[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84971-en":3,"doc-seo-84971-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84971,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","A General Reduction from Near-Additive Emulators to Near-Exact Hopsets","Graph emulators and hopsets both approximate distances in graphs but use different mechanisms. For (α,β)-emulators, a sparse graph on the same vertices preserves pairwise distances with multiplicative stretch α and additive stretch β. For (α,β)-hopsets, extra edges guarantee that approximate shortest paths use at most β hops. With α=1+ε for arbitrarily small ε, these become near-additive emulators and near-exact hopsets. The paper proves a reverse reduction: any near-additive emulator construction for undirected unweighted graphs can be used as a black box to build a hopset for undirected weighted graphs with comparable size, stretch, and hopbound, answering an open question for sparse graphs and advancing the formal connection between the two structures.","arXiv :2607 .07 190v 1 [ cs .DS] 8 Jul 2026  \nA General Reduction from Near-Additive Emulators to  \nNear-Exact Hopsets∗  \nJulian Aeri † Sebastian Forster † Mara Grilnberger †‡  \nAbstract  \nGraph emulators and hopsets are two fundamental concepts for distance approximation. For a given graph 􀀜, an (􀁕, 􀁖)-emulator is a sparse graph on the same vertex set that preserves the distances of 􀀜 up to a multiplicative stretch 􀁕 and additive stretch 􀁖 . In contrast, an (􀁕, 􀁖)-hopset is a set of additional edges that, when added to 􀀜, ensures that distances can be approximated up to a multiplicative stretch 􀁕, using paths containing at most 􀁖 edges. When 􀁕 = 1 + 􀁮 for arbitrarily small 􀁮 > 0, these structures are known as near-additive emulatorsand near-exact hopsets, respectively. Prior work showed that there is a remarkable similarity between the constructions and guarantees of these two objects. In their survey on this topic, Elkin and Neiman [Bull. EATCS 130, 2020] explicitly asked whether one can obtain a general reduction between near-additive emulators and near-exact hopsets. Following that, Kogan and Parter [FOCS, 2022] provided a general reduction from hopsets to emulators and spanners.  \nIn this paper, we address the reverse direction and show that any construction for a nearadditive emulator for undirected unweighted graphs can be leveraged as a black box to construct a hopset for an undirected weighted graph with comparable size, stretch, and a hopbound comparable to the emulator’s additive stretch. Specifically, we show that any algorithm that constructs a (1 + 􀁮′, 􀁖)-emulator, with 0 ≤ 􀁮′ ≤ 1 and 􀁖 ≥ 1, of size 􀀨A (􀀽, 􀁮′, 􀁖), can be used to obtain a (1 + 􀁮, 􀀤 ( 􀁖􀁮22 ln ( ~~􀀽~~􀁮 )))-hopset of size 􀀤 ((􀀨A (􀀽 + 􀀼 􀁖􀁮2 , 2~~􀁮~~94 , 􀁖) 1􀁮 + 􀀽) ln ( ~~􀀽~~􀁮 )), for any 0 \u003C 􀁮 ≤ 1. Therefore, our reduction answers the question of Elkin and Neiman [Bull. EATCS 130, 2020] for sparse graphs and further advances the understanding of the formal connection between these two structures. Designing a reduction resulting in a hopset size that does not depend on 􀀼 remains an intriguing open question.  \n∗This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 947702) .  \n†Department of Computer Science, University of Salzburg, Austria  \n‡This publication has been supported by the EXDIGIT (Excellence in Digital Sciences and Interdisciplinary Technologies) project, funded by Land Salzburg under grant number 20204-WISS/263/6-6022 .  \nContents  \n1 Introduction  \n1. 1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  \n1.2 Overview of Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  \n2 Hopset Reduction  \n2. 1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  \n2.2 Size Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  \n2.3 Hop Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  \n2.4 Projection to the Original Vertex Set . . . . . . . . . . . . . . . . . . . . . . . . . .  \nReferences  \n1 Introduction  \nGiven a graph 􀀜, a hopset is a set of additional edges that, when added to 􀀜, ensures that for any pair of vertices, an approximate shortest path exists with a bounded number of edges (or hops) . They were first formally introduced by Cohen [] as a tool for efficient parallel computation of approximate shortest paths. And since then they have been central in parallel algorithms [,  , , , ,  , ], dynamic distance maintenance [, , , ], and distributed shortest paths computations [ , , ] .  \nDefinition 1.1 (ℎ-hop distance). Given a graph 􀀜 = (􀀫 , 􀀚) . For a pair of vertices 􀁄, 􀁅 ∈ 􀀫 , the ℎ-hop distance dℎ􀀜 (􀁄, 􀁅 ) denotes the weight of the shortest path from 􀁄 to 􀁅 in 􀀜 that contains at most ℎ edges.  \nDefinition 1.2 ((􀁕, 􀁖)-hopset). Given a graph 􀀜 = (􀀫 , 􀀚), a mu","cbCaiu5nGVYKYL0C","https://ap.wps.com/l/cbCaiu5nGVYKYL0C","pdf",509270,1,29,"English","en",105,"# Introduction\n## Our Results\n## Overview of Techniques\n# Hopset Reduction\n## Construction\n## Size Analysis\n## Hop Reduction\n## Projection to the Original Vertex Set\n# References","[{\"question\":\"What problem does the paper address regarding emulators and hopsets?\",\"answer\":\"It studies whether a general reduction can convert constructions for near-additive emulators into near-exact hopsets, answering a question raised in prior work.\"},{\"question\":\"How does a (α,β)-emulator approximate distances?\",\"answer\":\"A (α,β)-emulator is a sparse graph on the same vertex set where each pair’s distance is preserved within multiplicative stretch α and additive stretch β.\"},{\"question\":\"What is the main reduction result of the paper?\",\"answer\":\"Any algorithm that builds a (1+ε',k)-emulator for undirected unweighted graphs can be used to construct a hopset for undirected weighted graphs with comparable size, multiplicative stretch, and a hopbound related to the emulator’s additive 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problem does the paper address regarding emulators and hopsets?","Question",{"text":75,"@type":76},"It studies whether a general reduction can convert constructions for near-additive emulators into near-exact hopsets, answering a question raised in prior work.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does a (α,β)-emulator approximate distances?",{"text":80,"@type":76},"A (α,β)-emulator is a sparse graph on the same vertex set where each pair’s distance is preserved within multiplicative stretch α and additive stretch β.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main reduction result of the paper?",{"text":84,"@type":76},"Any algorithm that builds a (1+ε',k)-emulator for undirected unweighted graphs can be used to construct a hopset for undirected weighted graphs with comparable size, multiplicative stretch, and a hopbound related to the emulator’s additive 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