3165. Maximum Sum of Subsequence With Non-adjacent Elements

You are given an array nums consisting of integers. You are also given a 2D array queries, where queries[i] = [posi, xi].

For query i, we first set nums[posi] equal to xi, then we calculate the answer to query i which is the maximum sum of a subsequence of nums where no two adjacent elements are selected.

Return the sum of the answers to all queries.

Since the final answer may be very large, return it modulo 109 + 7.

A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.

c++

#include <bits/stdc++.h>

#pragma optimization_level 3
#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math,O3")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx")
#pragma GCC optimize("Ofast")//Comment optimisations for interactive problems (use endl)
#pragma GCC target("avx,avx2,fma")
#pragma GCC optimization ("unroll-loops")

using namespace std;

struct PairHash {inline std::size_t operator()(const std::pair<long long, long long> &v) const { return v.first * 31ll + v.second; }};

// speed
#define Code ios_base::sync_with_stdio(false);
#define By ios::sync_with_stdio(0);
#define Sumfi cout.tie(NULL);

// alias
using ll = long long;
using ld = long double;
using ull = unsigned long long;

// constants
const ld PI = acosl(-1.0);  /* pi */
const ll INF = 1e18;
const ld EPS = 1e-9;
const ll MAX_N = 1010101;
const ll mod = 1e9+7;

// typedef
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
typedef vector<pll> vpll;
typedef array<int,3> ai3;
typedef array<ll,3> all3;
typedef array<ll,4> all4;
typedef array<ll,5> all5;
typedef vector<all3> vall3;
typedef vector<all4> vall4;
typedef vector<all5> vall5;
typedef pair<ld, ld> pld;
typedef vector<pld> vpld;
typedef vector<ld> vld;
typedef vector<ll> vll;
typedef vector<ull> vull;
typedef vector<vll> vvll;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<bool> vb;
typedef deque<ll> dqll;
typedef deque<pll> dqpll;
typedef pair<string, string> pss;
typedef vector<pss> vpss;
typedef vector<string> vs;
typedef vector<vs> vvs;
typedef unordered_set<ll> usll;
typedef unordered_set<pll, PairHash> uspll;
typedef unordered_map<ll, ll> umll;
typedef unordered_map<pll, ll, PairHash> umpll;

// macros
#define precision(x) cout<<fixed;cout.precision(x);
#define rep(i,m,n) for(ll i=m;i<n;i++)
#define rrep(i,m,n) for(ll i=n;i>=m;i--)
#define all(a) begin(a), end(a)
#define rall(a) rbegin(a), rend(a)
#define uniq(a) sort(all(a)), a.erase(unique(all(a)),end(a))
#define ZERO(a) memset(a,0,sizeof(a))
#define MINUS(a) memset(a,0xff,sizeof(a))
#define INF(a) memset(a,0x3f3f3f3f3f3f3f3fLL,sizeof(a))
#define NEGINF(a) memset(a,0xcf,sizeof(a))
#define ASCEND(a) iota(all(a),0)
#define sz(x) ll((x).size())
#define BIT(a,i) ((a>>i)&1)
#define BITSHIFT(a,i,n) (((a<<i) & ((1ll<<n) - 1)) | (a>>(n-i)))
#define MAXBIT(a) (64ll - __builtin_clzll(a) - 1ll)
#define MINBIT(a) (__builtin_ctzll(a))
#define pyes cout<<"YES\n";
#define pno cout<<"NO\n";
#define endl "\n"
#define pneg1 cout<<"-1\n";
#define ppossible cout<<"possible\n";
#define pimpossible cout<<"impossible\n";
#define TC(x) cout<<"Case #"<<x<<": ";
#define X first
#define Y second

