HackerRank Definite Random Walks problem solution

In this HackerRank Definite Random Walks problem solution, we have given n, m, k, the game board, and the probabilities for each die face, print n lines where each line I contains the probability that the chip is on the field I at the end of the game.

Problem solution in Java.

```import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.HashMap;
import java.util.InputMismatchException;
import java.util.Map;

public class DefiniteRandomWalks {
InputStream is;
PrintWriter out;
String INPUT = "";

void solve()
{
int n = ni(), m = ni(), K = ni();
int[] f = na(n);
for(int i = 0;i < n;i++)f[i]--;
long[] ps = new long[m];
for(int i = 0;i < m;i++)ps[i] = nl();
int mod = 998244353;
long[] made = make(ps, K, n+1, 1);

int H = (int)Math.sqrt(n)*8; // naive height limit
int B = (int)Math.sqrt(n)*8; // cycle split period

SplitResult sres = split(f);
int[] tclus = new int[n];
Arrays.fill(tclus, -1);
for(int i = n-1;i >= 0;i--){
int cur = sres.ord[i];
if(sres.incycle[cur]){
tclus[cur] = cur;
}else{
tclus[cur] = tclus[f[cur]];
}
}
//		tr("phase 1");
long[] rets = new long[n];
int[][] maps = makeBuckets(tclus, n);
for(int i = 0;i < n;i++){
if(maps[i].length > 0){
int[] map = maps[i];
int[] lpar = new int[map.length];
int p = 0;
for(int x : maps[i]){
if(sres.incycle[x]){
lpar[p++] = -1;
}else{
lpar[p++] = Arrays.binarySearch(map, f[x]);
}
}
long[] res = solve(parentToG(lpar), lpar, made, H, Arrays.binarySearch(map, i));
for(int j = 0;j < res.length;j++){
if(!sres.incycle[map[j]]){
rets[map[j]] += res[j];
}
}
}
}

int[] maxdep = new int[n];
for(int i = 0;i < n;i++){
int cur = sres.ord[i];
if(!sres.incycle[cur]){
maxdep[f[cur]] = Math.max(maxdep[f[cur]], maxdep[cur]+1);
}
}
int[] tdep = new int[n];
for(int i = n-1;i >= 0;i--){
int cur = sres.ord[i];
if(!sres.incycle[cur]){
tdep[cur] = tdep[f[cur]]+1;
}
}

boolean[] ved = new boolean[n];
int[] cycle = new int[n];
Map<Long, long[]> cache = new HashMap<>();
for(int i = 0;i < n;i++){
if(sres.incycle[i] && !ved[i]){
int p = 0;
ved[i] = true;
cycle[p++] = i;
int lmaxdep = maxdep[i];
for(int j = f[i];!ved[j];j = f[j]){
ved[j] = true;
cycle[p++] = j;
lmaxdep = Math.max(lmaxdep, maxdep[j]);
}
int tail = lmaxdep+p+1;
int fp = p;
long[] di = cache.computeIfAbsent((long)tail<<32|fp, (z) -> {
long[] res = make(ps, K, tail, fp);
for(int j = res.length-fp-1;j >= 0;j--){
res[j] += res[j+fp];
if(res[j] >= mod)res[j] -= mod;
}
return res;
});

if(p <= B){
for(int j = 0;j < p;j++){
for(int v : maps[cycle[j]]){
for(int k = tdep[v], l = j;k < tdep[v]+p;k++,l++){
if(l == p)l = 0;
rets[cycle[l]] += di[k];
}
}
}
}else{
inputed = null;
for(int b = 0;b < p;b+=B){
long[] ents = new long[tail+1];
for(int j = 0;j < b;j++){
for(int v : maps[cycle[j]]){
ents[tail-(b-j)-tdep[v]]++;
}
}
for(int j = b+B;j < p;j++){
for(int v : maps[cycle[j]]){
ents[tail-b-(p-j)-tdep[v]]++;
}
}
long[] ced = convoluteSimply(ents, di, mod, 3);
inputed = saved;
for(int k = b;k < p && k < b+B;k++){
rets[cycle[k]] += ced[tail+k-b];
}
}
inputed = null;

for(int j = 0;j < p;j++){
for(int v : maps[cycle[j]]){
for(int k = tdep[v], l = j;k < tdep[v]+p && l < p && l < j/B*B+B;k++,l++){
rets[cycle[l]] += di[k];
}
for(int k = tdep[v]+p-j+j/B*B, l = j/B*B;k < tdep[v]+p && l < j;k++,l++){
rets[cycle[l]] += di[k];
}
}
}
}
}
};

long RN = invl(n, mod);
for(long ret : rets){
out.println(ret%mod*RN%mod);
}
}

int mod = 998244353;

long[] make(long[] ps, int K, int tail, int period)
{
long[] ms = ps;
