In this HackerRank Circular Palindromes problem solution we have given N and S, find all N k-length rotations of s; for each rotated string, Sk, print the maximum possible length of any palindromic substring of Sk on a new line.

HackerRank Circular Palindromes problem solution


Problem solution in Python.

#!/bin/python3

import os
import sys
import random
import bisect

#
# Complete the circularPalindromes function below.
#
def get_len(s, ll, flag):
    # use flag = 0 for odd number of letters in palindrome, 1 for even
    maxlen = 1
    l1 = ll - 2
    l2 = ll + 1 + flag
    while l1>=0 and l2 < len(s) and s[l1] == s[l2]:
        maxlen += 1
        l1 -= 1
        l2 += 1
    return 2*maxlen + flag

def max_pal(s):
    # find the length of the longest palindrome in s
    ls = len(s)
    maxlen = 1 
    for ll in range(1, ls):
        if s[ll-1] == s[ll]:
            newlen = get_len(s, ll, 0)
            if newlen > maxlen:
                maxlen = newlen
    for ll in range(1, ls-1):
        if s[ll-1] == s[ll+1]:
            newlen = get_len(s, ll, 1)
            if newlen > maxlen:
                maxlen = newlen
    return maxlen

def get_len_round_fast(slist, ll, lens):
    ls = len(slist)
    if ls == 1:
        return (slist[0][1], slist[0][2])
    start = slist[ll][1]
    end = slist[ll][2]
    l1 = ll - 1
    l2 = ll + 1
    notdone = True
    while notdone and (end - start)<lens and slist[l1 % ls][0] == slist[l2 % ls][0]:
        lgth1 = slist[l1][2]-slist[l1][1]
        if lgth1 < 0:
            lgth1 += lens
        ls2 = l2 % ls
        lgth2 = slist[ls2][2]-slist[ls2][1]
        if lgth2 < 0:
            lgth2 += lens
        lmax = lens - (end-start)
        if lgth1 != lgth2:
            notdone = False
        # make lgth2 the smaller for subsequent calculations
        if lgth1 < lgth2:
            lgth2 = lgth1
        if lgth2 + lgth2 > lmax:
            lgth2 = lmax//2
            notdone = False
        end+= lgth2
        start -= lgth2
        l1 -= 1
        l2 += 1
#        print(l1, l2)
    return (start, end)
    
def compress_string(s):
    # replaces strings of contiguous identical characters with (char, #) pairs
    #   where # is the end of the string sequence
    ls = []
    cc = '.'
    start = 0
    for ss in range(len(s)):    
        if s[ss] != cc: # new char
            ls.append((cc, start, ss))
            start = ss
            cc = s[ss]
    ls.append((cc, start, len(s))) # append the last characters encountered
    ls.pop(0) # first value is a throwaway one
    if ls[0][0] == ls[-1][0]: # stitch the ends, move the start of sequence before 0
        ls[0] = (ls[0][0], ls[-1][1]-len(s), ls[0][2])
        ls.pop() # remove last element, now that it is combined with the first
    return ls

        
def make_pal_dict(slist, lens):
    ls = len(slist)
    dict1 = {}
    list1 = []
    for ll in range(ls):
        (start, stop) = get_len_round_fast(slist, ll, lens)
#        print(ll, start, stop)
        lgth = stop - start
        if lgth > 1:
            if start < 0:
                start, stop = start+lens, stop+lens
            if lgth in dict1:
                dict1[lgth].append((start, stop))
            else:
                dict1[lgth]= [(start, stop)]
                bisect.insort(list1, lgth)
    for (_, start, stop) in slist:
        lgth = stop - start
        if lgth > 1:
            if start < 0:
                start, stop = start+lens, stop+lens
            if lgth in dict1:
                if (start, stop) in dict1[lgth]:
                    dict1[lgth].remove((start, stop))
                dict1[lgth].append((-start, -stop))
            else:
                dict1[lgth]= [(-start, -stop)]
                bisect.insort(list1, lgth)        
    return (dict1, list1)

