pith. sign in

arxiv: 2605.03376 · v1 · submitted 2026-05-05 · 💻 cs.DC · cs.CC· cs.DS

On Solving Problems of Substantially Super-linear Complexity in N^(o(1)) Rounds in the MPC Model

Andrzej Lingas This is my paper

Pith reviewed 2026-05-07 14:32 UTC · model grok-4.3

classification 💻 cs.DC cs.CCcs.DS
keywords massively parallel computationMPC modelround complexitysuper-linear complexitylocal computationlower boundsparallel algorithmstime exponents
0
0 comments X

The pith

In the MPC model with at most N machines and relatively small local memory, any protocol solving substantially super-linear problems in N^{o(1)} rounds must use local computation whose time exponent exceeds that of the problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the feasibility of solving problems with super-linear sequential time complexity using very few rounds in the Massively Parallel Computation model. It proves that when the number of machines is bounded by the input size N and local memory is relatively small, the local computation in each round must have a higher time complexity exponent relative to the memory size than the original problem does. A sympathetic reader cares because this reveals inherent barriers to parallelizing complex computations efficiently under standard hardware limits in MPC. The result emphasizes trade-offs between round count, machine resources, and computation time in parallel settings.

Core claim

The central claim is that if the machines are not equipped with relatively large local memory and their number does not exceed N, then the exponent of the average time complexity of the local computation performed by a machine in a round (in terms of local memory size) in such protocols must be larger than the exponent of the time complexity of the given problem.

What carries the argument

The MPC model restrictions on machine count (at most N) and local memory size, which enforce a higher exponent in local per-round time complexity for sub-polynomial round protocols.

Load-bearing premise

The machines are not equipped with relatively large local memory and their number does not exceed N in the MPC setting.

What would settle it

Finding an N^{o(1)}-round MPC protocol with at most N machines and small local memory that solves a substantially super-linear problem while keeping the local time complexity exponent no larger than the problem's would falsify the claim.

read the original abstract

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity in the model of Massively Parallel Computation, where $N$ is the input size. We show that if the machines are not equipped with relatively large local memory and their number does not exceed $N$, then the exponent of the average time complexity of the local computation performed by a machine in a round (in terms of local memory size) in such protocols must be larger than the exponent of the time complexity of the given problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper studies the possibility of designing N^{o(1)}-round protocols for problems of substantially super-linear polynomial-time sequential complexity in the MPC model. It claims that if the machines are not equipped with relatively large local memory and their number does not exceed N, then the exponent of the average time complexity of the local computation performed by a machine in a round (in terms of local memory size) in such protocols must be larger than the exponent of the time complexity of the given problem.

Significance. If established with a correct proof, the conditional lower bound would clarify fundamental limitations on low-round MPC protocols for super-linear problems under the stated model restrictions (local memory not relatively large, at most N machines), helping delineate what can and cannot be efficiently parallelized in this setting.

major comments (1)
  1. [Abstract] Abstract: The abstract asserts the existence of a proof for the stated conditional lower bound but supplies no derivation, proof sketch, or technical details. This makes it impossible to assess whether the mathematics supports the claim or to verify the argument's soundness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The full paper contains the complete proof of the conditional lower bound in the body (Sections 3 and 4), while the abstract provides only a high-level statement as is standard. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract asserts the existence of a proof for the stated conditional lower bound but supplies no derivation, proof sketch, or technical details. This makes it impossible to assess whether the mathematics supports the claim or to verify the argument's soundness.

    Authors: Abstracts in theoretical papers are intentionally concise and do not include derivations or proof sketches; their purpose is to state the main result and context. The complete technical argument, including the model assumptions (local memory not relatively large, at most N machines), the definition of local computation time exponent, and the full proof of the lower bound, appears in the manuscript body. We are happy to append a brief high-level sketch to the abstract in revision if the editor prefers, but we maintain that the current structure is appropriate and the claim is supported by the detailed proof. revision: no

Circularity Check

0 steps flagged

No significant circularity; conditional lower bound is self-contained

full rationale

The paper presents a conditional lower bound on the local computation exponent in N^{o(1)}-round MPC protocols for super-linear problems, explicitly conditioned on the model restrictions (local memory not relatively large, ≤N machines). The abstract and claim formulation state the exponent comparison directly without defining any quantity in terms of itself, without fitting parameters to data and relabeling them as predictions, and without load-bearing self-citations that reduce the central result to prior unverified work by the same authors. No equations or derivation steps in the provided text exhibit self-definition, renaming of known results, or ansatz smuggling. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard domain assumptions of the MPC model (machine count ≤ N, modest local memory size, definition of round-based local time complexity as a function of memory size) that are not derived inside the paper.

