結果

問題 No.409 ダイエット
ユーザー ningenMe
提出日時 2021-05-03 04:42:02
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 1,089 ms / 2,000 ms
コード長 11,706 bytes
コンパイル時間 2,493 ms
コンパイル使用メモリ 215,412 KB
最終ジャッジ日時 2025-01-21 06:12:37
ジャッジサーバーID
（参考情報） 		judge4 / judge1
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ファイルパターン 結果
other AC * 92
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ソースコード

diff # 

#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using ull = unsigned long long;
 
#define REP(i,n) for(long long i = 0; i < (n); i++)
#define FOR(i, m, n) for(long long i = (m);i < (n); ++i)
#define ALL(obj) (obj).begin(),(obj).end()
#define SPEED cin.tie(0);ios::sync_with_stdio(false);
 
template<class T> using V = vector<T>;
template<class T, class U> using P = pair<T, U>;
template<class T> using PQ = priority_queue<T>;
template<class T> using PQR = priority_queue<T,vector<T>,greater<T>>;
 
constexpr long long MOD = (long long)1e9 + 7;
constexpr long long MOD2 = 998244353;
constexpr long long HIGHINF = (long long)1e18;
constexpr long long LOWINF = (long long)1e15;
constexpr long double PI = 3.1415926535897932384626433;
 
template <class T> vector<T> multivector(size_t N,T init){return vector<T>(N,init);}
template <class... T> auto multivector(size_t N,T... t){return vector<decltype(multivector(t...))>(N,multivector(t...));}
template <class T> void corner(bool flg, T hoge) {if (flg) {cout << hoge << endl; exit(0);}}
template <class T, class U>ostream &operator<<(ostream &o, const map<T, U>&obj) {o << "{"; for (auto &x : obj) o << " {" << x.first << " : " << x.second << "}" << ","; o << " }"; return o;}
template <class T>ostream &operator<<(ostream &o, const set<T>&obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr) o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
template <class T>ostream &operator<<(ostream &o, const multiset<T>&obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr) o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
template <class T>ostream &operator<<(ostream &o, const vector<T>&obj) {o << "{"; for (int i = 0; i < (int)obj.size(); ++i)o << (i > 0 ? ", " : "") << obj[i]; o << "}"; return o;}
template <class T, class U>ostream &operator<<(ostream &o, const pair<T, U>&obj) {o << "{" << obj.first << ", " << obj.second << "}"; return o;}
template <template <class tmp>  class T, class U> ostream &operator<<(ostream &o, const T<U> &obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr)o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
void print(void) {cout << endl;}
template <class Head> void print(Head&& head) {cout << head;print();}
template <class Head, class... Tail> void print(Head&& head, Tail&&... tail) {cout << head << " ";print(forward<Tail>(tail)...);}
template <class T> void chmax(T& a, const T b){a=max(a,b);}
template <class T> void chmin(T& a, const T b){a=min(a,b);}
void YN(bool flg) {cout << (flg ? "YES" : "NO") << endl;}
void Yn(bool flg) {cout << (flg ? "Yes" : "No") << endl;}
void yn(bool flg) {cout << (flg ? "yes" : "no") << endl;}

