From b2d1ec83f04a66d9d2d3f09150bcaf0cacb0f7ba Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Fri, 16 Jul 2021 11:51:53 +0200 Subject: [PATCH 001/168] Binary representation of floating-point numbers (#737) * Add "Binary representation of floating point numbers" section. * Adding a half-precision explanatory picture. * Binary representation of the floating-point numbers. --- README.md | 1 + .../math/binary-floating-point/README.md | 93 ++++++++++++++ .../__tests__/bitsToFloat.test.js | 32 +++++ .../__tests__/floatAsBinaryString.test.js | 20 +++ .../math/binary-floating-point/bitsToFloat.js | 119 ++++++++++++++++++ .../floatAsBinaryString.js | 61 +++++++++ ...cision-floating-point-number-explained.png | Bin 0 -> 352613 bytes .../images/03-scientific-notation.png | Bin 0 -> 76251 bytes .../math/binary-floating-point/testCases.js | 71 +++++++++++ 9 files changed, 397 insertions(+) create mode 100644 src/algorithms/math/binary-floating-point/README.md create mode 100644 src/algorithms/math/binary-floating-point/__tests__/bitsToFloat.test.js create mode 100644 src/algorithms/math/binary-floating-point/__tests__/floatAsBinaryString.test.js create mode 100644 src/algorithms/math/binary-floating-point/bitsToFloat.js create mode 100644 src/algorithms/math/binary-floating-point/floatAsBinaryString.js create mode 100644 src/algorithms/math/binary-floating-point/images/02-half-precision-floating-point-number-explained.png create mode 100644 src/algorithms/math/binary-floating-point/images/03-scientific-notation.png create mode 100644 src/algorithms/math/binary-floating-point/testCases.js diff --git a/README.md b/README.md index b8e2be362a..d00945e5a5 100644 --- a/README.md +++ b/README.md @@ -68,6 +68,7 @@ a set of rules that precisely define a sequence of operations. * **Math** * `B` [Bit Manipulation](src/algorithms/math/bits) - set/get/update/clear bits, multiplication/division by two, make negative etc. + * `B` [Binary Floating Point](src/algorithms/math/binary-floating-point) - binary representation of the floating-point numbers. * `B` [Factorial](src/algorithms/math/factorial) * `B` [Fibonacci Number](src/algorithms/math/fibonacci) - classic and closed-form versions * `B` [Prime Factors](src/algorithms/math/prime-factors) - finding prime factors and counting them using Hardy-Ramanujan's theorem diff --git a/src/algorithms/math/binary-floating-point/README.md b/src/algorithms/math/binary-floating-point/README.md new file mode 100644 index 0000000000..2851902845 --- /dev/null +++ b/src/algorithms/math/binary-floating-point/README.md @@ -0,0 +1,93 @@ +# Binary representation of floating-point numbers + +Have you ever wondered how computers store the floating-point numbers like `3.1415` (𝝿) or `9.109 × 10⁻³¹` (the mass of the electron in kg) in the memory which is limited by a finite number of ones and zeroes (aka bits)? + +It seems pretty straightforward for integers (i.e. `17`). Let's say we have 16 bits (2 bytes) to store the number. In 16 bits we may store the integers in a range of `[0, 65535]`: + +```text +(0000000000000000)₂ = (0)₁₀ + +(0000000000010001)₂ = + (1 × 2⁴) + + (0 × 2³) + + (0 × 2²) + + (0 × 2¹) + + (1 × 2⁰) = (17)₁₀ + +(1111111111111111)₂ = + (1 × 2¹⁵) + + (1 × 2¹⁴) + + (1 × 2¹³) + + (1 × 2¹²) + + (1 × 2¹¹) + + (1 × 2¹⁰) + + (1 × 2⁹) + + (1 × 2⁸) + + (1 × 2⁷) + + (1 × 2⁶) + + (1 × 2⁵) + + (1 × 2⁴) + + (1 × 2³) + + (1 × 2²) + + (1 × 2¹) + + (1 × 2⁰) = (65535)₁₀ +``` + +If we need a signed integer we may use [two's complement](https://en.wikipedia.org/wiki/Two%27s_complement) and shift the range of `[0, 65535]` towards the negative numbers. In this case, our 16 bits would represent the numbers in a range of `[-32768, +32767]`. + +As you might have noticed, this approach won't allow you to represent the numbers like `-27.15625` (numbers after the decimal point are just being ignored). + +We're not the first ones who have noticed this issue though. Around ≈36 years ago some smart folks overcame this limitation by introducing the [IEEE 754](https://en.wikipedia.org/wiki/IEEE_754) standard for floating-point arithmetic. + +The IEEE 754 standard describes the way (the framework) of using those 16 bits (or 32, or 64 bits) to store the numbers of wider range, including the small floating numbers (smaller than 1 and closer to 0). + +To get the idea behind the standard we might recall the [scientific notation](https://en.wikipedia.org/wiki/Scientific_notation) - a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. + +![Scientific number notation](images/03-scientific-notation.png) + +As you may see from the image, the number representation might be split into three parts: + +- **sign** +- **fraction (aka significand)** - the valuable digits (the meaning, the payload) of the number +- **exponent** - controls how far and in which direction to move the decimal point in the fraction + +The **base** part we may omit by just agreeing on what it will be equal to. In our case, we'll be using `2` as a base. + +Instead of using all 16 bits (or 32 bits, or 64 bits) to store the fraction of the number, we may share the bits and store a sign, exponent, and fraction at the same time. Depending on the number of bits that we're going to use to store the number we end up with the following splits: + +| Floating-point format | Total bits | Sign bits | Exponent bits | Fraction bits | Base | +| :-------------------- | :--------: | :-------: | :-----------: | :--------------: | :--: | +| [Half-precision](https://en.wikipedia.org/wiki/Half-precision_floating-point_format) | 16 | 1 | 5 | 10 | 2 | +| [Single-precision](https://en.wikipedia.org/wiki/Single-precision_floating-point_format) | 32 | 1 | 8 | 23 | 2 | +| [Double-precision](https://en.wikipedia.org/wiki/Double-precision_floating-point_format) | 64 | 1 | 11 | 52 | 2 | + +With this approach, the number of bits for the fraction has been reduced (i.e. for the 16-bits number it was reduced from 16 bits to 10 bits). It means that the fraction might take a narrower range of values now (losing some precision). However, since we also have an exponent part, it will actually increase the ultimate number range and also allow us to describe the numbers between 0 and 1 (if the exponent is negative). + +> For example, a signed 32-bit integer variable has a maximum value of 2³¹ − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of ≈ 3.4028235 × 10³⁸. + +To make it possible to have a negative exponent, the IEEE 754 standard uses the [biased exponent](https://en.wikipedia.org/wiki/Exponent_bias). The idea is simple - subtract the bias from the exponent value to make it negative. For example, if the exponent has 5 bits, it might take the values from the range of `[0, 31]` (all values are positive here). But if we subtract the value of `15` from it, the range will be `[-15, 16]`. The number `15` is called bias, and it is being calculated by the following formula: + +``` +exponent_bias = 2 ^ (k−1) − 1 + +k - number of exponent bits +``` + +I've tried to describe the logic behind the converting of floating-point numbers from a binary format back to the decimal format on the image below. Hopefully, it will give you a better understanding of how the IEEE 754 standard works. The 16-bits number is being used here for simplicity, but the same approach works for 32-bits and 64-bits numbers as well. + +![Half-precision floating point number format explained in one picture](images/02-half-precision-floating-point-number-explained.png) + +> Checkout the [interactive version of this diagram](https://trekhleb.dev/blog/2021/binary-floating-point/) to play around with setting bits on and off, and seeing how it would influence the final result + +## Code examples + +- See the [bitsToFloat.js](bitsToFloat.js) for the example of how to convert array of bits to the floating point number (the example is a bit artificial but still it gives the overview of how the conversion is going on) +- See the [floatAsBinaryString.js](floatAsBinaryString.js) for the example of how to see the actual binary representation of the floating-point number in JavaScript + +## References + +You might also want to check out the following resources to get a deeper understanding of the binary representation of floating-point numbers: + +- [Here is what you need to know about JavaScript’s Number type](https://indepth.dev/posts/1139/here-is-what-you-need-to-know-about-javascripts-number-type) +- [Float Exposed](https://float.exposed/) +- [IEEE754 Visualization](https://bartaz.github.io/ieee754-visualization/) diff --git a/src/algorithms/math/binary-floating-point/__tests__/bitsToFloat.test.js b/src/algorithms/math/binary-floating-point/__tests__/bitsToFloat.test.js new file mode 100644 index 0000000000..53afe691a2 --- /dev/null +++ b/src/algorithms/math/binary-floating-point/__tests__/bitsToFloat.test.js @@ -0,0 +1,32 @@ +import { testCases16Bits, testCases32Bits, testCases64Bits } from '../testCases'; +import { bitsToFloat16, bitsToFloat32, bitsToFloat64 } from '../bitsToFloat'; + +describe('bitsToFloat16', () => { + it('should convert floating point binary bits to floating point decimal number', () => { + for (let testCaseIndex = 0; testCaseIndex < testCases16Bits.length; testCaseIndex += 1) { + const [decimal, binary] = testCases16Bits[testCaseIndex]; + const bits = binary.split('').map((bitString) => parseInt(bitString, 10)); + expect(bitsToFloat16(bits)).toBeCloseTo(decimal, 4); + } + }); +}); + +describe('bitsToFloat32', () => { + it('should convert floating point binary bits to floating point decimal number', () => { + for (let testCaseIndex = 0; testCaseIndex < testCases32Bits.length; testCaseIndex += 1) { + const [decimal, binary] = testCases32Bits[testCaseIndex]; + const bits = binary.split('').map((bitString) => parseInt(bitString, 10)); + expect(bitsToFloat32(bits)).toBeCloseTo(decimal, 7); + } + }); +}); + +describe('bitsToFloat64', () => { + it('should convert floating point binary