axioms (1)
  • domain assumption Standard MPC model definitions with at most N machines and relatively small local memory per machine
    The lower bound is explicitly conditioned on these restrictions as stated in the abstract.

pith-pipeline@v0.9.0 · 5408 in / 1340 out tokens · 131742 ms · 2026-05-07T14:32:40.848980+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    A. Aho, J. Hopcroft, and J. Ullman.The Design and Analysis of Computer Algo- rithms. Addison-Wesley Publishing Company, Reading, 1974

    work page 1974

  2. [2]

    Behnezhad, M

    S. Behnezhad, M. Charikar, W. Ma, and L.-Y. Tan. Almost 3- approximate cor- relation clustering in constant rounds. InProceedings of the 63rd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 720–731. IEEE, 2022

    work page 2022

  3. [3]

    Censor-Hillel, P

    K. Censor-Hillel, P. Kaski, J. H. Korhonen, C. Lenzen, A. Paz, and J. Suomela. Algebraic methods in the congested clique.Distributed Computing, 32(6):461–478, 2019

    work page 2019

  4. [4]

    Gonzalez

    T. Gonzalez. Clustering to minimize the maximum intercluster distance.Theoret- ical Computer Science, 38:293–306, 1985

    work page 1985

  5. [5]

    Goodrich, N

    M. Goodrich, N. Sitchinava, and Q. Zhang. Sorting, searching, and simulation in the MapReduce framework. InProceedings of the 22nd Interna- tional Symposium on Algorithms and Computation (ISAAC), pages 374–383. Springer, 2011

    work page 2011

  6. [6]

    Haqi and H

    A. Haqi and H. Zarabi-Zadeh. Almost optimal massively parallel algorithms for k- center clustering and diversity maximization. InProceedings of the 35th ACM Sym- posium on Parallelism in Algorithms and Architectures (SPAA’23). ACM, 2023

    work page 2023

  7. [7]

    J. W. Hegeman and S. V. Pemmanraju. Lessons from the congested clique applied to mapreduce.Theoretical Computer Science, 608:268–281, 2015

    work page 2015

  8. [8]

    Jansson, C

    J. Jansson, C. Levcopoulos, A. Lingas, V. Polishchuk, and Q. Xue. Determinis- tic protocols for voronoi diagrams and triangulations of planar point sets on the congested clique.Theoretical Computer Science, 105:115491, 2025

    work page 2025

  9. [9]

    Karloff, S

    H. Karloff, S. Suri, and S. Vassilvitskii. A model of computation for mapre- duce. InProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pages 938–948. ACM-SIAM, 2010

    work page 2010

  10. [10]

    C. Lenzen. Optimal deterministic routing and sorting on the congested clique. In Proceedings of the 2013 ACM Symposium on Principles of Distributed Computing (PODC 2013), pages 42–50. ACM, 2013

    work page 2013

  11. [11]

    A. Lingas. A note on solving problems of substantially super-linear complexity in no(1) rounds of the congested clique.Parallel Processing Letters, 35, 2026

    work page 2026

  12. [12]

    Malkomes, M

    G. Malkomes, M. Kusner, W. Chen, K. Weinberger, and B.Moseley. Fast dis- tributed k-center clustering with outliers on massive data.Advances in Neural Information Processing Systems 28 (NIPS 2015), 2015

    work page 2015

  13. [13]

    K. Nowicki. A deterministic algorithm for the MST problem in constant rounds of congested clique. InProceedings of the Fifty-Third Annual ACM SIGACT Sympo- sium on Theory of Computing (STOC 2021), pages 1154–1165. ACM, 2021

    work page 2021

  14. [14]

    Roughgarden, S

    T. Roughgarden, S. Vassilvitskii, and J. Wang. Shuffles and circuits (on lower bounds for modern parallel computation).Journal of the ACM, 65(6), 2018. 7

    work page 2018

  15. [15]

    V. V. Williams and R. Williams. Subcubic equivalences between path, matrix, and triangle problems.Journal of the ACM, 65, 2008

    work page 2008

  16. [16]

    V. V. Williams, Y. Xu, Z. Xu, and R. Zhou. New bounds for matrix multiplication: from alpha to omega. InProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2024). ACM-SIAM, 2024. 8

    work page 2024