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test/yukicoder/4_2.test.cpp
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- Last update: 2023-11-09 00:59:01+09:00
- Problem: https://yukicoder.me/problems/no/4
Depends on
Code
#define PROBLEM "https://yukicoder.me/problems/no/4"
#include "my_template.hpp"
#include "other/io.hpp"
#include "knapsack/subset_sum.hpp"
void solve() {
LL(N);
VEC(int, A, N);
ll S = SUM<int>(A);
auto I = subset_sum<int>(A, S / 2);
bool can = (S == 0 || len(I) > 0);
if (can && S % 2 == 0) {
print("possible");
} else {
print("impossible");
}
}
signed main() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << setprecision(15);
ll T = 1;
// LL(T);
FOR(_, T) solve();
return 0;
}#line 1 "test/yukicoder/4_2.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/4"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;
template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;
using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;
#define vv(type, name, h, ...) \
vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) \
vector<vector<vector<type>>> name( \
h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...) \
vector<vector<vector<vector<type>>>> name( \
a, vector<vector<vector<type>>>( \
b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))
// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)
#define FOR_subset(t, s) \
for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if
#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second
#define stoi stoll
int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
template <typename T>
T floor(T a, T b) {
return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
T q = floor(x, y);
return {q, x - q * y};
}
template <typename T, typename U>
T SUM(const vector<U> &A) {
T sm = 0;
for (auto &&a: A) sm += a;
return sm;
}
#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()
template <typename T>
T POP(deque<T> &que) {
T a = que.front();
que.pop_front();
return a;
}
template <typename T>
T POP(pq<T> &que) {
T a = que.top();
que.pop();
return a;
}
template <typename T>
T POP(pqg<T> &que) {
T a = que.top();
que.pop();
return a;
}
template <typename T>
T POP(vc<T> &que) {
T a = que.back();
que.pop_back();
return a;
}
template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
if (check_ok) assert(check(ok));
while (abs(ok - ng) > 1) {
auto x = (ng + ok) / 2;
(check(x) ? ok : ng) = x;
}
return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
FOR(iter) {
double x = (ok + ng) / 2;
(check(x) ? ok : ng) = x;
}
return (ok + ng) / 2;
}
template <class T, class S>
inline bool chmax(T &a, const S &b) {
return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
return (a > b ? a = b, 1 : 0);
}
// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
vc<int> A(S.size());
FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
return A;
}
template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
int N = A.size();
vector<T> B(N + 1);
FOR(i, N) { B[i + 1] = B[i] + A[i]; }
if (off == 0) B.erase(B.begin());
return B;
}
// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
vector<int> ids(len(A));
iota(all(ids), 0);
sort(all(ids),
[&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
return ids;
}
// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
vc<T> B(len(I));
FOR(i, len(I)) B[i] = A[I[i]];
return B;
}
#endif
#line 1 "other/io.hpp"
#define FASTIO
#include <unistd.h>
// https://judge.yosupo.jp/submission/21623
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
char num[10000][4];
constexpr Pre() : num() {
for (int i = 0; i < 10000; i++) {
int n = i;
for (int j = 3; j >= 0; j--) {
num[i][j] = n % 10 | '0';
n /= 10;
}
}
}
} constexpr pre;
inline void load() {
memcpy(ibuf, ibuf + pil, pir - pil);
pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
pil = 0;
if (pir < SZ) ibuf[pir++] = '\n';
}
inline void flush() {
fwrite(obuf, 1, por, stdout);
por = 0;
}
void rd(char &c) {
do {
if (pil + 1 > pir) load();
c = ibuf[pil++];
} while (isspace(c));
}
void rd(string &x) {
x.clear();
char c;
do {
if (pil + 1 > pir) load();
c = ibuf[pil++];
} while (isspace(c));
do {
x += c;
if (pil == pir) load();
c = ibuf[pil++];
} while (!isspace(c));
}
template <typename T>
void rd_real(T &x) {
string s;
rd(s);
x = stod(s);
}
template <typename T>
void rd_integer(T &x) {
if (pil + 100 > pir) load();
char c;
do
c = ibuf[pil++];
while (c < '-');
bool minus = 0;
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (c == '-') { minus = 1, c = ibuf[pil++]; }
}
x = 0;
while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (minus) x = -x;
}
}
void rd(int &x) { rd_integer(x); }
void rd(ll &x) { rd_integer(x); }
void rd(i128 &x) { rd_integer(x); }
void rd(u32 &x) { rd_integer(x); }
void rd(u64 &x) { rd_integer(x); }
void rd(u128 &x) { rd_integer(x); }
void rd(double &x) { rd_real(x); }
void rd(long double &x) { rd_real(x); }
void rd(f128 &x) { rd_real(x); }
template <class T, class U>
void rd(pair<T, U> &p) {
return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T>
void rd_tuple(T &t) {
if constexpr (N < std::tuple_size<T>::value) {
auto &x = std::get<N>(t);
rd(x);
rd_tuple<N + 1>(t);
}
}
template <class... T>
void rd(tuple<T...> &tpl) {
rd_tuple(tpl);
}
template <size_t N = 0, typename T>
void rd(array<T, N> &x) {
for (auto &d: x) rd(d);
}
template <class T>
void rd(vc<T> &x) {
for (auto &d: x) rd(d);
}
void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
rd(h), read(t...);
}
void wt(const char c) {
if (por == SZ) flush();
obuf[por++] = c;
}
void wt(const string s) {
for (char c: s) wt(c);
}
void wt(const char *s) {
size_t len = strlen(s);
for (size_t i = 0; i < len; i++) wt(s[i]);
}
template <typename T>
void wt_integer(T x) {
if (por > SZ - 100) flush();
if (x < 0) { obuf[por++] = '-', x = -x; }
int outi;
for (outi = 96; x >= 10000; outi -= 4) {
memcpy(out + outi, pre.num[x % 10000], 4);
x /= 10000;
}
if (x >= 1000) {
memcpy(obuf + por, pre.num[x], 4);
por += 4;
} else if (x >= 100) {
memcpy(obuf + por, pre.num[x] + 1, 3);
por += 3;
} else if (x >= 10) {
int q = (x * 103) >> 10;
obuf[por] = q | '0';
obuf[por + 1] = (x - q * 10) | '0';
por += 2;
} else
obuf[por++] = x | '0';
memcpy(obuf + por, out + outi + 4, 96 - outi);
por += 96 - outi;
}
template <typename T>
void wt_real(T x) {
ostringstream oss;
oss << fixed << setprecision(15) << double(x);
string s = oss.str();
wt(s);
}
void wt(int x) { wt_integer(x); }
void wt(ll x) { wt_integer(x); }
void wt(i128 x) { wt_integer(x); }
void wt(u32 x) { wt_integer(x); }
void wt(u64 x) { wt_integer(x); }
void wt(u128 x) { wt_integer(x); }
void wt(double x) { wt_real(x); }
void wt(long double x) { wt_real(x); }
void wt(f128 x) { wt_real(x); }
template <class T, class U>
void wt(const pair<T, U> val) {
wt(val.first);
wt(' ');
wt(val.second);
}
template <size_t N = 0, typename T>
void wt_tuple(const T t) {
if constexpr (N < std::tuple_size<T>::value) {
if constexpr (N > 0) { wt(' '); }
const auto x = std::get<N>(t);
wt(x);
wt_tuple<N + 1>(t);
}
}
template <class... T>
void wt(tuple<T...> tpl) {
wt_tuple(tpl);
}
template <class T, size_t S>
void wt(const array<T, S> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i) wt(' ');
wt(val[i]);
}
}
template <class T>
void wt(const vector<T> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i) wt(' ');
wt(val[i]);
}
}
void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
wt(head);
if (sizeof...(Tail)) wt(' ');
print(forward<Tail>(tail)...);
}
// gcc expansion. called automaticall after main.
