Invers Natural Number System to Maintain User-defined Sequence of

Data Records

Seyfettin Öztürk

Nokia Solutions and Networks GmbH & Co. KG, Nürnberg, Germany

Keywords: Database Indexing, Storing User-defined Row Sequence, Avoiding Reordering, Increasing Performance,

Decreasing Computing Time and Energy Consumption, Reducing Internet Data Traffic.

Abstract: The objective of this paper is to present a method to insert, edit, and delete database records without affecting

the sequence of existing data. Typically, databases comprise integer data fields, in this paper named sequence

number, meant to determine the user-defined sequence of data records. Inserting new data records or editing

the sequence number of data records might cause a resequencing of the existing data records. This

resequencing can be avoided by using a numbering system that decreases the value of a number when a digit

is added to its end. Such a numbering system allows to insert an infinite quantity of additional sequence

numbers between two sequence numbers even if their difference is 1.

1 INTRODUCTION

Generally, the user-defined sequence of data records

in database applications, such as list elements, is

stored in a dedicated data field named “sequence

number”, “list order”, “position number” etc.

Inserting new data records or editing existing data

records normally requires the renumbering

respectively updating of subsequent data records.

The presented method in this paper allows user-

defined sequencing without the need of resequencing.

2 THE ISSUE WITH THE

USER-DEFINED SEQUENCES

The following use case is discussed to explain the

issue with the user-defined sequence: An application

like an internet browser processes a query for

bookmarks and displays the resulting list to the user

(Nelson, 2018). The user can easily rearrange the

positions of the bookmarks according to his needs.

The sequence of the bookmarks is determined by the

values in the column “sequence number” of the

corresponding data record in the database. Before

user’s rearrangement, it can be assumed that the

sequence of the available bookmarks is as shown in

table 1.

Table 1: Original sequence of bookmarks displayed to the

user.

Bookmar

Sequence Numbe

The issue of user-defined sequences appears when

the user rearranges the sequence in table 1 and moves,

for example, the bookmark “f” from position 4 to 1.

This leads to “write” activities for the data records

“d”, “g” and “b” to assign them the new sequence

numbers 2, 3 and 4. Write operation are resource- and

time-intensive, and software architects are interested

very much in minimizing these factors.

2.1 State of the Art of Maintaining the

User-defined Sequences

As outlined in the previous section, the sequence

number values of the bookmarks “d”, “g”, and “b”

must be updated when the user moves the bookmark

“f” from position 4 to position 1, as shown in table 2.

260

Öztürk, S.

Invers Natural Number System to Maintain User-deﬁned Sequence of Data Records.

DOI: 10.5220/0010571502600266

In Proceedings of the 10th International Conference on Data Science, Technology and Applications (DATA 2021), pages 260-266

ISBN: 978-989-758-521-0

Table 2: Updated sequence of bookmarks displayed to the

user.

Bookmar

Sequence Numbe

f 1

a 5

To implement the resequencing in table 2, the

state of the art uses a database API (Application

Programming Interface) or an SQL (Structured Query

Language, Bayer, 1970) update statement. The

following pseudocode is an example to illustrate the

database writing activities using an SQL statement

(Nelson, 2018):

Input:

Database table Bookmarks: bookmarks

Attribute Bookmark: bookmark

Attribute Sequence Number: s

SQL Update Statement:

Update bookmarks

Set s = s + 1

Where s >= 1 and s < (Select s

from bookmarks where bookmark = 'f')

Update bookmarks

Set s = 1

Where bookmark = 'f'

The SQL statement above will initiate and perform

four write operations to rearrange the database table

“bookmarks”; three operations will update the data

records “d”, “g” and “b” and one operation will set

“f” to the sequence number 1.

The quantity of necessary write operations to

update the sequence will depend on the position of the

inserted new data record or the updated data record.

The size of the required write operation in “Big O”

notation is O(n) (Wikipedia, 2021). With other words,

a given sequence with n records requires at worst “n”

write operations to perform the rearrangement.