if(ps.length > tail+period){
ms = Arrays.copyOf(ps, tail+period);
for(int j = tail+period, k = tail;j < ps.length;j++,k++){
if(k == tail+period)k -= period;
ms[k] += ps[j];
if(ms[k] >= mod)ms[k] -= mod;
}
}

long[] pps = new long[1];
pps[0] = 1;
for(int i = 0;1<<i <= K;i++){
if(K<<~i<0){
long[] res = convoluteSimply(pps, ms, mod, 3);
for(int j = res.length-1-period;j >= tail;j--){
res[j] += res[j+period];
res[j+period] = 0;
if(res[j] >= mod)res[j] -= mod;
}
pps = Arrays.copyOf(res, Math.min(tail+period, pps.length+ms.length));
}

if(1<<i+1 <= K){
long[] res = convoluteSimply(ms, ms, mod, 3);
for(int j = res.length-1-period;j >= tail;j--){
res[j] += res[j+period];
res[j+period] = 0;
if(res[j] >= mod)res[j] -= mod;
}
ms = Arrays.copyOf(res, Math.min(tail+period, ms.length+ms.length));
}
}
if(pps.length < tail+period)pps = Arrays.copyOf(pps, tail+period);
return pps;
}

long[] solve(int[][] g, int[] par, long[] di, int H, int root)
{
int n = g.length;
long[] ret = new long[n];
int[][] pars = parents3(g, root);
int[] des = new int[n];
long[] ws = new long[n];
int[] ord = pars[1], dep = pars[2];
int[] marked = new int[n];
for(int i = n-1;i >= 0;i--){
int cur = ord[i];
des[cur]++;
ws[cur] += des[cur];
if(marked[cur] == 0 && ws[cur] > (long)H*des[cur]){
marked[cur] = 1;
}
if(i > 0){
des[par[cur]] += des[cur];
ws[par[cur]] += ws[cur];
if(marked[cur] >= 1)marked[par[cur]] = 2;
}
}
for(int i = 0;i < n;i++){
if(marked[i] == 1){
int[] fdep = new int[n];
collect(i, par[i], g, dep, fdep);
for(int j = par[i];j != -1 && marked[j] == 2;j = par[j]){
fdep[dep[j]]++;
marked[j] = 3;
}

int lmaxdep = n;
for(int j = n-1;j >= 0;j--){
if(fdep[j] > 0){
lmaxdep = j;
break;
}
}
long[] rfdep = new long[lmaxdep+1];
for(int j = 0;j <= lmaxdep;j++){
rfdep[lmaxdep-j] = fdep[j];
}

long[] ced = convoluteSimply(rfdep, Arrays.copyOf(di, lmaxdep+1), mod, 3);
for(int j = i;j != -1;j = par[j]){
ret[j] += ced[lmaxdep-dep[j]];
}
}
}

for(int i = 0;i < n;i++){
if(marked[i] == 0){
for(int j = i;j != -1 && marked[j] != 1;j = par[j]){
ret[j] += di[dep[i]-dep[j]];
}
}
}

for(int i = 0;i < n;i++){
ret[i] %= mod;
}

return ret;
}

void collect(int cur, int par, int[][] g, int[] dep, int[] fdep)
{
fdep[dep[cur]]++;
for(int e : g[cur]){
if(e != par)collect(e, cur, g, dep, fdep);
}
}

public static int[][] parents3(int[][] g, int root) {
int n = g.length;
int[] par = new int[n];
Arrays.fill(par, -1);

int[] depth = new int[n];
depth[0] = 0;

int[] q = new int[n];
q[0] = root;
for (int p = 0, r = 1; p < r; p++) {
int cur = q[p];
for (int nex : g[cur]) {
if (par[cur] != nex) {
q[r++] = nex;
par[nex] = cur;
depth[nex] = depth[cur] + 1;
}
}
}
return new int[][] { par, q, depth };
}

public static final int[] NTTPrimes = {1053818881, 1051721729, 1045430273, 1012924417, 1007681537, 1004535809, 998244353, 985661441, 976224257, 975175681};
public static final int[] NTTPrimitiveRoots = {7, 6, 3, 5, 3, 3, 3, 3, 3, 17};

static long[] inputed;
static long[] saved;

public static long[] convoluteSimply(long[] a, long[] b, int P, int g)
{
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
long[] fa = nttmb(a, m, false, P, g);
long[] fb = a == b ? fa : inputed != null ? inputed : nttmb(b, m, false, P, g);
saved = fb;
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i]%P;
}
return nttmb(fa, m, true, P, g);
}

public static long[] convolute(long[] a, long[] b)
{
int USE = 2;
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
long[][] fs = new long[USE][];
for(int k = 0;k < USE;k++){
