Equal Number of Prefix Maximums CodeChef Solution – Queslers

Problem: Equal Number of Prefix Maximums CodeChef Solution

For an array [B1,B2,…,BK][B1,B2,…,BK] of distinct integers, let’s define its weight as the number of its prefix maximums −− that is, of such indexes ii, that Bi>BjBi>Bj for all 1≤j<i1≤j<i.

You are given a permutation [P1,P2,…,PN][P1,P2,…,PN] of integers from 11 to NN. Determine if it’s possible to split it into two nonempty subsequences with equal weights.

Input Format

• The first line will contain TT, the number of test cases. The description of test cases follows.
• The first line of each test case contains a single integer NN −− the length of the permutation.
• The second line of each test case contains NN integers P1,P2,…,PNP1,P2,…,PN −− the permutation.

Output Format

For each test case, if it’s possible to split the permutation into two nonempty subsequences with equal weight, output YES. Otherwise, output NO.

Constraints

• 1≤T≤2⋅1051≤T≤2⋅105.
• 2≤N≤2⋅1052≤N≤2⋅105
• The sum of NN over all test cases does not exceed 2⋅1052⋅105.
• 1≤Pi≤N1≤Pi≤N
• Pi≠PjPi≠Pj for i≠ji≠j

Sample Input 1 

5
6
1 2 3 4 5 6
7
1 2 3 4 5 6 7
6
1 4 2 5 3 6
8
5 6 7 8 1 2 3 4
9
5 6 7 8 9 1 2 3 4

Sample Output 1 

YES
NO
YES
YES
NO

Explanation

Explanation of test case 1: We can split the permutation into two subsequences [1,3,4][1,3,4] and [2,5,6][2,5,6], both of which have 33 prefix maximums.

Explanation of test case 2: No matter how we split, both sequences will be increasing, therefore the sum of their weights will be 77, so they can’t be equal.

Explanation of test case 3: One way to split is into [1,4,2][1,4,2] and [5,3,6][5,3,6]. 1,41,4 are prefix maximums of the first sequences, and 5,65,6 of the second one.

Explanation of test case 4: One way to split is into [5,6,7,8][5,6,7,8] and [1,2,3,4][1,2,3,4]. Both have 44 prefix maximums.

Explanation of test case 5: It’s possible to show that the required split does not exist.

Equal Number of Prefix Maximums CodeChef Solution in C++

#include <bits/stdc++.h>

using namespace std;

#define all(v) (v).begin(), (v).end()
#define val const auto&
#define st_mtimespec st_mtim
#define let auto

using ll = long long;
using ull = unsigned long long;
using li = __int128;
using uli = unsigned __int128;
using ld = long double;
using pii = pair<int, int>;
using vi = vector<int>;

void doReplace() {
}

template <typename T, typename U, typename...Vs>
void doReplace(T& t, const U& u, Vs&&...vs) {
    t = u;
    doReplace(vs...);
}

template <typename T, typename...Us>
T minim(T& was, const T& cand, Us&&...us) {
    if (was > cand) {
        was = cand;
        doReplace(us...);
    }
    return was;
}

template <typename T, typename...Us>
T maxim(T& was, const T& cand, Us&&...us) {
    if (was < cand) {
        was = cand;
        doReplace(us...);
    }
    return was;
}





template <typename D>
D dPower(D base, ll exponent) {
    if (exponent < 0) {
        return dPower(1 / base, -exponent);
    }
    if (exponent == 0) {
        return 1;
    }
    if ((exponent & 1) == 1) {
        return dPower(base, exponent - 1) * base;
    } else {
        D res = dPower(base, exponent >> 1);
        return res * res;
    }
}








class NumberIterator : iterator<forward_iterator_tag, int> {
public:
    int v;

