結果
| 問題 |
No.1510 Simple Integral
|
| コンテスト | |
| ユーザー |
 uwi
|
| 提出日時 | 2021-05-15 01:16:21 |
| 言語 | Java (openjdk 23) |
| 結果 |
AC
|
| 実行時間 | 184 ms / 2,000 ms |
| コード長 | 18,072 bytes |
| コンパイル時間 | 5,403 ms |
| コンパイル使用メモリ | 101,556 KB |
| 実行使用メモリ | 54,392 KB |
| 最終ジャッジ日時 | 2024-10-02 08:03:34 |
| 合計ジャッジ時間 | 13,562 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge5 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 43 |
ソースコード
package contest210514;
import java.io.*;
import java.util.*;
import java.util.function.IntUnaryOperator;
import java.util.function.LongUnaryOperator;
public class G {
InputStream is;
FastWriter out;
String INPUT = "";
static class FactorPower
{
int e;
long[] a;
public FactorPower(int e, long[] a) {
this.e = e;
this.a = a;
}
}
static final int mod = 998244353;
static class PartialFractionDecompositionResult
{
FactorPower fp;
long[][] nums;
public PartialFractionDecompositionResult(FactorPower fp, long[][] nums) {
this.fp = fp;
this.nums = nums;
}
}
static List<PartialFractionDecompositionResult> partialFractionDecomposition(FactorPower... fs)
{
long[] all = {1};
int esum = 0;
for(FactorPower fp : fs){
for(int i = 0;i < fp.e;i++) {
all = mulnaive(all, fp.a, mod);
esum += fp.a.length-1;
}
}
int[][] M = new int[esum+1][esum];
int gen = 0;
for(FactorPower fp : fs){
long[] my = all;
for(int i = 1;i <= fp.e;i++) {
my = divnaive(my, fp.a, mod)[0];
for(int j = 0;j < fp.a.length-1;j++){
for(int k = 0;k < my.length;k++){
M[k+j][gen] += my[k];
}
gen++;
}
}
}
assert gen == esum;
int[] v = new int[esum+1];
v[0] = 1;
Result res = gaussElimination(M, v, mod);
// tr(res.sol);
List<PartialFractionDecompositionResult> ret = new ArrayList<>();
gen = 0;
for(FactorPower fp : fs){
long[][] ps = new long[fp.e][];
for(int i = 1;i <= fp.e;i++) {
long[] lf = new long[fp.a.length-1];
for(int j = 0;j < fp.a.length-1;j++){
lf[j] = res.sol[gen++];
}
ps[i-1] = lf;
}
ret.add(new PartialFractionDecompositionResult(fp, ps));
}
return ret;
}
public static Result gaussElimination(int[][] M, int[] v, int mod)
{
int n = M.length, m = M[0].length;
int[] head = new int[n];
// if not needed, comment out.
