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arxiv: 2605.21400 · v3 · pith:BEJPTA5Hnew · submitted 2026-05-20 · 💻 cs.DS · cs.NI

Space-Time Trade-off in Integer Linear Scaling Rounded to the Nearest Integer through Multiplicative and Additive Decomposition

Kyeong Soo Kim This is my paper

Pith reviewed 2026-05-22 08:39 UTC · model grok-4.3

classification 💻 cs.DS cs.NI
keywords clock skew compensationinteger linear scalingmultiplicative decompositionadditive decompositionnearest integer solutionspace-time tradeofffloating-point precision
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The pith

Clock skew compensation reduces to integer linear scaling solved exactly for the nearest integer by two decomposition algorithms that avoid floating-point errors.

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

The paper reformulates clock skew compensation as the integer linear scaling problem iD/A and introduces two new algorithms to compute its nearest-integer solution. The multiplicative decomposition of integer division (MDID) and the additive decomposition of direct search (ADDS) operate entirely in integer arithmetic, making them immune to floating-point precision loss and non-incremental. The authors derive both methods from a unified formulation, then compare their computational complexity and overflow behavior on 32-bit and 64-bit integers. Numerical examples show that MDID runs in constant time when the scaling factor is small but can overflow, while ADDS handles every case without overflow at higher cost when the multiplier grows large; the integer methods also match the accuracy of 64-bit and 128-bit floating-point arithmetic respectively.

Core claim

By decomposing the integer division in the scaling expression iD/A either multiplicatively or additively, the nearest integer solution can be obtained using only integer operations that satisfy explicit non-overflow conditions and achieve the same compensation accuracy as higher-precision floating-point methods while using less storage in the tested cases.

What carries the argument

MDID (multiplicative decomposition of integer division) and ADDS (additive decomposition of direct search) applied to the integer linear scaling problem iD/A rounded to the nearest integer

If this is right

  • MDID delivers O(1) complexity whenever D is much smaller than the maximum integer value but overflows outside that regime.
  • ADDS handles every valid input without overflow yet increases in cost as i approaches the maximum integer value.
  • Both algorithms using 32-bit integers produce results equivalent to 64-bit double-precision floating-point compensation.
  • Both algorithms using 64-bit integers produce results equivalent to 128-bit quadruple-precision floating-point compensation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decompositions could be applied to other real-time scaling tasks that currently rely on floating-point units in embedded systems.
  • A hybrid scheduler might switch between MDID and ADDS at runtime based on current value ranges to combine speed and safety.
  • The equivalence to higher-precision floating point suggests these integer methods could replace floating-point code in environments where hardware floating-point support is absent or costly.

Load-bearing premise

The nearest-integer solution to the integer linear scaling iD/A can be obtained through the proposed multiplicative and additive decompositions while satisfying the stated non-overflow conditions for the chosen integer bit widths.

What would settle it

A concrete counterexample in which MDID or ADDS returns a value different from the true nearest integer to (i*D)/A or produces an overflow for inputs that the non-overflow analysis claims are safe within the given bit width.

Figures

Figures reproduced from arXiv: 2605.21400 by Kyeong Soo Kim.

Figure 1
Figure 1. Figure 1: Clock skew compensation based on (a) the original [2] and (b) the D [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the integer linear scaling errors resulting from setting ∆˜ n to zero for n=1, . . ., N−1 in (17) of the additive decomposition of direct search, where the integer linear scaling errors are calculated with respect to ⌊ fp(i D A )+0.5⌋ based on binary512 of IEEE 754-2008 [3, Table III]. We set i and D to 1×106 for 32-bit integers and 1×1012 for 64-bit integers, respectively. We also set A to D+u, wher… view at source ↗
read the original abstract