    NumberIterator(int v) : v(v) {}

    operator int &() { return v; }

    int operator*() { return v; }
};

class range : pii {
public:
    range(int begin, int end) : pii(begin, max(begin, end)) {}

    range(int n) : pii(0, max(0, n)) {}

    NumberIterator begin() {
        return first;
    }

    NumberIterator end() {
        return second;
    }
};

range rangeIncl(int begin, int end) {
    return range(begin, end + 1);
}

range rangeIncl(int n) {
    return range(n + 1);
}

template <typename T>
class arr {
    T* b;
    int n;

    void allocate(int sz) {
#ifdef LOCAL
        if (sz < 0) {
            throw "bad alloc";
        }
#endif
        n = sz;
        if (sz > 0) {
            b = (T*)(::operator new(sz * sizeof(T)));
        } else {
            b = nullptr;
        }
    }

public:
    arr(int n = 0) {
        allocate(n);
#ifdef LOCAL
        view();
#endif
    }

    arr(int n, const T& init) {
        allocate(n);
        for (int i : range(n)) {
            ::new((void*)(b + i)) T(init);
        }
#ifdef LOCAL
        view();
#endif
    }

    arr(T* b, int n) : arr(b, b + n) {}
    arr(T* b, T* e) : b(b), n(e - b) {}

    arr(initializer_list<T> l) : arr(l.size()) {
        if (n > 0) {
            memcpy(b, l.begin(), n * sizeof(T));
        }
    }

    int size() const {
        return n;
    }

    T* begin() {
        return b;
    }

    const T* begin() const {
        return b;
    }

    T* end() {
        return b + n;
    }

    const T* end() const {
        return b + n;
    }

    arr<T> clone() const {
        arr<T> res(n);
        copy(b, b + n, res.begin());
        return res;
    }

    T& operator[](int at) {
#ifdef LOCAL
        if (at < 0 || at >= n) {
            throw "Out of bounds";
        }
#endif
        return b[at];
    }

    const T& operator[](int at) const {
#ifdef LOCAL
        if (at < 0 || at >= n) {
            throw "Out of bounds";
        }
#endif
        return b[at];
    }

    bool operator ==(const arr& other) const {
        if (n != other.n) {
            return false;
        }
        for (int i : range(n)) {
            if (b[i] != other.b[i]) {
                return false;
            }
        }
        return true;
    }

    vector<T> view() {
        return vector<T>(b, b + n);
    }

    arr<T> subArray(int from, int to) const {
        return arr<T>(b + from, b + to);
    }
};

typedef arr<int> arri;

void decreaseByOne() {}

template <typename T, class...Vs>
void decreaseByOne(arr<T>& array, Vs&...vs) {
    int n = array.size();
    for (int i : range(n)) {
        array[i]--;
    }
    decreaseByOne(vs...);
}







template <typename T>
class arr2d {
    T* b;
    int d1;
    int d2;
    int sz;

    void allocate(int n) {
#ifdef LOCAL
        if (n < 0) {
            throw "bad alloc";
        }
#endif
        if (n > 0) {
            b = (T*)(::operator new(n * sizeof(T)));
        } else {
            b = nullptr;
        }
    }

public:
    explicit arr2d(int d1 = 0, int d2 = 0) : d1(d1), d2(d2), sz(d1 * d2) {
#ifdef LOCAL
        if (d1 < 0 || d2 < 0) {
            throw "bad alloc";
        }
#endif
        allocate(sz);
#ifdef LOCAL
        view();
#endif
    }

    arr2d(int d1, int d2, const T& init) : d1(d1), d2(d2), sz(d1 * d2) {
#ifdef LOCAL
        if (d1 < 0 || d2 < 0) {
            throw "bad alloc";
        }
#endif
        allocate(sz);
        for (int i : range(sz)) {
            ::new((void*)(b + i)) T(init);
        }
#ifdef LOCAL
        view();
#endif
    }

    arr2d(T* b, int d1, int d2) : b(b), d1(d1), d2(d2), sz(d1 * d2) {}

    arr2d<T> clone() {
        arr2d<T> res(d1, d2);
        copy(all(*this), res.b);
        return res;
    }

    int size() const {
        return sz;
    }

    int dim1() const {
        return d1;
    }

    int dim2() const {
        return d2;
    }

    T* begin() {
        return b;
    }

    T* end() {
        return b + sz;
    }

    T& operator()(int i1, int i2) {
#ifdef LOCAL
        if (i1 < 0 || i1 >= d1 || i2 < 0 || i2 >= d2) {
            throw "Out of bounds";
        }
#endif
        return b[i1 * d2 + i2];
    }

    const T& operator()(int i1, int i2) const {
#ifdef LOCAL
        if (i1 < 0 || i1 >= d1 || i2 < 0 || i2 >= d2) {
            throw "Out of bounds";
        }
#endif
        return b[i1 * d2 + i2];
    }

