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Add Polynomial Hash function.
This commit is contained in:
Oleksii Trekhleb
parent
98a44ea832
commit
d5be477bd8
@ -1,12 +1,14 @@
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const DEFAULT_PRIME = 37;
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const DEFAULT_BASE = 37;
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const DEFAULT_MODULUS = 101;
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export default class PolynomialHash {
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/**
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* @param {number} [prime] - A prime number used to create the hash representation of a word.
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* @param {number} [base] - Base number that is used to create the polynomial.
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* @param {number} [modulus] - Modulus number that keeps the hash from overflowing.
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*/
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constructor(prime = DEFAULT_PRIME) {
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this.prime = prime;
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this.primeModulus = 101;
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constructor({ base = DEFAULT_BASE, modulus = DEFAULT_MODULUS } = {}) {
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this.base = base;
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this.modulus = modulus;
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}
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/**
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@ -18,10 +20,15 @@ export default class PolynomialHash {
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* @return {number}
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*/
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hash(word) {
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const charCodes = Array.from(word).map(char => this.charToNumber(char));
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let hash = 0;
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for (let charIndex = 0; charIndex < word.length; charIndex += 1) {
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hash += word.charCodeAt(charIndex) * (this.prime ** charIndex);
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for (let charIndex = 0; charIndex < charCodes.length; charIndex += 1) {
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hash *= this.base;
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hash %= this.modulus;
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hash += charCodes[charIndex] % this.modulus;
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hash %= this.modulus;
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}
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return hash;
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@ -42,12 +49,45 @@ export default class PolynomialHash {
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* @return {number}
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*/
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roll(prevHash, prevWord, newWord) {
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const newWordLastIndex = newWord.length - 1;
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let hash = prevHash;
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let hash = prevHash - prevWord.charCodeAt(0);
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hash /= this.prime;
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hash += newWord.charCodeAt(newWordLastIndex) * (this.prime ** newWordLastIndex);
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const prevValue = this.charToNumber(prevWord[0]);
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const newValue = this.charToNumber(newWord[newWord.length - 1]);
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let prevValueMultiplier = 1;
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for (let i = 1; i < prevWord.length; i += 1) {
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prevValueMultiplier *= this.base;
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prevValueMultiplier %= this.modulus;
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}
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hash += this.modulus;
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hash -= (prevValue * prevValueMultiplier) % this.modulus;
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hash %= this.modulus;
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hash *= this.base;
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hash %= this.modulus;
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hash += newValue % this.modulus;
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hash %= this.modulus;
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return hash;
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}
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/**
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* Converts char to number.
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*
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* @param {string} char
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* @return {number}
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*/
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charToNumber(char) {
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let charCode = char.codePointAt(0);
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// Check if character has surrogate pair.
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const surrogate = char.codePointAt(1);
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if (surrogate !== undefined) {
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const surrogateShift = 2 ** 16;
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charCode += surrogate * surrogateShift;
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}
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return charCode;
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}
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}
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@ -37,23 +37,80 @@ The *Rabin–Karp string search algorithm* is often explained using a very simpl
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rolling hash function that only uses multiplications and
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additions - **polynomial rolling hash**:
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> H(s<sub>0</sub>, s<sub>1</sub>, ..., s<sub>k</sub>) = (s<sub>0</sub> * p<sup>0</sup> + s<sub>1</sub> * p<sup>1</sup> + ... + s<sub>k</sub> * p<sup>k</sup>) mod M
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> H(s<sub>0</sub>, s<sub>1</sub>, ..., s<sub>k</sub>) = s<sub>0</sub> * p<sup>k-1</sup> + s<sub>1</sub> * p<sup>k-2</sup> + ... + s<sub>k</sub> * p<sup>0</sup>
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where `p` is a constant, and *(s<sub>1</sub>, ... , s<sub>k</sub>)* are the input
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characters.
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A careful choice of the parameters `M`, `p` is important to obtain “good”
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properties of the hash function, i.e., low collision rate.
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For example we can convert short strings to key numbers by multiplying digit codes by
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powers of a constant. The three letter word `ace` could turn into a number
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by calculating:
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> key = 1 * 26<sup>2</sup> + 3 * 26<sup>1</sup> + 5 * 26<sup>0</sup>
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In order to avoid manipulating huge `H` values, all math is done modulo `M`.
