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Merged
NASU41 merged 3 commits into from
Jul 23, 2021
Merged

Feature/new convolution #65

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a649af5
fix https://github.com/atcoder/ac-library/issues/30
Hiromi-Ayase Jul 22, 2021
239c2b8
4-base NTT
Hiromi-Ayase Jul 22, 2021
d44d306
remove vscode files
Hiromi-Ayase Jul 22, 2021
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263 changes: 171 additions & 92 deletions Convolution/Convolution.java
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Original file line number Diff line number Diff line change
@@ -1,8 +1,8 @@
/**
* Convolution.
*
* @verified https://atcoder.jp/contests/practice2/tasks/practice2_f
* @verified https://judge.yosupo.jp/problem/convolution_mod_1000000007
* @verified https://atcoder.jp/contests/practice2/submissions/24449847
* @verified https://judge.yosupo.jp/submission/53841
*/
class Convolution {
/**
Expand Down Expand Up @@ -78,6 +78,74 @@ private static int ceilPow2(int n) {
return x;
}

private static class FftInfo {
private static int bsfConstexpr(int n) {
int x = 0;
while ((n & (1 << x)) == 0) x++;
return x;
}

private static long inv(long a, long mod) {
long b = mod;
long p = 1, q = 0;
while (b > 0) {
long c = a / b;
long d;
d = a;
a = b;
b = d % b;
d = p;
p = q;
q = d - c * q;
}
return p < 0 ? p + mod : p;
}

private final int rank2;
public final long[] root;
public final long[] iroot;
public final long[] rate2;
public final long[] irate2;
public final long[] rate3;
public final long[] irate3;

public FftInfo(int g, int mod) {
rank2 = bsfConstexpr(mod - 1);
root = new long[rank2 + 1];
iroot = new long[rank2 + 1];
rate2 = new long[Math.max(0, rank2 - 2 + 1)];
irate2 = new long[Math.max(0, rank2 - 2 + 1)];
rate3 = new long[Math.max(0, rank2 - 3 + 1)];
irate3 = new long[Math.max(0, rank2 - 3 + 1)];

root[rank2] = pow(g, (mod - 1) >> rank2, mod);
iroot[rank2] = inv(root[rank2], mod);
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1] % mod;
iroot[i] = iroot[i + 1] * iroot[i + 1] % mod;
}

{
long prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod % mod;
irate2[i] = iroot[i + 2] * iprod % mod;
prod = prod * iroot[i + 2] % mod;
iprod = iprod * root[i + 2] % mod;
}
}
{
long prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod % mod;
irate3[i] = iroot[i + 3] * iprod % mod;
prod = prod * iroot[i + 3] % mod;
iprod = iprod * root[i + 3] % mod;
}
}
}
};

/**
* Garner's algorithm.
*
Expand All @@ -104,87 +172,65 @@ private static long garner(long[] c, int[] mods) {
return cnst[n - 1];
}

/**
* Pre-calculation for NTT.
*
* @param mod NTT Prime.
* @param g Primitive root of mod.
* @return Pre-calculation table.
*/
private static long[] sumE(int mod, int g) {
long[] sum_e = new long[30];
long[] es = new long[30];
long[] ies = new long[30];
int cnt2 = Integer.numberOfTrailingZeros(mod - 1);
long e = pow(g, (mod - 1) >> cnt2, mod);
long ie = pow(e, mod - 2, mod);
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e;
ies[i - 2] = ie;
e = e * e % mod;
ie = ie * ie % mod;
}
long now = 1;
for (int i = 0; i < cnt2 - 2; i++) {
sum_e[i] = es[i] * now % mod;
now = now * ies[i] % mod;
}
return sum_e;
}

/**
* Pre-calculation for inverse NTT.
*
* @param mod Mod.
* @param g Primitive root of mod.
* @return Pre-calculation table.
*/
private static long[] sumIE(int mod, int g) {
long[] sum_ie = new long[30];
long[] es = new long[30];
long[] ies = new long[30];

int cnt2 = Integer.numberOfTrailingZeros(mod - 1);
long e = pow(g, (mod - 1) >> cnt2, mod);
long ie = pow(e, mod - 2, mod);
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e;
ies[i - 2] = ie;
e = e * e % mod;
ie = ie * ie % mod;
}
long now = 1;
for (int i = 0; i < cnt2 - 2; i++) {
sum_ie[i] = ies[i] * now % mod;
now = now * es[i] % mod;
}
return sum_ie;
}