    T& operator[](const pii& p) {
        return operator()(p.first, p.second);
    }

    const T& operator[](const pii& p) const {
        return operator()(p.first, p.second);
    }

    arr<T> operator[](int at) {
#ifdef LOCAL
        if (at < 0 || at >= d1) {
            throw "Out of bounds";
        }
#endif
        return arr<T>(b + at * d2, d2);
    }

    vector<vector<T>> view() {
        vector<vector<T>> res(min(d1, 50));
        for (int i : range(res.size())) {
            res[i] = (*this)[i].view();
        }
        return res;
    }

};



inline bool isWhitespace(int c) {
    return isspace(c) || c == EOF;
}

class Input {
private:
    bool exhausted = false;
    int bufSize = 4096;
    char* buf = new char[bufSize];
    int bufRead = 0;
    int bufAt = 0;

public:
    inline int get() {
        if (exhausted) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return EOF;
        }
        if (bufRead == bufAt) {
            bufRead = fread(buf, sizeof(char), bufSize, stdin);
            bufAt = 0;
        }
        if (bufRead == bufAt) {
            exhausted = true;
            return EOF;
        }
        return buf[bufAt++];
    }

    inline int peek() {
        if (exhausted) {
            return EOF;
        }
        if (bufRead == bufAt) {
            bufRead = fread(buf, sizeof(char), bufSize, stdin);
            bufAt = 0;
        }
        if (bufRead == bufAt) {
            exhausted = true;
            getchar();
            return EOF;
        }
        return buf[bufAt];
    }

private:
    template<typename T>
    inline T readInteger() {
        int c = skipWhitespace();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = get();
        } else if (c == '+') {
            c = get();
        }
        T res = 0;
        do {
            if (!isdigit(c)) {
#ifdef LOCAL
                throw "Number format error";
#endif
                return sgn * res;
            }
            res *= 10;
            res += c - '0';
            c = get();
        } while (!isWhitespace(c));
        return res * sgn;
    }

    void initArrays(int) {}

    template <typename T, class...Vs>
    void initArrays(int n, arr<T>& array, Vs&...vs) {
        array = arr<T>(n);
        initArrays(n, vs...);
    }

    template <typename T, class...Vs>
    void initArrays(int n, vector<T>&, Vs&...vs) {
        initArrays(n, vs...);
    }

    void readImpl(int) {}

    template <typename T, class...Vs>
    void readImpl(int i, arr<T>& arr, Vs&...vs) {
        arr[i] = readType<T>();
        readImpl(i, vs...);
    }

    template <typename T, class...Vs>
    void readImpl(int i, vector<T>& arr, Vs&...vs) {
        arr.push_back(readType<T>());
        readImpl(i, vs...);
    }

public:
    inline int skipWhitespace() {
        int c;
        while (isWhitespace(c = get()) && c != EOF);
        return c;
    }

    inline int peekNext() {
        while (isWhitespace(peek()) && peek() != EOF) {
            get();
        }
        return peek();
    }

    inline int readInt() {
        return readInteger<int>();
    }

    inline ll readLong() {
        return readInteger<ll>();
    }

    inline unsigned readUnsigned() {
        return readInteger<unsigned>();
    }

    string readString() {
        int c = skipWhitespace();
        if (c == EOF) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return "";
        }
        string res;
        do {
            res.push_back(c);
        } while (!isWhitespace(c = get()));
        return res;
    }

    arri readIntArray(int size) {
        return readArray<int>(size);
    }

    arr<ll> readLongArray(int size) {
        return readArray<ll>(size);
    }

    arr<double> readDoubleArray(int size) {
        return readArray<double>(size);
    }

