Algorithms/DiscreteFourierTransform.cpp at master · dhirajfx3/Algorithms · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
#define ve vector
#define complex num
#define float2 long double
#define pb push_back
template <typename y>
ve<y> operator*(const  ve<y> &A, const   ve<y> &B)
{
	ve<y> R(A);
	for (int i = 0; i < B.size(); i++)
		R[i] = A[i] * B[i];
	return R;
}
struct complex
{
	float2 re, im;
public:
	complex() :re(0), im(0) {}
	complex(float2 R, float2 I) :re(R), im(I) {}
	complex  operator* (const complex  &nu) const
	{
		return complex(re*nu.re - im*nu.im, im*nu.re + re*nu.im);
	}
	complex const operator+(const complex  &nu) const
	{
		return complex(re + nu.re, im + nu.im);
	}
	complex   operator-(const complex  &nu) const
	{
		return complex(re - nu.re, im - nu.im);
	}
	complex(float2 angle) :re(cos(angle)), im(sin(angle)) {}
	complex& operator*=(const complex &n)
	{
		float2 re_cpy = (re*n.re - im*n.im), im_cpy = (im*n.re + re*n.im);
		re = re_cpy;
		im = im_cpy;
		return *this;
	}
	complex& operator-=(const complex &n)
	{
		re -= n.re;
		im -= n.im;
		return *this;
	}
	complex& operator+=(const complex &n)
	{
		re += n.re;
		im += n.im;
		return *this;
	}
};
template<typename my_type>
class DiscreteFourierTransform
{
	int N;
	int revised()
	{
		int temp = N - 1;
		int ans = 1;
		while (temp)
		{
			ans *= 2;
			temp >>= 1;
		}
		return ans;
	}
public:
	ve<num> DFT(const ve<my_type> &v)
	{
		int R = revised();
		ve<num> a(R);
		for (int i = 0; i<N && i<v.size(); i++)
		{
			a[i].re = v[i];
		}
		return DFT(a);
	}
	ve<num> DFT(const ve<num> &a)
	{
		int n = a.size();
		if (n == 1)
			return a;
		num wn(2 * pi / n);
		num w(1, 0);
		ve<num> a0, a1;
		for (int i = 0; i<n;)
		{
			a0.pb(a[i++]);
			a1.pb(a[i++]);
		}
		ve<num> y0 = DFT(a0);
		ve<num> y1 = DFT(a1);
		ve<num> y(n);
		int n_2 = n / 2;
		for (int k = 0; k<n_2; k++)
		{
			y[k] = y0[k] + w*y1[k];
			y[k + n_2] = y0[k] - w*y1[k];
			w = w*wn;
		}
		return y;
	}
	ve<num> inv_DFT(const ve<num> &y)
	{
		int n = y.size();
		if (n == 1)
			return y;
		num wn(2 * pi / n);
		wn.im = -wn.im;
		num w(1, 0);
		ve<num> y0, y1;
		for (int i = 0; i<n;)
		{
			y0.pb(y[i++]);
			y1.pb(y[i++]);
		}
		ve<num> a0 = inv_DFT(y0);
		ve<num> a1 = inv_DFT(y1);
		ve<num> a(n);
		int n_2 = n / 2;
		for (int k = 0; k<n_2; k++)
		{
			a[k] = (a0[k] + w*a1[k]);
			a[k + n_2] = (a0[k] - w*a1[k]);
			w = w*wn;
		}
		return a;
	}
	ve<my_type> poly_mult(const ve<my_type> &a, const ve<my_type> &b)
	{
		N = a.size() + b.size();
		int val = revised();
		ve<num> R(inv_DFT(DFT(a)*DFT(b)));
		for (auto &x : R)
			x.re /= val, x.im /= val;
		ve<long long> Res;
		for (int i = 0; i < N - 1; i++)
			Res.pb(static_cast<long long>(R[i].re + 0.5));
		return Res;
	}
};
DiscreteFourierTransform<int> D;