math.interval-utils

This library provides a data structure and functions to do operations with intervals.

Version

0.3.0

Install

npm install math.inteval-utils --save

Index

• Interval
  • Data structure and valid values
  • Empty interval
• Functions
  • areDisjoint
  • areEqual
  • contains
  • intersection
  • isEmpty
  • isInterval
  • multiIntersection
  • numToInterval
  • parser
  • relativeComplement
• License

Interval

Data structure and valid values

Real interval can be represented with an pair of real numbers and two flags to indicate if these numbers are included or not in interval. This library defines the Interval type as the set of arrays of two objects with value and limit number properties:

[
    {
        value: (Number)
        limit: (0|1)
    },
    {
        value: (Number)
        limit: (-1|0)
    }
]

value properties correspond to the values of interval and limit properties correspond to if values of interval are included or not.

• If limit is 0, it indicates that value is included.
• If the limit of first item is 1 , it indicates that first value is not included in interval.
• If the limit of second item is -1, it indicates that second value is not included in interval.

For example:

• (1, 5] is represented with:

[
    {
        value: 1
        limit: 1
    },
    {
        value: 5
        limit: 0
    }
]

• [-1, 3] is represented with:

[
    {
        value: -1
        limit: 0
    },
    {
        value: 3
        limit: 0
    }
]

• (10, 12) is represented with:

[
    {
        value: 10
        limit: 1
    },
    {
        value: 12
        limit: -1
    }
]

Empty interval

An interval is empty in these cases:

• value of first element is greater than value of second element.
• value of first and second elements are equal but limit of first element is greater than limit of second element.

Functions

areDisjoint :: (Interval, Interval) -> Boolean

Given two interval inputs, it returns true or false depending on intervals are disjoint or not, respectively.

Example:

const { areDisjoint } = require('math.interval-utils')
 
// [1, 3)
const interval1 = [
    {value: 1, limit: 0},
    {value: 3, limit: -1}
]
 
// (1, 2)
const interval2 = [
    {value: 1, limit: 1},
    {value: 2, limit: -1}
]
 
// {1}
const interval3 = [
    {value: 1, limit: 0},
    {value: 1, limit: 0}
]
 
areDisjoint(interval1, interval2) // false
areDisjoint(interval2, interval3) // true

areEqual :: (Interval, Interval) -> Boolean

Given two interval inputs, it returns true or false depending on intervals are equal or not, respectively.

Example:

const { areEqual } = require('math.interval-utils')
 
// [1, 3)
const interval1 = [
    {value: 1, limit: 0},
    {value: 3, limit: -1}
]
 
// [1, 3)
const interval2 = [
    {value: 1, limit: 0},
    {value: 3, limit: -1}
]
 
// [1, 3]
const interval3 = [
    {value: 1, limit: 0},
    {value: 3, limit: 0}
]
 
areEqual(interval1, interval2) // true
areEqual(interval2, interval3) // false

contains :: (Interval, Interval) -> Boolean

Given two interval inputs, it returns true or false if first interval contains the second interval or not, respectively.

Example:

const { contains } = require('math.interval-utils')
 
// [1, 3)
const interval1 = [
    {value: 1, limit: 0},
    {value: 3, limit: -1}
]
 
// (1, 2)
const interval2 = [
    {value: 1, limit: 1},
    {value: 2, limit: -1}
]
 
contains(interval1, interval2) // true
contains(interval2, interval1) // false

intersection :: (Interval, Interval) -> Interval

Given two interval inputs, it returns the intersection of these intervals.

Example:

const { intersection } = require('math.interval-utils')
const I = {}
// [1, 3)
I['[1, 3)'] = [
    {value: 1, limit: 0},
    {value: 3, limit: -1}
]
 
// (1, 2)
I['(1, 2)'] = [
    {value: 1, limit: 1},
    {value: 2, limit: -1}
]
 
// (3, 4)
I['(3, 4)'] = [
    {value: 3, limit: 1},
    {value: 4, limit: -1}
]
 
intersection(I['[1, 3)'], I['(1, 2)']) // (1, 2)
intersection(I['(1, 2)'], I['(3, 4)']) // (3, 4)
intersection(I['[1, 3)'], I['(3, 4)']) // empty

isEmpty :: Interval -> Boolean

Given an interval, it returns true or false if interval is empty or not, respectively.

Example:

const { isEmpty } = require('math.interval-utils')
 
// [2, -2]
const interval1 = [
    {value: 2, limit: 0},
    {value: -2, limit: 0}
]
 
// (1, 1]
const interval2 = [
    {value: 1, limit: 1},
    {value: 1, limit: 0}
]
 
// [1, 1]
const interval3 = [
    {value: 1, limit: 0},
    {value: 1, limit: 0}
]
 
isEmpty(interval1) // true
isEmpty(interval2) // true
isEmpty(interval3) // false

isInterval :: Interval -> Boolean

It returns true or false if interval is an Interval.