        
def cp(s):
    ls = len(s)
    slist = compress_string(s)
#    print(slist)
    (dict1, list1) = make_pal_dict(slist, ls)
#    print(dict1, list1)
    maxes = [] # value for k = 0
    ll = len(list1)-1 # start here to look for longest palindrome 
    for k in range(ls):
        maxlgth = 1 
        done = False
        ks = k + ls
        for ind in range(ll, -1, -1): # go backwards through list of lengths
            if done: # max value already reached for a longer word
                break
            lgth = list1[ind]
            if lgth < maxlgth: # only shorter words available past here
                break
            for (start, stop) in dict1[lgth]:
                if start<=0 and stop <= 0: # same chars, no need to cut palindrome at both ends
#                    print(start, stop, k)
                    if -start <= k < -stop:
                        lgth1 = max(-stop - k, k+start) 
                    else:
                        lgth1 = lgth
                    if -start < ks <= -stop:
                        lgth2 = max(ks + start, -stop - ks)
                    else:
                        lgth2 = lgth
                    #print(lgth1, lgth2)
                else:
#                print(lgth, maxlgth, k, start, stop)
                    if start <= k <= stop:
                        lgth1 = abs(start+stop-k-k)
                    else:
                        lgth1 = lgth
                    if start <= ks <= stop:
                        lgth2 = abs(start+stop-ks-ks)
                    else:
                        lgth2 = lgth
                if lgth1 > lgth2:
                    lgth1 = lgth2
                if maxlgth < lgth1:
                    maxlgth = lgth1
#                    print(maxlgth)
                if lgth1 == lgth:
                    done = True
                    break
#        print("k=", k, "ml=", maxlgth)
        maxes.append(maxlgth)
    return maxes

def circularPalindromes(s):
    #
    # Write your code here.
    #
    debug = 0
    if debug == 0:
        return cp(s)
    elif debug == 1:
        for ii in range(1000):
            s = ""
            for jj in range(10):
                s += chr(97 + random.randrange(0, 26))
            r1 = cp(s) 
            r = cp_gold(s)
            if r1 != r:
                print(r1, "should be", r, "for", s)
        return [1, 1, 1]
    elif debug == 2:
        return cp_gold(s)
    else:
        (dict1, list1) = make_pal_list(s)
        print(dict1, list1)
        return cp_gold(s)
                
    
def rotate_s(s, val):
    val = val % len(s)
    return s[val:]+s[:val]
    
def cp_gold(s):
    #
    # Write your code here.
    #
    ls = len(s)
    maxes = [max_pal(s)]
    for ll in range(1, ls):
        s = rotate_s(s, 1)
        maxes.append(max_pal(s))
    return maxes

    
if __name__ == '__main__':
    fptr = open(os.environ['OUTPUT_PATH'], 'w')

    n = int(input())

    s = input()

    result = circularPalindromes(s)

    fptr.write('\n'.join(map(str, result)))
    fptr.write('\n')

    fptr.close()

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


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.InputMismatchException;

public class E2 {
    InputStream is;
    PrintWriter out;
    String INPUT = "";
    
    void solve()
    {
        int n = ni();
        char[] s = ns(n);
        char[] s2 = new char[2*n];
        for(int i = 0;i < n;i++){
            s2[i] = s2[i+n] = s[i];
        }
        int[] pal = palindrome(s2);
//        tr(pal, pal.length, n);
        long[] es = new long[16*n];
        int p = 0;
        for(int i = 0;i < 4*n;i+=2){
            pal[i] = Math.min(pal[i], n-((n&1)^1));
            es[p++] = (long)(i/2)<<32|i;
            es[p++] = (long)(i/2+pal[i]/2)<<32|i;
            es[p++] = (long)(i/2+n-pal[i]/2-1)<<32|i;
            es[p++] = (long)(i/2+n)<<32|i;
        }
        for(int i = 1;i < 4*n;i+=2){
            pal[i] = Math.min(pal[i], n-((n&1)));
            es[p++] = (long)(i/2)<<32|i;
            es[p++] = (long)(i/2+pal[i]/2)<<32|i;
            es[p++] = (long)(i/2+n-pal[i]/2)<<32|i;
            es[p++] = (long)(i/2+n)<<32|i;
        }
        
        Arrays.sort(es, 0, p);
        MaxHeap inc = new MaxHeap(4*n+1);
        MaxHeap dec = new MaxHeap(4*n+1);
        MaxHeap flat = new MaxHeap(4*n+1);
        