/*
 * @title RandomizedBinarySearchTree - 平衡二分探索木
 * @docs md/data-structure/binary-search-tree/RandomizedBinarySearchTree.md
 */
template<class Monoid> class RandomizedBinarySearchTree {
    using TypeNode = typename Monoid::TypeNode;
    unsigned int x = 123456789, y = 362436069, z = 521288629, w = 88675123;
    unsigned int xor_shift() {
        unsigned int t = (x ^ (x << 11)); x = y; y = z; z = w;
        return (w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)));
    }
    struct Node {
    private:
        void build() {left = right = nullptr;size = 1;}
    public:
        Node *left, *right;
        TypeNode value, range_value;
        int size;
        Node() : value(Monoid::unit_node), range_value(Monoid::unit_node) {build();}
        Node(TypeNode v) : value(v), range_value(v) {build();}
        friend ostream &operator<<(ostream &os, const Node* node) {return os << "{" << node->value << ", " << node->range_value << ", " << node->size << "}";}
    };
    Node* root;
    inline int size(Node *node) {return node==nullptr ? 0 : node->size;}
    inline TypeNode range_value(Node *node) {return node==nullptr ? Monoid::unit_node : node->range_value;}
    inline TypeNode get(Node *node, size_t k) {
        if (node==nullptr) return Monoid::unit_node;
        if (k == size(node->left)) return node->value;
        if (k < size(node->left)) return get(node->left, k);
        else return get(node->right, k-1 - size(node->left));
    }
    inline Node* update(Node *node) {
        node->size = size(node->left) + size(node->right) + 1;
        node->range_value = Monoid::func_fold(Monoid::func_fold(range_value(node->left),node->value),range_value(node->right));
        return node;
    }
    inline Node* merge_impl(Node *left, Node *right) {
        if (left==nullptr)  return right;
        if (right==nullptr) return left;
        if (xor_shift() % (left->size + right->size) < left->size) {
            left->right = merge_impl(left->right, right);
            return update(left);
        }
        else {
            right->left = merge_impl(left, right->left);
            return update(right);
        }
    }
    inline pair<Node*, Node*> split_impl(Node* node, int k) {
        if (node==nullptr) return make_pair(nullptr, nullptr);
        if (k <= size(node->left)) {
            pair<Node*, Node*> sub = split_impl(node->left, k);
            node->left = sub.second;
            return make_pair(sub.first, update(node));
        }
        else {
            pair<Node*, Node*> sub = split_impl(node->right, k - 1 - size(node->left));
            node->right = sub.first;
            return make_pair(update(node), sub.second);
        }
    }
    inline TypeNode fold_impl(Node *node, int l, int r) {
        if (l < 0 || size(node) <= l || r<=0 || r-l <= 0) return Monoid::unit_node;
        if (l == 0 && r == size(node)) return range_value(node);
        TypeNode value = Monoid::unit_node;
        int sl = size(node->left);
        if(sl > l) value = Monoid::func_fold(value,fold_impl(node->left,l,min(sl,r)));
        l = max(l-sl,0), r -= sl;
        if(l == 0 && r > 0) value = Monoid::func_fold(value,node->value);
        l = max(l-1,0), r -= 1;
        if(l >= 0 && r > l) value = Monoid::func_fold(value,fold_impl(node->right,l,r));
        return value;
    }
    inline int lower_bound(Node *node, TypeNode value) {
        if (node==nullptr) return 0;
        if (value <= node->value) return lower_bound(node->left, value);
        else return size(node->left) + lower_bound(node->right, value) + 1;
    }
    inline int upper_bound(Node *node, TypeNode value) {
        if (node==nullptr) return 0;
        if (value < node->value) return upper_bound(node->left, value);
        else return size(node->left) + upper_bound(node->right, value) + 1;
    }
    inline void insert_impl(const TypeNode value) {
        pair<Node*, Node*> sub = split_impl(this->root, lower_bound(this->root,value)); 
        this->root = this->merge_impl(this->merge_impl(sub.first, new Node(value)), sub.second);
    }
    inline void erase_impl(const TypeNode value) {
        int k1 = lower_bound(value), k2 = upper_bound(value);
        if(k1==k2) return;
        auto sub = split_impl(this->root,k1);
        this->root = merge_impl(sub.first, split_impl(sub.second, 1).second);
    }
public:
    RandomizedBinarySearchTree() : root(nullptr) {}
    inline int size() {return size(this->root);}
    inline int empty(void) {return bool(size()==0);}
    inline Node* merge(Node *left, Node *right) {return merge_impl(left,right);}
    inline pair<Node*, Node*> split(int k) {return split_impl(this->root,k);}
    inline void insert(const TypeNode value) {insert_impl(value);}
    inline void erase(const TypeNode value) {erase_impl(value);}
    inline TypeNode get(size_t k) {return get(this->root, k);}
    inline TypeNode fold(int l, int r) {return fold_impl(this->root,l,r);}
    inline int lower_bound(TypeNode value) {return lower_bound(this->root,value);}
    inline int upper_bound(TypeNode value) {return upper_bound(this->root,value);}
    inline int count(TypeNode value) {return upper_bound(value) - lower_bound(value);}
    void print() {int m = size(this->root); for(int i=0;i<m;++i) cout << get(i) << " \n"[i==m-1];}
};

//最大値クエリ
template<class T> struct ValueMax {
	using TypeValue = T;
	inline static constexpr TypeValue unit_value = -3e18;
	inline static constexpr bool func_compare(TypeValue l,TypeValue r){return l>r;}
};

