3013. Divide an Array Into Subarrays With Minimum Cost II

You are given a 0-indexed array of integers nums of length n, and two positive integers k and dist.

The cost of an array is the value of its first element. For example, the cost of [1,2,3] is 1 while the cost of [3,4,1] is 3.

You need to divide nums into k disjoint contiguous subarrays, such that the difference between the starting index of the second subarray and the starting index of the kth subarray should be less than or equal to dist. In other words, if you divide nums into the subarrays nums[0..(i1 - 1)], nums[i1..(i2 - 1)], ..., nums[ik-1..(n - 1)], then ik-1 - i1 <= dist.

Return the minimum possible sum of the cost of these subarrays.

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-15;
const ll MAX_N = 303030;
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(),"r",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 XOR(ll l, ll r) {return __xor(r)^__xor(l-1);}

struct Seg {
    ll mi, ma, cnt, sum;
    Seg *left, *right;
    Seg(vll& A, ll l, ll r) : mi(A[l]), ma(A[r]), cnt(0), sum(0), left(nullptr), right(nullptr) {
        if(l^r) {
            ll m = l + (r - l) / 2;
            left = new Seg(A,l,m);
            right = new Seg(A,m+1,r);
        }
    }
    void update(ll n, ll k) {
        if(mi <= n and n <= ma) {
            cnt += k;
            sum += n * k;
            if(left) left->update(n,k);
            if(right) right->update(n,k);
        }
    }
    ll query(ll k) {
        if(cnt <= k) return sum;
        if(mi == ma) return k * mi;
        if(left->cnt >= k) return left->query(k);
        return left->sum + right->query(k - left->cnt);
    }
};
struct DS {
    multiset<ll> left, right;
    ll sum = 0,k;
    DS(ll k) : k(k), sum(0) {}
    void add(ll x) {
        if(sz(left) < k or *prev(end(left)) > x) {
            sum += x;
            left.insert(x);
            if(sz(left) > k) {
                sum -= *prev(end(left));
                right.insert(*prev(end(left)));
                left.erase(*prev(end(left)));
            }
        } else right.insert(x);
    }
    void del(ll x) {
        if(right.count(x)) {
            right.erase(right.find(x));
            return;
        }
        left.erase(left.find(x));
        sum -= x;
        ll mi = *right.begin();
        left.insert(mi);
        sum += mi;
        right.erase(begin(right));
    }
};
class Solution {
public:
    long long minimumCost(vector<int>& nums, int k, int dist) {
        if(k == nums.size()) {
            ll res = 0;
            rep(i,0,sz(nums)) res += nums[i];
            return res;
        }
        if(dist == sz(nums) - 2) {
            ll res = nums[0];
            sort(begin(nums) + 1, end(nums));
            rep(i,1,k) res += nums[i];
            return res;
        }
        ll res = INF;
        vll S;
        rep(i,1,sz(nums)) S.push_back(nums[i]);
        uniq(S);
        if(sz(S) == 1) return nums[0] + (k - 1) * S[0];
        Seg* seg = new Seg(S,0,sz(S) - 1);
        rep(i,1,sz(nums)) {
            seg->update(nums[i], 1);
            if(i - dist - 1 >= 1) {
                seg->update(nums[i-dist-1], -1);
            }
            if(i >= k - 1) {
                res = min(res, seg->query(k-1) + nums[0]);
            }
        }
        return res;
    }
};