        int[] st = new int[4*n];
        int q = 0;
        for(int i = 0;i < 2*n-1;i++){
            while(q < p && es[q]>>>32 <= i){
                int ind = (int)es[q];
                if(st[ind] == 0){
                    inc.add(ind, (pal[ind]&1)-2*i);
                }else if(st[ind] == 1){
                    inc.remove(ind);
                    flat.add(ind, pal[ind]);
                }else if(st[ind] == 2){
                    flat.remove(ind);
                    dec.add(ind, pal[ind]+2*i);
                }else if(st[ind] == 3){
                    dec.remove(ind);
                }
                st[ind]++;
                q++;
            }
            if(i >= n-1){
//                tr("i", i);
                int max = 0;
                if(inc.size() > 0)max = Math.max(inc.max()+2*i, max);
//                tr(max);
                if(dec.size() > 0)max = Math.max(dec.max()-2*i, max);
//                tr(max);
                max = Math.max(flat.max(), max);
//                tr(max);
                out.println(max);
            }
        }
    }
    public static class MaxHeap {
        public int[] a;
        public int[] map;
        public int[] imap;
        public int n;
        public int pos;
        public static int INF = Integer.MIN_VALUE;
        
        public MaxHeap(int m)
        {
            n = m+2;
            a = new int[n];
            map = new int[n];
            imap = new int[n];
            Arrays.fill(a, INF);
            Arrays.fill(map, -1);
            Arrays.fill(imap, -1);
            pos = 1;
        }
        
        public int add(int ind, int x)
        {
            int ret = imap[ind];
            if(imap[ind] < 0){
                a[pos] = x; map[pos] = ind; imap[ind] = pos;
                pos++;
                up(pos-1);
            }
            return ret != -1 ? a[ret] : x;
        }
        
        public int update(int ind, int x)
        {
            int ret = imap[ind];
            if(imap[ind] < 0){
                a[pos] = x; map[pos] = ind; imap[ind] = pos;
                pos++;
                up(pos-1);
            }else{
                int o = a[ret];
                a[ret] = x;
                up(ret);
                down(ret);
//                if(a[ret] < o){
//                    up(ret);
//                }else{
//                    down(ret);
//                }
            }
            return x;
        }
        
        public int remove(int ind)
        {
            if(pos == 1)return INF;
            if(imap[ind] == -1)return INF;
            
            pos--;
            int rem = imap[ind];
            int ret = a[rem];
            map[rem] = map[pos];
            imap[map[pos]] = rem;
            imap[ind] = -1;
            a[rem] = a[pos];
            a[pos] = INF;
            map[pos] = -1;
            
            up(rem);
            down(rem);
            return ret;
        }
        
        public int max() { return a[1]; }
        public int argmax() { return map[1]; }
        public int size() {    return pos-1; }
        
        private void up(int cur)
        {
            for(int c = cur, p = c>>>1;p >= 1 && a[p] < a[c];c>>>=1, p>>>=1){
                int d = a[p]; a[p] = a[c]; a[c] = d;
                int e = imap[map[p]]; imap[map[p]] = imap[map[c]]; imap[map[c]] = e;
                e = map[p]; map[p] = map[c]; map[c] = e;
            }
        }
        
        private void down(int cur)
        {
            for(int c = cur;2*c < pos;){
                int b = a[2*c] > a[2*c+1] ? 2*c : 2*c+1;
                if(a[b] > a[c]){
                    int d = a[c]; a[c] = a[b]; a[b] = d;
                    int e = imap[map[c]]; imap[map[c]] = imap[map[b]]; imap[map[b]] = e;
                    e = map[c]; map[c] = map[b]; map[b] = e;
                    c = b;
                }else{
                    break;
                }
            }
        }
    }
    
    public static int[] palindrome(char[] str)
    {
        int n = str.length;
        int[] r = new int[2*n];
        int k = 0;
        for(int i = 0, j = 0;i < 2*n;i += k, j = Math.max(j-k, 0)){
            // normally
            while(i-j >= 0 && i+j+1 < 2*n && str[(i-j)/2] == str[(i+j+1)/2])j++;
            r[i] = j;
            
            // skip based on the theorem
            for(k = 1;i-k >= 0 && r[i]-k >= 0 && r[i-k] != r[i]-k;k++){
                r[i+k] = Math.min(r[i-k], r[i]-k);
            }
        }
        return r;
    }

    
    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 E2().run(); }
    
    private byte[] inbuf = new byte[1024];
    private 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 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);
            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 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();
        }
    }
    
    private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
}

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


Problem solution in C++.

