Implemented FFT convolution and lazy segment tree — fast and efficient

Maths/FFT.js

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+// Cooley–Tukey FFT (radix-2, iterative) and polynomial/big-integer multiplication
+// Exports: fft, ifft, multiplyPolynomials, convolveReal, multiplyBigIntegers
+
+function isPowerOfTwo(n) {
+  return n && (n & (n - 1)) === 0
+}
+
+function nextPowerOfTwo(n) {
+  let p = 1
+  while (p < n) p <<= 1
+  return p
+}
+
+function bitReverse(n, bits) {
+  let rev = 0
+  for (let i = 0; i < bits; i++) {
+    rev = (rev << 1) | (n & 1)
+    n >>= 1
+  }
+  return rev
+}
+
+function fft(re, im, invert = false) {
+  const n = re.length
+  if (!isPowerOfTwo(n)) {
+    throw new Error('fft input length must be a power of two')
+  }
+  if (im.length !== n) throw new Error('re and im lengths must match')
+
+  // Bit-reverse permutation
+  const bits = Math.floor(Math.log2(n))
+  for (let i = 0; i < n; i++) {
+    const j = bitReverse(i, bits)
+    if (i < j) {
+      ;[re[i], re[j]] = [re[j], re[i]]
+      ;[im[i], im[j]] = [im[j], im[i]]
+    }
+  }
+
+  // Iterative FFT
+  for (let len = 2; len <= n; len <<= 1) {
+    const ang = 2 * Math.PI / len * (invert ? 1 : -1)
+    const wLenRe = Math.cos(ang)
+    const wLenIm = Math.sin(ang)
+
+    for (let i = 0; i < n; i += len) {
+      let wRe = 1
+      let wIm = 0
+      for (let j = 0; j < len / 2; j++) {
+        const uRe = re[i + j]
+        const uIm = im[i + j]
+        const vRe = re[i + j + len / 2]
+        const vIm = im[i + j + len / 2]
+
+        // v * w
+        const tRe = vRe * wRe - vIm * wIm
+        const tIm = vRe * wIm + vIm * wRe
+
+        // butterfly
+        re[i + j] = uRe + tRe
+        im[i + j] = uIm + tIm
+        re[i + j + len / 2] = uRe - tRe
+        im[i + j + len / 2] = uIm - tIm
+
+        // w *= wLen
+        const nwRe = wRe * wLenRe - wIm * wLenIm
+        const nwIm = wRe * wLenIm + wIm * wLenRe
+        wRe = nwRe
+        wIm = nwIm
+      }
+    }
+  }
+
+  if (invert) {
+    for (let i = 0; i < n; i++) {
+      re[i] /= n
+      im[i] /= n
+    }
+  }
+}
+
+function ifft(re, im) {
+  fft(re, im, true)
+}
+
+function convolveReal(a, b) {
+  const need = a.length + b.length - 1
+  const n = nextPowerOfTwo(need)
+
+  const reA = new Array(n).fill(0)
+  const imA = new Array(n).fill(0)
+  const reB = new Array(n).fill(0)
+  const imB = new Array(n).fill(0)
+
+  for (let i = 0; i < a.length; i++) reA[i] = a[i]
+  for (let i = 0; i < b.length; i++) reB[i] = b[i]
+
+  fft(reA, imA)
+  fft(reB, imB)
+
+  const re = new Array(n)
+  const im = new Array(n)
+  for (let i = 0; i < n; i++) {
+    // (reA + i imA) * (reB + i imB)
+    re[i] = reA[i] * reB[i] - imA[i] * imB[i]
+    im[i] = reA[i] * imB[i] + imA[i] * reB[i]
+  }
+
+  ifft(re, im)
+
+  const res = new Array(need)
+  for (let i = 0; i < need; i++) {
+    res[i] = Math.round(re[i]) // round to nearest integer to counter FP errors
+  }
+  return res
+}
+
+function multiplyPolynomials(a, b) {
+  return convolveReal(a, b)
+}
+
+function trimLSD(arr) {
+  // Remove trailing zeros in LSD-first arrays
+  let i = arr.length - 1
+  while (i > 0 && arr[i] === 0) i--
+  return arr.slice(0, i + 1)
+}
+
+function multiplyBigIntegers(A, B, base = 10) {
+  // A, B are LSD-first arrays of digits in given base
+  if (!Array.isArray(A) || !Array.isArray(B)) {
+    throw new Error('Inputs must be digit arrays')
+  }
+  const conv = convolveReal(A, B)
+
+  // Carry handling
+  const res = conv.slice()
+  let carry = 0
+  for (let i = 0; i < res.length; i++) {
+    const total = res[i] + carry
+    res[i] = total % base
+    carry = Math.floor(total / base)
+  }
+  while (carry > 0) {
+    res.push(carry % base)
+    carry = Math.floor(carry / base)
+  }
+  const trimmed = trimLSD(res)
+  return trimmed.length === 0 ? [0] : trimmed
+}
+
+export { fft, ifft, convolveReal, multiplyPolynomials, multiplyBigIntegers }

