p-adic度量空间：整数距离的流处理算法

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"p-adic度量空间中整数距离测量的流处理方法" 这篇论文探讨了在p-adic度量空间中对整数进行距离测量的流处理技术。p-adic数值系统是一种非十进制的数系，它在数学、物理以及密码学等领域有广泛应用。p-adic度量空间是一个基于p-adic数定义的距离概念，对于理解和解决非欧几里得几何问题具有重要意义。首先，作者使用欧几里得算法来推导出正整数在多项式表示下的系数。欧几里得算法是计算两个整数最大公约数的高效方法，这里的应用是为了解析整数的p-adic表示。通过这个算法，作者能够确定正整数的系数，同时提出了一种类似二进制补码的方法来获取负整数的系数。二进制补码是计算机中表示负数的标准方式，这里被用来转换负整数的p-adic表示。其次，文章介绍了一个将系数从模p映射到模p^n的过程，n可以取0到接近无穷大的整数。p^n表示p的n次幂，这种映射使得p-adic数的表示更加精确，因为它允许更细粒度的距离测量。在p-adic数系中，每个整数可以用无限p进小数表示，而这些系数的变化就反映了这个小数的位值。最后，论文通过Java 8编程语言和Javaslang库实现了一种基于多路树的数据结构来处理和存储这些系数。多路树（multi-way tree）是一种数据结构，能有效处理多个分支的情况，适应于表示和操作p-adic数的无限小数部分。这种数据结构结合流处理技术，能够实时地计算和处理大量整数间的p-adic距离，提高了计算效率，尤其适合大数据量或连续性的分析任务。论文的结果不仅提供了在p-adic度量空间中计算整数距离的有效方法，还为p-adic数字表示和非欧几里得空间问题的解决方案提供了实用工具。这在理论研究和实际应用中都具有重要的价值，比如在数论、几何、编码理论以及现代密码系统的设计中。关键词：距离测量；p-adic；度量空间；多路树；流处理

617

A Stream Processing Approach to Distance

Measurement of Integers in p-adic Metric Space

Wenlong Yi

#*1

, I. V. Gerasimov

Saint Petersburg Electrotechnical University "LETI"

St. Petersburg, Russia

Jiangxi Agricultural University

Nanchang, China

yiwenlong@jxau.edu.cn,

IVGerasimov-45@yandex.ru

S. A. Kuzmin

, Huojiao He

Saint Petersburg Electrotechnical University "LETI"

St. Petersburg, Russia

Jiangxi Agricultural University

Nanchang, China

Abstract—This paper takes the p-adic representation of

integers as the research object to realize the distance

measurement of integers in the p-adic metric space. The authors

firstly apply the Euclidean algorithm to infer the coefficients of a

positive integer in polynomial representation, whose

corresponding negative integer can be obtained with the help of

the similar solution method of binary complement; secondly, the

coefficients are respectively mapped into from mod p to mod the

n-th power of p layers, where n can be 0 to approaching infinite

integer; lastly, the distance measurement method and the data

storage structure based on multi-way tree are achieved by Java 8

programming and the library Javaslang. The research results

can be applied in p-adic numbers representation and non-

Euclidean space task solutions based on p-adic metrics.

Keywords—distance measure; p-adic; metric space; multi-way

tree; stream processing

I. I

NTRODUCTION

It is 120 years since the concept of p-adic numbers was put

forward in the paper by the Germany mathematician Kurt

Hensel in 1897 [1]. It is equipped with not only excellent

algebraic properties, for example, the integral algebraic system

composed of it is the finite commutative ring to satisfy the

addition and multiplication operation, but also the properties

like intuitively describing the nonlinear topological structure.

Therefore, it is widely applied in various fields such as quantum

physics, biology, optimization theory and error-free

computation. For instance, Vladimirov and Volovich introduced

formalized the probability measurement of p-adic quantum

mechanics in the generalized function space[2]. After

establishing a 5-adic model to simulate the DNA sequences,

Dragovich et al. found that the gene encoding degradation was

related to the p-adic distance between codons[3]. Gregory and

Krishnamurthy introduced p-adic-based error-free computation

theory in details in their books, indicating that it enabled the

computer to eliminate round-off error in the calculation [4].

Murtagh adopted p-adic to encode the sparse data, and the

experiment revealed that the time complexity of the content

search was a constant [5]. Starting from a programming point of

view, this paper firstly designs a distance algorithm of integers

in the p-adic metric space, then creates a suitable data structure

for p-adic numbers; and finally realizes the above program

through the data stream processing method.

II. B

ASIC CONCEPT AND METHOD

A. Metric space

Definition 1: given a nonempty set S, where d

is a

function to measure the distance between any two elements in

this set

:dS S×→

. For

,,xyz S∀∈

, the function needs to

satisfy the below axioms:

(i) Non-negativity:

(, ) 0dxy>=

, and

(, )dxy

=0, iff

xy=

;

(ii) Symmetry:

(, ) (,)dxy dyx=

;

(iii) Triangle inequality:

(,) (, ) (,)dxz dxy dyz≤+

Strong triangle inequality:

{

}

(,) max (, ), (,)dxz dxy dyz≤

In which, the triangle inequality in the axiom (iii) is

applicable to the Euclidean space, while the strong triangle

inequality is suitable for the non-Euclidean space, such as the

p-adic metric space.

Definition 2:

in the rational number field



, if there is a

nonzero element

, where

,{0}xy∈−

and they

are co-prime, then

can be uniquely expressed by (1) [4].

. (1)

Where,

is a given prime number;

,,cdn∈

;

0d ≠

; c

and d do not divide

, then a rational number

is measured

, as shown in (2).

pif

−





≠



. (2)

The distance function defined in (2) can satisfy to the

above axiom (i) to axiom (iii). In addition, Koblitz specifically

deduced (2) in [6], and composed it as a proposition, where

is a norm in



The metric space in (



, d) is not complete. For example,

taking two arbitrary limited series of the irrational number

their distance not always smaller than any given

. In the

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