for(int[] row : M){
for(int i = 0;i < row.length;i++){
row[i] %= mod;
if(row[i] < 0)row[i] += mod;
}
}
// Forward Elimination
int row = 0;
for(int col = 0;col < m;col++){
// select pivot
boolean pivotFound = false;
out:
for(int prow = row;prow < n;prow++){
if(M[prow][col] != 0){
// pivot found
if(prow != row){
// swap rows
for(int k = 0;k < m;k++){
int u = M[prow][k]; M[prow][k] = M[row][k]; M[row][k] = u;
}
int dum = v[prow]; v[prow] = v[row]; v[row] = dum;
}
pivotFound = true;
break out;
}
}
if(!pivotFound)continue;
head[row] = col;
// diag to 1
long imul = invl(M[row][col], mod);
for(int k = 0;k < m;k++){
M[row][k] = (int)(M[row][k] * imul % mod);
}
v[row] = (int)(v[row] * imul % mod);
for(int j = row+1;j < n;j++){
if(M[j][col] != 0){
long mul = mod-M[j][col];
for(int k = col;k < m;k++){
M[j][k] = (int)((M[j][k] + M[row][k] * mul) % mod);
}
v[j] = (int)((v[j] + v[row] * mul) % mod);
}
}
row++;
}
Result ret = new Result();
ret.mat = M;
for(int i = row;i < n;i++){
if(v[i] != 0){
ret.rank = row;
ret.exists = false;
return ret;
}
}
for(int i = row-1;i >= 0;i--){
for(int j = i-1;j >= 0;j--){
if(M[j][head[i]] != 0){
long mul = mod-M[j][head[i]];
for(int k = head[i];k < m;k++){
M[j][k] = (int)((M[j][k] + M[i][k] * mul) % mod);
}
v[j] = (int)((v[j] + v[i] * mul) % mod);
}
}
}
int[] retv = new int[m];
for(int i = 0;i < row;i++){
retv[head[i]] = v[i];
}
ret.sol = retv;
ret.rank = row;
ret.exists = true;
return ret;
}
public static class Result
{
public int[][] mat;
public int[] sol;
public int rank;
public boolean exists;
}
public static int[][] uniqcount(int[] a)
{
int n = a.length;
int p = 0;
int[][] b = new int[n][];
for(int i = 0;i < n;i++){
if(i == 0 || a[i] != a[i-1])b[p++] = new int[]{a[i], 0};
b[p-1][1]++;
}
return Arrays.copyOf(b, p);
}
public void solve()
{
int n = ni();
int[] a = na(n);
Arrays.sort(a);
int[][] u = uniqcount(a);
FactorPower[] fps = new FactorPower[u.length];
for(int i = 0;i < u.length;i++) {
fps[i] = new FactorPower(u[i][1], new long[]{(long) u[i][0] * u[i][0] % mod, 0, 1});
}
List<PartialFractionDecompositionResult> res = partialFractionDecomposition(fps);
// 1 1: pi*1/1/a
// 1 2: pi*1/2/a^3
// 1 3: pi:3/8/a^5
// (2n)!/2^{2n}/(n!)^2 <- n-1
int[][] fif = enumFIF(1000, mod);
long[] cos = new long[500];
for(int i = 1;i <= 499;i++){
cos[i] = fif[0][2*i-2] * invl(pow(2, 2*(i-1), mod), mod) % mod
* fif[1][i-1] % mod * fif[1][i-1] % mod;
}
long ans = 0;
for(int j = 0;j < res.size();j++){
PartialFractionDecompositionResult re = res.get(j);
for(int i = 1;i <= re.fp.e;i++){
ans += cos[i] * invl(pow(u[j][0], 2*i-1, mod), mod) % mod * re.nums[i-1][0];
ans %= mod;
}
}
out.println(ans);
}
public static long pow(long a, long n, long mod) {
// a %= mod;
long ret = 1;
int x = 63 - Long.numberOfLeadingZeros(n);
for (; x >= 0; x--) {
ret = ret * ret % mod;
if (n << 63 - x < 0) ret = ret * a % mod;
}
return ret;
}
public static int[][] enumFIF(int n, int mod) {
int[] f = new int[n + 1];
int[] invf = new int[n + 1];
f[0] = 1;
for (int i = 1; i <= n; i++) {
f[i] = (int) ((long) f[i - 1] * i % mod);
}
long a = f[n];
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;
}
invf[n] = (int) (p < 0 ? p + mod : p);
for (int i = n - 1; i >= 0; i--) {
invf[i] = (int) ((long) invf[i + 1] * (i + 1) % mod);
}
return new int[][]{f, invf};
}
public static long invl(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;
}
public static long[][] divnaive(long[] a, long[] b, int mod)
{
int n = a.length, m = b.length;