bits to floating point decimal number', () => { + for (let testCaseIndex = 0; testCaseIndex < testCases64Bits.length; testCaseIndex += 1) { + const [decimal, binary] = testCases64Bits[testCaseIndex]; + const bits = binary.split('').map((bitString) => parseInt(bitString, 10)); + expect(bitsToFloat64(bits)).toBeCloseTo(decimal, 14); + } + }); +}); diff --git a/src/algorithms/math/binary-floating-point/__tests__/floatAsBinaryString.test.js b/src/algorithms/math/binary-floating-point/__tests__/floatAsBinaryString.test.js new file mode 100644 index 0000000000..efb8a2ba5e --- /dev/null +++ b/src/algorithms/math/binary-floating-point/__tests__/floatAsBinaryString.test.js @@ -0,0 +1,20 @@ +import { floatAs32BinaryString, floatAs64BinaryString } from '../floatAsBinaryString'; +import { testCases32Bits, testCases64Bits } from '../testCases'; + +describe('floatAs32Binary', () => { + it('should create a binary representation of the floating numbers', () => { + for (let testCaseIndex = 0; testCaseIndex < testCases32Bits.length; testCaseIndex += 1) { + const [decimal, binary] = testCases32Bits[testCaseIndex]; + expect(floatAs32BinaryString(decimal)).toBe(binary); + } + }); +}); + +describe('floatAs64Binary', () => { + it('should create a binary representation of the floating numbers', () => { + for (let testCaseIndex = 0; testCaseIndex < testCases64Bits.length; testCaseIndex += 1) { + const [decimal, binary] = testCases64Bits[testCaseIndex]; + expect(floatAs64BinaryString(decimal)).toBe(binary); + } + }); +}); diff --git a/src/algorithms/math/binary-floating-point/bitsToFloat.js b/src/algorithms/math/binary-floating-point/bitsToFloat.js new file mode 100644 index 0000000000..6d1ef0d851 --- /dev/null +++ b/src/algorithms/math/binary-floating-point/bitsToFloat.js @@ -0,0 +1,119 @@ +/** + * Sequence of 0s and 1s. + * @typedef {number[]} Bits + */ + +/** + * @typedef {{ + * signBitsCount: number, + * exponentBitsCount: number, + * fractionBitsCount: number, + * }} PrecisionConfig + */ + +/** + * @typedef {{ + * half: PrecisionConfig, + * single: PrecisionConfig, + * double: PrecisionConfig + * }} PrecisionConfigs + */ + +/** + * ┌───────────────── sign bit + * │ ┌───────────── exponent bits + * │ │ ┌───── fraction bits + * │ │ │ + * X XXXXX XXXXXXXXXX + * + * @type {PrecisionConfigs} + */ +const precisionConfigs = { + // @see: https://en.wikipedia.org/wiki/Half-precision_floating-point_format + half: { + signBitsCount: 1, + exponentBitsCount: 5, + fractionBitsCount: 10, + }, + // @see: https://en.wikipedia.org/wiki/Single-precision_floating-point_format + single: { + signBitsCount: 1, + exponentBitsCount: 8, + fractionBitsCount: 23, + }, + // @see: https://en.wikipedia.org/wiki/Double-precision_floating-point_format + double: { + signBitsCount: 1, + exponentBitsCount: 11, + fractionBitsCount: 52, + }, +}; + +/** + * Converts the binary representation of the floating point number to decimal float number. + * + * @param {Bits} bits - sequence of bits that represents the floating point number. + * @param {PrecisionConfig} precisionConfig - half/single/double precision config. + * @return {number} - floating point number decoded from its binary representation. + */ +function bitsToFloat(bits, precisionConfig) { + const { signBitsCount, exponentBitsCount } = precisionConfig; + + // Figuring out the sign. + const sign = (-1) ** bits[0]; // -1^1 = -1, -1^0 = 1 + + // Calculating the exponent value. + const exponentBias = 2 ** (exponentBitsCount - 1) - 1; + const exponentBits = bits.slice(signBitsCount, signBitsCount + exponentBitsCount); + const exponentUnbiased = exponentBits.reduce( + (exponentSoFar, currentBit, bitIndex) => { + const bitPowerOfTwo = 2 ** (exponentBitsCount - bitIndex - 1); + return exponentSoFar + currentBit * bitPowerOfTwo; + }, + 0, + ); + const exponent = exponentUnbiased - exponentBias; + + // Calculating the fraction value. + const fractionBits = bits.slice(signBitsCount + exponentBitsCount); + const fraction = fractionBits.reduce( + (fractionSoFar, currentBit, bitIndex) => { + const bitPowerOfTwo = 2 ** -(bitIndex + 1); + return fractionSoFar + currentBit * bitPowerOfTwo; + }, + 0, + ); + + // Putting all parts together to calculate the final number. + return sign * (2 ** exponent) * (1 + fraction); +} + +/** + * Converts the 16-bit binary representation of the floating point number to decimal float number. + * + * @param {Bits} bits - sequence of bits that represents the floating point number. + * @return {number} - floating point number decoded from its binary representation. + */ +export function bitsToFloat16(bits) { + return bitsToFloat(bits, precisionConfigs.half); +} + +/** + * Converts the 32-bit binary representation of the floating point number to decimal float number. + * + * @param {Bits} bits - sequence of bits that represents the floating point number. + * @return {number} - floating point number decoded from its binary representation. + */ +export function