    arr<string> readStringArray(int size) {
        return readArray<string>(size);
    }

    arr<char> readCharArray(int size) {
        return readArray<char>(size);
    }

    template<typename T>
    T readType() {
        throw "Operation not supported";
    }

    template<typename U, typename V>
    pair<U, V> readType() {
        U first = readType<U>();
        V second = readType<V>();
        return make_pair(first, second);
    }

    template<typename T>
    arr<T> readArray(int n) {
        arr<T> res(n, T());
        for (int i : range(n)) {
            res[i] = readType<T>();
        }
        return res;
    }



    template <class...Vs>
    void readArrays(int n, Vs&...vs) {
        initArrays(n, vs...);
        for (int i : range(n)) {
            readImpl(i, vs...);
        }
    }

    template<typename U, typename V>
    arr<pair<U, V> > readArray(int n) {
        arr<pair<U, V> > res(n);
        for (int i : range(n)) {
            res[i] = readType<U, V>();
        }
        return res;
    }

    template<typename T>
    arr2d<T> readTable(int rows, int cols) {
        arr2d<T> result(rows, cols);
        for (int i : range(rows)) {
            for (int j : range(cols)) {
                result(i, j) = readType<T>();
            }
        }
        return result;
    }

    arr2d<int> readIntTable(int rows, int cols) {
        return readTable<int>(rows, cols);
    }

    arr2d<char> readCharTable(int rows, int cols) {
        return readTable<char>(rows, cols);
    }

    string readLine() {
        int c = get();
        if (c == EOF) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return "";
        }
        string res;
        while (c != '\n' && c != '\r' && c != EOF) {
            res.push_back(c);
            c = get();
        }
        if (c == '\r' && peek() == '\n') {
            get();
        }
        return res;
    }

    inline double readDouble() {
        int c = skipWhitespace();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = get();
        }
        double res = 0;
        do {
            if (tolower(c) == 'e') {
                return sgn * res * dPower(double(10), readInt());
            }
            if (!isdigit(c)) {
#ifdef LOCAL
                throw "Number format error";
#endif
                return sgn * res;
            }
            res *= 10;
            res += c - '0';
            c = get();
        } while (!isWhitespace(c) && c != '.');
        if (c == '.') {
            double add = 0;
            int length = 0;
            c = get();
            do {
                if (tolower(c) == 'e') {
                    return sgn * (res + add * dPower(double(10), -length)) * dPower(double(10), readInt());
                }
                if (!isdigit(c)) {
#ifdef LOCAL
                    throw "Number format error";
#endif
                    res += add * dPower(10, -length);
                    return res * sgn;
                }
                add *= 10;
                add += c - '0';
                length++;
                c = get();
            } while (!isWhitespace(c));
            res += add * dPower(double(10), -length);
        }
        return res * sgn;
    }

    inline char readChar() {
        int c = skipWhitespace();
        if (c == EOF) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return 0;
        }
        return c;
    }

    inline bool isExhausted() { return exhausted; }

    inline void setBufSize(int newBufSize) {
        if (newBufSize > bufSize) {
            char* newBuf = new char[newBufSize];
            memcpy(newBuf, buf, bufSize);
            buf = newBuf;
        }
        bufSize = newBufSize;
    }
};

template<>
inline double Input::readType() {
    return readDouble();
}

template<>
inline int Input::readType() {
    return readInt();
}

template<>
inline ll Input::readType() {
    return readLong();
}

template<>
inline unsigned Input::readType() {
    return readUnsigned();
}

template<>
inline char Input::readType() {
    return readChar();
}

template<>
inline string Input::readType() {
    return readString();
}

Input in;







class Output {
private:
    ostream* out;

    template<typename T>
    inline void printSingle(const T& value) {
        *out << value;
    }

    template<typename T>
    void printSingle(const vector<T>& array) {
        size_t n = array.size();
        for (int i : range(n)) {
            *out << array[i];
            if (i + 1 != n) {
                *out << ' ';
            }
        }
    }