Within comprehensive databases, such

rearrangements might cause numerous read and write

actions, consuming considerable computing capacity.

In the worst-case scenario, all existing data records in

the sequence must be updated, as e.g. when inserting

a new data record in the first position of the sequence.

The implementation of such data access

transactions, in particular via the internet, is usually

complex and time-consuming.

Using the innovative method presented in this

paper, this comprehensive updating can be avoided

since the insertion will not affect the remaining

sequences anymore and the size of the write

operations will be limited to O(1).

3 OPTIONS FOR USER-DEFINED

SEQUENCES WITHOUT THE

NEED OF RESEQUENCING

3.1 The Consideration of Fixed-point

or Floating-point Arithmetic

In theory, it is possible to generate sequence numbers

in fixed-point or floating-point format to avoid the

resequencing (Schäfer, 1989). However, these data

types have limited precision and after certain iteration

the rightmost digits will be skipped, with the effect

that the accuracy and resolution of results might be

affected. This is due to the well-known machine

“overflows” respectively “underflows” that can occur

in electronic fixed- and floating arithmetic.

Furthermore, electronic computers might have

serious performance limitations when fixed-point and

floating-point arithmetic is used (Schmid, 1974).

Therefore, fixed- or floating-point numbers are rarely

used in applications to maintain the user-defined

sequences.

In contrast, the presented alternative method is

based on integer data type that is interpreted as

inverse natural numbers. The inverse natural number

system keeps the original sequence of data records

valid after new data record insertion and edition. The

renumbering is avoided even if two sequence

numbers only differ by 1 and a new data record is

inserted in between.

3.2 The Inverse Natural Number

System

When using the inverse natural numbering system,

resequencing is avoided by interpreting the value of

the sequence number inversely. In this context

inverse means that the maximum value of the number

is absolutely limited to the value of the leftmost digit.

Consequently, the value of a sequence number

decreases as the number of its digits increases. For

example, if the leftmost digit of a number is 9, the

value of this number will never exceed 9, no matter

how many digits are appended at the right of the

number.

With other words, the value of a sequence number

can be reduced by adding further digits to the right of

the existing digits of the number. Consequently, the

sequence number 100 is interpreted as being smaller

Invers Natural Number System to Maintain User-deﬁned Sequence of Data Records

261

than the sequence number 10. Accordingly, the

sequence number 10 is assigned a smaller value than

the sequence number 1.

The value of positive integer numbers, i.e. natural

numbers, results from adding the place value

multiplied with the digit, as shown below in the

summation equation (1) (Sleator, 1988).

∑

∗10







 a



n-1

… a



a Є ℕ,a



0,1,…9





(1)

This leads to the commonly known inequation (2):

10 < 100 (2)

In contrast, the place value assignment of the inverse

natural numbering system is determined using the

following summation equation in (3):

∗10



1∗











9a



∗10



9

9∗ 10



9

(3)

 a



n-1

… a



a Є ℕ,a



0,1,…9





This leads to the inequation of the inverse natural

numbers 10 and 100 (4):

10 > 100 (4)

Due to this formula, it is possible to insert an infinite

quantity of sequence numbers between two numbers

that differ by 1. This is because the value of a

sequence number decreases as the number of its digits

increases.

4 THE COMPARATOR OF THE

INVERSE NATURAL NUMBER

SYSTEM

For the purpose of the inverse natural number system,

the comparator can be used to determine the greater

respectively the smaller of two given inverse natural

numbers without applying the summation formula in

(3). Hereinafter, the comparator’s algorithm to

compare two given inverse natural numbers will be

presented.

The first inverse natural number’s value is

interpreted as being greater than the second number’s

value if:

a. all digits of the first number - from left to right -

are identical to the corresponding digits of the

second number and

b. the second number has at least one additional

digit. In other words, the second number has more

digits than the first number.

See table 3 for the comparison between 10 and 100.