bitsToFloat32(bits) { + return bitsToFloat(bits, precisionConfigs.single); +} + +/** + * Converts the 64-bit binary representation of the floating point number to decimal float number. + * + * @param {Bits} bits - sequence of bits that represents the floating point number. + * @return {number} - floating point number decoded from its binary representation. + */ +export function bitsToFloat64(bits) { + return bitsToFloat(bits, precisionConfigs.double); +} diff --git a/src/algorithms/math/binary-floating-point/floatAsBinaryString.js b/src/algorithms/math/binary-floating-point/floatAsBinaryString.js new file mode 100644 index 0000000000..aa907ac049 --- /dev/null +++ b/src/algorithms/math/binary-floating-point/floatAsBinaryString.js @@ -0,0 +1,61 @@ +// @see: https://en.wikipedia.org/wiki/Single-precision_floating-point_format +const singlePrecisionBytesLength = 4; // 32 bits + +// @see: https://en.wikipedia.org/wiki/Double-precision_floating-point_format +const doublePrecisionBytesLength = 8; // 64 bits + +const bitsInByte = 8; + +/** + * Converts the float number into its IEEE 754 binary representation. + * @see: https://en.wikipedia.org/wiki/IEEE_754 + * + * @param {number} floatNumber - float number in decimal format. + * @param {number} byteLength - number of bytes to use to store the float number. + * @return {string} - binary string representation of the float number. + */ +function floatAsBinaryString(floatNumber, byteLength) { + let numberAsBinaryString = ''; + + const arrayBuffer = new ArrayBuffer(byteLength); + const dataView = new DataView(arrayBuffer); + + const byteOffset = 0; + const littleEndian = false; + + if (byteLength === singlePrecisionBytesLength) { + dataView.setFloat32(byteOffset, floatNumber, littleEndian); + } else { + dataView.setFloat64(byteOffset, floatNumber, littleEndian); + } + + for (let byteIndex = 0; byteIndex < byteLength; byteIndex += 1) { + let bits = dataView.getUint8(byteIndex).toString(2); + if (bits.length < bitsInByte) { + bits = new Array(bitsInByte - bits.length).fill('0').join('') + bits; + } + numberAsBinaryString += bits; + } + + return numberAsBinaryString; +} + +/** + * Converts the float number into its IEEE 754 64-bits binary representation. + * + * @param {number} floatNumber - float number in decimal format. + * @return {string} - 64 bits binary string representation of the float number. + */ +export function floatAs64BinaryString(floatNumber) { + return floatAsBinaryString(floatNumber, doublePrecisionBytesLength); +} + +/** + * Converts the float number into its IEEE 754 32-bits binary representation. + * + * @param {number} floatNumber - float number in decimal format. + * @return {string} - 32 bits binary string representation of the float number. + */ +export function floatAs32BinaryString(floatNumber) { + return floatAsBinaryString(floatNumber, singlePrecisionBytesLength); +} diff --git a/src/algorithms/math/binary-floating-point/images/02-half-precision-floating-point-number-explained.png b/src/algorithms/math/binary-floating-point/images/02-half-precision-floating-point-number-explained.png new file mode 100644 index 0000000000000000000000000000000000000000..1bc0a5b83596b9479525362d67af1dfb5168ef70 GIT binary patch literal 352613 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Subject: [PATCH 002/168] Binary representation of the floating-point numbers. --- src/algorithms/math/binary-floating-point/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/algorithms/math/binary-floating-point/README.md b/src/algorithms/math/binary-floating-point/README.md index 2851902845..32b41f642a 100644 --- a/src/algorithms/math/binary-floating-point/README.md +++ b/src/algorithms/math/binary-floating-point/README.md @@ -1,6 +1,6 @@ # Binary representation of floating-point numbers -Have you ever wondered how computers store the floating-point numbers like `3.1415` (𝝿) or `9.109 × 10⁻³¹` (the mass of the electron in kg) in the memory which is limited by a finite number of ones and zeroes (aka bits)? +Have you ever wondered how computers store the floating-point numbers like `3.1416` (𝝿) or `9.109 × 10⁻³¹` (the mass of the electron in kg) in the memory which is limited by a finite number of ones and zeroes (aka bits)? It seems pretty straightforward for integers (i.e. `17`). Let's say we have 16 bits (2 bytes) to store the number. In 16 bits we may store the integers in a range of `[0, 65535]`: From 71da6df63312096964eaaf8f538c174d1c39014b Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Mon, 26 Jul 2021 07:43:07 +0200 Subject: [PATCH 003/168] Binary representation of the floating-point numbers. --- src/algorithms/math/binary-floating-point/README.