int P = NTTPrimes[k], g = NTTPrimitiveRoots[k];
long[] fa = nttmb(a, m, false, P, g);
long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i]%P;
}
fs[k] = nttmb(fa, m, true, P, g);
}

int[] mods = Arrays.copyOf(NTTPrimes, USE);
long[] gammas = garnerPrepare(mods);
int[] buf = new int[USE];
for(int i = 0;i < fs[0].length;i++){
for(int j = 0;j < USE;j++)buf[j] = (int)fs[j][i];
long[] res = garnerBatch(buf, mods, gammas);
long ret = 0;
for(int j = res.length-1;j >= 0;j--)ret = ret * mods[j] + res[j];
fs[0][i] = ret;
}
return fs[0];
}

public static long[] convolute(long[] a, long[] b, int USE, int mod)
{
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
long[][] fs = new long[USE][];
for(int k = 0;k < USE;k++){
int P = NTTPrimes[k], g = NTTPrimitiveRoots[k];
long[] fa = nttmb(a, m, false, P, g);
long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i]%P;
}
fs[k] = nttmb(fa, m, true, P, g);
}

int[] mods = Arrays.copyOf(NTTPrimes, USE);
long[] gammas = garnerPrepare(mods);
int[] buf = new int[USE];
for(int i = 0;i < fs[0].length;i++){
for(int j = 0;j < USE;j++)buf[j] = (int)fs[j][i];
long[] res = garnerBatch(buf, mods, gammas);
long ret = 0;
for(int j = res.length-1;j >= 0;j--)ret = (ret * mods[j] + res[j]) % mod;
fs[0][i] = ret;
}
return fs[0];
}

private static long[] nttmb(long[] src, int n, boolean inverse, int P, int g)
{
long[] dst = Arrays.copyOf(src, n);

int h = Integer.numberOfTrailingZeros(n);
long K = Integer.highestOneBit(P)<<1;
int H = Long.numberOfTrailingZeros(K)*2;
long M = K*K/P;

int[] wws = new int[1<<h-1];
long dw = inverse ? pow(g, P-1-(P-1)/n, P) : pow(g, (P-1)/n, P);
long w = (1L<<32)%P;
for(int k = 0;k < 1<<h-1;k++){
wws[k] = (int)w;
w = modh(w*dw, M, H, P);
}
long J = invl(P, 1L<<32);
for(int i = 0;i < h;i++){
for(int j = 0;j < 1<<i;j++){
for(int k = 0, s = j<<h-i, t = s|1<<h-i-1;k < 1<<h-i-1;k++,s++,t++){
long u = (dst[s] - dst[t] + 2*P)*wws[k];
dst[s] += dst[t];
if(dst[s] >= 2*P)dst[s] -= 2*P;
//					long Q = (u&(1L<<32)-1)*J&(1L<<32)-1;
long Q = (u<<32)*J>>>32;
dst[t] = (u>>>32)-(Q*P>>>32)+P;
}
}
if(i < h-1){
for(int k = 0;k < 1<<h-i-2;k++)wws[k] = wws[k*2];
}
}
for(int i = 0;i < n;i++){
if(dst[i] >= P)dst[i] -= P;
}
for(int i = 0;i < n;i++){
int rev = Integer.reverse(i)>>>-h;
if(i < rev){
long d = dst[i]; dst[i] = dst[rev]; dst[rev] = d;
}
}

if(inverse){
long in = invl(n, P);
for(int i = 0;i < n;i++)dst[i] = modh(dst[i]*in, M, H, P);
}

return dst;
}

// Modified Shoup + Barrett
private static long[] nttsb(long[] src, int n, boolean inverse, int P, int g)
{
long[] dst = Arrays.copyOf(src, n);

int h = Integer.numberOfTrailingZeros(n);
long K = Integer.highestOneBit(P)<<1;
int H = Long.numberOfTrailingZeros(K)*2;
long M = K*K/P;

long dw = inverse ? pow(g, P-1-(P-1)/n, P) : pow(g, (P-1)/n, P);
long[] wws = new long[1<<h-1];
long[] ws = new long[1<<h-1];
long w = 1;
for(int k = 0;k < 1<<h-1;k++){
wws[k] = (w<<32)/P;
ws[k] = w;
w = modh(w*dw, M, H, P);
}
for(int i = 0;i < h;i++){
for(int j = 0;j < 1<<i;j++){
for(int k = 0, s = j<<h-i, t = s|1<<h-i-1;k < 1<<h-i-1;k++,s++,t++){
long ndsts = dst[s] + dst[t];
if(ndsts >= 2*P)ndsts -= 2*P;
long T = dst[s] - dst[t] + 2*P;
long Q = wws[k]*T>>>32;
dst[s] = ndsts;
dst[t] = ws[k]*T-Q*P&(1L<<32)-1;
}
}
if(i < h-1){
for(int k = 0;k < 1<<h-i-2;k++){
wws[k] = wws[k*2];
ws[k] = ws[k*2];
}
}
}
for(int i = 0;i < n;i++){
if(dst[i] >= P)dst[i] -= P;
}
for(int i = 0;i < n;i++){
int rev = Integer.reverse(i)>>>-h;
if(i < rev){
