#line 1 "test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/409" #include using namespace std; #line 4 "https://drken1215.github.io/algorithm/template.hpp.html" using namespace std; // My library is in https://github.com/drken1215/algorithm #line 4 "https://drken1215.github.io/algorithm/Others/macro.hpp.html" using namespace std; using ll = long long; using i128 = __int128_t; using u128 = __uint128_t; using pint = pair; using pll = pair; using tll = array; using fll = array; using vint = vector; using vll = vector; using dint = deque; using dll = deque; using vvint = vector>; using vvll = vector>; using vpll = vector>; template using min_priority_queue = priority_queue, greater>; template inline bool chmax(S &a, T b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(S &a, T b) { return (a > b ? a = b, 1 : 0); } template inline auto maxll(S a, T b) { return max(ll(a), ll(b)); } template inline auto minll(S a, T b) { return min(ll(a), ll(b)); } template auto max(const T &a) { return *max_element(a.begin(), a.end()); } template auto min(const T &a) { return *min_element(a.begin(), a.end()); } template auto argmax(const T &a) { return max_element(a.begin(), a.end()) - a.begin(); } template auto argmin(const T &a) { return min_element(a.begin(), a.end()) - a.begin(); } template auto accum(const vector &a) { return accumulate(a.begin(), a.end(), T()); } template auto accum(const deque &a) { return accumulate(a.begin(), a.end(), T()); } #define REP(i, a) for (long long i = 0; i < (long long)(a); i++) #define REP2(i, a, b) for (long long i = a; i < (long long)(b); i++) #define RREP(i, a) for (long long i = (a)-1; i >= (long long)(0); --i) #define RREP2(i, a, b) for (long long i = (b)-1; i >= (long long)(a); --i) #define EB emplace_back #define PF push_front #define PB push_back #define MP make_pair #define FI first #define SE second #define ALL(x) x.begin(), x.end() #line 3 "https://drken1215.github.io/algorithm/Others/stl_container_stream.hpp.html" using namespace std; // input stream template istream& operator >> (istream &is, vector &P) { for (int i = 0; i < (int)P.size(); ++i) cin >> P[i]; return is; } template istream& operator >> (istream &is, deque &P) { for (int i = 0; i < (int)P.size(); ++i) cin >> P[i]; return is; } template istream& operator >> (istream &is, vector> &P) { for (int i = 0; i < (int)P.size(); ++i) cin >> P[i]; return is; } // output stream #define COUT(x) cout << #x << " = " << (x) << " (L" << __LINE__ << ")" << endl template ostream& operator << (ostream &s, const pair &P) { return s << '<' << P.first << ", " << P.second << '>'; } template ostream& operator << (ostream &s, const array &P) { return s << '<' << P[0] << "," << P[1] << '>'; } template ostream& operator << (ostream &s, const array &P) { return s << '<' << P[0] << "," << P[1] << "," << P[2] << '>'; } template ostream& operator << (ostream &s, const array &P) { return s << '<' << P[0] << "," << P[1] << "," << P[2] << "," << P[3] << '>'; } template ostream& operator << (ostream &s, const vector &P) { for (int i = 0; i < P.size(); ++i) { s << P[i] << endl; } return s; } template ostream& operator << (ostream &s, const vector &P) { for (int i = 0; i < P.size(); ++i) { if (i > 0) { s << " "; } s << P[i]; } return s; } template ostream& operator << (ostream &s, const deque &P) { for (int i = 0; i < P.size(); ++i) { if (i > 0) { s << " "; } s << P[i]; } return s; } template ostream& operator << (ostream &s, const vector> &P) { for (int i = 0; i < P.size(); ++i) { s << endl << P[i]; } return s << endl; } template ostream& operator << (ostream &s, const set &P) { for (auto it : P) { s << "<" << it << "> "; } return s; } template ostream& operator << (ostream &s, const multiset &P) { for (auto it : P) { s << "<" << it << "> "; } return s; } template ostream& operator << (ostream &s, const unordered_set &P) { for (auto it : P) { s << "<" << it << "> "; } return s; } template ostream& operator << (ostream &s, const map &P) { for (auto it : P) { s << "<" << it.first << "->" << it.second << "> "; } return s; } template ostream& operator << (ostream &s, const unordered_map &P) { for (auto it : P) { s << "<" << it.first << "->" << it.second << "> "; } return s; } // Yes/No void yes(bool a) { cout << (a ? "yes" : "no") << endl; } void YES(bool a) { cout << (a ? "YES" : "NO") << endl; } void Yes(bool a) { cout << (a ? "Yes" : "No") << endl; } #line 4 "https://drken1215.github.io/algorithm/Optimization/monge_shortest_path_by_cht.hpp.html" using namespace std; #line 4 "https://drken1215.github.io/algorithm/Optimization/general_li_chao_tree.hpp.html" using namespace std; // General Li-Chao Tree (Dynamic ver.) for Pairwise Single Crossing Functions template struct DynamicGeneralLiChaoTree { struct Node { int id = -1; // function id Node *left = nullptr, *right = nullptr; }; // inner values using chunk-based memory pool vector> blocks; vector funcs; Node *root = nullptr; X XL, XR; VAL INF = numeric_limits::max() / 2; // constructors explicit DynamicGeneralLiChaoTree(X xl, X xr, VAL inf = numeric_limits::max() / 2) : XL(xl), XR(xr), INF(inf) { assert(xl < xr); } // basic functions void reserve_functions(size_t n) { funcs.reserve(n); } VAL eval(int id, X x) const { return funcs[id](x); } bool better(int fid, int gid, X x) const { VAL a = eval(fid, x), b = eval(gid, x); return (a < b) || (a == b && fid < gid); } void clear() { funcs.clear(); blocks.clear(); used = BLOCK; root = nullptr; } // insert(f), in O(log X) int insert(FUNC f) { int id = (int)funcs.size(); funcs.push_back(std::move(f)); insert_at(root, XL, XR, id); return id; } // insert_segment(l, r, f), in O(log^2 X) int insert_segment(FUNC f, X l, X r) { assert(XL <= l && l <= r && r <= XR); int id = (int)funcs.size(); funcs.push_back(std::move(f)); if (l != r) insert_segment_at(root, XL, XR, l, r, id); return id; } // query(x), in O(log X) pair query(X x) const { assert(XL <= x && x < XR); VAL best = INF; int best_id = -1; Node *p = root; X left = XL, right = XR; while (p) { if (p->id != -1) { int fid = p->id; VAL y = eval(fid, x); if (best_id == -1 || y < best || (y == best && fid < best_id)) { best = y, best_id = fid; } } if (right - left == 1) break; X mid = (left + right) / 2; if (x < mid) p = p->left, right = mid; else p = p->right, left = mid; } return {best, best_id}; } private: // chunk-based node allocation static constexpr int BLOCK = 4096; int used = BLOCK; Node *new_node() { if (used == BLOCK) { blocks.emplace_back(make_unique(BLOCK)); used = 0; } return &blocks.back()[used++]; } // inner insert void insert_at(Node *&p, X left, X right, int fid) { Node **cur = &p; while (true) { if (!*cur) *cur = new_node(); Node *v = *cur; int gid = v->id; if (gid == -1) { v->id = fid; return; } bool left_better = better(fid, gid, left), right_better = better(fid, gid, right - 1); if (!left_better && !right_better) return; if (left_better && right_better) { v->id = fid; return; } X mid = (left + right) / 2; bool mid_better = better(fid, gid, mid); if (mid_better) v->id = fid, fid = gid; if (left_better != mid_better) cur = &v->left, right = mid; else cur = &v->right, left = mid; } } // inner insert_segment void insert_segment_at(Node *&p, X left, X right, X l, X r, int fid) { if (r <= left || right <= l) return; if (l <= left && right <= r) { insert_at(p, left, right, fid); return; } if (!p) p = new_node(); X mid = (left + right) / 2; if (l < mid) insert_segment_at(p->left, left, mid, l, r, fid); if (mid < r) insert_segment_at(p->right, mid, right, l, r, fid); } }; // General Li-Chao Tree (static ver.) for Pairwise Single Crossing Functions template struct GeneralLiChaoTree { // inner values int N, XL, SIZ; bool compressed = true; VAL INF = numeric_limits::max() / 2; vector xs; vector ids; vector funcs; // constructors explicit GeneralLiChaoTree(vector xs_, VAL inf = numeric_limits::max() / 2, bool sorted = true) : N((int)xs_.size()), XL(0), SIZ(1), compressed(true), INF(inf), xs(std::move(xs_)) { if (!sorted) { sort(xs.begin(), xs.end()); xs.erase(unique(xs.begin(), xs.end()), xs.end()); } while (SIZ < N) SIZ <<= 1; ids.assign(SIZ << 1, -1); } explicit GeneralLiChaoTree(int xl, int xr, VAL inf = numeric_limits::max() / 2) : N(xr - xl), XL(xl), SIZ(1), compressed(false), INF(inf) { assert(N >= 0); while (SIZ < N) SIZ <<= 1; ids.assign(SIZ << 1, -1); } // basic functions void reserve_functions(size_t n) { funcs.reserve(n); } VAL eval(int id, X x) const { return funcs[id](x); } bool better(int fid, int gid, X x) const { VAL a = eval(fid, x), b = eval(gid, x); return (a < b) || (a == b && fid < gid); } void clear() { funcs.clear(); fill(ids.begin(), ids.end(), -1); } // insert(f), in