void __attribute__((destructor)) _d() { flush(); }
} // namespace fastio
using fastio::read;
using fastio::print;
using fastio::flush;
#define INT(...) \
int __VA_ARGS__; \
read(__VA_ARGS__)
#define LL(...) \
ll __VA_ARGS__; \
read(__VA_ARGS__)
#define U32(...) \
u32 __VA_ARGS__; \
read(__VA_ARGS__)
#define U64(...) \
u64 __VA_ARGS__; \
read(__VA_ARGS__)
#define STR(...) \
string __VA_ARGS__; \
read(__VA_ARGS__)
#define CHAR(...) \
char __VA_ARGS__; \
read(__VA_ARGS__)
#define DBL(...) \
double __VA_ARGS__; \
read(__VA_ARGS__)
#define VEC(type, name, size) \
vector<type> name(size); \
read(name)
#define VV(type, name, h, w) \
vector<vector<type>> name(h, vector<type>(w)); \
read(name)
void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 2 "ds/my_bitset.hpp"
// https://codeforces.com/contest/914/problem/F
// https://yukicoder.me/problems/no/142
// わずかに普通の bitset より遅いときもあるようだが，
// 固定長にしたくないときや slice 操作が必要なときに使う
struct My_Bitset {
using T = My_Bitset;
int N;
vc<u64> dat;
// x で埋める
My_Bitset(int N = 0, int x = 0) : N(N) {
assert(x == 0 || x == 1);
u64 v = (x == 0 ? 0 : -1);
dat.assign((N + 63) >> 6, v);
if (N) dat.back() >>= (64 * len(dat) - N);
}
int size() { return N; }
void resize(int size) {
dat.resize((size + 63) >> 6);
int remainingBits = size & 63;
if (remainingBits != 0) {
u64 mask = (u64(1) << remainingBits) - 1;
dat.back() &= mask;
}
N = size;
}
// thanks to chatgpt!
class Proxy {
public:
Proxy(vc<u64> &d, int i) : dat(d), index(i) {}
operator bool() const { return (dat[index >> 6] >> (index & 63)) & 1; }
Proxy &operator=(u64 value) {
dat[index >> 6] &= ~(u64(1) << (index & 63));
dat[index >> 6] |= (value & 1) << (index & 63);
return *this;
}
void flip() {
dat[index >> 6] ^= (u64(1) << (index & 63)); // XOR to flip the bit
}
private:
vc<u64> &dat;
int index;
};
Proxy operator[](int i) { return Proxy(dat, i); }
T &operator&=(const T &p) {
assert(N == p.N);
FOR(i, len(dat)) dat[i] &= p.dat[i];
return *this;
}
T &operator|=(const T &p) {
assert(N == p.N);
FOR(i, len(dat)) dat[i] |= p.dat[i];
return *this;
}
T &operator^=(const T &p) {
assert(N == p.N);
FOR(i, len(dat)) dat[i] ^= p.dat[i];
return *this;
}
T operator&(const T &p) const { return T(*this) &= p; }
T operator|(const T &p) const { return T(*this) |= p; }
T operator^(const T &p) const { return T(*this) ^= p; }
int count() {
int ans = 0;
for (u64 val: dat) ans += popcnt(val);
return ans;
}
int next(int i) {
chmax(i, 0);
if (i >= N) return N;
int k = i >> 6;
{
u64 x = dat[k];
int s = i & 63;
x = (x >> s) << s;
if (x) return (k << 6) | lowbit(x);
}
FOR(idx, k + 1, len(dat)) {
if (dat[idx] == 0) continue;
return (idx << 6) | lowbit(dat[idx]);
}
return N;
}
int prev(int i) {
chmin(i, N - 1);
if (i <= -1) return -1;
int k = i >> 6;
if ((i & 63) < 63) {
u64 x = dat[k];