long d = dst[i]; dst[i] = dst[rev]; dst[rev] = d;
}
}

if(inverse){
long in = invl(n, P);
for(int i = 0;i < n;i++){
dst[i] = modh(dst[i] * in, M, H, P);
}
}

return dst;
}

static final long mask = (1L<<31)-1;

public static long modh(long a, long M, int h, int mod)
{
return r < mod ? r : r-mod;
}

private static long[] garnerPrepare(int[] m)
{
int n = m.length;
assert n == m.length;
if(n == 0)return new long[0];
long[] gamma = new long[n];
for(int k = 1;k < n;k++){
long prod = 1;
for(int i = 0;i < k;i++){
prod = prod * m[i] % m[k];
}
gamma[k] = invl(prod, m[k]);
}
return gamma;
}

private static long[] garnerBatch(int[] u, int[] m, long[] gamma)
{
int n = u.length;
assert n == m.length;
long[] v = new long[n];
v[0] = u[0];
for(int k = 1;k < n;k++){
long temp = v[k-1];
for(int j = k-2;j >= 0;j--){
temp = (temp * m[j] + v[j]) % m[k];
}
v[k] = (u[k] - temp) * gamma[k] % m[k];
if(v[k] < 0)v[k] += m[k];
}
return v;
}

private 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;
}

private 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 int[][] parentToG(int[] par)
{
int n = par.length;
int[] ct = new int[n];
for(int i = 0;i < n;i++){
if(par[i] >= 0){
ct[i]++;
ct[par[i]]++;
}
}
int[][] g = new int[n][];
for(int i = 0;i < n;i++){
g[i] = new int[ct[i]];
}
for(int i = 0;i < n;i++){
if(par[i] >= 0){
g[par[i]][--ct[par[i]]] = i;
g[i][--ct[i]] = par[i];
}
}
return g;
}

public static int[][] makeBuckets(int[] a, int sup)
{
int n = a.length;
int[][] bucket = new int[sup+1][];
int[] bp = new int[sup+1];
for(int i = 0;i < n;i++)bp[a[i]]++;
for(int i = 0;i <= sup;i++)bucket[i] = new int[bp[i]];
for(int i = n-1;i >= 0;i--)bucket[a[i]][--bp[a[i]]] = i;
return bucket;
}

public static class SplitResult
{
public boolean[] incycle;
public int[] ord;
}

public static SplitResult split(int[] f)
{
int n = f.length;
boolean[] incycle = new boolean[n];
Arrays.fill(incycle, true);
int[] indeg = new int[n];
for(int i = 0;i < n;i++)indeg[f[i]]++;
int[] q = new int[n];
int qp = 0;
for(int i = 0;i < n;i++){
if(indeg[i] == 0)q[qp++] = i;
}
for(int r = 0;r < qp;r++){
int cur = q[r];
indeg[cur] = -9999999;
incycle[cur] = false;
int e = f[cur];
indeg[e]--;
if(indeg[e] == 0)q[qp++] = e;
}
for(int i = 0;i < n;i++){
if(indeg[i] == 1){
q[qp++] = i;
}
}
assert qp == n;
SplitResult ret = new SplitResult();
ret.incycle = incycle;
ret.ord = q;
return ret;
}

void run() throws Exception
{
is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
out = new PrintWriter(System.out);

long s = System.currentTimeMillis();
solve();
out.flush();
if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
}

public static void main(String[] args) throws Exception { new DefiniteRandomWalks().run(); }

private byte[] inbuf = new byte[1024];
public int lenbuf = 0, ptrbuf = 0;

{
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 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))){ // when nextLine, (isSpaceChar(b) && b != ' ')
sb.appendCodePoint(b);
}
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;
}
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 int ni()
{
int num = 0, b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
}

while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
}
}

private long nl()
{
long num = 0;
int b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
}

while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
}
}

private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
}```

{"mode":"full","isActive":false}

Problem solution in C++.

