TY - GEN
T1 - Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs
AU - Levi, Reut
AU - Medina, Moti
AU - Tubul, Omer
N1 - Publisher Copyright: © Reut Levi, Moti Medina, and Omer Tubul.
PY - 2024/9/1
Y1 - 2024/9/1
N2 - In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO’20). In this problem, the goal is to locally decide for each e ∈ E if it is in G′ where G′ is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ2). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n2/3) queries for ϕ = Ω(1/n1/12). We then extend our result for (k, ϕin, ϕout)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕoutn) for constant k and ϕin. This bound is almost optimal when ϕout = O(1/√n).
AB - In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO’20). In this problem, the goal is to locally decide for each e ∈ E if it is in G′ where G′ is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ2). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n2/3) queries for ϕ = Ω(1/n1/12). We then extend our result for (k, ϕin, ϕout)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕoutn) for constant k and ϕin. This bound is almost optimal when ϕout = O(1/√n).
KW - Clusterbale Graphs
KW - Locally Computable Algorithms
KW - Spanning Subgraphs
KW - Sublinear algorithms
UR - http://www.scopus.com/inward/record.url?scp=85204460551&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.APPROX/RANDOM.2024.60
DO - 10.4230/LIPIcs.APPROX/RANDOM.2024.60
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024
A2 - Kumar, Amit
A2 - Ron-Zewi, Noga
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 27th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2024 and the 28th International Conference on Randomization and Computation, RANDOM 2024
Y2 - 28 August 2024 through 30 August 2024
ER -