// debug
void __print(int x) { cerr << x; }
void __print(long x) { cerr << x; }
void __print(long long x) { cerr << x; }
void __print(unsigned x) { cerr << x; }
void __print(unsigned long x) { cerr << x; }
void __print(unsigned long long x) { cerr << x; }
void __print(float x) { cerr << x; }
void __print(double x) { cerr << x; }
void __print(long double x) { cerr << x; }
void __print(char x) { cerr << '\'' << x << '\''; }
void __print(const char *x) { cerr << '\"' << x << '\"'; }
void __print(const string &x) { cerr << '\"' << x << '\"'; }
void __print(bool x) { cerr << (x ? "true" : "false"); }
template <typename A>
void __print(const A &x);
template <typename A, typename B>
void __print(const pair<A, B> &p);
template <typename... A>
void __print(const tuple<A...> &t);
template <typename T>
void __print(stack<T> s);
template <typename T>
void __print(queue<T> q);
template <typename T, typename... U>
void __print(priority_queue<T, U...> q);
template <typename A>
void __print(const A &x) {
    bool first = true;
    cerr << '{';
    for (const auto &i : x) {
        cerr << (first ? "" : ","), __print(i);
        first = false;
    }
    cerr << '}';
}
template <typename A, typename B>
void __print(const pair<A, B> &p) {
    cerr << '(';
    __print(p.first);
    cerr << ',';
    __print(p.second);
    cerr << ')';
}
template <typename... A>
void __print(const tuple<A...> &t) {
    bool first = true;
    cerr << '(';
    apply([&first](const auto &...args) { ((cerr << (first ? "" : ","), __print(args), first = false), ...); }, t);
    cerr << ')';
}
template <typename T>
void __print(stack<T> s) {
    vector<T> debugVector;
    while (!s.empty()) {
        T t = s.top();
        debugVector.push_back(t);
        s.pop();
    }
    reverse(debugVector.begin(), debugVector.end());
    __print(debugVector);
}
template <typename T>
void __print(queue<T> q) {
    vector<T> debugVector;
    while (!q.empty()) {
        T t = q.front();
        debugVector.push_back(t);
        q.pop();
    }
    __print(debugVector);
}
template <typename T, typename... U>
void __print(priority_queue<T, U...> q) {
    vector<T> debugVector;
    while (!q.empty()) {
        T t = q.top();
        debugVector.push_back(t);
        q.pop();
    }
    __print(debugVector);
}
void _print() { cerr << "]\n"; }
template <typename Head, typename... Tail>
void _print(const Head &H, const Tail &...T) {
    __print(H);
    if (sizeof...(T))
        cerr << ", ";
    _print(T...);
}
#ifndef ONLINE_JUDGE
#define debug(...) cerr << "Line:" << __LINE__ << " [" << #__VA_ARGS__ << "] = ["; _print(__VA_ARGS__);
#else
#define debug(...)
#endif

// utility functions
template <typename T>
void print(T &&t)  { cout << t << "\n"; }
template<typename T>
void printv(vector<T>v){ll n=v.size();rep(i,0,n){cout<<v[i];if(i+1!=n)cout<<' ';}cout<<endl;}
template<typename T>
void printvv(vector<vector<T>>v){ll n=v.size();rep(i,0,n)printv(v[i]);}
template<typename T>
void printvln(vector<T>v){ll n=v.size();rep(i,0,n)cout<<v[i]<<endl;}
void fileIO(string in = "input.txt", string out = "output.txt") {freopen(in.c_str(),"ma",stdin); freopen(out.c_str(),"w",stdout);}
void hackercupIO(string in) {fileIO("/Users/hayoungsong/Downloads/" + in, "/Users/hayoungsong/Downloads/solution.txt");}
void readf() {freopen("", "rt", stdin);}
template <typename... T>
void in(T &...a) { ((cin >> a), ...); }
template<typename T>
void in(vector<T>& v){rep(i,0,sz(v)) in(v[i]);}
template<typename T>
void in(deque<T>& v){rep(i,0,sz(v)) in(v[i]);}
template<typename T, typename U>
void in(pair<T,U>& A) {in(A.first, A.second);}
template<typename T, typename U>
void in(vector<pair<T,U>>& A) {rep(i,0,sz(A)) in(A[i]); }
template<typename T, std::size_t N>
void in(vector<array<T,N>>& A) {rep(i,0,sz(A)) rep(j,0,sz(A[i])) in(A[i][j]); }
template<typename T>
void in(vector<vector<T>>& A) {rep(i,0,sz(A)) in(A[i]);}

struct Combination {
    vll fac, inv;
    ll n, MOD;

    ll modpow(ll n, ll x, ll MOD = mod) { if(!x) return 1; ll res = modpow(n,x>>1,MOD); res = (res * res) % MOD; if(x&1) res = (res * n) % MOD; return res; }

