# Build a Palindrome

### Problem Statement :

```You have two strings,  and . Find a string, , such that:

can be expressed as  where  is a non-empty substring of  and  is a non-empty substring of .
is a palindromic string.
The length of  is as long as possible.
For each of the  pairs of strings ( and ) received as input, find and print string  on a new line. If you're able to form more than one valid string , print whichever one comes first alphabetically. If there is no valid answer, print  instead.

Input Format

The first line contains a single integer, , denoting the number of queries. The subsequent lines describe each query over two lines:

The first line contains a single string denoting a.
The second line contains a single string denoting  b.

Output Format

For each pair of strings ( ai and bi ), find some si satisfying the conditions above and print it on a new line. If there is no such string, print  - 1  instead.```

### Solution :

```                            ```Solution in C :

In    C++  :

#include <bits/stdc++.h>
using namespace std;

inline void wssert(bool b) { if(!b) exit(0); }

const int MAXN = 2e5;
const int MAXM = 2e5;
const int MAXL = MAXN * 2 + MAXM * 2 + 20;
int N;
char A[MAXN];
int M;
char B[MAXM];
int L;
char V[MAXL];
int S;

int P[MAXL]; // length of palindrome - 1 / 2
int C[MAXL];

void manacher() {
int c = 0, r = 0;
memset(P, 0, sizeof(P));
for(int i = 0; i < L; ) {
assert(i - P[i] >= 0);
assert(i + P[i] < L);
assert(r == c + P[c]);
assert(i >= c);
assert(r >= i + P[i]);
assert(i == c || r > i + P[i]);
if(i == c) {
assert(r == i + P[i]);
if(i - P[i] - 1 >= 0 && i + P[i] + 1 < L && V[i - P[i] - 1] == V[i + P[i] + 1]) {
P[i] ++;
assert(i - P[i] >= 0);
assert(i + P[i] < L);
} else {
i++;
assert(P[i] == 0);
assert(i - P[i] >= 0);
assert(i == L || i + P[i] < L);
}
} else {
assert(i > c);
assert(r > i + P[i]);
assert(c - P[c] >= 0);
assert(c - (i - c) >= 0);
int v = min(P[c - (i - c)], r - i);
assert(v >= P[i]);
if(v > P[i]) {
P[i] = v;
assert(i - P[i] >= 0);
assert(i + P[i] < L);
} else if (v == r - i) {
assert(false);
} else {
i++;
assert(P[i] == 0);
assert(i - P[i] >= 0);
assert(i == L || i + P[i] < L);
}
}
if(i == L) break;
if(i + P[i] >= r) {
c = i;
r = i + P[i];
}
assert(i - P[i] >= 0);
assert(i + P[i] < L);
}
}

#define REP(i, n) for(int i = 0; i < int(n); i++)
int gap;
int sa[MAXL], pos[MAXL], tmp[MAXL], lcp[MAXL];

bool sufCmp(int i, int j)
{
if (pos[i] != pos[j])
return pos[i] < pos[j];
i += gap;
j += gap;
return (i < L && j < L) ? pos[i] < pos[j] : i > j;
}

void buildSA()
{
REP(i, L) sa[i] = i, pos[i] = V[i];
for (gap = 1;; gap *= 2)
{
sort(sa, sa + L, sufCmp);
REP(i, L - 1) tmp[i + 1] = tmp[i] + sufCmp(sa[i], sa[i + 1]);
REP(i, L) pos[sa[i]] = tmp[i];
if (tmp[L - 1] == L - 1) break;
}
}

void buildLCP()
{
for (int i = 0, k = 0; i < L; ++i) if (pos[i] != L - 1)
{
for (int j = sa[pos[i] + 1]; V[i + k] == V[j + k];)
++k;
lcp[pos[i]] = k;
if (k)--k;
}
}