/**
* Inverse NTT.
*
* @param a Target array.
* @param sumIE Pre-calculation table.
* @param g Primitive root of mod.
* @param mod NTT Prime.
*/
private static void butterflyInv(long[] a, long[] sumIE, int mod) {
private static void butterflyInv(long[] a, int g, int mod) {
int n = a.length;
int h = ceilPow2(n);

for (int ph = h; ph >= 1; ph--) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
long inow = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
long l = a[i + offset];
long r = a[i + offset + p];
a[i + offset] = (l + r) % mod;
a[i + offset + p] = (mod + l - r) * inow % mod;
FftInfo info = new FftInfo(g, mod);

int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len > 0) {
if (len == 1) {
int p = 1 << (h - len);
long irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
long l = a[i + offset];
long r = a[i + offset + p];
a[i + offset] = (l + r) % mod;
a[i + offset + p] = (mod + l - r) % mod * irot % mod;
}
if (s + 1 != (1 << (len - 1))) {
irot *= info.irate2[Integer.numberOfTrailingZeros(~s)];
irot %= mod;
}
}
int x = Integer.numberOfTrailingZeros(~s);
inow = inow * sumIE[x] % mod;
len--;
} else {
// 4-base
int p = 1 << (h - len);
long irot = 1, iimag = info.iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
long irot2 = irot * irot % mod;
long irot3 = irot2 * irot % mod;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
long a0 = 1L * a[i + offset + 0 * p];
long a1 = 1L * a[i + offset + 1 * p];
long a2 = 1L * a[i + offset + 2 * p];
long a3 = 1L * a[i + offset + 3 * p];

long a2na3iimag = 1L * (mod + a2 - a3) % mod * iimag % mod;

a[i + offset] = (a0 + a1 + a2 + a3) % mod;
a[i + offset + 1 * p] = (a0 + (mod - a1) + a2na3iimag) % mod * irot % mod;
a[i + offset + 2 * p] = (a0 + a1 + (mod - a2) + (mod - a3)) % mod * irot2 % mod;
a[i + offset + 3 * p] = (a0 + (mod - a1) + (mod - a2na3iimag)) % mod * irot3 % mod;
}
if (s + 1 != (1 << (len - 2))) {
irot *= info.irate3[Integer.numberOfTrailingZeros(~s)];
irot %= mod;
}
}
len -= 2;
}
}
}
Expand All @@ -193,26 +239,61 @@ private static void butterflyInv(long[] a, long[] sumIE, int mod) {
* Inverse NTT.
*
* @param a Target array.
* @param sumE Pre-calculation table.
* @param g Primitive root of mod.
* @param mod NTT Prime.
*/
private static void butterfly(long[] a, long[] sumE, int mod) {
private static void butterfly(long[] a, int g, int mod) {
int n = a.length;
int h = ceilPow2(n);

for (int ph = 1; ph <= h; ph++) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
long now = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
long l = a[i + offset];
long r = a[i + offset + p] * now % mod;
a[i + offset] = (l + r) % mod;
a[i + offset + p] = (l - r + mod) % mod;
FftInfo info = new FftInfo(g, mod);

int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
long rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
long l = a[i + offset];
long r = a[i + offset + p] * rot % mod;
a[i + offset] = (l + r) % mod;
a[i + offset + p] = (l + mod - r) % mod;
}
if (s + 1 != (1 << len)) {
rot *= info.rate2[Integer.numberOfTrailingZeros(~s)];
rot %= mod;
}
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
long rot = 1, imag = info.root[2];
for (int s = 0; s < (1 << len); s++) {
long rot2 = rot * rot % mod;
long rot3 = rot2 * rot % mod;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
long mod2 = 1L * mod * mod;
long a0 = 1L * a[i + offset];
long a1 = 1L * a[i + offset + p] * rot % mod;
long a2 = 1L * a[i + offset + 2 * p] * rot2 % mod;
long a3 = 1L * a[i + offset + 3 * p] * rot3 % mod;
long a1na3imag = 1L * (a1 + mod2 - a3) % mod * imag % mod;
long na2 = mod2 - a2;
a[i + offset] = (a0 + a2 + a1 + a3) % mod;
a[i + offset + 1 * p] = (a0 + a2 + (2 * mod2 - (a1 + a3))) % mod;
a[i + offset + 2 * p] = (a0 + na2 + a1na3imag) % mod;
a[i + offset + 3 * p] = (a0 + na2 + (mod2 - a1na3imag)) % mod;
}
if (s + 1 != (1 << len)) {
rot *= info.rate3[Integer.numberOfTrailingZeros(~s)];
rot %= mod;
}
}
int x = Integer.numberOfTrailingZeros(~s);
now = now * sumE[x] % mod;
len += 2;
}
}
}
Expand Down Expand Up @@ -241,15 +322,13 @@ public static long[] convolution(long[] a, long[] b, int mod) {
}

int g = primitiveRoot(mod);
long[] sume = sumE(mod, g);
long[] sumie = sumIE(mod, g);

butterfly(a, sume, mod);
butterfly(b, sume, mod);
butterfly(a, g, mod);
butterfly(b, g, mod);
for (int i = 0; i < z; i++) {
a[i] = a[i] * b[i] % mod;
}
butterflyInv(a, sumie, mod);
butterflyInv(a, g, mod);
a = java.util.Arrays.copyOf(a, n + m - 1);

long iz = pow(z, mod - 2, mod);
Expand Down