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/src/algorithms/math/binary-floating-point/README.md b/src/algorithms/math/binary-floating-point/README.md index 32b41f642a..c0c6fce046 100644 --- a/src/algorithms/math/binary-floating-point/README.md +++ b/src/algorithms/math/binary-floating-point/README.md @@ -79,6 +79,8 @@ I've tried to describe the logic behind the converting of floating-point numbers > Checkout the [interactive version of this diagram](https://trekhleb.dev/blog/2021/binary-floating-point/) to play around with setting bits on and off, and seeing how it would influence the final result +Be aware that this is by no means a complete and sufficient overview of the IEEE 754 standard. It is rather a simplified and basic overview. Several corner cases were omitted in the examples above for simplicity of presentation (i.e. `-0`, `-∞`, `+∞` and `NaN` (not a number) values) + ## Code examples - See the [bitsToFloat.js](bitsToFloat.js) for the example of how to convert array of bits to the floating point number (the example is a bit artificial but still it gives the overview of how the conversion is going on) From 2ad5617a4bcf05a59c650d91ab950fd054d0f69a Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Mon, 26 Jul 2021 07:44:21 +0200 Subject: [PATCH 004/168] Binary representation of the floating-point numbers. --- src/algorithms/math/binary-floating-point/README.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/src/algorithms/math/binary-floating-point/README.md b/src/algorithms/math/binary-floating-point/README.md index c0c6fce046..0952d99f5b 100644 --- a/src/algorithms/math/binary-floating-point/README.md +++ b/src/algorithms/math/binary-floating-point/README.md @@ -79,6 +79,13 @@ I've tried to describe the logic behind the converting of floating-point numbers > Checkout the [interactive version of this diagram](https://trekhleb.dev/blog/2021/binary-floating-point/) to play around with setting bits on and off, and seeing how it would influence the final result +Here is the number ranges that different floating-point formats support: + +| Floating-point format | Exp min | Exp max | Range | Min positive | +| :-------------------- | :------ | :------ | :--------------- | :----------- | +| Half-precision | −14 | +15 | ±65,504 | 6.10 × 10⁻⁵ | +| Single-precision | −126 | +127 | ±3.4028235 × 10³⁸| 1.18 × 10⁻³⁸ | + Be aware that this is by no means a complete and sufficient overview of the IEEE 754 standard. It is rather a simplified and basic overview. Several corner cases were omitted in the examples above for simplicity of presentation (i.e. `-0`, `-∞`, `+∞` and `NaN` (not a number) values) ## Code examples From 4548296affb227c29ead868309e48667f8280c55 Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Mon, 26 Jul 2021 11:55:40 +0200 Subject: [PATCH 005/168] Binary representation of the floating-point numbers. --- src/algorithms/math/binary-floating-point/README.md | 1 + 1 file changed, 1 insertion(+) diff --git a/src/algorithms/math/binary-floating-point/README.md b/src/algorithms/math/binary-floating-point/README.md index 0952d99f5b..d1ae72731e 100644 --- a/src/algorithms/math/binary-floating-point/README.md +++ b/src/algorithms/math/binary-floating-point/README.md @@ -97,6 +97,7 @@ Be aware that this is by no means a complete and sufficient overview of the IEEE You might also want to check out the following resources to get a deeper understanding of the binary representation of floating-point numbers: +- [Interactive version of this article](https://trekhleb.dev/blog/2021/binary-floating-point/) (allows setting the bits manually and seeing the resulting floating number) - [Here is what you need to know about JavaScript’s Number type](https://indepth.dev/posts/1139/here-is-what-you-need-to-know-about-javascripts-number-type) - [Float Exposed](https://float.exposed/) - [IEEE754 Visualization](https://bartaz.github.io/ieee754-visualization/) From d0576a2594f843cae01abfdb9a2aa08663bdee54 Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Mon, 18 Oct 2021 18:02:40 +0200 Subject: [PATCH 006/168] Upgrade dependencies. (#790) --- package-lock.json | 6881 +++++++++++++++++---------------------------- package.json | 20 +- 2 files changed, 2597 insertions(+), 4304 deletions(-) diff --git a/package-lock.json b/package-lock.json index 93de7f66c9..ea195beb9b 100644 --- a/package-lock.json +++ b/package-lock.json @@ -5,53 +5,52 @@ "requires": true, "dependencies": { "@babel/cli": { - "version": "7.12.10", - "resolved": "https://registry.npmjs.org/@babel/cli/-/cli-7.12.10.tgz", - "integrity": "sha512-+y4ZnePpvWs1fc/LhZRTHkTesbXkyBYuOB+5CyodZqrEuETXi3zOVfpAQIdgC3lXbHLTDG9dQosxR9BhvLKDLQ==", + "version": "7.15.7", + "resolved": "https://registry.npmjs.org/@babel/cli/-/cli-7.15.7.tgz", + "integrity": "sha512-YW5wOprO2LzMjoWZ5ZG6jfbY9JnkDxuHDwvnrThnuYtByorova/I0HNXJedrUfwuXFQfYOjcqDA4PU3qlZGZjg==", "dev": true, "requires": { - "@nicolo-ribaudo/chokidar-2": "2.1.8-no-fsevents", + "@nicolo-ribaudo/chokidar-2": "2.1.8-no-fsevents.3", "chokidar": "^3.4.0", "commander": "^4.0.1", "convert-source-map": "^1.1.0", "fs-readdir-recursive": "^1.1.0", "glob": "^7.0.0", - 