x &= (u64(1) << ((i & 63) + 1)) - 1;
if (x) return (k << 6) | topbit(x);
--k;
}
FOR_R(idx, k + 1) {
if (dat[idx] == 0) continue;
return (idx << 6) | topbit(dat[idx]);
}
return -1;
}
My_Bitset range(int L, int R) {
assert(L <= R);
My_Bitset p(R - L);
int rm = (R - L) & 63;
FOR(rm) {
p[R - L - 1] = bool((*this)[R - 1]);
--R;
}
int n = (R - L) >> 6;
int hi = L & 63;
int lo = 64 - hi;
int s = L >> 6;
if (hi == 0) {
FOR(i, n) { p.dat[i] ^= dat[s + i]; }
} else {
FOR(i, n) { p.dat[i] ^= (dat[s + i] >> hi) ^ (dat[s + i + 1] << lo); }
}
return p;
}
int count_range(int L, int R) {
assert(L <= R);
int cnt = 0;
while ((L < R) && (L & 63)) cnt += (*this)[L++];
while ((L < R) && (R & 63)) cnt += (*this)[--R];
int l = L >> 6, r = R >> 6;
FOR(i, l, r) cnt += popcnt(dat[i]);
return cnt;
}
// [L,R) に p を代入
void assign_to_range(int L, int R, My_Bitset &p) {
assert(p.N == R - L);
int a = 0, b = p.N;
while (L < R && (L & 63)) { (*this)[L++] = bool(p[a++]); }
while (L < R && (R & 63)) { (*this)[--R] = bool(p[--b]); }
// p[a:b] を [L:R] に
int l = L >> 6, r = R >> 6;
int s = a >> 6, t = b >> t;
int n = r - l;
if (!(a & 63)) {
FOR(i, n) dat[l + i] = p.dat[s + i];
} else {
int hi = a & 63;
int lo = 64 - hi;
FOR(i, n) dat[l + i] = (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
}
}
// [L,R) に p を xor
void xor_to_range(int L, int R, My_Bitset &p) {
assert(p.N == R - L);
int a = 0, b = p.N;
while (L < R && (L & 63)) {
dat[L >> 6] ^= u64(p[a]) << (L & 63);
++a, ++L;
}
while (L < R && (R & 63)) {
--b, --R;
dat[R >> 6] ^= u64(p[b]) << (R & 63);
}
// p[a:b] を [L:R] に
int l = L >> 6, r = R >> 6;
int s = a >> 6, t = b >> t;
int n = r - l;
if (!(a & 63)) {
FOR(i, n) dat[l + i] ^= p.dat[s + i];
} else {
int hi = a & 63;
int lo = 64 - hi;
FOR(i, n) dat[l + i] ^= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
}
}
// [L,R) に p を and
void and_to_range(int L, int R, My_Bitset &p) {
assert(p.N == R - L);
int a = 0, b = p.N;
while (L < R && (L & 63)) {
if (!p[a++]) (*this)[L++] = 0;
}
while (L < R && (R & 63)) {
if (!p[--b]) (*this)[--R] = 0;
}
// p[a:b] を [L:R] に
int l = L >> 6, r = R >> 6;
int s = a >> 6, t = b >> t;
int n = r - l;
if (!(a & 63)) {
FOR(i, n) dat[l + i] &= p.dat[s + i];
} else {
int hi = a & 63;
int lo = 64 - hi;
FOR(i, n) dat[l + i] &= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
}
}
// [L,R) に p を or
void or_to_range(int L, int R, My_Bitset &p) {
assert(p.N == R - L);
int a = 0, b = p.N;
while (L < R && (L & 63)) {
dat[L >> 6] |= u64(p[a]) << (L & 63);
++a, ++L;
}
while (L < R && (R & 63)) {
--b, --R;
dat[R >> 6] |= u64(p[b]) << (R & 63);
}
// p[a:b] を [L:R] に
int l = L >> 6, r = R >> 6;
int s = a >> 6, t = b >> t;
int n = r - l;
if (!(a & 63)) {
FOR(i, n) dat[l + i] |= p.dat[s + i];
} else {
int hi = a & 63;
int lo = 64 - hi;
