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- W3208230342 abstract "In this work, we develop the low-space Massively Parallel Computation (MPC) complexity landscape for a family of fundamental graph problems on trees. We present a general method that solves most locally checkable labeling (LCL) problems exponentially faster in the low-space MPC model than in the LOCAL message passing model. In particular, we show that all solvable LCL problems on trees can be solved in $O(log n)$ time (high-complexity regime) and that all LCL problems on trees with deterministic complexity $n^{o(1)}$ in the LOCAL model can be solved in $O(log log n)$ time (mid-complexity regime). We observe that obtaining a greater speed-up than from $n^{o(1)}$ to $Theta(log log n)$ is conditionally impossible, since the problem of 3-coloring trees, which is a LCL problem with LOCAL time complexity $n^{o(1)}$, has a conditional MPC lower bound of $Omega(log log n)$ [Linial, FOCS'87; Ghaffari, Kuhn and Uitto, FOCS'19]. We emphasize that we solve LCL problems on constant-degree trees, and that our algorithms are deterministic, component-stable, and work in the low-space MPC model, where local memory is $O(n^delta)$ for $delta in (0,1)$ and global memory is $O(m)$. For the high-complexity regime, there are two key ingredients. One is a novel $O(log n)$-time tree rooting algorithm, which may be of independent interest. The other is a novel pointer-chain technique and analysis that allows us to solve any solvable LCL problem on trees in $O(log n)$ time. For the mid-complexity regime, we adapt the approach by Chang and Pettie [FOCS'17], who gave a canonical LOCAL algorithm for solving LCL problems on trees." @default.
- W3208230342 created "2021-11-08" @default.
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- W3208230342 date "2021-12-17" @default.
- W3208230342 modified "2023-09-23" @default.
- W3208230342 title "Towards a Complexity Classification of LCL Problems in Massively Parallel Computation" @default.
- W3208230342 doi "https://doi.org/10.4230/lipics.disc.2021.50" @default.
- W3208230342 hasPublicationYear "2021" @default.
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