Table 3: Example inverse natural numbers to compare.

Comparison 1: a

n-1

n-2

n-3

First Number 1 0

Second Number 1 0 0

The digits a

and a

n-1

are identical, but second number

has additionally the digit a

n-2

, therefore: 10 > 100

Comparison 2: a

n-1

n-2

n-3

First Number 1 2 3

Second Number 1 2 4 7

The digit a

n-2

of the first number is smaller than the a

n-2

of the second number, therefore: 123 < 1247

The first two digits of 10 and 100 from left are

identical in table 3. Since the integer number 100 has

a third digit 0, it has, according to the new methods’

comparator, less value than the integer number 10.

In summary, if all digits of the first number are

processed and identical to the corresponding digits of

the second number, and the second number still has

further digits, then the second number is smaller. This

is explained with the shown additional examples in

table 4.

Table 4: Comparison of two inverse natural numbers P1 and

P2 by the comparator.

P1 P2 Com

arato

134 1350 P2 > P1

2783 2785 P2 > P1

221 22022 P1 > P2

66 667 P1 > P2

2595 25 P2 > P1

1330 133 P2 > P1

As presented in table 4, the algorithm enables to

interpret the greater value between two of given

inverse natural numbers just by comparing their digits

from the left to the right.

5 MAINTAINING THE

USER-DEFINED SEQUENCE

USING THE INVERSE

NATURAL NUMBER SYSTEM

This section briefly describes the insertion algorithm

of a new section number:

 In the first position of a sequence

 Between two existing sequence numbers

 In the last position of the sequence.

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262

The proposed insertion algorithm is also able to

preserve the uniqueness of sequence numbers in the

list order. This is even more relevant when the inverse

natural number system is utilised for indexing and for

key fields of the database records.

In order to maintain the uniqueness of sequence

numbers, the method in this section avoids 9 as last

rightmost digit of the newly inserted sequence

numbers.

This is a key functionality to create index files that

allow to insert new index items without the need to

update the index tree (Bayer, 1970, Schäfer, 1989).

5.1 Insertion of Data Records in the

First Position of the Sequence

This section outlines the methodology to add a new

row in the beginning of the ordered list.

An additional data record is inserted in the first

position of the sequence by calculating the sequence

number P

(n stands for new) of the additional data

record as 10 times the sequence number P

of the

subsequent data record in the sequence (see equation

(5)):

= 10 * P

(5)

Here, the subsequent data record in the sequence is

the data record that originally occupied the first place

in the sequence of the data records.

This enables insertion of an additional data record

in the first position of the sequence by reading just the

sequence number of one further data record. The

sequence numbers of the already existing data records

do not need to be changed (see table 5).

Table 5: Inserting a row in the first position of the sequence.

Before insertion After insertion

1 (P

) 10 (P

)

2 1 (P

)

3 2

etc. 3

etc.

5.2 Insertion of Data Records in the

Sequence between a Preceding Data

Record and a Succeeding One

To insert an additional data record in the sequence

between a preceding data record and a succeeding

one, the sequence numbers of the preceding and

succeeding data record must be read.

In the event that the sequence numbers of the

preceding data record and the succeeding data record

differ by 1, the sequence number P

of the additional

data record is calculated as being 10 times the

sequence number P

i+1

of the succeeding data record

(see equation (6) and table 6).

= 10 * P

i+1

(6)

Table 6: Inserting a row between sequence numbers 1 and

Before insertion After insertion

(

)

(

)

(

i+1

)

(

)

3 2 (P

i+1

)

etc. 3

etc.

If the sequence numbers of the preceding data

record and the succeeding data record differ by more

than 1, a number of steps need to be performed in

order to determine the sequence number of the

additional data record.

Firstly, the sequence number P

of the additional

data record is calculated as being 1 plus the sequence

number P

of the preceding data record (see equation

(7)):

= P

+1 (7)

If the last digit of the preceding sequence number is

equal to 8, then – in addition thereto – the obtained

sequence number of the additional data record needs

to be multiplied by 10 (see equation (8)).