FOR(i, n) dat[l + i] |= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
}
}
string to_string() const {
string S;
FOR(i, N) S += '0' + (dat[i >> 6] >> (i & 63) & 1);
return S;
}
// bitset に仕様を合わせる
void set(int i) { (*this)[i] = 1; }
void reset(int i) { (*this)[i] = 0; }
void flip(int i) { (*this)[i].flip(); }
void set() {
fill(all(dat), u64(-1));
resize(N);
}
void reset() { fill(all(dat), 0); }
int _Find_first() { return next(0); }
int _Find_next(int p) { return next(p + 1); }
};
#line 1 "enumerate/bits.hpp"
template <typename F>
void enumerate_bits_32(u32 s, F f) {
while (s) {
int i = __builtin_ctz(s);
f(i);
s ^= 1 << i;
}
}
template <typename F>
void enumerate_bits_64(u64 s, F f) {
while (s) {
int i = __builtin_ctzll(s);
f(i);
s ^= u64(1) << i;
}
}
template <typename BS, typename F>
void enumerate_bits_bitset(BS& b, int L, int R, F f) {
int p = (b[L] ? L : b._Find_next(L));
while (p < R) {
f(p);
p = b._Find_next(p);
}
}
#line 3 "knapsack/subset_sum.hpp"
// O(N MAX(vals))
template <typename T>
vc<int> subset_sum_solution_1(vc<T>& vals, int target) {
int n = len(vals);
if (n == 0) return {};
int mx = MAX(vals);
int b = 0, sb = 0;
while (b < n && sb + vals[b] <= target) { sb += vals[b++]; }
if (b == n && sb != target) return {};
int off = target - mx + 1;
vc<int> dp(2 * mx, -1);
vv(int, PAR, n, 2 * mx, -1);
dp[sb - off] = b;
FOR3(i, b, n) {
auto newdp = dp;
auto& par = PAR[i];
int a = vals[i];
FOR(j, mx) {
if (chmax(newdp[j + a], dp[j])) { par[j + a] = -2; }
}
FOR3_R(j, mx, 2 * mx) {
FOR3_R(k, max(dp[j], 0), newdp[j]) {
if (chmax(newdp[j - vals[k]], k)) par[j - vals[k]] = k;
}
}
swap(dp, newdp);
}
if (dp[mx - 1] == -1) return {};
vc<bool> use(n);
int i = n - 1, j = mx - 1;
while (i >= b) {
int p = PAR[i][j];
if (p == -2) {
use[i] = !use[i];
j -= vals[i--];
}
elif (p == -1) { --i; }
else {
use[p] = !use[p];
j += vals[p];
}
}
while (i >= 0) {
use[i] = !use[i];
--i;
}
vc<int> I;
FOR(i, n) if (use[i]) I.eb(i);
ll sm = 0;
for (auto&& i: I) sm += vals[i];
assert(sm == target);
return I;
}
// O(N target / w)
template <typename T>
vc<int> subset_sum_solution_2(vc<T>& vals, int target) {
int n = len(vals);
auto I = argsort(vals);
My_Bitset dp(1, 1);
vc<int> last(target + 1, -1);
FOR(k, n) {
int v = vals[I[k]];
if (v > target) continue;
My_Bitset newdp = dp;
int new_size = len(dp) + v;
newdp.resize(new_size);
newdp.or_to_range(v, new_size, dp);
if (len(newdp) > target + 1) newdp.resize(target + 1);
// update したところをメモ
FOR(i, len(newdp.dat)) {
u64 upd = (i < len(dp.dat) ? dp.dat[i] : u64(0)) ^ newdp.dat[i];
enumerate_bits_64(upd, [&](int p) -> void { last[(i << 6) | p] = I[k]; });
}
swap(dp, newdp);
}
if (target >= len(dp) || !dp[target]) return {};
vc<int> ANS;
while (target > 0) {
int i = last[target];
ANS.eb(i);
target -= vals[i];
}
return ANS;
}
// O(sum^{1.5} / w)