    template<typename T>
    void printSingle(const arr<T>& array) {
        int n = array.size();
        for (int i : range(n)) {
            *out << array[i];
            if (i + 1 != n) {
                *out << ' ';
            }
        }
    }

    template<typename T>
    void printSingle(const arr2d<T>& array) {
        size_t n = array.dim1();
        size_t m = array.dim2();
        for (int i : range(n)) {
            for (int j : range(m)) {
                *out << array(i, j);
                if (j + 1 != m) {
                    *out << ' ';
                }
            }
            if (i + 1 != n) {
                *out << '\n';
            }
        }
    }

    template<typename T, typename U>
    inline void printSingle(const pair<T, U>& value) {
        *out << value.first << ' ' << value.second;
    }

public:
    bool autoflush;

    Output(ostream& out, bool autoflush) : out(&out), autoflush(autoflush) {
        setPrecision(20);
    }

    void setOut(ostream& nOut) {
        out = &nOut;
        setPrecision(20);
    }

    inline void print() {}

    template<typename T, typename...Targs>
    inline void print(const T& first, const Targs... args) {
        printSingle(first);
        if (sizeof...(args) != 0) {
            *out << ' ';
            print(args...);
        }
        if (autoflush) {
            flush();
        }
    }

    template<typename...Targs>
    inline void printLine(const Targs... args) {
        print(args...);
        *out << '\n';
        if (autoflush) {
            flush();
        }
    }

    inline void flush() {
        out->flush();
    }

    inline void setPrecision(int digs) {
        *out << fixed << setprecision(digs);
    }
};

Output out(cout, false);
Output err(cerr, true);

#ifdef LOCAL
#define debug(x) err.printLine(#x, '=', x);
#else
#define debug(x)
#endif







template <typename T>
inline void unique(vector<T>& v) {
    v.resize(unique(all(v)) - v.begin());
}

arri createOrder(int n) {
    arri order(n);
    for (int i = 0; i < n; i++) {
        order[i] = i;
    }
    return order;
}

template <typename T>
arri createOrder(const T& v) {
    auto order = createOrder(v.size());
    sort(all(order), [&](int a, int b) -> bool {
        return v[a] < v[b];
    });
    return order;
}

arri inverse(const arri& p) {
    arri res(p.size());
    for (int i : range(p.size())) {
        res[p[i]] = i;
    }
    return res;
}

template <typename T, typename Iterator>
inline void addAll(vector<T>& v, Iterator begin, Iterator end) {
    v.insert(v.end(), begin, end);
}

template <class Collection, typename Iterator>
inline void addAll(Collection& v, Iterator begin, Iterator end) {
    v.insert(begin, end);
}

template <typename Iterator>
arri getQty(Iterator begin, Iterator end, int length) {
    arri res(length, 0);
    for (Iterator it = begin; it != end; it++) {
        res[*it]++;
    }
    return res;
}

template <typename Iterator>
arri getQty(Iterator begin, Iterator end) {
    return getQty(begin, end, *max_element(begin, end) + 1);
}

template <class Collection>
void collect(Collection&) {}

template <class Collection, class Other, class ...Vs>
void collect(Collection& all, Other& a, Vs&...vs) {
    addAll(all, all(a));
    collect(all, vs...);
}

void replace(const vi&) {}

template <class ...Vs>
void replace(const vi& all, vi& a, Vs&...vs) {
    for (int& i : a) {
        i = lower_bound(all(all), i) - all.begin();
    }
    replace(all, vs...);
}

template <class ...Vs>
void replace(const vi& all, arri& a, Vs&...vs) {
    for (int& i : a) {
        i = lower_bound(all(all), i) - all.begin();
    }
    replace(all, vs...);
}

template <class ...Vs>
vi compress(Vs&...vs) {
    vi vals;
    collect(vals, vs...);
    sort(all(vals));
    unique(vals);
    replace(vals, vs...);
    return vals;
}





template <typename T>
class RecursiveFunction {
    T t;

public:
    RecursiveFunction(T&& t) : t(forward<T>(t)) {}

    template <typename... Args>
    auto operator() (Args&&... args) const {
        return t(*this, forward<Args>(args)...);
    }
};





class ReverseNumberIterator : public NumberIterator {
public:
    ReverseNumberIterator(int v) : NumberIterator(v) {}