if(n-m+1 <= 0)return new long[][]{new long[0], Arrays.copyOf(a, n)};
long[] r = Arrays.copyOf(a, n);
long[] q = new long[n-m+1];
long ib = invl(b[m-1], mod);
for(int i = n-1;i >= m-1;i--){
long x = ib * r[i] % mod;
q[i-(m-1)] = x;
for(int j = m-1;j >= 0;j--){
r[i+j-(m-1)] -= b[j]*x;
r[i+j-(m-1)] %= mod;
if(r[i+j-(m-1)] < 0)r[i+j-(m-1)] += mod;
// r[i+j-(m-1)] = modh(r[i+j-(m-1)]+(long)mod*mod - b[j]*x, MM, HH, mod);
}
}
return new long[][]{q, Arrays.copyOf(r, m-1)};
}
public static long[] mulnaive(long[] a, long[] b, int mod)
{
final long big = (Long.MAX_VALUE/mod/mod-1)*mod*mod;
long[] c = new long[a.length+b.length-1];
for(int i = 0;i < a.length;i++){
for(int j = 0;j < b.length;j++){
c[i+j] += a[i]*b[j];
if(c[i+j] >= big)c[i+j] -= big;
}
}
for(int i = 0;i < c.length;i++)c[i] %= mod;
return c;
}
public static void main(String[] args) {
new G().run();
}
public void run()
{
long S = System.currentTimeMillis();
is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
out = new FastWriter(System.out);
solve();
out.flush();
long G = System.currentTimeMillis();
tr(G-S+"ms");
// Thread t = new Thread(null, null, "~", Runtime.getRuntime().maxMemory()){
// @Override
// public void run() {
// long s = System.currentTimeMillis();
// solve();
// out.flush();
// if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
// }
// };
// t.start();
// t.join();
}
private boolean eof()
{
if(lenbuf == -1)return true;
int lptr = ptrbuf;
while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
try {
is.mark(1000);
while(true){
int b = is.read();
if(b == -1){
is.reset();
return true;
}else if(!isSpaceChar(b)){
is.reset();
return false;
}
}
} catch (IOException e) {
return true;
}
}
private final byte[] inbuf = new byte[1024];
public int lenbuf = 0, ptrbuf = 0;
private int readByte()
{
if(lenbuf == -1)throw new InputMismatchException();
if(ptrbuf >= lenbuf){
ptrbuf = 0;
try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
if(lenbuf <= 0)return -1;
}
return inbuf[ptrbuf++];
}
private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
// private boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
private double nd() { return Double.parseDouble(ns()); }
private char nc() { return (char)skip(); }
private String ns()
{
int b = skip();
StringBuilder sb = new StringBuilder();
while(!(isSpaceChar(b))){
sb.appendCodePoint(b);
b = readByte();
}
return sb.toString();
}
private char[] ns(int n)
{
char[] buf = new char[n];
int b = skip(), p = 0;
while(p < n && !(isSpaceChar(b))){
buf[p++] = (char)b;
b = readByte();
}
return n == p ? buf : Arrays.copyOf(buf, p);
}
private char[][] nm(int n, int m)
{
char[][] map = new char[n][];
for(int i = 0;i < n;i++)map[i] = ns(m);
return map;
}
private int[] na(int n)
{
int[] a = new int[n];
for(int i = 0;i < n;i++)a[i] = ni();
return a;
}
private long[] nal(int n)
{
long[] a = new long[n];
for(int i = 0;i < n;i++)a[i] = nl();
return a;
}
private int ni()
{
int num = 0, b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
b = readByte();
}
while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
b = readByte();
}
}
private long nl()
{
long num = 0;
int b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
b = readByte();
}
while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
b = readByte();
}
}
public static class FastWriter
{
private static final int BUF_SIZE = 1<<13;
private final byte[] buf = new byte[BUF_SIZE];
private final OutputStream out;
private int ptr = 0;
