Submission #49422940

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Source Code Expand

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <numeric>
#include <optional>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

#ifdef RUTHEN_LOCAL
#include <debug.hpp>
#else
#define show(x) true
#endif

// type definition
using i64 = long long;
using i128 = __int128_t;
using u32 = unsigned int;
using u64 = unsigned long long;
using u128 = __uint128_t;
using f32 = float;
using f64 = double;
using f128 = long double;
template <class T> using pque = std::priority_queue<T>;
template <class T> using pqueg = std::priority_queue<T, std::vector<T>, std::greater<T>>;

#define overload4(_1, _2, _3, _4, name, ...) name
#define overload3(_1, _2, _3, name, ...) name
#define overload2(_1, _2, name, ...) name

// loop
#define REP1(a) for (long long _ = 0; _ < (a); _++)
#define REP2(i, a) for (long long i = 0; i < (a); i++)
#define REP3(i, a, b) for (long long i = (a); i < (b); i++)
#define REP4(i, a, b, c) for (long long i = (a); i < (b); i += (c))
#define REP(...) overload4(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
#define RREP1(a) for (long long _ = (a)-1; _ >= 0; _--)
#define RREP2(i, a) for (long long i = (a)-1; i >= 0; i--)
#define RREP3(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define RREP(...) overload3(__VA_ARGS__, RREP3, RREP2, RREP1)(__VA_ARGS__)
#define FORE1(x, a) for (auto&& x : a)
#define FORE2(x, y, a) for (auto&& [x, y] : a)
#define FORE3(x, y, z, a) for (auto&& [x, y, z] : a)
#define FORE(...) overload4(__VA_ARGS__, FORE3, FORE2, FORE1)(__VA_ARGS__)
#define FORSUB(t, s) for (long long t = (s); t; t = (t - 1) & (s))

// function
#define ALL(a) (a).begin(), (a).end()
#define RALL(a) (a).rbegin(), (a).rend()
#define SORT(a) std::sort((a).begin(), (a).end())
#define RSORT(a) std::sort((a).rbegin(), (a).rend())
#define REV(a) std::reverse((a).begin(), (a).end())
#define UNIQUE(a)                      \
    std::sort((a).begin(), (a).end()); \
    (a).erase(std::unique((a).begin(), (a).end()), (a).end())
#define LEN(a) int((a).size())
#define MIN(a) *std::min_element((a).begin(), (a).end())
#define MAX(a) *std::max_element((a).begin(), (a).end())
#define SUM1(a) std::accumulate((a).begin(), (a).end(), 0LL)
#define SUM2(a, x) std::accumulate((a).begin(), (a).end(), (x))
#define SUM(...) overload2(__VA_ARGS__, SUM2, SUM1)(__VA_ARGS__)
#define LB(a, x) std::distance((a).begin(), std::lower_bound((a).begin(), (a).end(), (x)))
#define UB(a, x) std::distance((a).begin(), std::upper_bound((a).begin(), (a).end(), (x)))
template <class T, class U> inline bool chmin(T& a, const U& b) { return (a > T(b) ? a = b, 1 : 0); }
template <class T, class U> inline bool chmax(T& a, const U& b) { return (a < T(b) ? a = b, 1 : 0); }
template <class T, class S> inline T floor(const T x, const S y) {
    assert(y);
    return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1)));
}
template <class T, class S> inline T ceil(const T x, const S y) {
    assert(y);
    return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y));
}
template <class T, class S> std::pair<T, T> inline divmod(const T x, const S y) {
    T q = floor(x, y);
    return {q, x - q * y};
}

// bit operation
// (0, 1, 2, 3, 4) -> (0, 1, 1, 2, 1)
int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(i64 x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(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(i64 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(i64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

// binary search
template <class T, class F> T bin_search(T ok, T ng, F& f) {
    while ((ok > ng ? ok - ng : ng - ok) > 1) {
        T md = (ng + ok) >> 1;
        (f(md) ? ok : ng) = md;
    }
    return ok;
}
template <class T, class F> T bin_search_real(T ok, T ng, F& f, const int iter = 100) {
    for (int _ = 0; _ < iter; _++) {
        T md = (ng + ok) / 2;
        (f(md) ? ok : ng) = md;
    }
    return ok;
}

// rotate matrix counterclockwise by pi / 2
template <class T> void rot(std::vector<std::vector<T>>& a) {
    if (int(a.size()) == 0) return;
    if (int(a[0].size()) == 0) return;
    int n = int(a.size()), m = int(a[0].size());
    std::vector res(m, std::vector<T>(n));
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            res[m - 1 - j][i] = a[i][j];
        }
    }
    a.swap(res);
}