/*
 * @title ConvexHullTrick - 非単調CHT
 * @docs md/data-structure/convex-hull-trick/ConvexHullTrick.md
 */
template<class Operator> class ConvexHullTrick {
private:
	using TypeValue = typename Operator::TypeValue;
	using Line = pair<TypeValue,TypeValue>;
	struct Monoid {
		using TypeNode = Line;
		inline static constexpr TypeNode unit_node = {0,0};
		inline static constexpr TypeNode func_fold(TypeNode l,TypeNode r){return {0,0};}
	};
	RandomizedBinarySearchTree<Monoid> lines;

	//3直線に関してline2が必要か確認 (このとき a1 < a2 < a3が必要=rbstの順そのまま)
	inline int is_required(const Line& line1, const Line& line2, const Line& line3) {
		return Operator::func_compare((line2.second-line3.second)*(line2.first-line1.first),(line1.second-line2.second)*(line3.first-line2.first));
	}
	
	//y=ax+bの値
	inline TypeValue y(const Line line, TypeValue x) {
		return line.first * x + line.second;
	}

public:
	ConvexHullTrick() {
		// do nothing
	} 

	//ax+bを追加
	void insert(const TypeValue a, const TypeValue b) {
		insert({a,b});
	}
	//ax+bを追加 armortized O(log(N))
	void insert(const Line line) {
		int k=lines.lower_bound(line), flg=1;
		Line l,r;
		if(0 <= k-1) {
			l = lines.get(k-1);
			//lと傾きが同じなら、どちらかをerase
			if(l.first==line.first) {
				if(Operator::func_compare(l.second,line.second)) return;
				else lines.erase(l);
			}
		}
		if(k < lines.size()) {
			r = lines.get(k);
			//rと傾きが同じなら、どちらかをerase
			if(r.first==line.first) {
				if(Operator::func_compare(r.second,line.second)) return;
				else lines.erase(r);
			}	
		}
		//自身が必要か判定
		if(0 <= k-1 && k < lines.size() && !is_required(l,line,r)) return;
		//傾きが小さい側の不必要な直線を取り除く
		for(k=lines.lower_bound(line);k>=2&&!is_required(lines.get(k-2), lines.get(k-1), line);k=lines.lower_bound(line)) lines.erase(lines.get(k-1));
		//傾きが大きい側の不必要な直線を取り除く
		for(k=lines.lower_bound(line);k+1<lines.size()&&!is_required(line,lines.get(k),lines.get(k+1));k=lines.lower_bound(line)) lines.erase(lines.get(k));
		lines.insert(line);
	}
	
	//O(log(N)^2)
	TypeValue get(TypeValue x) {
		int ng = -1, ok = (int)lines.size()-1, md;
		while (ok - ng > 1) {
			md = (ok + ng) >> 1;
			( Operator::func_compare(y(lines.get(md),x),y(lines.get(md+1),x)) ?ok:ng)=md;
		}
		return y(lines.get(ok),x);
	}

	//O(log(N)^2)
	Line get_line(TypeValue x) {
		int ng = -1, ok = (int)lines.size()-1, md;
		while (ok - ng > 1) {
			md = (ok + ng) >> 1;
			( Operator::func_compare(y(lines.get(md),x),y(lines.get(md+1),x)) ?ok:ng)=md;
		}
		return lines.get(ok);
	}

	void print() {
		lines.print();
	}
};

int main() {
	ll N,A,B,W; cin >> N >> A >> B >> W;
	vector<ll> D(N+2,0);
	for(int i = 1; i <= N; ++i) cin >> D[i];
	// dp[i]=min{j:[0,i)} -> dp[j]+B*k*(k+1)/2-k*A+D[i] (k=i-j-1)
	//                    -> dp[j]+B*(i-j-1)*(i-j)/2-(i-j-1)*A+D[i]
	//                    -> dp[j]+B/2*(i*i-2*i*j+j*j-i+j)-A*(i-j-1)+D[i]
	//                    -> (-B*j)*i  +  dp[j]+B/2*(j*j+j)+A*j  +  B/2*(i*i-i)-A*(i-1)+D[i] 
	// dp[i]=-max{j:[0,i)}-> (B*j)*i  +  -{dp[j]+B/2*(j*j+j)+A*j} 
	//                    ->
	vector<ll> dp(N+2,1e15);
	dp[0]=W;
	ConvexHullTrick<ValueMax<ll>> cht;
	cht.insert(0,-dp[0]);
	for(ll i=1;i<=N+1;++i){
		dp[i]=-cht.get(i) + B*(i*i-i)/2-A*(i-1)+D[i];
		cht.insert(B*i,-(dp[i]+B*(i*i+i)/2+A*i));
	}
	cout << dp[N+1] << endl;
	return 0;
}
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