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Removing and adding characters simply involves adding or subtracting the first or
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last term. Shifting all characters by one position to the right requires multiplying
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the entire sum `H` by `a`. Shifting all characters by one position to the left
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requires dividing the entire sum `H` by `a`.
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> H(s<sub>0</sub>, s<sub>1</sub>, ..., s<sub>k</sub>) = (s<sub>0</sub> * p<sup>k-1</sup> + s<sub>1</sub> * p<sup>k-2</sup> + ... + s<sub>k</sub> * p<sup>0</sup>) mod M
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A careful choice of the parameters `M`, `p` is important to obtain “good”
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properties of the hash function, i.e., low collision rate.
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This approach has the desirable attribute of involving all the characters in the
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input string. The calculated key value can then be hashed into an array index in
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the usual way:
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```javascript
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function hash(key, arraySize) {
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const base = 13;
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let hash = 0;
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for (let charIndex = 0; charIndex < key.length; charIndex += 1) {
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const charCode = key.charCodeAt(charIndex);
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hash += charCode * (base ** (key.length - charIndex - 1));
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}
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return hash % arraySize;
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}
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```
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The `hash()` method is not as efficient as it might be. Other than the
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character conversion, there are two multiplications and an addition inside
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the loop. We can eliminate one multiplication by using **Horner's method*:
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> a<sub>4</sub> * x<sup>4</sup> + a<sub>3</sub> * x<sup>3</sup> + a<sub>2</sub> * x<sup>2</sup> + a<sub>1</sub> * x<sup>1</sup> + a<sub>0</sub> = (((a<sub>4</sub> * x + a<sub>3</sub>) * x + a<sub>2</sub>) * x + a<sub>1</sub>) * x + a<sub>0</sub>
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In other words:
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> H<sub>i</sub> = (P * H<sub>i-1</sub> + S<sub>i</sub>) mod M
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The `hash()` cannot handle long strings because the hashVal exceeds the size of
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int. Notice that the key always ends up being less than the array size.
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In Horner's method we can apply the modulo (%) operator at each step in the
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calculation. This gives the same result as applying the modulo operator once at
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the end, but avoids the overflow.
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```javascript
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function hash(key, arraySize) {
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const base = 13;
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let hash = 0;
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for (let charIndex = 0; charIndex < key.length; charIndex += 1) {
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const charCode = key.charCodeAt(charIndex);
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hash = (hash * base + charCode) % arraySize;
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}
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return hash;
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}
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```
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Polynomial hashing has a rolling property: the fingerprints can be updated
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efficiently when symbols are added or removed at the ends of the string
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(provided that an array of powers of p modulo M of sufficient length is stored).
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The popular Rabin–Karp pattern matching algorithm is based on this property
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## References
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- [Where to Use Polynomial String Hashing](https://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf)
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- [Hashing on uTexas](https://www.cs.utexas.edu/~mitra/csSpring2017/cs313/lectures/hash.html)
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- [Hash Function on Wikipedia](https://en.wikipedia.org/wiki/Hash_function)
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- [Rolling Hash on Wikipedia](https://en.wikipedia.org/wiki/Rolling_hash)
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@ -2,22 +2,22 @@ import PolynomialHash from '../PolynomialHash';
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describe('PolynomialHash', () => {
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it('should calculate new hash based on previous one', () => {
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// const primes = [3, 79, 101, 3251, 13229, 122743, 3583213];
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// const frameSizes = [5, 20];
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const primes = [3];
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const frameSizes = [20];
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const bases = [3, 79, 101, 3251, 13229, 122743, 3583213];
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const mods = [79, 101];
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const frameSizes = [5, 20];
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// @TODO: Provide Unicode support.
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const text = 'Lorem Ipsum is simply dummy text of the printing and '
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+ 'typesetting industry. Lorem Ipsum has been the industry\'s standard '
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+ 'galley of type and \u{ffff} scrambled it to make a type specimen book. It '
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+ 'electronic 耀 typesetting, remaining essentially unchanged. It was '
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+ 'popularised in the \u{20005} \u{20000}1960s with the release of Letraset sheets '
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// + 'popularised in the \u{20005} \u{20000}1960s with the release of Letraset sheets '
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+ 'publishing software like Aldus PageMaker 耀 including versions of Lorem.';
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// Check hashing for different prime base.