```#include <bits/stdc++.h>
using namespace std;
#define sz(x) ((int) (x).size())
#define forn(i,n) for (int i = 0; i < int(n); ++i)
typedef long long ll;
typedef long long i64;
typedef long double ld;
typedef pair<int, int> pii;
const int inf = int(1e9) + int(1e5);
const ll infl = ll(2e18) + ll(1e10);

const int mod = 998244353;
const int root = 787603194;
const int LG = 19;

int add(int a, int b) {
a += b;
if (a >= mod)
a -= mod;
return a;
}

int sub(int a, int b) {
}

void addTo(int &a, int b) {
a += b;
if (a >= mod)
a -= mod;
}

int mul(ll a, ll b) {
return a * b % mod;
}

int binpow(int a, int deg) {
int res = 1;
while (deg) {
if (deg & 1)
res = mul(res, a);
deg >>= 1;
a = mul(a, a);
}
return res;
}

int rev(int x) {
assert(x != 0);
return binpow(x, mod - 2);
}

int divide(int a, int b) {
return mul(a, rev(b));
}

vector<int> ang[2][LG + 1];

void initFFT() {
int rroot = rev(root);
int x0 = 1, x1 = 1;
ang[0][LG].resize(1 << LG);
ang[1][LG].resize(1 << LG);
forn (i, 1 << LG) {
ang[0][LG][i] = x0;
ang[1][LG][i] = x1;
x0 = mul(x0, root);
x1 = mul(x1, rroot);
}
for (int lg = LG - 1; lg >= 0; --lg) {
forn (q, 2) {
ang[q][lg].resize(1 << lg);
forn (i, 1 << lg)
ang[q][lg][i] = ang[q][lg + 1][i * 2];
}
}
}

void recFFT(int *a, int lg, bool inv) {
if (lg == 0)
return;
int hlen = (1 << lg) >> 1;
recFFT(a, lg - 1, inv);
recFFT(a + hlen, lg - 1, inv);
forn (i, hlen) {
int u = a[i];
int v = mul(ang[inv][lg][i], a[i + hlen]);
a[i + hlen] = sub(u, v);
}
}

void FFT(int *a, int n, bool inv) {
int lg = 0;
while ((1 << lg) < n)
++lg;
assert(n == (1 << lg));
int j = 0, bit;
for (int i = 1; i < n; ++i) {
for (bit = n >> 1; j & bit; bit >>= 1)
j ^= bit;
j ^= bit;
if (i < j)
swap(a[i], a[j]);
}
recFFT(a, lg, inv);
if (inv) {
int rn = rev(n);
forn (i, n)
a[i] = mul(a[i], rn);
}
}

void mul(int *a, int *b, int n) {
FFT(a, n, false);
if (a != b)
FFT(b, n, false);
forn (i, n)
a[i] = mul(a[i], b[i]);
FFT(a, n, true);
}

void cyclicMul(int *a, int *b, int n, int pre, int len) {
mul(a, b, n);
int j = 0;
forn (i, n) {
if (j < i) {
a[i] = 0;
}
++j;
if (j == pre + len)
j -= len;
}
}

const int maxn = 100100;
int ans[maxn];
int to[maxn];
vector<int> g[maxn];
int p[maxn];
bool used[maxn];
map<int, vector<int>> byLen;
int a[1 << LG];
int _a[1 << LG];
int cur[1 << LG];
int N, M, K;

int calcSize(int u, int root) {
int cnt = 1;
for (int v: g[u])
if (v != root)
cnt += calcSize(v, root);
return cnt;
}