    Combination(ll _n, ll MOD = mod): n(_n + 1), MOD(MOD) {
        inv = fac = vll(n,1);
        rep(i,1,n) fac[i] = fac[i-1] * i % MOD;
        inv[n - 1] = modpow(fac[n - 1], MOD - 2, MOD);
        rrep(i,1,n - 2) inv[i] = inv[i + 1] * (i + 1) % MOD;
    }

    ll fact(ll n) {return fac[n];}
    ll nCr(ll n, ll r) {
        if(n < r or n < 0 or r < 0) return 0;
        return fac[n] * inv[r] % MOD * inv[n-r] % MOD;
    }
};

struct Matrix {
    ll r,c,MOD;
    vvll matrix;
    Matrix(ll r, ll c, ll v = 0, ll MOD = mod): r(r), c(c), matrix(vvll(r,vll(c,v))), MOD(MOD) {}
    Matrix(vvll m, ll MOD = mod) : r(sz(m)), c(sz(m[0])), matrix(m), MOD(MOD) {}

    vector<ll>& operator[](ll pos) {return matrix[pos];}
    Matrix operator*(const Matrix& B) const {
        Matrix res(r, B.c, 0,MOD);
        rep(i,0,r) rep(j,0,B.c) rep(k,0,B.r) {
                    res[i][j] = (res[i][j] + matrix[i][k] * B.matrix[k][j] % MOD) % MOD;
                }
        return res;
    }

    Matrix copy() {
        Matrix copy(r,c,0,MOD);
        copy.matrix = matrix;
        return copy;
    }


    Matrix pow(ll n) {
        assert(r == c);
        Matrix res(r,r, 0,MOD);
        Matrix now = copy();
        rep(i,0,r) res[i][i] = 1;
        while(n) {
            if(n & 1) res = res * now;
            now = now * now;
            n /= 2;
        }
        return res;
    }

    ll det() {
        if(r == 1) return matrix[0][0];
        if(r == 2) return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
        ll res = 0;
        rep(p,0,c) {
            Matrix mat(c-1,c-1);
            rep(i,1,r) rep(j,0,c) {
                    if(j == p) continue;
                    mat[i][j - (j >= p)] = matrix[i][j];
                }
            res += matrix[0][p] * (p & 1 ? -1 : 1) * mat.det();
        }
        return res;
    }
};

// geometry data structures
template <typename T>
struct Point {
    T y,x;
    Point(T y, T x) : y(y), x(x) {}
    Point(pair<T,T> p) : y(p.first), x(p.second) {}
    Point() {}
    void input() {cin>>y>>x;}
    friend ostream& operator<<(ostream& os, const Point<T>& p) { os<<p.y<<' '<<p.x<<'\n'; return os;}
    Point<T> operator+(Point<T>& p) {return Point<T>(y + p.y, x + p.x);}
    Point<T> operator-(Point<T>& p) {return Point<T>(y - p.y, x - p.x);}
    Point<T> operator*(ll n) {return Point<T>(y*n,x*n); }
    Point<T> operator/(ll n) {return Point<T>(y/n,x/n); }
    bool operator==(const Point<T> &a) {return x == a.x && y == a.y;}
    bool operator!=(const Point<T> &a) {return x != a.x or y != a.y;}
    bool operator<(const Point &other) const {if (x == other.x) return y < other.y;return x < other.x;}
    Point<T> rotate(Point<T> center, ld angle) {
        ld si = sin(angle * PI / 180.), co = cos(angle * PI / 180.);
        ld y = this->y - center.y;
        ld x = this->x - center.x;

        return Point<T>(y * co - x * si + center.y, y * si + x * co + center.x);
    }
    ld distance(Point<T> other) {
        T dy = abs(this->y - other.y);
        T dx = abs(this->x - other.x);
        return sqrt(dy * dy + dx * dx);
    }

    T norm() { return x * x + y * y; }
};

template<typename T>
struct Line {
    Point<T> A, B;
    Line(Point<T> A, Point<T> B) : A(A), B(B) {}
    Line() {}

    void input() {
        A = Point<T>();
        B = Point<T>();
        A.input();
        B.input();
    }

    T ccw(Point<T> &a, Point<T> &b, Point<T> &c) {
        T res = a.x * b.y + b.x * c.y + c.x * a.y;
        res -= (a.x * c.y + b.x * a.y + c.x * b.y);
        return res;
    }

    bool on(Point<T> x) {
        return ccw(A,x,B) == 0;
    }

    bool isIntersect(Line<T> o) {
        T p1p2 = ccw(A,B,o.A) * ccw(A,B,o.B);
        T p3p4 = ccw(o.A,o.B,A) * ccw(o.A,o.B,B);
        if (p1p2 == 0 && p3p4 == 0) {
            pair<T,T> p1(A.y, A.x), p2(B.y,B.x), p3(o.A.y, o.A.x), p4(o.B.y, o.B.x);
            if (p1 > p2) swap(p2, p1);
            if (p3 > p4) swap(p3, p4);
            return p3 <= p2 && p1 <= p4;
        }
        return p1p2 <= 0 && p3p4 <= 0;
    }

    pair<bool,Point<ld>> intersection(Line<T> o) {
        if(!this->intersection(o)) return {false, {}};
        ld det = 1. * (o.B.y-o.A.y)*(B.x-A.x) - 1.*(o.B.x-o.A.x)*(B.y-A.y);
        ld t = ((o.B.x-o.A.x)*(A.y-o.A.y) - (o.B.y-o.A.y)*(A.x-o.A.x)) / det;
        return {true, {A.y + 1. * t * (B.y - A.y), B.x + 1. * t * (B.x - A.x)}};
    }