We formulate the problem of clock skew compensation as a special case of the integer linear scaling in the form of iD/A and propose two algorithms -- i.e., the multiplicative decomposition of integer division (MDID) and the additive decomposition of direct search (ADDS) -- for its nearest integer solution, which are not only immune to floating-point precision loss but also non-incremental unlike our prior approaches based on Bresenham's algorithm. Having theoretically established both decomposition algorithms based on a unified and rigorous formulation of the problem of the integer linear scaling rounded to the nearest integer, we discuss the space-time trade-off through the analysis of their computational complexities and non-overflow conditions. Through the numerical examples in a practical context of clock skew compensation under two different scenarios based on 32-bit and 64-bit integers, we observe that MDID can obtain the nearest integer solutions with the complexity of O(1) when D is much smaller than the maximum value of the underlying integer type but overflows otherwise; in comparison, ADDS can handle all the cases under both scenarios without overflows but at the expense of increased computational complexity when i approaches the maximum value of the underlying integer type. We also observe that ADDS based on 32-bit integers is equivalent to the clock skew compensation based on 64-bit double-precision floating-point arithmetic, while both algorithms based on 64-bit integers are equivalent to the clock skew compensation based on 128-bit quadruple-precision floating-point arithmetic, which highlights another trade-off between the bounded compensation errors and lower space complexity of the integer-based decomposition algorithms and the lower chances of overflows resulting from the wide ranges of numbers of the clock skew compensation based on floating-point arithmetic.

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

0 major / 3 minor

Summary. The paper formulates clock skew compensation as a special case of integer linear scaling in the form of iD/A. It proposes two algorithms—multiplicative decomposition of integer division (MDID) and additive decomposition of direct search (ADDS)—for the nearest-integer solution. These are theoretically established via a unified formulation, with analysis of computational complexities, non-overflow conditions, and space-time trade-offs. Numerical validation under 32-bit and 64-bit integer scenarios shows MDID achieves O(1) complexity when D is small but risks overflow, while ADDS handles all cases without overflow at higher complexity; both yield equivalences to double-precision (32-bit) and quadruple-precision (64-bit) floating-point arithmetic.

Significance. If the derivations hold, the work supplies integer-only, precision-loss-immune alternatives to floating-point methods for clock skew compensation, with explicit non-incremental algorithms and concrete trade-offs between time, space, and overflow risk. The bit-width equivalences to higher-precision floating-point and the provision of non-overflow conditions are practically useful strengths; the manuscript also supplies complexity bounds and reproducible numerical examples.

minor comments (3)
  1. [Abstract] Abstract: the claim of equivalence to 128-bit quadruple-precision floating-point for the 64-bit integer versions should be supported by an explicit error-bound derivation or reference to the bit-width analysis section.
  2. [Complexity and non-overflow analysis] The non-overflow conditions for MDID and ADDS are stated but would benefit from a compact summary table listing the exact bit-width thresholds for each algorithm under the two scenarios.
  3. [Introduction] Prior work based on Bresenham’s algorithm is referenced but not cited; include the relevant bibliographic entries to clarify the incremental vs. non-incremental distinction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the practical strengths of the MDID and ADDS algorithms, and recommendation for minor revision. The referee's description accurately captures the unified formulation, complexity analysis, non-overflow conditions, and numerical equivalences to higher-precision floating-point arithmetic.

Circularity Check

0 steps flagged

Minor self-citation to prior work but central derivations remain independent

full rationale

The paper references its own prior Bresenham-based approaches only to contrast the new non-incremental property of MDID and ADDS. The unified formulation of integer linear scaling rounded to nearest integer, the derivations of the two decomposition algorithms, complexity bounds, and non-overflow conditions are presented as self-contained and supported by bit-width analysis and numerical examples. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input. The work is internally consistent and externally falsifiable via the stated integer bit-width conditions and equivalence observations to floating-point arithmetic.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that the integer linear scaling formulation accurately models clock skew compensation.

pith-pipeline@v0.9.0 · 5838 in / 1128 out tokens · 27831 ms · 2026-05-22T08:39:25.788127+00:00 · methodology

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