    ReverseNumberIterator& operator++() {
        --v;
        return *this;
    }
};

class RevRange : pii {
public:
    RevRange(int begin, int end) : pii(begin - 1, min(begin, end) - 1) {}

    RevRange(int n) : pii(n - 1, min(n, 0) - 1) {}

    ReverseNumberIterator begin() {
        return first;
    }

    ReverseNumberIterator end() {
        return second;
    }
};

RevRange RevRangeIncl(int begin, int end) {
    return RevRange(begin + 1, end);
}

RevRange RevRangeIncl(int n) {
    return RevRange(n + 1);
}

class EqualNumberOfPrefixMaximums {
public:
    void solve() {
        let n = in.readInt();
        let p = in.readIntArray(n);

        decreaseByOne(p);
        let q = inverse(p);
        arr<bool> special(n, false);
        let so_far = -1;
        let num_special = 0;
        for (let i : range(n)) {
            if (so_far < p[i]) {
                so_far = p[i];
                special[i] = true;
                num_special++;
            }
        }
        if (num_special % 2 == 0) {
            out.printLine("YES");
            return;
        }
        arri max_tail(4 * n, 0);
        RecursiveFunction get_max_tail = [&](const auto& self, int root, int left, int right, int from, int to) -> int {
            if (left >= to || right <= from) {
                return 0;
            }
            if (left >= from && right <= to) {
                return max_tail[root];
            }
            int mid = (left + right) / 2;
            auto lRes = self(2 * root + 1, left, mid, from, to);
            auto rRes = self(2 * root + 2, mid, right, from, to);
            return max(lRes, rRes);
        };
        RecursiveFunction update_max_tail = [&](const auto& self, int root, int left, int right, int at, int v) -> void {
            maxim(max_tail[root], v);
            int mid = (left + right) / 2;
            if (left + 1 == right) {
                return;
            }
            if (at < mid) {
                self(2 * root + 1, left, mid, at, v);
            } else {
                self(2 * root + 2, mid, right, at, v);
            }
        };
        arri tail(n);
        for (let i : RevRange(n)) {
            tail[q[i]] = get_max_tail(0, 0, n, q[i], n);
            update_max_tail(0, 0, n, q[i], tail[q[i]] + (special[q[i]] ? 2 : 1));
        }
        so_far = -1;
        int qty = 0;
        arri head(n);
        for (let i : range(n)) {
            if (special[i]) {
                head[i] = qty;
                continue;
            }
            if (so_far < p[i]) {
                qty++;
                so_far = p[i];
            }
        }
        arri special_to_right(n);
        qty = 0;
        for (let i : RevRange(n)) {
            if (special[i]) {
                qty++;
            } else {
                special_to_right[i] = qty;
            }
        }
        arri special_less(n);
        qty = 0;
        for (let i : q) {
            if (special[i]) {
                qty++;
            } else {
                special_less[i] = qty;
            }
        }
        let to_left = 0;
        for (let i : range(n)) {
            if (!special[i]) {
                if (2 * special_less[i] + tail[i] + 1 >= num_special) {
                    out.printLine("YES");
                    return;
                }
//            } else {
//                let to_right = num_special - to_left;
//                if (to_left >= to_right && head[i] + to_right >= to_left) {
//                    out.printLine("YES");
//                    return;
//                }
//                to_left++;
            }
        }
        out.printLine("NO");
    }
};


int main() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
#ifdef LOCAL_RELEASE
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
    auto time = clock();
#endif
    EqualNumberOfPrefixMaximums solver;


    int n;
scanf("%d", &n);
for(int i = 0; i < n; ++i) {
	solver.solve();
}

    fflush(stdout);
#ifdef LOCAL_RELEASE
    cerr << setprecision(3) << double(clock() - time) / CLOCKS_PER_SEC << endl;
#endif
    return 0;
}

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