Continuum

Generates compact strings that behave as a continuum -- ordered points, for which between any two points is always another point.

Strings are kept as short as possible, unlike some traditional implementations of this data type in which strings grow quickly with normal usage.

Both integers and floating point numbers have representations as compact strings, for example the number 3 is "aD". This allows for a direct mapping between numeric continuums to this string-typed continuum. The reason to use strings instead of numbers for this purpose, is that floating point numbers do not have arbitrary precision, and floating point numbers can't be used as keys in maps.

Uses:

• GUIDs/UUIDs: With the additional property that they can easily be ordered with regard to each other or with regard to time, while being shorter than UUID5.
• Distributed, multi-writer arrays: Using these as keys in a map, writers can insert/prepend/append arbitrarily, maintaining ordering without coordination.
• CRDTs & Operational Transforms: These data structures need unique, orderable strings for things like keys in maps and arrays, and to resolve last-write-wins. Specifically, this solves the same problem as famous structures like LOGOOT, WOOT, and Treedoc, but more efficient in both time and space.

Features:

• Compatible: Strings only use the characters matching [A-Za-z]+, to be compatible with every kind of key/value map or data store, and they're easy to read while debugging.
• Compact: Strings converted from numbers are short, with no padding, and strings generated between other strings often collapse to shorter strings in the process.
• Typescript: Written in Typescript, but distributed as regular Javascript, and includes Typescript declarations

Usage

Use through npm, or get the source from Github.

fromInteger() / toInteger()

Generates a Continuum string from a Javascript number (integers only). The strings are designs to be as short as possible for integers with small absolute value, and still quite compact even with large ones. Ordering is preserved; that is, given two integers where i1 < i2, then also fromInteger(i1) < fromInteger(i2). See discussion below for the algorithm.

import * as C from '@asmartbear/continuum';

console.log(C.fromInteger( 0 ));  // -> a
console.log(C.fromInteger( 1 ));  // -> aC
console.log(C.fromInteger( 2 ));  // -> aD
console.log(C.fromInteger( 1234 ));  // -> bZj
console.log(C.fromInteger( Number.MAX_SAFE_INTEGER ));  // -> jFfeLmfmvUq
console.log(C.fromInteger( -1 ));  // -> Zx
console.log(C.fromInteger( -2 ));  // -> Zw
console.log(C.fromInteger( Number.MIN_SAFE_INTEGER ));  // -> QuUVoNUNEfJ

console.log(C.toInteger("Zw")); // -> -2

fromString()

Generates a valid Continuum string from an arbitrary string (including empty and unicode characters). Ordering is preserved; given two Javascript strings where s1 < s2, you will always also have fromString(s1) < fromString(s2).

import * as C from '@asmartbear/continuum';

console.log(C.fromString( "" ));  // -> a
console.log(C.fromString( "\u0000" ));  // -> aAa
console.log(C.fromString( "\uffff" ));  // -> aAcbLk
console.log(C.fromString( "Hello, World!" ));  // -> aAbCXAbD[...]DBAai

between()

Given two Continuum strings, generates a new string that sorts strictly between them.

import * as C from '@asmartbear/continuum';

console.log(C.between( 'abc', 'abg' ));  // -> abe
console.log(C.between( 'abc', 'abd' ));  // -> abccn

next()

Given a Continuum string, generates a string that sorts "next," but sufficiently distant from the first string that any number of other strings can be inserted between them, without causing very long strings. Includes an algorithm preventing a common problem with other libraries in which strings grow very long if next() is used thousands or millions of times.

You can also provide an increment for next(), though the minimum is 2.

import * as C from '@asmartbear/continuum';

console.log(C.next( 'abc' ));  // -> abe  (leaves space for abd)
console.log(C.next( 'abc', 3  ));  // -> abf
console.log(C.next( 'abc', 10 ));  // -> abp
console.log(C.next( 'abz' ));  // -> acB
console.log(C.next( 'zzz' ));  // -> zzzBBB  (string increases with pure 'z's prefixed)

getStarterForNext()

Strings can end up growing long when next() is invoked many times. By pre-allocating a longer string to begin with, you can prevent that from happening. Relatively few letters are needed for huge numbers of invocations, so this can save lots of space in the long run.

import * as C from '@asmartbear/continuum';

// Only need 5 characters to call `next()` 100,000,000 times without growing the string
console.log(C.getStarterForNext(100000000));  // -> BBBBB

random()