// const value
constexpr int dx[8] = {1, 0, -1, 0, 1, -1, -1, 1};
constexpr int dy[8] = {0, 1, 0, -1, 1, 1, -1, -1};

// infinity
template <class T> constexpr T INF = 0;
template <> constexpr int INF<int> = 1'000'000'000;                 // 1e9
template <> constexpr i64 INF<i64> = i64(INF<int>) * INF<int> * 2;  // 2e18
template <> constexpr i128 INF<i128> = i128(INF<i64>) * INF<i64>;   // 4e36
template <> constexpr u32 INF<u32> = INF<int>;                      // 1e9
template <> constexpr u64 INF<u64> = INF<i64>;                      // 2e18
template <> constexpr u128 INF<u128> = INF<i128>;                   // 4e36
template <> constexpr f32 INF<f32> = INF<i64>;                      // 2e18
template <> constexpr f64 INF<f64> = INF<i64>;                      // 2e18
template <> constexpr f128 INF<f128> = INF<i64>;                    // 2e18

// input
std::istream& operator>>(std::istream& is, __int128_t& x) {
    std::string s;
    is >> s;
    x = 0;
    int i = s[0] == '-' ? 1 : 0;
    while (i < (int)(s.size())) x = 10 * x + s[i++] - '0';
    if (s[0] == '-') x = -x;
    return is;
}
std::istream& operator>>(std::istream& is, __uint128_t& x) {
    std::string s;
    is >> s;
    assert(s[0] != '-');
    x = 0;
    int i = 0;
    while (i < (int)(s.size())) x = 10 * x + s[i++] - '0';
    return is;
}
template <class T> std::istream& operator>>(std::istream& is, std::vector<T>& v) {
    for (auto&& i : v) is >> i;
    return is;
}
template <class... T> void in(T&... a) { (std::cin >> ... >> a); }
void scan() {}
template <class Head, class... Tail> void scan(Head& head, Tail&... tail) {
    in(head);
    scan(tail...);
}

// definition & input
#define INT(...)     \
    int __VA_ARGS__; \
    scan(__VA_ARGS__)
#define I64(...)     \
    i64 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define I128(...)     \
    i128 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define U32(...)     \
    u32 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define U64(...)     \
    u64 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define U128(...)     \
    u128 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define F32(...)     \
    f32 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define F64(...)     \
    f64 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define F128(...)     \
    f128 __VA_ARGS__; \
    scan(__VA_ARGS__)
#define STR(...)             \
    std::string __VA_ARGS__; \
    scan(__VA_ARGS__)
#define CHR(...)      \
    char __VA_ARGS__; \
    scan(__VA_ARGS__)
#define VEC(type, name, size)     \
    std::vector<type> name(size); \
    scan(name)
#define VEC2(type, name1, name2, size)          \
    std::vector<type> name1(size), name2(size); \
    for (int i = 0; i < size; i++) scan(name1[i], name2[i])
#define VEC3(type, name1, name2, name3, size)                \
    std::vector<type> name1(size), name2(size), name3(size); \
    for (int i = 0; i < size; i++) scan(name1[i], name2[i], name3[i])
#define VEC4(type, name1, name2, name3, name4, size)                      \
    std::vector<type> name1(size), name2(size), name3(size), name4(size); \
    for (int i = 0; i < size; i++) scan(name1[i], name2[i], name3[i], name4[i])
#define VV(type, name, h, w)                       \
    std::vector name((h), std::vector<type>((w))); \
    scan(name)

// output
std::ostream& operator<<(std::ostream& os, const __int128_t& x) {
    if (x == 0) {
        return os << 0;
    }
    __int128_t y = (x > 0 ? x : -x);
    std::string res;
    while (y) {
        res += y % 10 + '0';
        y /= 10;
    }
    if (x < 0) res += '-';
    std::reverse(res.begin(), res.end());
    return os << res;
}
std::ostream& operator<<(std::ostream& os, const __uint128_t& x) {
    if (x == 0) {
        return os << 0;
    }
    __uint128_t y = x;
    std::string res;
    while (y) {
        res += y % 10 + '0';
        y /= 10;
    }
    std::reverse(res.begin(), res.end());
    return os << res;
}
template <class T> std::ostream& operator<<(std::ostream& os, const std::vector<T>& v) {
    auto n = v.size();
    for (size_t i = 0; i < n; i++) {
        if (i) os << ' ';
        os << v[i];
    }
    return os;
}
template <class... T> void out(const T&... a) { (std::cout << ... << a); }
void print() {
    out('\n');
    // std::cout.flush();
}
template <class Head, class... Tail> void print(Head&& head, Tail&&... tail) {
    out(head);
    if (sizeof...(Tail)) out(' ');
    print(tail...);
}