= 10 * P

(8)

This ensures that the last, i.e. the rightmost digit of

the sequence number of the additional data record is

not equal to 9 and the uniqueness of the sequence is

guaranteed (see table 7).

Table 7: Inserting a row between sequence numbers 28 and

Before insertion After insertion

(

)

(

)

(

i+1

)

290

(

)

3 2

(

i+1

)

etc. 3

etc.

Then, the sequence number of the additional data

record is compared by means of the comparator (see

Section 4) with the sequence number of the

succeeding data record. The sequence number of the

new data record must be smaller than the sequence

number of the succeeding data record.

i+1

(9)

Invers Natural Number System to Maintain User-deﬁned Sequence of Data Records

263

If the sequence number of the succeeding data record

is not interpreted as being greater than the sequence

number of the additional data record as in inequation

(9), then the sequence number of the additional data

record must be multiplied by 10 and compared with

the sequence number of the succeeding data record

again until the sequence number of the succeeding

data record is interpreted as being greater than the

resulting sequence number of the additional data

record (see equation (10) and table 8).

= 10 * P

(10)

Table 8: Inserting a row between sequence numbers 28 and

290.

Before insertion Intermediate After insertion

28 (P

) 28 (P

)

290 (Pi+1) 290 (P

) 2900 (P

)

3 290

(

i+1

)

290

(

i+1

)

etc. 3 3

etc. etc.

5.3 Insertion of Data Records in the

Last Position of the Sequence

For insertion of additional data record in the last

position of the sequence, it is necessary to distinguish

whether the last, i.e. the rightmost, digit of the

sequence number of the preceding data record is

equal or not to 8.

In the event that the last digit of the sequence

number of the preceding data record is not equal to 8,

then the sequence number P

of the additional data

record is calculated as being 1 plus the sequence

number Pi of the preceding data record (see equation

(11) and table 9):

= P

(11)

Table 9: Inserting a data record at the end of the sequence.

Before insertion After insertion

83 83

84 84

85 (P

) 85 (P

)

86 (P

)

If the last digit of the sequence number of the

preceding data record is 8, then the sequence number

of the additional data record is determined in such

a way that 1 is added to the sequence number of the

preceding data record, and the sum obtained is

multiplied by 10 (see equation (12)).

= 10 * (P

+1) (12)

This is to ensure that the rightmost digit of the

sequence number of the additional data record is not

equal to 9, thus guaranteeing the uniqueness of the

sequence.

The preceding data record here is the data record

that occupied the last position in the sequence of data

records originally, i.e. before the addition step (see

table 10).

Table 10: Inserting a data record at the end of the sequence.

Before insertion After insertion

6 6

(

)

(

)

90 (P

)

6 THE NEXT VALUE

ALGORITHM FOR THE

INVERSE NATURAL NUMBER

SYSTEM

Software applications require a standard functionality

that track the actual sequence and provide the last

sequence number increment as the next value (IBM,

2021). The next value of integer numbers Pi is

calculated as being 1 plus its selves (see equation

(13)).

nextValue

= 1 + P

(13)

As a matter of fact, the equation (14) will not result in

the correct next value considering the summation

equation (3) of the inverse natural number system.

For instance, the next value of 1 applying equation

(14) will result in 2. According to new method’s

comparator (see section 4), 2 as the next value of 1

would waste all numbers between 1 and 2 (see

inequation (14) and table 11).

1 < 2000 < 2 (14)

Table 11: Skipping sequence numbers between 1 and 2.

List Order Unused se

uence numbers

(

)

…

2000

…

etc.

2 (P

nextValue

)

To ensure that the range of values of the new

method are utilized effectively, it is proposed to start

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the next value at a so-called turning number if the

current sequence number is a single-digit number.

The turning number is calculated by multiplying the

current sequence number with a default power of 10.