int tmp[1 << LG];
void powk(int n, int pre, int len) {
forn (i, n)
a[i] = 0;
a[0] = rev(N);
int deg = K;
while (deg) {
if (deg & 1) {
forn (i, n)
tmp[i] = cur[i];
cyclicMul(a, tmp, n, pre, len);
}
deg >>= 1;
cyclicMul(cur, cur, n, pre, len);
}
}

int ROOT;
int in[maxn];
int out[maxn];
int ord[maxn];
int byh[maxn];
int h[maxn];
int timer;

void dfs1(int u, int ch) {
ord[timer] = u;
in[u] = timer++;
h[u] = ch;
for (int v: g[u]) {
if (v == ROOT)
continue;
dfs1(v, ch + 1);
}
out[u] = timer;
}

const int bs = 3500;

const int blocks = maxn / bs + 2;
int bl[blocks][1 << LG];

void calcBlock(int l, int r, int cnt, int lg, int *to) {
forn (i, 1 << lg)
to[i] = 0;
for (int i = l; i < r; ++i) {
int ch = h[ord[i]];
to[cnt - ch]++;
}
FFT(to, 1 << lg, false);
forn (i, 1 << lg)
to[i] = mul(to[i], _a[i]);
FFT(to, 1 << lg, true);
}

void solveComponent(int root, int m) {
ROOT = root, timer = 0;
dfs1(root, 0);
const int cnt = timer;
forn (i, cnt)
byh[i] = 0;
forn (i, cnt) {
int u = ord[i];
byh[h[u]]++;
}
int lg = 0;
while ((1 << lg) < m + cnt)
++lg;
assert(lg <= LG);
forn (i, m)
_a[i] = a[i];
for (int i = m; i < (1 << lg); ++i)
_a[i] = 0;
FFT(_a, 1 << lg, false);

for (int l = 0; l + bs <= cnt; l += bs)
calcBlock(l, l + bs, cnt, lg, bl[l / bs]);
forn (ii, cnt) {
int u = ord[ii];
int ch = h[u];

int l = in[u];
int r = out[u];
int lto = min(r, ((l + bs - 1) / bs) * bs);
for (; l < lto; ++l)
int rto = max(l, (r / bs) * bs);
for (; r > rto; --r)
addTo(ans[u], a[h[ord[r - 1]] - ch]);
while (l < r) {
addTo(ans[u], bl[l / bs][cnt - ch]);
l += bs;
}
}
calcBlock(0, cnt, cnt, lg, bl[blocks - 1]);
int u = root;
for (int i = cnt + 1; i < (1 << lg); ++i) {
u = to[u];
}
}

void solveLen(int len) {
//cerr << "solveLen " << len << ":";
//for (int x: byLen[len])
//cerr << ' ' << x + 1;
//cerr << '\n';

vector<pii> v;
for (int u: byLen[len])
v.emplace_back(calcSize(u, u), u);
sort(v.begin(), v.end());
reverse(v.begin(), v.end());
int pre = v[0].first;

int lg = 0;
while ((1 << lg) < (pre + len) * 2)
++lg;
assert(lg <= LG);
forn (i, 1 << lg)
cur[i] = 0;
int j = 0;
forn (i, M) {
++j;
if (j == pre + len)
j -= len;
}
powk(1 << lg, pre, len);
for (auto p: v) {
while (p.first <= pre - len) {
for (int i = pre - len; i < pre; ++i) {
a[i + len] = 0;
}
pre -= len;
}
solveComponent(p.second, pre + len);
}
}

void solve() {
forn (i, N) {
int u = i;
vector<int> st;
while (!used[u]) {
st.push_back(u);
used[u] = true;
u = to[u];
}
int len = 1;
while (!st.empty() && st.back() != u)
st.pop_back(), ++len;
if (!st.empty())
byLen[len].push_back(u);
}
for (const auto &p: byLen)
solveLen(p.first);
}

int main() {
#ifdef LOCAL
assert(freopen("test.in", "r", stdin));
#else
#endif
initFFT();

scanf("%d%d%d", &N, &M, &K);
forn (i, N) {
scanf("%d", &to[i]);
--to[i];
g[to[i]].push_back(i);
}
int sum = 0;
forn (i, M) {
scanf("%d", &p[i]);
}
assert(sum == 1);
solve();
sum = 0;
forn (i, N) {
cout << ans[i] << '\n';