private FastWriter(){out = null;}
public FastWriter(OutputStream os)
{
this.out = os;
}
public FastWriter(String path)
{
try {
this.out = new FileOutputStream(path);
} catch (FileNotFoundException e) {
throw new RuntimeException("FastWriter");
}
}
public FastWriter write(byte b)
{
buf[ptr++] = b;
if(ptr == BUF_SIZE)innerflush();
return this;
}
public FastWriter write(char c)
{
return write((byte)c);
}
public FastWriter write(char[] s)
{
for(char c : s){
buf[ptr++] = (byte)c;
if(ptr == BUF_SIZE)innerflush();
}
return this;
}
public FastWriter write(String s)
{
s.chars().forEach(c -> {
buf[ptr++] = (byte)c;
if(ptr == BUF_SIZE)innerflush();
});
return this;
}
private static int countDigits(int l) {
if (l >= 1000000000) return 10;
if (l >= 100000000) return 9;
if (l >= 10000000) return 8;
if (l >= 1000000) return 7;
if (l >= 100000) return 6;
if (l >= 10000) return 5;
if (l >= 1000) return 4;
if (l >= 100) return 3;
if (l >= 10) return 2;
return 1;
}
public FastWriter write(int x)
{
if(x == Integer.MIN_VALUE){
return write((long)x);
}
if(ptr + 12 >= BUF_SIZE)innerflush();
if(x < 0){
write((byte)'-');
x = -x;
}
int d = countDigits(x);
for(int i = ptr + d - 1;i >= ptr;i--){
buf[i] = (byte)('0'+x%10);
x /= 10;
}
ptr += d;
return this;
}
private static int countDigits(long l) {
if (l >= 1000000000000000000L) return 19;
if (l >= 100000000000000000L) return 18;
if (l >= 10000000000000000L) return 17;
if (l >= 1000000000000000L) return 16;
if (l >= 100000000000000L) return 15;
if (l >= 10000000000000L) return 14;
if (l >= 1000000000000L) return 13;
if (l >= 100000000000L) return 12;
if (l >= 10000000000L) return 11;
if (l >= 1000000000L) return 10;
if (l >= 100000000L) return 9;
if (l >= 10000000L) return 8;
if (l >= 1000000L) return 7;
if (l >= 100000L) return 6;
if (l >= 10000L) return 5;
if (l >= 1000L) return 4;
if (l >= 100L) return 3;
if (l >= 10L) return 2;
return 1;
}
public FastWriter write(long x)
{
if(x == Long.MIN_VALUE){
return write("" + x);
}
if(ptr + 21 >= BUF_SIZE)innerflush();
if(x < 0){
write((byte)'-');
x = -x;
}
int d = countDigits(x);
for(int i = ptr + d - 1;i >= ptr;i--){
buf[i] = (byte)('0'+x%10);
x /= 10;
}
ptr += d;
return this;
}
public FastWriter write(double x, int precision)
{
if(x < 0){
write('-');
x = -x;
}
x += Math.pow(10, -precision)/2;
// if(x < 0){ x = 0; }
write((long)x).write(".");
x -= (long)x;
for(int i = 0;i < precision;i++){
x *= 10;
write((char)('0'+(int)x));
x -= (int)x;
}
return this;
}
public FastWriter writeln(char c){ return write(c).writeln(); }
public FastWriter writeln(int x){ return write(x).writeln(); }
public FastWriter writeln(long x){ return write(x).writeln(); }
public FastWriter writeln(double x, int precision){ return write(x, precision).writeln(); }
public FastWriter write(int... xs)
{
boolean first = true;
for(int x : xs) {
if (!first) write(' ');
first = false;
write(x);
}
return this;
}
public FastWriter write(long... xs)
{
boolean first = true;
for(long x : xs) {
if (!first) write(' ');
first = false;
write(x);
}
return this;
}
public FastWriter write(IntUnaryOperator f, int... xs)
{
boolean first = true;
for(int x : xs) {
if (!first) write(' ');
first = false;
write(f.applyAsInt(x));
}
return this;
}
public FastWriter write(LongUnaryOperator f, long... xs)
{
boolean first = true;
for(long x : xs) {
if (!first) write(' ');
first = false;
write(f.applyAsLong(x));
}
return this;
}
public FastWriter writeln()
{
return write((byte)'\n');
}