    //@formula for : y = ax + pre
    //@return {a,pre};
    pair<ld, ld> formula() {
        T y1 = A.y, y2 = B.y;
        T x1 = A.x, x2 = B.x;
        if(y1 == y2) return {0, (ld)y1};
        if(x1 == x2) return {INF, (ld)x1};
        ld a = 1. * (y2 - y1) / (x2 - x1);
        ld b = -x1 * a + y1;
        return {a, b};
    }
};

template<typename T>
struct Circle {
    Point<T> center;
    T radius;
    Circle(T y, T x, T radius) : center(Point<T>(y,x)), radius(radius) {}
    Circle(Point<T> center, T radius) : center(center), radius(radius) {}
    Circle() {}

    void input() {
        center = Point<T>();
        center.input();
        cin>>radius;
    }

    bool circumference(Point<T> p) {
        return (center.x - p.x) * (center.x - p.x) + (center.y - p.y) * (center.y - p.y) == radius * radius;
    }

    bool intersect(Circle<T> c) {
        T d = (center.x - c.center.x) * (center.x - c.center.x) + (center.y - c.center.y) * (center.y - c.center.y);
        return (radius - c.radius) * (radius - c.radius) <= d and d <= (radius + c.radius) * (radius + c.radius);
    }

    bool include(Circle<T> c) {
        T d = (center.x - c.center.x) * (center.x - c.center.x) + (center.y - c.center.y) * (center.y - c.center.y);
        return d <= radius * radius;
    }
};

ll __gcd(ll x, ll y) { return !y ? x : __gcd(y, x % y); }
all3 __exgcd(ll x, ll y) { if(!y) return {x,1,0}; auto [g,x1,y1] = __exgcd(y, x % y); return {g, y1, x1 - (x/y) * y1}; }
ll __lcm(ll x, ll y) { return x / __gcd(x,y) * y; }
ll modpow(ll n, ll x, ll MOD = mod) {if(x<0){return modpow(modpow(n,-x,MOD),MOD-2,MOD);}n%=MOD;ll res=1;while(x){if(x&1){res=res*n%MOD;}n=n*n%MOD;x>>=1;}return res;}
ll __xor(ll n) {return n%4==0?n:n%4==1?1:n%4==2?n+1:0;}
ll __rangexor(ll l, ll r) {return __xor(r)^__xor(l-1);}

struct Seg {
    ll mi,ma;
    all4 best;
    Seg* left, *right;
    void pushUp() {
        best = {0,0,0,0};
        rep(i,0,4) rep(j,0,4) {
            if(BIT(i,0) and BIT(j,1)) continue;
            ll k = j % 2 + (BIT(i,1)<<1);
            best[k] = max(best[k], left->best[i] + right->best[j]);
        }
    }
    Seg(vi& A, ll l, ll r) : mi(l), ma(r), left(nullptr), right(nullptr), best({0,0,0,0}) {
        if(l^r) {
            ll m = l + (r - l) / 2;
            left = new Seg(A,l,m);
            right = new Seg(A,m+1,r);

            pushUp();
        } else {
            best[3] = max(0,A[l]);
        }
    }
    void update(ll n, ll k) {
        if(mi <= n and n <= ma) {
            if(n == ma and n == mi) best = {0,0,0,max(0ll,k)};
            else {
                left->update(n,k);
                right->update(n,k);
                pushUp();
            }
        }
    }
    ll pick() {
        return max({best[0], best[1], best[2], best[3]});
    }
};
class Solution {
public:
    int maximumSumSubsequence(vector<int>& A, vector<vector<int>>& Q) {
        ll res = 0;
        Seg* seg = new Seg(A,0,sz(A) - 1);
        rep(i,0,sz(Q)) {
            ll p = Q[i][0], x = Q[i][1];
            seg->update(p,x);
            res = (res + seg->pick()) % mod;
        }
        return res;
    }
};