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Problem solution · C++

Multiply Two Polynomials

Multiply Two Polynomials: a C++ 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
77 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

  • 77 lines of C++ from the credited upstream file 3549.cpp.
  • The implementation visibly relies on sequence storage.
  • 10 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 · C++C++
Use this to learn the idea, then write your own version.
class Solution { public:  vector<long long> multiply(vector<int>& poly1, vector<int>& poly2) {    const unsigned n1 = poly1.size();    const unsigned n2 = poly2.size();    const unsigned n = n1 + n2 - 1;    const unsigned sz = 1 << std::bit_width(n - 1);     // Prepare input arrays for FFT.    vector<complex<double>> a(sz);    vector<complex<double>> b(sz);     // Copy polynomial coefficients to complex arrays.    for (int i = 0; i < n1; ++i)      a[i] = poly1[i];     for (int i = 0; i < n2; ++i)      b[i] = poly2[i];     // Transform polynomials to frequency domain.    fft(a, false);    fft(b, false);     // Multiply in frequency domain.    for (int i = 0; i < sz; ++i)      a[i] *= b[i];     // Transform back to coefficient domain.    fft(a, true);     // Extract real parts as polynomial coefficients.    vector<long long> ans(n);     for (int i = 0; i < n; ++i)      ans[i] = round(a[i].real());     return ans;  }  private:  // Fast Fourier Transform.  void fft(vector<complex<double>>& a, bool inverse) {    const unsigned n = a.size();     // Bit-reversal permutation.    for (int i = 1, j = 0; i < n; ++i) {      int bit = n >> 1;      for (; j & bit; bit >>= 1)        j ^= bit;      j ^= bit;      if (i < j)        swap(a[i], a[j]);    }     // FFT computation.    for (int len = 2; len <= n; len *= 2) {      const double angle = 2 * M_PI / len * (inverse ? -1 : 1);      const complex<double> w_len(cos(angle), sin(angle));      for (int i = 0; i < n; i += len) {        complex<double> w(1);        for (int j = 0; j < len / 2; ++j) {          const complex<double> u = a[i + j];          const complex<double> v = a[i + j + len / 2] * w;          a[i + j] = u + v;          a[i + j + len / 2] = u - v;          w *= w_len;        }      }    }     // Normalize if inverse transform.    if (inverse)      for (complex<double>& x : a)        x /= n;  }};

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