public FastWriter writeln(int... xs) { return write(xs).writeln(); }
public FastWriter writeln(long... xs) { return write(xs).writeln(); }
public FastWriter writeln(IntUnaryOperator f, int... xs) { return write(f, xs).writeln(); }
public FastWriter writeln(LongUnaryOperator f, long... xs) { return write(f, xs).writeln(); }
public FastWriter writeln(char[] line) { return write(line).writeln(); }
public FastWriter writeln(char[]... map) { for(char[] line : map)write(line).writeln();return this; }
public FastWriter writeln(String s) { return write(s).writeln(); }
private void innerflush()
{
try {
out.write(buf, 0, ptr);
ptr = 0;
} catch (IOException e) {
throw new RuntimeException("innerflush");
}
}
public void flush()
{
innerflush();
try {
out.flush();
} catch (IOException e) {
throw new RuntimeException("flush");
}
}
public FastWriter print(byte b) { return write(b); }
public FastWriter print(char c) { return write(c); }
public FastWriter print(char[] s) { return write(s); }
public FastWriter print(String s) { return write(s); }
public FastWriter print(int x) { return write(x); }
public FastWriter print(long x) { return write(x); }
public FastWriter print(double x, int precision) { return write(x, precision); }
public FastWriter println(char c){ return writeln(c); }
public FastWriter println(int x){ return writeln(x); }
public FastWriter println(long x){ return writeln(x); }
public FastWriter println(double x, int precision){ return writeln(x, precision); }
public FastWriter print(int... xs) { return write(xs); }
public FastWriter print(long... xs) { return write(xs); }
public FastWriter print(IntUnaryOperator f, int... xs) { return write(f, xs); }
public FastWriter print(LongUnaryOperator f, long... xs) { return write(f, xs); }
public FastWriter println(int... xs) { return writeln(xs); }
public FastWriter println(long... xs) { return writeln(xs); }
public FastWriter println(IntUnaryOperator f, int... xs) { return writeln(f, xs); }
public FastWriter println(LongUnaryOperator f, long... xs) { return writeln(f, xs); }
public FastWriter println(char[] line) { return writeln(line); }
public FastWriter println(char[]... map) { return writeln(map); }
public FastWriter println(String s) { return writeln(s); }
public FastWriter println() { return writeln(); }
}
public static void trnz(int... o)
{
for(int i = 0;i < o.length;i++)if(o[i] != 0)System.out.print(i+":"+o[i]+" ");
System.out.println();
}
// print ids which are 1
public static void trt(long... o)
{
Queue<Integer> stands = new ArrayDeque<>();
for(int i = 0;i < o.length;i++){
for(long x = o[i];x != 0;x &= x-1)stands.add(i<<6|Long.numberOfTrailingZeros(x));
}
System.out.println(stands);
}
public static void tf(boolean... r)
{
for(boolean x : r)System.out.print(x?'#':'.');
System.out.println();
}
public static void tf(boolean[]... b)
{
for(boolean[] r : b) {
for(boolean x : r)System.out.print(x?'#':'.');
System.out.println();
}
System.out.println();
}
public void tf(long[]... b)
{
if(INPUT.length() != 0) {
for (long[] r : b) {
for (long x : r) {
for (int i = 0; i < 64; i++) {
System.out.print(x << ~i < 0 ? '#' : '.');
}
}
System.out.println();
}
System.out.println();
}
}
public void tf(long... b)
{
if(INPUT.length() != 0) {
for (long x : b) {
for (int i = 0; i < 64; i++) {
System.out.print(x << ~i < 0 ? '#' : '.');
}
}
System.out.println();
}
}
private void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
// private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
}