New Classes of Distributed Time Complexity Alkida Balliu Joint work - PowerPoint PPT Presentation

New Classes of Distributed Time Complexity Alkida Balliu Joint work with: Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Oliveti, Jukka Suomela HALG 2018 New Classes of Distributed Time Complexity 1 / 9

LOCAL Model Distributed Unlimited bandwidth Unlimited computational power Nodes have IDs HALG 2018 New Classes of Distributed Time Complexity 2 / 9

Locally Checkable Labellings (LCLs) Introduced by Naor and Stockmeyer in 1995 Constant-size input labels Constant-size output labels The maximum degree is constant Validity of the output is locally checkable HALG 2018 New Classes of Distributed Time Complexity 3 / 9

LCLs on Cycles log ∗ n 1 n HALG 2018 New Classes of Distributed Time Complexity 4 / 9

LCLs on General Graphs n o ( 1 ) n 1 / 4 n 1 / 3 n 1 / 2 1 loglog ∗ n log ∗ n log n n . . . ? ? ? ? ? ? ? HALG 2018 New Classes of Distributed Time Complexity 5 / 9

LCLs on General Graphs? n o ( 1 ) n 1 / 4 n 1 / 3 n 1 / 2 1 loglog ∗ n log ∗ n log n n . . . HALG 2018 New Classes of Distributed Time Complexity 6 / 9

Motivation ∆ –colouring in general graphs can be done in O ( polylog n ) rounds [Panconesi, Srinivasan 1995] 4–colouring in 2–dimensional balanced grids can be done in O ( polylog n ) rounds √ n 1 polylog n HALG 2018 New Classes of Distributed Time Complexity 7 / 9

Motivation ∆ –colouring in general graphs can be done in O ( polylog n ) rounds [Panconesi, Srinivasan 1995] 4–colouring in 2–dimensional balanced grids can be done in O ( polylog n ) rounds √ n log ∗ n 1 polylog n [Brandt et al. 2017] HALG 2018 New Classes of Distributed Time Complexity 7 / 9

Motivation ∆ –colouring in general graphs can be done in O ( polylog n ) rounds [Panconesi, Srinivasan 1995] 4–colouring in 2–dimensional balanced grids can be done in O ( polylog n ) rounds √ n log ∗ n 1 polylog n HALG 2018 New Classes of Distributed Time Complexity 7 / 9

LCLs on General Graphs (Our Results) n o ( 1 ) n 1 / 4 n 1 / 3 n 1 / 2 loglog ∗ n log ∗ n 1 log n . . . n New (unpublished) results HALG 2018 New Classes of Distributed Time Complexity 8 / 9

