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#define PROBLEM "https://yukicoder.me/problems/no/409" #include <iostream> #include <vector> #include <queue> using namespace std; #include "../../lib/12-binary-search-tree/RandomizedBinarySearchTreeSet.cpp" #include "../../lib/99-operator/operator/ValueMax.cpp" #include "../../lib/16-convex-hull-trick/ConvexHullTrick.cpp" using ll = long long; int main(void){ 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; }
#line 1 "test/convex-hull-trick/ConvexHullTrick-max.test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/409" #include <iostream> #include <vector> #include <queue> using namespace std; #line 1 "lib/12-binary-search-tree/RandomizedBinarySearchTreeSet.cpp" /* * @title RandomizedBinarySearchTree - ランダム平衡二分探索木set * @docs md/binary-search-tree/RandomizedBinarySearchTree.md */ template<class Monoid> class RandomizedBinarySearchTreeSet { 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: RandomizedBinarySearchTreeSet() : 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];} }; #line 1 "lib/99-operator/operator/ValueMax.cpp" //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;} }; #line 1 "lib/16-convex-hull-trick/ConvexHullTrick.cpp" /* * @title ConvexHullTrick - 非単調CHT * @docs md/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};} }; RandomizedBinarySearchTreeSet<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(); } }; #line 10 "test/convex-hull-trick/ConvexHullTrick-max.test.cpp" using ll = long long; int main(void){ 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; }