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Rubinfeld, Ronitt Cao, Ruidi Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science 2022-08-29T16:23:11Z 2022-08-29T16:23:11Z 2022-05 2022-05-27T16:19:36.411Z https://hdl.handle.net/1721.1/144952 Given an input graph 𝐺, a Local Computation Algorithm for sparse spanning graphs provides query access to a sparse subgraph 𝐺′ ⊆ 𝐺, where 𝐺′ maintains the connectivity and/or distances in 𝐺, by making a sublinear number of probes to the input 𝐺 for each query to 𝐺′ . It is known that worst-case graphs require Ω(√ 𝑛) probes in order to detect whether a specific edge 𝑒 ∈ 𝐺′ . We want to show that, in expectation, this task can be accomplished much faster, by considering average-case graphs such as Erdos-Renyi random graphs and the Preferential Attachment model. We first present an LCA algorithm which, on an Erdos-Renyi graph input 𝐺 with edge parameter 𝑝 ≥ Ω(log(𝑛) 𝑛 ), gives fast access to a sparsification 𝐺′ of 𝐺, such that 𝐺′ is connected and has 𝑛+𝑜(𝑛) edges. Queries to 𝐺′ are answered 𝒪(∆ log2 (𝑛)) probes to 𝐺 (where ∆ = 𝒪(𝑝𝑛) is the maximum degree). We then show an LCA algorithm that, for an Erdos-Renyi graph 𝐺 with edge parameter 𝑝 ≥ Ω(log(𝑛)/√ 𝑛 ), gives access to a 4-spanner 𝐺′ of 𝐺 in 𝒪(log2/(𝑛)) probes in expectation per query, such that 𝐺′ has at most 2𝑛 edges. Finally, we give an LCA that runs on a Preferential Attachment graph 𝐺 with edge parameter Θ(log(𝑛)), which gives fast access to a sparsification 𝐺′ of 𝐺 where 𝐺′ is connected and has 𝑛 + 𝑜(𝑛) edges. Each query to 𝐺′ takes an expected 𝒪(log3 (𝑛)) probes to 𝐺. M.Eng. Massachusetts Institute of Technology In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ Local Algorithms for Sparsification of Average-case Graphs Thesis application/pdf Master Master of Engineering in Electrical Engineering and Computer Science Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="5e605bc4-c8a4-4d78-8247-ca1a51f8dced"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Title>Local Algorithms for Sparsification of Average-case Graphs</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2022-05</PublicationDate> <Authors> <Author> <DisplayName>Cao, Ruidi</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://rightsstatements.org/page/InC-EDU/1.0/</License> <Abstract>Given an input graph 𝐺, a Local Computation Algorithm for sparse spanning graphs provides query access to a sparse subgraph 𝐺′ ⊆ 𝐺, where 𝐺′ maintains the connectivity and/or distances in 𝐺, by making a sublinear number of probes to the input 𝐺 for each query to 𝐺′ . It is known that worst-case graphs require Ω(√ 𝑛) probes in order to detect whether a specific edge 𝑒 ∈ 𝐺′ . We want to show that, in expectation, this task can be accomplished much faster, by considering average-case graphs such as Erdos-Renyi random graphs and the Preferential Attachment model. We first present an LCA algorithm which, on an Erdos-Renyi graph input 𝐺 with edge parameter 𝑝 ≥ Ω(log(𝑛) 𝑛 ), gives fast access to a sparsification 𝐺′ of 𝐺, such that 𝐺′ is connected and has 𝑛+𝑜(𝑛) edges. Queries to 𝐺′ are answered 𝒪(∆ log2 (𝑛)) probes to 𝐺 (where ∆ = 𝒪(𝑝𝑛) is the maximum degree). We then show an LCA algorithm that, for an Erdos-Renyi graph 𝐺 with edge parameter 𝑝 ≥ Ω(log(𝑛)/√ 𝑛 ), gives access to a 4-spanner 𝐺′ of 𝐺 in 𝒪(log2/(𝑛)) probes in expectation per query, such that 𝐺′ has at most 2𝑛 edges. Finally, we give an LCA that runs on a Preferential Attachment graph 𝐺 with edge parameter Θ(log(𝑛)), which gives fast access to a sparsification 𝐺′ of 𝐺 where 𝐺′ is connected and has 𝑛 + 𝑜(𝑛) edges. Each query to 𝐺′ takes an expected 𝒪(log3 (𝑛)) probes to 𝐺.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>