Example:

const { isInterval } = require('math.interval-utils')
 
const interval1 = [
    {value: 2, limit: 0},
    {value: -2, limit: 0}
]
 
const interval2 = [
    {value: -2, limit: 0}
]
 
const interval3 = [
    {value: 2, limit: 0},
    {value: Infinity, limit: -1}
]
 
isInterval(interval1) // true
isInterval(interval2) // false
isInterval(interval3) // true

multiIntersection :: ([Interval], [Interval]) -> [Interval]

Given two lists of disjoint sorted intervals, it returns a new list of disjoint sorted intervals that represent the intersection of these sets.

Example:

// (1, 3) U {4} U [5, 6)
const listIntervals1 = [[
  {value: 1, limit: 1},
  {value: 3, limit: -1}
], [
  {value: 4, limit: 0},
  {value: 4, limit: 0}
], [
  {value: 5, limit: 0},
  {value: 6, limit: -1}
]]
 
// {0} U [1, 2] U (3, 4] U (5, 7)
const listIntervals2 = [[
  {value: 0, limit: 0},
  {value: 0, limit: 0}
], [
  {value: 3, limit: 1},
  {value: 4, limit: 0}
], [
  {value: 5, limit: 1},
  {value: 7, limit: -1}
]]
 
// returns (1, 2] U {4} U (5, 6)
multiIntersection(listIntervals1, listIntervals2) // [[
  {value: 1, limit: 1},
  {value: 2, limit: 0}
], [
  {value: 4, limit: 0},
  {value: 4, limit: 0}
], [
  {value: 5, limit: 1},
  {value: 6, limit: -1}
]]

This method has better perfomance than recolecting the intersections of each interval of each lists and intersecting one by one.

numToInterval :: Number -> Interval

Given a number input, it returns an singleton interval that contains this number.

Example:

const { numToInterval } = require('math.interval-utils')
 
// returns {5} or [5, 5]
numToInterval(5) /* returns [
    {value: 5, limit: 0},
    {value: 5, limit: 0}
]*/

parser :: string -> Either Interval String

Given a string that represents an interval, it returns an Either.Right value that saves an interval. If the string does not represent an interval it returns and Either.Left value that saves an error.

Example:

const { Right, Left } = require('data.either')
const { parser } = require('math.interval-utils')
 
parser('(2, 3]') /* Right [
    {value: 2, limit: 1},
    {value: 3, limit: 0}
] */
 
parser('{5}') /* Right([
    {value: 5, limit: 0},
    {value: 5, limit: 0}
]) */
 
parser('(2, 5(') // Left('"(2, 5(" does not match to interval expression')

relativeComplement :: (Interval, Interval) -> [Interval]

Given two interval inputs, it returns a list of intevals that represents the relative complement. It means, the set of numbers that belongs to the first interval but not the second.

Example:

const { relativeComplement } = require('math.interval-utils')
const I = {}
// [1, 5)
I['[1, 5)'] = [
    {value: 1, limit: 0},
    {value: 5, limit: -1}
]
 
// (2, 3)
I['(2, 3)'] = [
    {value: 2, limit: 1},
    {value: 3, limit: -1}
]
 
// [4, 6]
I['[4, 6]'] = [
    {value: 4, limit: 0},
    {value: 6, limit: 0}
]
 
// returns [1, 2] U [3, 5)
relativeComplement(I['[1, 5)'], I['(2, 3)']) /* [[
  {value: 1, limit: 0},
  {value: 2, limit: 0}
], [
  {value: 3, limit: 0},
  {value: 5, limit: -1}
]] */
 
// returns [1, 4)
relativeComplement(I['[1, 5)'], I['[4, 6]']) /* [[
  {value: 1, limit: 0},
  {value: 4, limit: -1}
]] */
 
// returns (2, 3)
relativeComplement(I['(2, 3)'], I['[4, 6]']) /* [[
    {value: 2, limit: 1},
    {value: 3, limit: -1}
]] */

// returns empty array relativeComplement(I['(2, 3)'], I['[1, 5)']) // []

union :: [Interval] -> [Interval]

Given an array of intervals, it returns an array of sorted disjoint intervals that represents the union of these intervals.

Example:

const { union } = require('math.interval-utils').union
const I = {}
// [1, 3)
I['[1, 3)'] = [
    {value: 1, limit: 0},
    {value: 3, limit: -1}
]
 
// (2, 4)
I['(2, 4)'] = [
    {value: 2, limit: 1},
    {value: 4, limit: -1}
]
 
// [5, 5]
I['[5, 5]'] = [
    {value: 5, limit: 0},
    {value: 5, limit: 0}
]
 
// (5, 6)
I['(5, 6)'] = [
    {value: 5, limit: 1},
    {value: 6, limit: -1}
]
 
// returns [1, 4) U [5, 6)
union([interval1, interval2 interval3, interval4]) /* [[
    {value: 1, limit: 0},
    {value: 4, limit: -1}
], [
    {value: 5, limit: 0},
    {value: 6, limit: -1}
]] */

LICENSE

MIT