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"@babel/cli": "7.12.10", - "@babel/preset-env": "7.12.11", - "@types/jest": "26.0.19", - "canvas": "^2.7.0", - "eslint": "7.16.0", + "@babel/cli": "7.15.7", + "@babel/preset-env": "7.15.8", + "@types/jest": "27.0.2", + "canvas": "2.8.0", + "eslint": "8.0.1", "eslint-config-airbnb": "18.2.1", - "eslint-plugin-import": "2.22.1", - "eslint-plugin-jest": "24.1.3", + "eslint-plugin-import": "2.25.2", + "eslint-plugin-jest": "25.2.2", "eslint-plugin-jsx-a11y": "6.4.1", - "eslint-plugin-react": "7.21.5", - "husky": "6.0.0", - "jest": "26.6.3" + "eslint-plugin-react": "7.26.1", + "husky": "7.0.2", + "jest": "27.3.0" }, "engines": { "node": ">=12.0.0", From 99945f30deba9fb0ec5077e07ed093d0a531f1c1 Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Fri, 22 Oct 2021 09:56:44 +0200 Subject: [PATCH 007/168] Add the "Weighted Random" algorithm (#792) * Add the link to the Weighted Random algorithm to the main README. * Add Weighted Random implementation and tests. * Add Weighted Random README. * Add Weighted Random README. * Add Weighted Random README. --- README.md | 2 + .../statistics/weighted-random/README.md | 141 ++++++++++++++++++ .../__test__/weightedRandom.test.js | 64 ++++++++ .../weighted-random/weightedRandom.js | 52 +++++++ 4 files changed, 259 insertions(+) create mode 100644 src/algorithms/statistics/weighted-random/README.md create mode 100644 src/algorithms/statistics/weighted-random/__test__/weightedRandom.test.js create mode 100644 src/algorithms/statistics/weighted-random/weightedRandom.js diff --git a/README.md b/README.md index d00945e5a5..826a3a45b8 100644 --- a/README.md +++ b/README.md @@ -156,6 +156,8 @@ a set of rules that precisely define a sequence of operations. * `B` [k-Means](src/algorithms/ml/k-means) - k-Means clustering algorithm * **Image Processing** * `B` [Seam Carving](src/algorithms/image-processing/seam-carving) - content-aware image resizing algorithm +* **Statistics** + * `B` [Weighted Random](src/algorithms/statistics/weighted-random) - select the random item from the list based on items' weights * **Uncategorized** * `B` [Tower of Hanoi](src/algorithms/uncategorized/hanoi-tower) * `B` [Square Matrix Rotation](src/algorithms/uncategorized/square-matrix-rotation) - in-place algorithm diff --git a/src/algorithms/statistics/weighted-random/README.md b/src/algorithms/statistics/weighted-random/README.md new file mode 100644 index 0000000000..1c700d41d6 --- /dev/null +++ b/src/algorithms/statistics/weighted-random/README.md @@ -0,0 +1,141 @@ +# Weighted Random + +## What is "Weighted Random" + +Let's say you have a list of **items**. Item could be anything. For example, we may have a list of fruits and vegetables that you like to eat: `[ '🍌', '🍎', '🥕' ]`. + +The list of **weights** represent the weight (or probability, or importance) of each item. Weights are numbers. For example, the weights like `[3, 7, 1]` would say that: + +- you would like to eat `🍎 apples` more often (`7` out of `3 + 7 + 1 = 11` times), +- then you would like to eat `bananas 🍌` less often (only `3` out of `11` times), +- and the `carrots 🥕` you really don't like (want to eat it only `1` out of `11` times). + +> If we speak in terms of probabilities than the weights list might be an array of floats that sum up to `1` (i.e. `[0.1, 0.5, 0.2, 0.2]`). + +The **Weighted Random** in this case will be the function that will randomly return you the item from the list, and it will take each item's weight into account, so that items with the higher weight will be picked more often. + +Example of the function interface: + +```javascript +const items = [ '🍌', '🍎', '🥕' ]; +const weights = [ 3, 7, 1 ]; + +function weightedRandom(items, weights) { + // implementation goes here ... +} + +const nextSnackToEat = weightedRandom(items, weights); // Could be '🍎' +``` + +## Applications of Weighted Random + +- In [Genetic Algorithm](https://en.wikipedia.org/wiki/Genetic_algorithm) the weighted random is used during the "Selection" phase, when we need to select the fittest/strongest individuums based on their fitness score for mating and for producing the next stronger generation. You may find an **example** in the [Self-Parking Car in 500 Lines of Code](https://trekhleb.dev/blog/2021/self-parking-car-evolution/) article. +- In [Recurrent Neural Networks (RNN)](https://en.wikipedia.org/wiki/Recurrent_neural_network) when trying to decide what letter to choose next (to form the sentence) based on the next letter probability. You may find an **example** in the [Recipe Generation using Recurrent Neural Network (RNN)](https://nbviewer.org/github/trekhleb/machine-learning-experiments/blob/master/experiments/recipe_generation_rnn/recipe_generation_rnn.ipynb) Jupyter notebook. +- In [Nginx Load Balancing](https://docs.nginx.com/nginx/admin-guide/load-balancer/http-load-balancer/) to send HTTP requests more often to the servers with the higher weights. +- And more... + +## The Algorithm + +The **straightforward approach** would be to: + +1. Repeat each item in the list according to its weight. +2. Pick