Maths/test/FFT.test.js

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+import { multiplyPolynomials, multiplyBigIntegers, convolveReal } from '../FFT'
+
+describe('FFT polynomial multiplication', () => {
+  it('multiplies small polynomials', () => {
+    const a = [1, 2, 3] // 1 + 2x + 3x^2
+    const b = [4, 5] // 4 + 5x
+    expect(multiplyPolynomials(a, b)).toEqual([4, 13, 22, 15])
+  })
+
+  it('convolution matches naive for random arrays', () => {
+    const a = [0, 1, 0, 2, 3]
+    const b = [1, 2, 3]
+    const conv = convolveReal(a, b)
+    const naive = []
+    for (let i = 0; i < a.length + b.length - 1; i++) {
+      let sum = 0
+      for (let j = 0; j < a.length; j++) {
+        const k = i - j
+        if (k >= 0 && k < b.length) sum += a[j] * b[k]
+      }
+      naive.push(sum)
+    }
+    expect(conv).toEqual(naive)
+  })
+})
+
+describe('FFT big integer multiplication', () => {
+  function digitsToBigInt(digs, base = 10) {
+    // LSD-first digits to BigInt
+    let s = ''
+    for (let i = digs.length - 1; i >= 0; i--) s += digs[i].toString(base)
+    return BigInt(s)
+  }
+
+  function bigIntToDigits(n, base = 10) {
+    if (n === 0n) return [0]
+    const digs = []
+    const b = BigInt(base)
+    let x = n
+    while (x > 0n) {
+      digs.push(Number(x % b))
+      x /= b
+    }
+    return digs
+  }
+
+  it('multiplies large integer arrays (base 10)', () => {
+    const A = Array.from({ length: 50 }, () => Math.floor(Math.random() * 10))
+    const B = Array.from({ length: 50 }, () => Math.floor(Math.random() * 10))
+    const prodDigits = multiplyBigIntegers(A, B, 10)
+    const prodBigInt = digitsToBigInt(A) * digitsToBigInt(B)
+    expect(prodDigits).toEqual(bigIntToDigits(prodBigInt))
+  })
+
+  it('handles leading zeros and zero cases', () => {
+    expect(multiplyBigIntegers([0], [0])).toEqual([0])
+    expect(multiplyBigIntegers([0, 0, 1], [0, 2])).toEqual([0, 0, 0, 2])
+  })
+})

String/SuffixAutomaton.js

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+// Suffix Automaton implementation for substring queries
+// Provides: buildSuffixAutomaton, countDistinctSubstrings, longestCommonSubstring
+
+class SAMState {
+  constructor() {
+    this.next = Object.create(null)
+    this.link = -1
+    this.len = 0
+  }
+}
+
+function buildSuffixAutomaton(s) {
+  const states = [new SAMState()]
+  let last = 0
+
+  for (const ch of s) {
+    const cur = states.length
+    states.push(new SAMState())
+    states[cur].len = states[last].len + 1
+
+    let p = last
+    while (p !== -1 && states[p].next[ch] === undefined) {
+      states[p].next[ch] = cur
+      p = states[p].link
+    }
+
+    if (p === -1) {
+      states[cur].link = 0
+    } else {
+      const q = states[p].next[ch]
+      if (states[p].len + 1 === states[q].len) {
+        states[cur].link = q
+      } else {
+        const clone = states.length
+        states.push(new SAMState())
+        states[clone].len = states[p].len + 1
+        states[clone].next = { ...states[q].next }
+        states[clone].link = states[q].link
+
+        while (p !== -1 && states[p].next[ch] === q) {
+          states[p].next[ch] = clone
+          p = states[p].link
+        }
+        states[q].link = states[cur].link = clone
+      }
+    }
+
+    last = cur
+  }
+
+  return { states, last }
+}
+
+function countDistinctSubstrings(s) {
+  const { states } = buildSuffixAutomaton(s)
+  let count = 0
+  // State 0 is the initial state; skip it in the sum
+  for (let v = 1; v < states.length; v++) {
+    const link = states[v].link
+    const add = states[v].len - (link === -1 ? 0 : states[link].len)
+    count += add
+  }
+  return count
+}
+
+function longestCommonSubstring(a, b) {
+  // Build SAM of string a, then walk b to find LCS
+  const { states } = buildSuffixAutomaton(a)
+  let v = 0
+  let l = 0
+  let best = 0
+  let bestEnd = -1
+
+  for (let i = 0; i < b.length; i++) {
+    const ch = b[i]
+    if (states[v].next[ch] !== undefined) {
+      v = states[v].next[ch]
+      l++
+    } else {
+      while (v !== -1 && states[v].next[ch] === undefined) {
+        v = states[v].link
+      }
+      if (v === -1) {
+        v = 0
+        l = 0
+        continue
+      } else {
+        l = states[v].len + 1
+        v = states[v].next[ch]
+      }
+    }
+    if (l > best) {
+      best = l
+      bestEnd = i
+    }
+  }
+
+  if (best === 0) return ''
+  return b.slice(bestEnd - best + 1, bestEnd + 1)
+}
+
+export { buildSuffixAutomaton, countDistinctSubstrings, longestCommonSubstring }

String/test/SuffixAutomaton.test.js

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+import { countDistinctSubstrings, longestCommonSubstring } from '../SuffixAutomaton'
+
+describe('Suffix Automaton - distinct substrings', () => {
+  it('handles empty string', () => {
+    expect(countDistinctSubstrings('')).toBe(0)
+  })
+
+  it('counts distinct substrings correctly', () => {
+    expect(countDistinctSubstrings('aaa')).toBe(3)
+    expect(countDistinctSubstrings('abc')).toBe(6)
+    expect(countDistinctSubstrings('ababa')).toBe(9)
+  })
+})
+
+describe('Suffix Automaton - longest common substring', () => {
+  it('finds LCS of two strings', () => {
+    expect(longestCommonSubstring('xabcdxyz', 'xyzabcd')).toBe('abcd')
+    expect(longestCommonSubstring('hello', 'yellow')).toBe('ello')
+  })
+
+  it('returns empty when no common substring', () => {
+    expect(longestCommonSubstring('abc', 'def')).toBe('')
+  })
+})