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primes.forEach((prime) => {
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const polynomialHash = new PolynomialHash(prime);
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bases.forEach((base) => {
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mods.forEach((modulus) => {
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const polynomialHash = new PolynomialHash({ base, modulus });
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// Check hashing for different word lengths.
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frameSizes.forEach((frameSize) => {
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@ -39,65 +39,21 @@ describe('PolynomialHash', () => {
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});
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});
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});
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});
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// it('should calculate new hash based on previous one', () => {
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// const polynomialHash = new PolynomialHash();
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//
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// const wordLength = 3;
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// const string = 'Hello World!';
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//
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// const word1 = string.substr(0, wordLength);
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// const word2 = string.substr(1, wordLength);
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// const word3 = string.substr(2, wordLength);
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// const word4 = string.substr(3, wordLength);
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//
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// const directHash1 = polynomialHash.hash(word1);
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// const directHash2 = polynomialHash.hash(word2);
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// const directHash3 = polynomialHash.hash(word3);
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// const directHash4 = polynomialHash.hash(word4);
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//
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// const rollingHash2 = polynomialHash.roll(directHash1, word1, word2);
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// const rollingHash3 = polynomialHash.roll(directHash2, word2, word3);
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// const rollingHash4 = polynomialHash.roll(directHash3, word3, word4);
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//
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// expect(directHash1).toBe(151661);
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// expect(directHash2).toBe(151949);
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// expect(directHash3).toBe(156063);
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// expect(directHash4).toBe(48023);
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//
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// expect(rollingHash2).toBe(directHash2);
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// expect(rollingHash3).toBe(directHash3);
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// expect(rollingHash4).toBe(directHash4);
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// });
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//
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// it('should calculate new hash based on previous one with 3 as a primeModulus', () => {
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// const PRIME = 3;
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// const polynomialHash = new PolynomialHash(PRIME);
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//
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// const wordLength = 3;
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// const string = 'Hello World!';
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//
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// const word1 = string.substr(0, wordLength);
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// const word2 = string.substr(1, wordLength);
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// const word3 = string.substr(2, wordLength);
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// const word4 = string.substr(3, wordLength);
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//
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// const directHash1 = polynomialHash.hash(word1);
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// const directHash2 = polynomialHash.hash(word2);
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// const directHash3 = polynomialHash.hash(word3);
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// const directHash4 = polynomialHash.hash(word4);
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//
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// const rollingHash2 = polynomialHash.roll(directHash1, word1, word2);
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// const rollingHash3 = polynomialHash.roll(directHash2, word2, word3);
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// const rollingHash4 = polynomialHash.roll(directHash3, word3, word4);
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//
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// expect(directHash1).toBe(1347);
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// expect(directHash2).toBe(1397);
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// expect(directHash3).toBe(1431);
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// expect(directHash4).toBe(729);
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//
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// expect(rollingHash2).toBe(directHash2);
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// expect(rollingHash3).toBe(directHash3);
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// expect(rollingHash4).toBe(directHash4);
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// });
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it('should generate numeric hashed less than 100', () => {
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const polynomialHash = new PolynomialHash({ modulus: 100 });
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expect(polynomialHash.hash('Some long text that is used as a key')).toBe(41);
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expect(polynomialHash.hash('Test')).toBe(92);
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expect(polynomialHash.hash('a')).toBe(97);
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expect(polynomialHash.hash('b')).toBe(98);
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expect(polynomialHash.hash('c')).toBe(99);
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expect(polynomialHash.hash('d')).toBe(0);
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expect(polynomialHash.hash('e')).toBe(1);
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expect(polynomialHash.hash('ab')).toBe(87);
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// @TODO: Provide Unicode support.
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expect(polynomialHash.hash('\u{20000}')).toBe(92);
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});
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});
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@ -37,7 +37,7 @@ describe('rabinKarp', () => {
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it('should work with UTF symbols', () => {
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expect(rabinKarp('a\u{ffff}', '\u{ffff}')).toBe(1);
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expect(rabinKarp('\u0000耀\u0000', '耀\u0000')).toBe(1);
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expect(rabinKarp('a\u{20000}', '\u{20000}')).toBe(1);
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expect(rabinKarp('ab\u{20005}a', '\u{20005}a')).toBe(2);
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// @TODO: Provide Unicode support.