As an example, the possible valuation flow of the

inverse natural numbering system using 3 as default

power of 10 for the turning number is shown in table

12:

Table 12: Next value series of sequence numbers using the

inverse natural number system.

Next value series of numbers using 3 as default power

of the 10 for the turning numbe

2000, 2001, 2002, …, 2008,

200

2010, 2011, 2012, …, 2018,

201

2020, 2021, 2022, …, 2028,

…

2090, 2091, 2092, …, 2098

2100, 2101, 2102, …, 2108

…

2190, 2191, 2192, …, 2198

2200, 2201, 2102, …, 2208

…

2290, 2291, 2292, …, 2298

…

2990, 2991, 2992, …, 2998

3000, 3001, 3002, …, 3008

…

The pseudocode to calculate the next sequence

number of the inverse natural number system is

illustrated below:

Program Module to get the next value

for a given sequence number, written

in pseudocode

Input:

Given sequence number: v Є ℕ

Output:

Next sequence number value: x Є ℕ

Algorithm:

Initialize default power of 10 of the

most-left digit of the turning

number: p Є ℕ

Initialize power of 10 of the

most-left digit of the given

sequence number value v. This is

calculated using the logarithm base

10 to get the exponent and the

function Floor()to get the integer

part of a given decimal number:

q = Floor(Log

(v))

If rightmost digit of v is 8 and

p smaller or equal q Then

v ← (v - 8) / 10

While rightmost digit of v is 9

v ← (v - 9) / 10

Loop

Else

If p greater q Then

v ← (v + 1) * 10

(p – q)

Else

v ← (v + 1)

Return x ← v

7 CONCLUSIONS

As a result, insertion of further data records in any

position within the sequence of data records is

performed effectively and easily when using the

described method.

The new method allows management of databases

without the need for recalculating already allocated

sequence numbers of data records when inserting

further data records in any position into the sequence

of these records.

Using this method, a sequential renumbering of

the sequence numbers of all succeeding data records

in the sequence is avoided. It remains valid and

relevant, even following changes of the sequence

numbers or deletion of any data records.

Consequently, the computation time for managing

a database is reduced and the data volume required

when accessing the database, particularly via the

internet, is minimized.

REFERENCES

Bayer R. and McCreight E. (1970). Organization and

Maintenance of Ordered Indices, Boeing Scientific

Research Laboratories.

Cowlishaw, M. F. (2021). Decimal Arithmetic FAQ.

http://speleotrove.com/decimal/decifaq.html.

Invers Natural Number System to Maintain User-deﬁned Sequence of Data Records

265

Cowlishaw, M. F. (2003). Decimal Floating-Point:

Algorism for Computers. IBM UK.

IBM (2021). db2 product hub. https://www.ibm.com/

support/producthub/db2/docs/content/SSEPGG_11.5.0

/com.ibm.db2.luw.sql.ref.doc/doc/r0023464.html.

IBM (2021). Sequence objects. https://www.ibm.com/docs/

en/db2-for-zos/11?topic=programs-sequence-objects.

Microsoft (2021). ALTER SEQUENCE (Transact-SQL).

https://docs.microsoft.com/de-de/sql/t-sql/statements/

alter-sequence-transact-sql?view=sql-server-ver15.

Nelson, J (2018). User-defined Order in SQL.

https://begriffs.com/posts/2018-03-20-user-defined-

order.html.

Sleator, D. D. and Dietz, P. F. (1988). Two Algorithms for

Maintaining Order in a List.

Schäfer, G. (1989). Datenstrukturen und Datenbanken.

Schmid, H. (1974). Decimal Computation, General Electric

Company, New York.

Horn, T. (2007). SQL-Grundlagen. https://www.torsten-

horn.de/techdocs/sql.htm.

Wikipedia (2021). Decimal. https://en.wikipedia.org/wiki/

Decimal.

Wikipedia (2021). Big O Notation. https://en.wikipedia.

org/wiki/Big_O_notation.

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