For More Details... New Classes of Distributed Time Complexity Alkida Balliu, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Olivetti, Jukka Suomela Aalto University, Finland Context and Goals LCLs on General Graphs A Valid LCL • Study locally checkable labelling (LCL) problems in the • There are problems with complexity Θ ( log n ) An LCL problem must be defined on any graph, not just on LOCAL model • Any o ( log log ∗ n ) rounds algorithm can be converted to an some “relevant” instances • Understanding the complexity landscape of LCL problems on O ( 1 ) rounds algorithm (same techniques of [2]) general graphs • Any o ( log n ) rounds algorithm can be converted to an O ( log ∗ n ) rounds algorithm [5] Local Checkability of the Input Graph • Many problems require Ω ( log n ) and O ( poly log n ) rounds The LOCAL Model Landscape of Complexities on General Graphs • Synchronous model 1 loglog ∗ n log ∗ n log n n o ( 1 ) n 1 4 / n 1 / 3 n 1 / 2 n • Nodes have IDs . . . ? ? ? ? ? ? ? • No limits on bandwidth or computational power On Correct Instances • T ( n ) = Θ ( log ∗ n ) for 3 –vertex colouring on cycles Conjectures 1 loglog ∗ n log ∗ n log n n o ) ( 1 n 1 / 4 n 1 / 3 n 1 / 2 • T ( n ) = Θ ( n ) for 2 –vertex colouring on cycles . . . n • Problem Π can be solved in o ( T ( n )) rounds using the shortcuts A Motivating Example • ∆ –colouring in general graphs can be done in O ( polylog n ) Locally Checkable Labellings rounds • 4 –colouring a 2 –dimensional balanced grid can be done in • Introduced by Naor and Stockmeyer in 1995 [2] On Incorrect Instances O ( polylog n ) rounds • ∆ –bounded degree graphs (where ∆ is a constant) • In 2 –dimensional grids, there is a gap between ω ( log ∗ n ) • Constant-size input and output labels and o ( √ n ) [6] • Validity of the output is locally checkable • Implication : 4 –colouring a 2 –dimensional balanced grid can be done in O ( log ∗ n ) rounds Example: Vertex Colouring Hardness Balance Our Results • On incorrect instances, it should be easy to prove that there is an error Complexities on General Graphs [1] • On correct instances, it should be impossible, or hard, to prove that there is an error 1 loglog ∗ n log ∗ n log n n o ) 1 ( n 1 / 4 n 1 / 3 n 1 / 2 n . . . ? LCLs on Cycles and Paths • Θ ( 1 ) : trivial problems Latest (Unpublished) News [4] • Θ ( log ∗ n ) : local problems (symmetry breaking) 1 loglog ∗ n log ∗ n log n n o ) ( 1 n 1 4 / n 1 / 3 n 1 2 / n . . . • Θ ( n ) : global problems Low vs High Complexities Open Problems Landscape of Complexities on Cycles and Paths 1 loglog ∗ n log ∗ n 1 log ∗ n • What happens between Ω ( log log ∗ n ) and O ( log ∗ n ) on n log p / q log ∗ n 2 log q p log ∗ n / (log ∗ n ) q/ p trees? • What are meaningful subclasses of LCL problems worth v studying? log ∗ n log n n o 1 ( ) n 1 4 / n 1 / 3 n 1 / 2 . . . n LCLs on Trees log p / q n 2 log q / p n n q/ p • Any n o ( 1 ) rounds algorithm can be converted to an O ( log n ) References rounds algorithm [3] • There are problems of complexity Θ ( n 1/ k ) [3] [1] A. Balliu, J. Hirvonen, J. H. Korhonen, T. Lempiäinen, Proof Ideas D. Olivetti, and J. Suomela, “New classes of distributed time Landscape of Complexities on Trees complexity,” in STOC 2018 (to appear) . • Start from an LCL problem Π on cycles [2] M. Naor and L. Stockmeyer, “What can be computed 1 loglog ∗ n log ∗ n log n n o ( 1 ) n 1 / 4 n 1 3 / n 1 2 / n . . . locally?,” SIAM Journal on Computing , 1995. ? ? ? ? ? ? [3] Y. Chang and S. Pettie, “A time hierarchy theorem for the LOCAL model,” in FOCS 2017 . • Build a speed-up construction Conjecture on Trees • Example: exponential speed-up function ( 2 ℓ , where ℓ is the [4] A. Balliu, S. Brandt, D. Olivetti, and J. Suomela, “Almost n o ( ) 1 n 1 / 4 n 1 / 3 n 1 2 / level of the grid-like structure) global problems in the LOCAL model,” 2018 (unpublished). 1 loglog ∗ n log ∗ n log n n . . . https://arxiv.org/abs/1805.04776. ? [5] Y. Chang, T. Kopelowitz, and S. Pettie, “An exponential separation between randomized and deterministic Towards Proving the Conjecture on Trees [4] complexity in the LOCAL model,” in FOCS 2016 . [6] S. Brandt, J. Hirvonen, J. H. Korhonen, T. Lempiäinen, P. R. 1 loglog ∗ n log ∗ n log n n o ( 1 ) n 1 / 4 n 1 / 3 n 1 2 / . . . n Östergård, C. Purcell, J. Rybicki, J. Suomela, and P. Uznański, ? ? ? ? ? “LCL problems on grids,” in PODC 2017 . HALG 2018 New Classes of Distributed Time Complexity 9 / 9