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// expect(rabinKarp('a\u{20000}', '\u{20000}')).toBe(1);
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});
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});
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@ -1,51 +0,0 @@
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/**
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* Generates fingerprints using Rabin scheme with x = 2 (for potential compiler optimizations).
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* Guaranteed not to over or underflow if function assumptions are met.
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*/
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export default class RabinFingerprint {
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/**
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* @param { function() : number } [primeGenerator]
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* @assumes Output from any function call is prime less than Number.MAX_SAFE_INTEGER / 2.
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*/
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constructor(primeGenerator) {
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this.prime = primeGenerator();
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}
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/**
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* @param { array[number] } [values]
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* @returns {number} - The hash value after digesting input.
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* @assumes All array elements are non-negative.
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* @note First element in array is considered to be oldest value.
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*/
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init(values) {
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this.val = 0;
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this.len = values.length;
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for (let i = 0; i < values.length; i += 1) {
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this.val = (((this.val * 2) % this.prime) + (values[i] % this.prime)) % this.prime;
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}
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return this.val;
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}
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/*
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* @param {number} [oldValue]
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* @param {number} [newValue]
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* @returns {number} - The hash value after removing the oldest value & inserting the newest.
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* @assumes Instance has already been initialized.
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* @assumes oldValue is the oldest value still processed by the hash.
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* @assumes newValue is non-negative.
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*/
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roll(oldValue, newValue) {
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let oldVal = oldValue % this.prime;
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for (let i = 1; i < this.len; i += 1) {
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oldVal = (oldVal * 2) % this.prime;
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}
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this.val = (this.val + this.prime - (oldVal % this.prime)) % this.prime;
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const newVal = newValue % this.prime;
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this.val = (((this.val * 2) % this.prime) + (newVal % this.prime)) % this.prime;
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return this.val;
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}
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}
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@ -1,59 +0,0 @@
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import RabinFingerprint from '../Rabin_Fingerprint';
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describe('Rabin fingerprint Hash Family', () => {
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it('should hash deterministically', () => {
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const primeVals = [3, 5, 19, 53, 97, 401, 7039, 193939];
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for (let primeIdx = 0; primeIdx < primeVals.length; primeIdx += 1) {
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const primeVal = primeVals[primeIdx];
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const hasher = new RabinFingerprint(() => primeVal);
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// Test basic values
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expect(hasher.init([])).toEqual(0);
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expect(hasher.init([1])).toEqual(1);
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// Test overflow
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const largeVal = Number.MAX_SAFE_INTEGER;
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expect(hasher.init([primeVal])).toEqual(0);
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expect(hasher.init([largeVal])).toEqual(largeVal % primeVal);
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const numLargeVal = 2; // 2 ^ numLargeVal fits in javascript number
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const largeValues = new Array(numLargeVal).fill(largeVal);
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const expVal = ((largeVal % primeVal) * ((2 ** numLargeVal) - 1)) % primeVal;
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expect(hasher.init(largeValues)).toEqual(expVal);
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// Test using Fermat's little theorem
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const fermatValues = new Array(primeVal).fill(primeVal);
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const numFermatTests = 100;
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for (let i = 0; i < numFermatTests; i += 1) {
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const randomValue = Math.floor(Math.random() * largeVal);
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fermatValues[0] = randomValue;
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expect(hasher.init(fermatValues)).toEqual(randomValue % primeVal);
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}
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}
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});
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it('should roll appropriately', () => {
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const primeVals = [3, 5, 19, 53, 97, 401, 7039, 193939];
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for (let primeIdx = 0; primeIdx < primeVals.length; primeIdx += 1) {
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const primeVal = primeVals[primeIdx];
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const hasher = new RabinFingerprint(() => primeVal);
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// Test basic values
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const largeVal = Number.MAX_SAFE_INTEGER;
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expect(hasher.init([0])).toEqual(0);
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expect(hasher.roll(0, 1)).toEqual(1);
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expect(hasher.roll(1, primeVal)).toEqual(0);
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expect(hasher.roll(primeVal, largeVal)).toEqual(largeVal % primeVal);
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const numRollTest = 100;
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let previousValue = largeVal;
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for (let i = 0; i < numRollTest; i += 1) {
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const randomVal = Math.floor(Math.random() * largeVal);
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expect(hasher.roll(previousValue, randomVal)).toEqual(randomVal % primeVal);
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previousValue = randomVal;
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}
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}
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});
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});
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