Generates a valid Continuum string using random characters. Use to seed a new distributed array, or as a sort of GUID (even just 10 characters is enough for a 1-in-100-quadrillion chance of collision!). This is not cryptographically secure!

import * as C from '@asmartbear/continuum';

console.log(C.random( 20 ));  // -> PyTkqGaJKkGjrsnUSIrq

isWellFormed()

Reports whether the given object is a well-formed Continuum string.

import * as C from '@asmartbear/continuum';

console.log(C.isWellFormed( null ));   // -> false   (same with any non-string)
console.log(C.isWellFormed( "" ));     // -> false   (empty is not orderable!)
console.log(C.isWellFormed( "nBq" ));  // -> true
console.log(C.isWellFormed( "RtA" ));  // -> false   (cannot end with first letter of the alphabet)

Technical Discussion

Alphabet

The "Alphabet" is the ordered list of characters that can make up a string. Currently these are A-Z followed by a-z, but code should reference the exported ALPHABET string for compatibility if the Alphabet changes.

The rule of the last letter of a string

Strings cannot end with the first letter of the alphabet, because then you can construct another string that won't sort between the two. For example, consider S1="BCD" and S2="BCDA", where "A" is the first letter of the alphabet. It's not possible to create a string between these two. You cannot increment "D" in S1, because the result would sort after S2. The lowest-value character you can append to S1 is "A"; the result is already equal to S2 so we don't have space to create something "in between!" This is not a problem if we require the last character to be anything else. For example, with S1="BCD" and S2="BCDB", we can build "BCDAN". The ability to use the first letter of the alphabet to "get past" the "B", then allows us to further append something in the middle of the alphabet, giving us space for more insertions before or after this new string and S1 or S2.

Converting from integers

The traditional way of representing integers as strings that sort the same way is to pad the numbers with zeros; for example, 37 might become "000000037". This has three drawbacks: (1) small numbers use far more memory than necessary; (2) numbers are limited in size (this example fails just before one billion); (3) can't represent negative numbers (because prepending a negative sign doesn't result in correct sort order, e.g. "-0001" < "-0002" but -1 > -2).

This system has none of those problems. Consider positive integers first. "Digits" are stored in base-50, using all but the first and last characters of the Alphabet. (This is done both for well-formed strings, and to allow space for operations like next() and between().) Digits do have to come in the usual order, with the largest place-values first, but then you have the usual string sorting problem, e.g. "2" > "12", or in our base-50 encoding, starting with the second Alphabet letter "B", that problem looks like "C" > "BC". Therefore, we prefix the digits with a character that encodes the number of digits, i.e. 'a' for one digit, "b" for two, etc.. So our immediate example becomes "aC" < "bBC" and now we're sorting correctly. (Actually, 2 would be aC and 12 would be aM because of base-50, but the principle is correct when you get multiple digits.) This does place a limit on the maximum size of an integer at 26 base-50 digits, about 7 * 10^45 or 146 bits.

Negative integers are represented the same way as negative binary integers are in machine code: in "50's complement," the equivalent of "two's complement." The idea is that each digit d is stored instead as 50-d. Thus, if 2 in base-50 would be the digit "C", then -2 would be the digit "x". We still need a prefix character to do a job for us, however, because just "x" alone might be the digit for -2 if the number is negative, but it's the digit for 48 if the number is positive, so without the prefix character, -2 would sort after 2. The prefix character works the same way as with positive integers -- encoding the number of digits -- but counts backwards from "Z". Therefore, -2 is encoded as "Zx" and -52 would be "Yyx".

Zero is special, and is represented simply as "a", which sorts after all negative numbers (because -1 is "Zy") and before all positive numbers (because 1 is aB).

Converting from floating point

Starting with our integer representation, it is apparent that we can continue adding digits to the end of the string to represent fractional place values. Because the leading character already encodes the number of integral place-values, it implies that anything beyond that count is fractional, and will already sort correctly given the previous rules.

There is a question of how many digits to use when the fraction is repeating in base-50. For example, if the fractional-part is just 1/2, there's no problem as this is the (single digit) [25] in base 50, and as the remaining digits would be zero, there's nothing else we need to append to the integer-part. But if the fraction is 1/3, then just as in base-10 the decimal representation repeats forever 0.333333..., so in base-50 it repeats forever in the two-digit pattern .[16][33][16][33]....

Of course, it is at our discretion when to stop appending digits. Therefore, we take as an input to floating-point conversions the maximum number of fractional-part digits, and simply stop there.