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Problem solution · Java

Multiply Two Polynomials

Multiply Two Polynomials: a Java solution using direct simulation. Learn the idea, check the complexity, and read the full code, with credit to walkccc LeetCode Solutions.

Technique
Direct simulation
Source
walkccc LeetCode Solutions
Length
117 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Direct simulation

For Multiply Two Polynomials, the implementation follows the problem’s operations directly while maintaining only the state needed for the next decision.

  1. Translate each rule into one explicit state update.
  2. Maintain the invariant after every processed item.
  3. Return the accumulated state once all relevant input has been handled.

Code notes

  • 117 lines of Java from the credited upstream file 3549.java.
  • The implementation visibly relies on sequence storage.
  • 11 loop blocks detected.

Complexity

Count the number and nesting of passes over the input, then include the maintained containers in the memory estimate.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from walkccc LeetCode Solutions by P.-Y. Chen (walkccc) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMultiply Two Polynomials · JavaJava
Use this to learn the idea, then write your own version.
class Complex {  public double real;  public double imag;   public Complex(double real, double imag) {    this.real = real;    this.imag = imag;  }   public Complex add(Complex other) {    return new Complex(this.real + other.real, this.imag + other.imag);  }   public Complex subtract(Complex other) {    return new Complex(this.real - other.real, this.imag - other.imag);  }   public Complex multiply(Complex other) {    return new Complex(this.real * other.real - this.imag * other.imag,                       this.real * other.imag + this.imag * other.real);  }   public Complex divide(double scalar) {    return new Complex(this.real / scalar, this.imag / scalar);  }} class Solution {  public long[] multiply(int[] poly1, int[] poly2) {    final int n1 = poly1.length;    final int n2 = poly2.length;    final int n = n1 + n2 - 1;    final int sz = 1 << bitLength(n - 1);     // Prepare input arrays for FFT.    Complex[] a = new Complex[sz];    Complex[] b = new Complex[sz];     // Initialize arrays with Complex objects    for (int i = 0; i < sz; ++i) {      a[i] = new Complex(0, 0);      b[i] = new Complex(0, 0);    }     // Copy polynomial coefficients to complex arrays.    for (int i = 0; i < n1; ++i)      a[i] = new Complex(poly1[i], 0);     for (int i = 0; i < n2; ++i)      b[i] = new Complex(poly2[i], 0);     // Transform polynomials to frequency domain.    fft(a, false);    fft(b, false);     // Multiply in frequency domain.    for (int i = 0; i < sz; ++i)      a[i] = a[i].multiply(b[i]);     // Transform back to coefficient domain.    fft(a, true);     // Extract real parts as polynomial coefficients.    long[] ans = new long[n];     for (int i = 0; i < n; ++i)      ans[i] = Math.round(a[i].real);     return ans;  }   private void fft(Complex[] a, boolean inverse) {    final int n = a.length;     // Bit-reversal permutation.    for (int i = 1, j = 0; i < n; ++i) {      int bit = n >> 1;      for (; (j & bit) != 0; bit >>= 1)        j ^= bit;      j ^= bit;      if (i < j)        swap(a, i, j);    }     // FFT computation.    for (int len = 2; len <= n; len *= 2) {      final double angle = 2 * Math.PI / len * (inverse ? -1 : 1);      final Complex wLen = new Complex(Math.cos(angle), Math.sin(angle));      for (int i = 0; i < n; i += len) {        Complex w = new Complex(1, 0);        for (int j = 0; j < len / 2; ++j) {          final Complex u = a[i + j];          final Complex v = a[i + j + len / 2].multiply(w);          a[i + j] = u.add(v);          a[i + j + len / 2] = u.subtract(v);          w = w.multiply(wLen);        }      }    }     // Normalize if inverse transform.    if (inverse)      for (int i = 0; i < n; ++i)        a[i] = a[i].divide(n);  }   private void swap(Complex[] a, int i, int j) {    Complex temp = a[i];    a[i] = a[j];    a[j] = temp;  }   private int bitLength(int n) {    return Integer.SIZE - Integer.numberOfLeadingZeros(n);  }}

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