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
#include <stdio.h>
#include <string.h>

using namespace std; 

const int MAXN = 5e5, MAXX = MAXN << 1, MAXLEAVES = 1 << 19; 
char S[MAXX + 1]; int R[2][MAXX + 1], fsize, T[2][MAXX + 1]; 

void manacher(const int length, const int rx) { 
    register int i, j, k; int *table = R[rx]; 
    for (i = j = 0; i < length; i += k, j = max(j - k, 0)) { 
        while (i - j >= 0 && i + j + rx < length && S[i - j] == S[i + j + rx]) 
            ++j; 
        table[i] = j; 
        for (k = 1; k < j && table[i - k] != table[i] - k; ++k) { 
            table[i + k] = min(table[i - k], table[i] - k);
        } 
    } 
} 

int tree[(MAXLEAVES << 1) + 1], rval, ileft, iright, leaves;

void update_tree(const int x, const int left, const int right) { 
    if (ileft > right || iright < left) { return; } if (ileft <= left && right <= iright) { 
        //node completely inside the update interval 
        tree[x] = max(tree[x], rval); 
    } else { 
        int mid = (left + right) >> 1; 
        update_tree(x << 1, left, mid); 
        update_tree((x << 1) + 1, mid + 1, right); 
    } 
} 

void adjust_and_gather(const int pos, const int parity) { 
    int &ptr = R[parity][pos]; 
    int diff = (ptr << 1) - (!parity) - fsize, lx, rx, flen; 
    if (diff > 0) { 
        diff += (diff & 1);
        ptr -= diff >> 1; 
    } 
    lx = pos - ptr + 1;
    rx = pos + ptr - (parity == 0); flen = (ptr << 1) - (!parity); 
    if ((parity == 0 && ptr > 1) || (parity == 1 && ptr > 0)) { 
        T[0][lx] = max(T[0][lx], flen); T[1][rx] = max(T[1][rx], flen);
        ileft = max(0, rx - fsize + 1), iright = min(fsize - 1, lx); 
        rval = flen; 
        update_tree(1, 0, leaves - 1); 
    } 
} 

inline void process_manacher_table(const int length) { 
    for (register int i = 0; i < length; ++i) { 
        adjust_and_gather(i, 0); //odd length; 
        adjust_and_gather(i, 1); //even length; 
    } 
}

inline int read_tree(const int pos) { 
    register int x = pos + leaves; 
    int result = 1; 
    while (x) { 
        result = max(result, tree[x]);
        x >>= 1; 
    } 
    return result; 
}

inline void init_leaves(const int n) { 
    leaves = 1; while (leaves < n) 
    { 
        leaves <<= 1; 
    } 
}

int main() { 
    register int N, i, answer; 
    scanf("%d%s", &N, S); 
    memcpy(S + N, S, (N - 1) * sizeof(char)); 
    init_leaves(N); 
    fsize = N, N = (N << 1) - 1; 
    manacher(N, 0); //odd length;
    manacher(N, 1); //even length; 
    process_manacher_table(N); 
    for (i = 1; i < N; ++i) {
        T[0][i] = max(T[0][i], T[0][i - 1] - 2); 
        T[1][N - i - 1] = max(T[1][N - i - 1], T[1][N - i] - 2); 
    } 
    for (i = 0; i < fsize; ++i) { 
        answer = read_tree(i); 
        answer = max(answer, T[0][i]);
        answer = max(answer, T[1][i + fsize - 1]); 
        printf("%d\n", answer); 
    } 
    return 0;
}

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


Problem solution in C.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
void solve(char *str,int *a);
void init( int n );
void range_increment( int i, int j, int val );
int query( int i );
int max(int x,int y);
void update(int x,int y,int z);
void sort_a2(int*a,int*b,int size);
void merge2(int*a,int*left_a,int*right_a,int*b,int*left_b,int*right_b,int left_size, int right_size);
char str[1000001]={0};
int N,NN,a[2000004],tree[2000000],ans[500000],b[500000],c[500000];

