[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83209-en":3,"doc-seo-83209-105":30,"detail-sidebar-cat-0-en-105":83},{"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},83209,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Sparse Relaxed Broadcast Graphs","Sparse relaxed broadcast graphs study the graph broadcasting problem in the synchronous telephone model, where each informed node can contact at most one neighbor per round. The work seeks minimum-edge network constructions whose broadcast time is close to the lower bound (1+ε)log2 n. Using the golden ratio ϕ and constant α≈0.44, it characterizes the edge additions needed for ε below α, improves prior asymptotic bounds, and proves tightness near the interval endpoints. It also refines known upper bounds on edge counts for near-optimal broadcast time.","Sparse Relaxed Broadcast Graphs ∗  \narXiv :2607 .07260v 1 [ cs .DM] 8 Jul 2026  \nPierre Fraigniaud† Hovhannes Harutyunyan‡  \nAbstract  \nBroadcasting in graphs refers to the information dissemination problem in which a source node has an atomic piece of information to be distributed to all the nodes of a graph. In the standard telephone model, broadcasting proceeds as a sequence of synchronous rounds, where, at each round, every informed node can transfer the information to at most one of its neighbors. The broadcast time of a graph G is the maximum, taken over every node v ∈ V (G), of the minimum number of rounds required for broadcasting from v in G. Since the number of informed nodes can at most double at each round, the broadcast time of any n-node graph is at least ⌈log2 n⌉ . We study the network design problem that, for every ϵ > 0, asks for the minimum number of edges of n-node graphs with broadcast time close to optimal, i.e., at most (1 + ϵ)log2 n.  \nLet ϕ = (1 +√5)/2 be the golden ratio, and let α = 1/log2 ϕ − 1 ≃ 0.44. The aforementioned problem is solved for ϵ ≥ α as Labahn (1989), and Khachatrian and Haroutunian (1989) independently proved that, for every n ≥ 1, there are n-node trees with broadcast time at most log2 n/log2 ϕ . We address the problem for ϵ \u003C α . We show that, for every n ≥ 1, and for every ϵ ∈ (0,α), it suffices to add O(n1−ϵ/α) edges to a well chosen n-node tree for designing an n-node graph with broadcast time (1 + ϵ)log2 n. This asymptotic bound on the additional number of edges improves the previsouly known bound O (n1−ϵ) by Averbuch, Peeri, and Roditty (2017), and has implications to the design of graphs with minimum broadcast cost, defined as number of edges times broadcast time.  \nMoreover, we show that the upper bound 2n−⌈log2 n⌉−2 by Grigni and Peleg (1991) on the minimum number of edges of an n-node graph with broadcast time ⌈log2 n⌉ +1 is of the correct order of magnitude in the sense that, for infinitely many values of n, Ω(n) edges must be added to some tree for designing an n-node graph with broadcast time ⌈log2 n⌉ + 1 . Specifically, we show that, for every k, all graphs with n = 2k nodes and broadcast time k+1 must have at least ~~9~~8n edges. Therefore, our bound O (n1−ϵ/α) on the additional number of edges for 0 \u003C ϵ \u003C α is asymptotically tight at the two extremities of the interval (0,α], as it is O (n) when ϵ → 0, and O(1) when ϵ = α .  \nFinally, we (slightly) improve the aforementioned upper bound 2n − ⌈log2 n⌉ − 2 by Grigni and Peleg (1991) by showing that, for every n, there exists an n-node graph with broadcast time ⌈log2 n⌉ + 1 and at most 2n − 4⌈log2 n⌉ + O(1) edges.  \n∗ This work was done while the first author was visiting the Department of Computer Science and Software Engineering at Concordia University, Montr´eal.  \n†Institut de Recherche en Informatique Fondamentale (IRIF), CNRS and Universit´e Paris Cit´e, France. Additional supports from ANR Projects ENEDISC (ANR-24-CE48-7768-01), and PREDICTIONS (ANR-23-CE48-0010), and from the InIDEX Project METALG.  \n‡Department of Computer Science & Software Engineering, Concordia University, Montr´eal, Canada  \n1 Introduction  \nThis paper is studying the construction of sparse networks (i.e., networks with few communication links) still capable to support efficient communication primitives such as routing, or one-to-all and all-to-all primitives. The paper specifically focuses on a one-to-all communication primitive commonly referred to as broadcasting, in which a source node has to transmit a message to all the other nodes of the network. Our goal is, for any given number of nodes in the network, to identify the tradeoff between the minimum number of communication links of the network and the broadcast time of that network.  \nIn other words, the question addressed in this paper is: given a number n of nodes, and a time bound t, what is the minimum number of edges m = m (n, t) such that there exists an n-node network with m edg","cbCaijRfJluaJZJt","https://ap.wps.com/l/cbCaijRfJluaJZJt","pdf",1207016,3,1,21,"English","en",105,"# Introduction\n## Context and Objective\n## Minimizing the Broadcast Time\n## Decision and Optimization Questions","[{\"question\":\"What is the main network design objective regarding edges and broadcast time?\",\"answer\":\"For given n and time bound t, the paper studies the minimum number of edges m(n,t) needed so that some n-node graph achieves broadcast time at most t, focusing on constructions whose time is at most (1+ε)log2 n.\"}]",1784185956,53,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"sparse-relaxed-broadcast-graphs","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/sparse-relaxed-broadcast-graphs/83209/",4,{"url":51,"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","2026-07-22","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main network design objective regarding edges and broadcast time?","Question",{"text":75,"@type":76},"For given n and time bound t, the paper studies the minimum number of edges m(n,t) needed so that some n-node graph achieves broadcast time at most t, focusing on constructions whose time is at most (1+ε)log2 n.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]