// bool output
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); }
void POSSIBLE(bool t = 1) { return print(t ? "POSSIBLE" : "IMPOSSIBLE"); }
void Possible(bool t = 1) { return print(t ? "Possible" : "Impossible"); }
void possible(bool t = 1) { return print(t ? "possible" : "impossible"); }

// I/O speed up
struct SetUpIO {
    SetUpIO() {
        std::ios::sync_with_stdio(false);
        std::cin.tie(0);
        std::cout << std::fixed << std::setprecision(15);
    }
} set_up_io;

// find s that sum s[i] * a[i] = x (s[i] = 1 or -1)
// O(n 2 ^ n) (n: size of a)
template <class T> std::vector<int> meet_in_the_middle(const std::vector<T>& a, T x) {
    const int n = int(a.size());
    const int n1 = n / 2, n2 = n - n1;
    std::unordered_map<T, int> mp;
    std::vector<int> res(n, 0);
    for (int bit = 0; bit < (1 << n1); bit++) {
        T c = T(0);
        for (int i = 0; i < n1; i++) {
            if (bit >> i & 1) {
                c += a[i];
            } else {
                c -= a[i];
            }
        }
        mp[c] = bit;
    }
    for (int bit = 0; bit < (1 << n2); bit++) {
        T d = T(0);
        for (int i = 0; i < n2; i++) {
            if (bit >> i & 1) {
                d += a[i + n1];
            } else {
                d -= a[i + n1];
            }
        }
        if (mp.count(x - d)) {
            for (int i = 0; i < n1; i++) res[i] = (mp[x - d] >> i & 1) * 2 - 1;
            for (int i = 0; i < n2; i++) res[i + n1] = (bit >> i & 1) * 2 - 1;
            return res;
        }
    }
    return res;
}

// calculate all possible sorted { sum s[i] * a[i], bit } (using std::inplace_merge)
template <class T> std::vector<std::pair<T, int>> calc_half(const std::vector<T>& a) {
    int n = int(a.size());
    std::vector<std::pair<T, int>> half(1 << n);
    int size = 1;
    half[0] = {T(0), 0};
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < size; j++) {
            half[j + size] = {half[j].first + a[i], half[j].second | (1 << i)};
        }
        for (int j = 0; j < size; j++) {
            half[j].first -= a[i];
        }
        std::inplace_merge(half.begin(), half.begin() + size, half.begin() + size * 2);
        size <<= 1;
    }
    return half;
}

// O(2 ^ n) (using std::inplace_merge, Two Pointer Approach)
template <class T> std::vector<int> meet_in_the_middle_faster(const std::vector<T>& a, T x) {
    const int n = int(a.size());
    const int n1 = n / 2, n2 = n - n1;

    std::vector<T> a1(n1), a2(n2);
    for (int i = 0; i < n1; i++) a1[i] = a[i];
    for (int i = 0; i < n2; i++) a2[i] = a[i + n1];

    auto half1 = calc_half(a1);
    auto half2 = calc_half(a2);

    int r = (1 << n2) - 1;
    std::vector<int> res(n, 0);
    for (int i = 0; i < (1 << n1); i++) {
        while (r > 0 and half1[i].first + half2[r].first > x) r--;
        if (half1[i].first + half2[r].first == x) {
            for (int j = 0; j < n1; j++) res[j] = (half1[i].second >> j & 1) * 2 - 1;
            for (int j = 0; j < n2; j++) res[j + n1] = (half2[r].second >> j & 1) * 2 - 1;
            return res;
        }
    }
    return res;
}

using namespace std;

int main() {
    INT(N, X, Y);
    VEC(int, A, N);
    vector<int> x, y;
    REP(i, N) {
        if (i % 2 == 1) x.push_back(A[i]);
        if (i % 2 == 0) y.push_back(A[i]);
    }
    auto rx = meet_in_the_middle_faster(x, X);
    auto ry = meet_in_the_middle_faster(y, Y);
    int ok = 1, dir = 1;
    string ans, LR = "LR";
    REP(i, N) {
        if (i % 2 == 0) {
            if (ry[i / 2] == 0) {
                ok = 0;
                break;
            }
            ans += LR[dir ^ ((ry[i / 2] + 1) / 2)];
            dir = ((ry[i / 2] + 1) / 2);
        } else {
            if (rx[i / 2] == 0) {
                ok = 0;
                break;
            }
            ans += LR[dir ^ ((rx[i / 2] + 1) / 2) ^ 1];
            dir = ((rx[i / 2] + 1) / 2);
        }
    }
    if (ok == 0) {
        No();
    } else {
        Yes();
        print(ans);
    }
    return 0;
}

Submission Info

Judge Result