int main(){
  int i,j;
  scanf("%d%s",&NN,str);
  strncpy(str+NN,str,NN);
  solve(str,a);
  init(NN);
  for(i=0;i<4*NN;i++)
    if(a[i])
      if(i%2)
        update(i/2-a[i]/2,i/2+a[i]/2,a[i]);
      else
        update(i/2-a[i]/2,i/2+a[i]/2-1,a[i]);
  for(i=0;i<NN;i++){
    ans[i]=query(i);
    b[i]=ans[i];
    c[i]=i;
  }
  sort_a2(b,c,NN);
  for(i=NN;i>=0;i--){
    for(j=c[i];1;j=(j-1+NN)%NN)
      if(ans[j]-ans[(j-1+NN)%NN]>2)
        ans[(j-1+NN)%NN]=ans[j]-2;
      else
        break;
    for(j=c[i];1;j=(j+1)%NN)
      if(ans[j]-ans[(j+1)%NN]>2)
        ans[(j+1)%NN]=ans[j]-2;
      else
        break;
  }
  for(i=0;i<NN;i++)
    printf("%d\n",ans[i]);
  return 0;
}
void solve(char *str,int *a){
  char *p;
  int len,R,Ri,i,j,mi;
  len=strlen(str);
  p=(char*)malloc(2*(len+1)*sizeof(char));
  for(i=0;i<len;i++){
    p[2*i]='#';
    p[2*i+1]=str[i];
  }
  p[2*i]='#';
  p[2*i+1]=0;
  a[0]=R=Ri=0;
  for(i=1;i<=len*2;i++)
    if(i>=R){
      if(p[i]!='#')
        a[i]=1;
      else
        a[i]=0;
      for(j=i+1;1;j++)
        if(j<=2*len && 2*i-j>=0 && p[j]==p[2*i-j]){
          if(p[j]!='#')
            a[i]+=2;
        }
        else{
          Ri=i;
          R=j-1;
          break;
        }
    }
    else{
      mi=2*Ri-i;
      if(i+a[mi]>=R || mi==a[mi]){
        a[i]=R-i;
        for(j=R+1;1;j++)
          if(j<=2*len && 2*i-j>=0 && p[j]==p[2*i-j]){
            if(p[j]!='#')
              a[i]+=2;
          }
          else{
            Ri=i;
            R=j-1;
            break;
          }
      }
      else
        a[i]=a[mi];
    }
  free(p);
  return;
}
void init( int n ){
  N = 1;
  while( N < n ) N *= 2;
  int i;
  for( i = 1; i < N + n; i++ ) tree[i] = 0;
}
void range_increment( int i, int j, int val ){
  for( i += N, j += N; i <= j; i = ( i + 1 ) / 2, j = ( j - 1 ) / 2 )
  {
    if( i % 2 == 1 ) tree[i] = max(tree[i],val);
    if( j % 2 == 0 ) tree[j] = max(tree[j],val);
  }
}
int query( int i ){
  int ans = 0,j;
  for( j = i + N; j; j /= 2 ) ans = max(ans,tree[j]);
  return ans;
}
int max(int x,int y){
  return (x>y)?x:y;
}
void update(int x,int y,int z){
  if(z>NN){
    int m=x+z/2;
    if(z%2)
      if(NN%2)
        update(m-NN/2,m+NN/2,NN);
      else
        update(m-NN/2+1,m+NN/2-1,NN-1);
    else
      if(NN%2)
        update(m-NN/2,m+NN/2-1,NN-1);
      else
        update(m-NN/2,m+NN/2-1,NN);
  }
  if(y<NN){
    range_increment(0,x,z);
    range_increment(y+1,NN-1,z);
  }
  else
    range_increment(y-NN+1,x,z);
  return;
}
void sort_a2(int*a,int*b,int size){
  if (size < 2)
    return;
  int m = (size+1)/2,i;
  int*left_a,*left_b,*right_a,*right_b;
  left_a=(int*)malloc(m*sizeof(int));
  right_a=(int*)malloc((size-m)*sizeof(int));
  left_b=(int*)malloc(m*sizeof(int));
  right_b=(int*)malloc((size-m)*sizeof(int));
  for(i=0;i<m;i++){
    left_a[i]=a[i];
    left_b[i]=b[i];
  }
  for(i=0;i<size-m;i++){
    right_a[i]=a[i+m];
    right_b[i]=b[i+m];
  }
  sort_a2(left_a,left_b,m);
  sort_a2(right_a,right_b,size-m);
  merge2(a,left_a,right_a,b,left_b,right_b,m,size-m);
  free(left_a);
  free(right_a);
  free(left_b);
  free(right_b);
  return;
}
void merge2(int*a,int*left_a,int*right_a,int*b,int*left_b,int*right_b,int left_size, int right_size){
  int i = 0, j = 0;
  while (i < left_size|| j < right_size) {
    if (i == left_size) {
      a[i+j] = right_a[j];
      b[i+j] = right_b[j];
      j++;
    } else if (j == right_size) {
      a[i+j] = left_a[i];
      b[i+j] = left_b[i];
      i++;
    } else if (left_a[i] <= right_a[j]) {
      a[i+j] = left_a[i];
      b[i+j] = left_b[i];
      i++;
    } else {
      a[i+j] = right_a[j];
      b[i+j] = right_b[j];
      j++;
    }
  }
  return;
}

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