string process(int l, int c) {
string res;
for(int i = l; i < c; i++) {
if(V[i] != 124) res += V[i];
}
for(int i = c; i >= l; i--) {
if(V[i] != 124) res += V[i];
}
return res;
}

string go() {
manacher();
for(int i = 0; i < L; i++) {
assert(!(i - P[i] - 1 >= 0 && i + P[i] + 1 < L && V[i - P[i] - 1] == V[i + P[i] + 1]));
//for(int j = 0; j <= P[i]; j++) assert(V[i - j] == V[i + j]);
}
memset(C, 0, sizeof(C));
for(int i = 0; i < L; i++) {
C[i - P[i]] = max(C[i - P[i]], i);
}
for(int i = 1; i < L; i++) {
C[i] = max(C[i], C[i - 1]);
}
buildSA();
buildLCP();
for(int i = 0; i < L; i++) {
//cerr << (V + sa[i]) << '\n';
}
for(int i = 0; i + 1 < L; i++) {
//for(int j = 0; j < lcp[i]; j++) assert(V[sa[i] + j] == V[sa[i + 1] + j]);
assert(V[sa[i] + lcp[i]] < V[sa[i + 1] + lcp[i]]);
}
assert(V[sa[N + M]] == 123);
int p = {-1, -1};
int l = {0, 0};
pair<int, int> res (0, 0);
for(int i = 0; i < N + M; i++) {
assert(V[sa[i]] < 123);
bool d = (sa[i] >= S);
p[d] = sa[i];
l[d] = L;
if(p[!d] != -1) {
int match = l[!d];
assert(match % 2 == 0);
//cerr << p[!d] << ' ' << p[d] << ' ' << match << '\n';
if(match) res = max(res, make_pair(C[sa[i] + match - 1] - sa[i], -i));
}
if(i + 1 < L) l = min(lcp[i], l), l = min(lcp[i], l);
}
p = p = -1;
l = l = -1;
for(int i = N + M - 1; i >= 0; i--) {
bool d = (sa[i] >= S);
p[d] = sa[i];
l[d] = L;
if(p[!d] != -1) {
int match = l[!d];
assert(match % 2 == 0);
if(match) res = max(res, make_pair(C[sa[i] + match - 1] - sa[i], -i));
}
if(i > 0) l = min(lcp[i - 1], l), l = min(lcp[i-1], l);
}
//cerr << res << '\n';
return process(sa[-res.second], sa[-res.second] + res.first);
}

int main() {
int Q; cin >> Q;
while(Q --> 0) {
cin >> A >> B;
N = strlen(A), M = strlen(B);
L = 0;
for(int i = 0; i < N; i++) {
V[L++] = A[i];
V[L++] = 124;
}
V[L++] = 123;
V[L++] = 125;
S = L;
for(int i = M - 1; i >= 0; i--) {
V[L++] = B[i];
V[L++] = 124;
}
V[L] = 0;
assert(L == 2 * N + 2 * M + 2);
cout << go() << '\n';
}
return 0;
}

In   Java  :

import java.util.*;

public class PalindromeBuilder {
public static class State {
int length;
int[] next = new int;
int endpos;

public State()
{
Arrays.fill(next, -1);
}
}

public static State[] buildSuffixAutomaton(String s) {
int n = s.length();
State[] st = new State[Math.max(2, 2 * n - 1)];
st = new State();
st.endpos = -1;
int last = 0;
int size = 1;
for (char c : s.toCharArray()) {
int cur = size++;
st[cur] = new State();
st[cur].length = st[last].length + 1;
st[cur].endpos = st[last].length;

int p = go(st, last, -1, c, cur);
if (p == -1) {
} else {
int q = st[p].next[c];
if (st[p].length + 1 == st[q].length)
else {
int clone = size++;
st[clone] = new State();
st[clone].length = st[p].length + 1;
st[clone].next = st[q].next.clone();
go(st, p, q, c, clone);
st[clone].endpos = -1;
}
}
last = cur;
}
for (int i = 1; i < size; i++) {
}
return Arrays.copyOf(st, size);
}

