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.

翻译：我们将时钟偏差补偿问题形式化为一种特殊形式的整数量化线性缩放（iD/A），并提出了两种算法——即整数除法的乘法分解（MDID）与直接搜索的加性分解（ADDS）——以求解其最近整数解。这两种算法不仅规避了浮点数精度损失问题，而且不同于我们先前基于 Bresenham 算法的方法，它们属于非增量式算法。基于对整数量化四舍五入线性缩放问题的统一严格理论构建，我们通过分析两种算法的计算复杂度与非溢出条件，讨论了其中存在的时空权衡。在基于 32 位与 64 位整数的两种实际时钟偏差补偿场景的数值实验中观察到：当除数 D 远小于底层整数类型最大值时，MDID 能以 O(1) 复杂度获得最近整数解，否则将发生溢出；相比之下，ADDS 在两种场景下均能无溢出处理所有情况，但代价是当被除数 i 接近底层整数类型最大值时计算复杂度显著增加。我们还发现：基于 32 位整数的 ADDS 等价于基于 64 位双精度浮点数的时钟偏差补偿方法，而基于 64 位整数的两种算法则等价于基于 128 位四精度浮点数的时钟偏差补偿方法。这揭示了基于整数的分解算法在补偿误差有界性与低空间复杂度方面，与基于浮点数的时钟偏差补偿算法因宽数值范围带来的低溢出风险之间的另一重权衡。