the random item from the list. + +For example in our case with fruits and vegetables we could generate the following list of size `3 + 7 + 1 = 11`: + +```javascript +const items = [ '🍌', '🍎', '🥕' ]; +const weights = [ 3, 7, 1 ]; + +// Repeating the items based on weights. +const weightedItems = [ + '🍌', '🍌', '🍌', + '🍎', '🍎', '🍎', '🍎', '🍎', '🍎', '🍎', + '🥕', +]; + +// And now just pick the random item from weightedItems array. +``` + +However, as you may see, this approach may require a lot of memory, in case if the objects are heavy, and in case if we have a lot of them to repeat in `weightedItems` list. + +The **more efficient approach** would be to: + +1. Prepare the list of cumulative weights for each item (i.e. the `cumulativeWeights` list which will have the same number of elements as the original `weights` list). In our case it will look like this: `cumulativeWeights = [3, 3 + 7, 3 + 7 + 1] = [3, 10, 11]` +2. Generate the random number `randomNumber` from `0` to the highest cumulative weight value. In our case the random number will be in a range of `[0..11]`. Let's say that we have `randomNumber = 8`. +3. Go through the `cumulativeWeights` list from left to right and pick the first element which is higher or equal to the `randomNumber`. The index of such element we will use to pick the item from the `items` array. + +The idea behind this approach is that the higher weights will "occupy" more numeric space. Therefore, there is a higher chance that the random number will fall into the "higher weight numeric bucket". + +```javascript +const weights = [3, 7, 1 ]; +const cumulativeWeights = [3, 10, 11]; + +// In a pseudo-representation we may think about the cumulativeWeights array like this. +const pseudoCumulativeWeights = [ + 1, 2, 3, // <-- [3] numbers + 4, 5, 6, 7, 8, 9, 10, // <-- [7] numbers + 11, // <-- [1] number +]; +``` + +Here is an example of how the `weightedRandom` function might be implemented: + +```javascript +/** + * Picks the random item based on its weight. + * The items with higher weight will be picked more often (with a higher probability). + * + * For example: + * - items = ['banana', 'orange', 'apple'] + * - weights = [0, 0.2, 0.8] + * - weightedRandom(items, weights) in 80% of cases will return 'apple', in 20% of cases will return + * 'orange' and it will never return 'banana' (because probability of picking the banana is 0%) + * + * @param {any[]} items + * @param {number[]} weights + * @returns {{item: any, index: number}} + */ +export default function weightedRandom(items, weights) { + if (items.length !== weights.length) { + throw new Error('Items and weights must be of the same size'); + } + + if (!items.length) { + throw new Error('Items must not be empty'); + } + + // Preparing the cumulative weights array. + // For example: + // - weights = [1, 4, 3] + // - cumulativeWeights = [1, 5, 8] + const cumulativeWeights = []; + for (let i = 0; i < weights.length; i += 1) { + cumulativeWeights[i] = weights[i] + (cumulativeWeights[i - 1] || 0); + } + + // Getting the random number in a range of [0...sum(weights)] + // For example: + // - weights = [1, 4, 3] + // - maxCumulativeWeight = 8 + // - range for the random number is [0...8] + const maxCumulativeWeight = cumulativeWeights[cumulativeWeights.length - 1]; + const randomNumber = maxCumulativeWeight * Math.random(); + + // Picking the random item based on its weight. + // The items with higher weight will be picked more often. + for (let itemIndex = 0; itemIndex < items.length; itemIndex += 1) { + if (cumulativeWeights[itemIndex] >= randomNumber) { + return { + item: items[itemIndex], + index: itemIndex, + }; + } + } +} +``` + +## Implementation + +- Check the [weightedRandom.js](weightedRandom.js) file for the implementation of the `weightedRandom()` function. +- Check the [weightedRandom.test.js](__test__/weightedRandom.test.js) file for the tests-cases. diff --git a/src/algorithms/statistics/weighted-random/__test__/weightedRandom.test.js b/src/algorithms/statistics/weighted-random/__test__/weightedRandom.test.js new file mode 100644 index 0000000000..1b0b267627 --- /dev/null +++ b/src/algorithms/statistics/weighted-random/__test__/weightedRandom.test.js @@ -0,0 +1,64 @@ +import weightedRandom from '../weightedRandom'; + +describe('weightedRandom', () => { + it('should throw an error when the number of weights does not match the number of items', () => { + const getWeightedRandomWithInvalidInputs = () => { + weightedRandom(['a', 'b', 'c'], [10, 0]); + }; + expect(getWeightedRandomWithInvalidInputs).toThrow('Items and weights must be of the same size'); + }); + + it('should throw an error when the number of weights or items are empty', () => { + const getWeightedRandomWithInvalidInputs = () => { + weightedRandom([], []); + }; + expect(getWeightedRandomWithInvalidInputs).toThrow('Items must not be empty'); + }); + + it('should correctly do random selection based on wights in straightforward