private static int go(State[] st,
int p, int q, char c, int ns) {
while (p != -1 && st[p].next[c] == q) {
st[p].next[c] = ns;
}
return p;
}

public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
int n = sc.nextInt();
for (int i = 0; i < n; ++i) {
String a = sc.next();
String b = sc.next();
System.out.println(solve(a, b));
}
}

static String candidate(String a, String b) {
State[] as = buildSuffixAutomaton(a);
int[] l = buildPalindromeLookup(b);

int len = 0;

int bestHalf = 0;
int bestMid = 0;
int bestTotal = 0;
int start = -1;
for (int i = 0, aPos = 0; i < b.length(); ++i) {
char c = b.charAt(i);
if (as[aPos].next[c] == -1) {
while (aPos != -1 && as[aPos].next[c] == -1) {
}
if (aPos == -1) {
aPos = 0;
len = 0;
continue;
}
len = as[aPos].length;
}
++len;
aPos = as[aPos].next[c];

int nStart = i - len + 1;
int nMid = 0;
if (i + 1 < b.length()) {
nMid = l[i + 1];
}
int nTotal = 2*len + nMid;

if (bestTotal < nTotal || (
bestTotal == nTotal && gt(
b, start, nStart, len + nMid))) {
bestHalf = len;
bestMid = nMid;
bestTotal = nTotal;
start = nStart;
}
}
StringBuilder sb = new StringBuilder();
for (int i = 0; i < bestHalf + bestMid; ++i) {
sb.append(b.charAt(start + i));
}
for (int i = bestHalf - 1; i >= 0; --i) {
sb.append(sb.charAt(i));
}
return sb.toString();
}

static String solve(String a, String b) {
String rb = rev(b);
String res = candidate(a, rb);
String c1 = candidate(rb, a);
if (c1.length() > res.length() || (
c1.length() == res.length() && c1.compareTo(res) < 0)) {
res = c1;
}
if (res.length() == 0) {
res = "-1";
}
return res;
}

static String rev(String s) {
StringBuilder sb = new StringBuilder();
for (int i = s.length() - 1; i >= 0; --i) {
sb.append(s.charAt(i));
}
return sb.toString();
}

static boolean gt(String s,
int start, int nStart, int size) {
int cmp = 0;
for (int i = 0; i < size; ++i) {
cmp = Character.compare(
s.charAt(start + i), s.charAt(nStart + i));
if (cmp != 0) {
break;
}
}
return cmp > 0;
}

static int[] buildPalindromeLookup(String s) {
int[] p = new int[s2.length];
int c = 0, r = 0;
int m = 0, n = 0;
for (int i = 1; i < s2.length; i++) {
if (i > r) {
p[i] = 0;
m = i - 1;
n = i + 1;
} else {
int i2 = c * 2 - i;
if (p[i2] < (r-i)) {
p[i] = p[i2];
m = -1;
} else {
p[i] = r - i;
n = r + 1;
m = i * 2 - n;
}
}
while (m >= 0 && n < s2.length && s2[m] == s2[n]) {
p[i]++;
m--;
n++;
}
if ((i + p[i]) > r) {
c = i;
r = i + p[i];
}
}
int[] res = new int[s.length()];
for (int i = 1; i < s2.length - 1; i++) {
int idx = (i - p[i])/2;
res[idx] = Math.max(res[idx], p[i]);
}
return res;
}

private static char[] addBoundaries(char[] cs) {
if (cs == null || cs.length == 0)
return "||".toCharArray();

char[] cs2 = new char[cs.length * 2 + 1];
for (int i = 0; i < cs2.length - 1; i += 2) {
cs2[i] = '|';
cs2[i + 1] = cs[i / 2];
}
cs2[cs2.length - 1] = '|';
return cs2;
}
}```
```

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