// sum=10^6 で 150ms：https://codeforces.com/contest/755/problem/F
template <typename T>
vc<int> subset_sum_solution_3(vc<T>& vals, int target) {
int SM = SUM<int>(vals);
int N = len(vals);
vvc<int> IDS(SM + 1);
FOR(i, N) IDS[vals[i]].eb(i);
vc<pair<int, int>> par(N, {-1, -1});
vc<int> grp_vals;
vc<int> raw_idx;
FOR(x, 1, SM + 1) {
auto& I = IDS[x];
while (len(I) >= 3) {
int a = POP(I), b = POP(I);
int c = len(par);
par.eb(a, b);
IDS[2 * x].eb(c);
}
for (auto& i: I) {
grp_vals.eb(x);
raw_idx.eb(i);
}
}
auto I = subset_sum_solution_2<int>(grp_vals, target);
vc<int> ANS;
for (auto& i: I) {
vc<int> st = {raw_idx[i]};
while (len(st)) {
auto c = POP(st);
if (c < N) {
ANS.eb(c);
continue;
}
auto [a, b] = par[c];
st.eb(a), st.eb(b);
}
}
return ANS;
}
template <typename T>
vc<int> subset_sum_solution_4(vc<T>& vals, T target) {
if (target <= 0) return {};
int N = len(vals);
int M = N / 2;
auto calc = [&](int L, int R) -> vc<T> {
int n = R - L;
vc<T> dp = {0};
FOR(i, n) {
T a = vals[L + i];
vc<T> dp1(len(dp));
FOR(k, len(dp)) dp1[k] = dp[k] + a;
vc<T> newdp;
merge(all(dp), all(dp1), back_inserter(newdp));
swap(dp, newdp);
}
return dp;
};
auto restore = [&](int L, int R, T v) -> vc<int> {
int n = R - L;
vc<T> dp(1 << n);
FOR(i, n) FOR(s, 1 << i) dp[s | 1 << i] = dp[s] + vals[L + i];
FOR(s, 1 << n) {
if (dp[s] == v) {
vc<int> I;
FOR(i, n) if (s >> i & 1) I.eb(L + i);
return I;
}
}
assert(0);
return {};
};
auto dp1 = calc(0, M);
auto dp2 = calc(M, N);
int t = 0;
FOR_R(s, len(dp1)) {
while (t + 1 < len(dp2) && dp1[s] + dp2[t + 1] <= target) { ++t; }
if (dp1[s] + dp2[t] == target) {
vc<int> I1 = restore(0, M, dp1[s]);
vc<int> I2 = restore(M, N, dp2[t]);
I1.insert(I1.end(), all(I2));
return I1;
}
}
return {};
}
template <typename T>
vc<int> subset_sum(vc<T>& vals, T target) {
if (target <= 0) return {};
int n = len(vals);
if (n == 0) return {};
int mx = MAX(vals);
// しきい値の調整をしていない
// solution 1: O(N mx))
// solution 2: O(N target / w)
// solution 3: O(sum^1.5 / w)
// solution 4: O(2^(N/2))
double x1 = double(len(vals)) * mx;
double x2 = double(len(vals)) * target / 500.0;
double x3 = pow(SUM<double>(vals), 1.5) / 500.0;
double x4 = pow(2.0, 0.5 * len(vals));
double mi = min({x1, x2, x3, x4});
if (x1 == mi) return subset_sum_solution_1(vals, target);
if (x2 == mi) return subset_sum_solution_2(vals, target);
if (x3 == mi) return subset_sum_solution_3(vals, target);
return subset_sum_solution_4(vals, target);
}
#line 5 "test/yukicoder/4_2.test.cpp"
void solve() {
LL(N);
VEC(int, A, N);
ll S = SUM<int>(A);
auto I = subset_sum<int>(A, S / 2);
bool can = (S == 0 || len(I) > 0);
if (can && S % 2 == 0) {
print("possible");
} else {
print("impossible");
}
}
signed main() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << setprecision(15);
ll T = 1;
// LL(T);
FOR(_, T) solve();
return 0;
}