O(log N) int insert(FUNC f) { int id = (int)funcs.size(); funcs.push_back(std::move(f)); if (N) insert_at(1, 0, SIZ, id); return id; } // insert_segment(f, l, r), in O(log^2 N) int insert_segment(FUNC f, X l, X r) { assert(l <= r); int id = (int)funcs.size(); funcs.push_back(std::move(f)); if (N == 0 || l == r) return id; int L = lower_bound_index(l); int R = lower_bound_index(r); if (L == R) return id; int a = L + SIZ, b = R + SIZ; while (a < b) { if (a & 1) { int v = a++; int depth = 31 - __builtin_clz((unsigned)v); int length = SIZ >> depth; int left = (v - (1 << depth)) * length; insert_at(v, left, left + length, id); } if (b & 1) { int v = --b; int depth = 31 - __builtin_clz((unsigned)v); int length = SIZ >> depth; int left = (v - (1 << depth)) * length; insert_at(v, left, left + length, id); } a >>= 1, b >>= 1; } return id; } // query(x), x must be in xs, in O(log N) pair query(X x) const { int pos = 0; if (compressed) { auto it = lower_bound(xs.begin(), xs.end(), x); assert(it != xs.end() && *it == x); pos = int(it - xs.begin()); } else { assert(X(XL) <= x && x < X(XL + N)); pos = int(x - XL); } VAL best = INF; int best_id = -1; int v = pos + SIZ; while (v) { int fid = ids[v]; if (fid != -1) { VAL y = eval(fid, x); if (best_id == -1 || y < best || (y == best && fid < best_id)) { best = y, best_id = fid; } } v >>= 1; } return {best, best_id}; } private: // coordinate compression int lower_bound_index(X x) const { if (compressed) { return int(std::lower_bound(xs.begin(), xs.end(), x) - xs.begin()); } else { if (x <= X(XL)) return 0; else if (x >= X(XL) + X(N)) return N; else return int(x - X(XL)); } } // inner insert void insert_at(int v, int left, int right, int fid) { auto coord = [&](int i) -> X { i = min(i, N - 1); return (compressed ? xs[i] : X(i) + X(XL)); }; while (true) { int gid = ids[v]; if (gid == -1) { ids[v] = fid; return; } bool left_better = better(fid, gid, coord(left)); bool right_better = better(fid, gid, coord(right - 1)); if (!left_better && !right_better) return; if (left_better && right_better) { ids[v] = fid; return; } X mid = (left + right) / 2; bool mid_better = better(fid, gid, coord(mid)); if (mid_better) ids[v] = fid, fid = gid; if (left_better != mid_better) { v <<= 1; right = mid; } else { v = v << 1 | 1; left = mid; } } } }; #line 7 "https://drken1215.github.io/algorithm/Optimization/monge_shortest_path_by_cht.hpp.html" // find shortest path on DAG with Pairwise Single Crossing cost, by Li Chao Tree, O(N log N) // vertex: 0, 1, 2, ..., N // A(j, i) = dp[i] + f(i, j) must be Pairwise Single Crossing (must be in all 0 <= i, j <= N, including i >= j) // Attention: Even if f is Pairwise Single Crossing, A is not necessarily Pairwise Single Crossing // edges: (i, j), j >= start[i] (i + 1 <= start[i] <= N + 1), start is non-decreasing template vector> MongeShortestPathByCHT (int N, const FUNC &f, const vector &start, bool use_segment_insertion = true) { assert((int)start.size() == N); for (int i = 0; i < N; i++) { assert(i + 1 <= start[i] && start[i] <= N + 1); if (i > 0) assert(start[i - 1] <= start[i]); } VAL INF = numeric_limits::max() / 2; vector> dp(N + 1, make_pair(numeric_limits::max() / 2, -1)); auto make_func = [&](int i) { return [&, i](long long j) -> VAL { return dp[i].first + f(i, j); }; }; GeneralLiChaoTree cht(1, N + 1, INF); dp[0].first = 0; int i = 0; for (int j = 1; j <= N; j++) { while (i < N && j >= start[i]) { if (use_segment_insertion) { // 挿入する関数群が j >= start[i] の領域でのみ PSC であるとき、use_segment_insertion = true にする cht.insert_segment(make_func(i), start[i], N + 1); } else { // 挿入する関数群が全ての i, j に対して PSC であるとき、use_segment_insertion = false にする cht.insert(make_func(i)); } i++; } dp[j] = cht.query(j); }; return dp; } template vector> MongeShortestPathByCHT (int N, const FUNC &f, bool use_segment_insertion = true) { vector start(N); for (int i = 0; i < N; i++) start[i] = i + 1; return MongeShortestPathByCHT(N, f, start, use_segment_insertion); } #line 8 "test.cpp" /* dp[j] = min_{j < i} (dp[i] + func(i, j)) func(i, j) = (j - i)^2 + X */ void solve() { ll X, A; cin >> X >> A; auto func = [&](ll i, ll j) -> ll { return (j - i) * (j - i) + X; }; auto res = MongeShortestPathByCHT(A, func); cout << res.back().first << endl; } int main() { int T; cin >> T; while (T--) solve(); }