cases', () => { + expect(weightedRandom(['a', 'b', 'c'], [1, 0, 0])).toEqual({ index: 0, item: 'a' }); + expect(weightedRandom(['a', 'b', 'c'], [0, 1, 0])).toEqual({ index: 1, item: 'b' }); + expect(weightedRandom(['a', 'b', 'c'], [0, 0, 1])).toEqual({ index: 2, item: 'c' }); + expect(weightedRandom(['a', 'b', 'c'], [0, 1, 1])).not.toEqual({ index: 0, item: 'a' }); + expect(weightedRandom(['a', 'b', 'c'], [1, 0, 1])).not.toEqual({ index: 1, item: 'b' }); + expect(weightedRandom(['a', 'b', 'c'], [1, 1, 0])).not.toEqual({ index: 2, item: 'c' }); + }); + + it('should correctly do random selection based on wights', () => { + // Number of times we're going to select the random items based on their weights. + const ATTEMPTS_NUM = 1000; + // The +/- delta in the number of times each item has been actually selected. + // I.e. if we want the item 'a' to be selected 300 times out of 1000 cases (30%) + // then 267 times is acceptable since it is bigger that 250 (which is 300 - 50) + // ans smaller than 350 (which is 300 + 50) + const THRESHOLD = 50; + + const items = ['a', 'b', 'c']; // The actual items values don't matter. + const weights = [0.1, 0.3, 0.6]; + + const counter = []; + for (let i = 0; i < ATTEMPTS_NUM; i += 1) { + const randomItem = weightedRandom(items, weights); + if (!counter[randomItem.index]) { + counter[randomItem.index] = 1; + } else { + counter[randomItem.index] += 1; + } + } + + for (let itemIndex = 0; itemIndex < items.length; itemIndex += 1) { + /* + i.e. item with the index of 0 must be selected 100 times (ideally) + or with the threshold of [100 - 50, 100 + 50] times. + + i.e. item with the index of 1 must be selected 300 times (ideally) + or with the threshold of [300 - 50, 300 + 50] times. + + i.e. item with the index of 2 must be selected 600 times (ideally) + or with the threshold of [600 - 50, 600 + 50] times. + */ + expect(counter[itemIndex]).toBeGreaterThan(ATTEMPTS_NUM * weights[itemIndex] - THRESHOLD); + expect(counter[itemIndex]).toBeLessThan(ATTEMPTS_NUM * weights[itemIndex] + THRESHOLD); + } + }); +}); diff --git a/src/algorithms/statistics/weighted-random/weightedRandom.js b/src/algorithms/statistics/weighted-random/weightedRandom.js new file mode 100644 index 0000000000..0d58a504f8 --- /dev/null +++ b/src/algorithms/statistics/weighted-random/weightedRandom.js @@ -0,0 +1,52 @@ +/** + * Picks the random item based on its weight. + * The items with higher weight will be picked more often (with a higher probability). + * + * For example: + * - items = ['banana', 'orange', 'apple'] + * - weights = [0, 0.2, 0.8] + * - weightedRandom(items, weights) in 80% of cases will return 'apple', in 20% of cases will return + * 'orange' and it will never return 'banana' (because probability of picking the banana is 0%) + * + * @param {any[]} items + * @param {number[]} weights + * @returns {{item: any, index: number}} + */ +/* eslint-disable consistent-return */ +export default function weightedRandom(items, weights) { + if (items.length !== weights.length) { + throw new Error('Items and weights must be of the same size'); + } + + if (!items.length) { + throw new Error('Items must not be empty'); + } + + // Preparing the cumulative weights array. + // For example: + // - weights = [1, 4, 3] + // - cumulativeWeights = [1, 5, 8] + const cumulativeWeights = []; + for (let i = 0; i < weights.length; i += 1) { + cumulativeWeights[i] = weights[i] + (cumulativeWeights[i - 1] || 0); + } + + // Getting the random number in a range of [0...sum(weights)] + // For example: + // - weights = [1, 4, 3] + // - maxCumulativeWeight = 8 + // - range for the random number is [0...8] + const maxCumulativeWeight = cumulativeWeights[cumulativeWeights.length - 1]; + const randomNumber = maxCumulativeWeight * Math.random(); + + // Picking the random item based on its weight. + // The items with higher weight will be picked more often. + for (let itemIndex = 0; itemIndex < items.length; itemIndex += 1) { + if (cumulativeWeights[itemIndex] >= randomNumber) { + return { + item: items[itemIndex], + index: itemIndex, + }; + } + } +} From 17bfb551c0d1d38de5ff3ec4ef9c587678ce6ea9 Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Fri, 22 Oct 2021 10:16:37 +0200 Subject: [PATCH 008/168] Add Weighted Random cover. --- .../statistics/weighted-random/README.md | 2 ++ .../weighted-random/images/cover.png | Bin 0 -> 303430 bytes 2 files changed, 2 insertions(+) create mode 100644 src/algorithms/statistics/weighted-random/images/cover.png diff --git a/src/algorithms/statistics/weighted-random/README.md b/src/algorithms/statistics/weighted-random/README.md index 1c700d41d6..8b9c8890d7 100644 --- a/src/algorithms/statistics/weighted-random/README.md +++ b/src/algorithms/statistics/weighted-random/README.md @@ -1,5 +1,7 @@ # Weighted Random +![Weighted Random](images/cover.png) + ## What is "Weighted Random" Let's say you have a list of **items**. Item could be anything. 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