude about catalytic memory and one-pass streaming algorithms?\",\"answer\":\"By relating the catalytic streaming model to a catalytic communication model, it shows catalytic memory is not useful for any one-pass streaming algorithm, while stronger lower bounds for multipass settings require additional restrictions and techniques.\"}]","Streaming with Catalytic Memory | 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is the key idea of the catalytic streaming model in this paper?","Question",{"text":77,"@type":78},"The model augments standard multi-pass streaming with a large catalytic memory that can be used during computation but must be restored to its original content by the end.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How are frequency moments computed exactly, and what are the memory and pass requirements?",{"text":82,"@type":78},"Frequency moments can be computed exactly with logarithmic regular memory bits and polynomial catalytic memory bits using a constant number of passes over the stream.",{"name":84,"@type":75,"acceptedAnswer":85},"What does the paper conclude about catalytic memory and one-pass streaming algorithms?",{"text":86,"@type":78},"By relating the catalytic streaming model to a catalytic communication model, it shows catalytic memory is not useful for any one-pass streaming algorithm, while stronger lower bounds for multipass settings require additional 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