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1fd0050 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 | #include <bits/stdc++.h>
using namespace std;
struct TestCase {
int n; long long k;
vector<vector<long long>> A; long long answer;
};
mt19937_64 rng_gen(42);
TestCase gen_matrix(int n, long long k, function<long long(int,int)> valfn) {
TestCase tc; tc.n = n; tc.k = k;
tc.A.assign(n+1, vector<long long>(n+1, 0));
vector<long long> all;
for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) { tc.A[i][j] = valfn(i, j); all.push_back(tc.A[i][j]); }
sort(all.begin(), all.end()); tc.answer = all[k-1]; return tc;
}
TestCase gen_multiplicative(int n, long long k) { return gen_matrix(n, k, [](int i, int j) -> long long { return (long long)i * j; }); }
TestCase gen_shifted(int n, long long k) { return gen_matrix(n, k, [n](int i, int j) -> long long { return (long long)(i + n) * (j + n); }); }
TestCase gen_additive(int n, long long k) { return gen_matrix(n, k, [](int i, int j) -> long long { return i + j; }); }
TestCase gen_random_sorted(int n, long long k) {
TestCase tc; tc.n = n; tc.k = k;
tc.A.assign(n+1, vector<long long>(n+1, 0));
for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) tc.A[i][j] = (long long)i * 1000000 + (long long)j * 1000 + (rng_gen() % 500);
for (int i = 1; i <= n; i++) for (int j = 2; j <= n; j++) tc.A[i][j] = max(tc.A[i][j], tc.A[i][j-1]);
for (int j = 1; j <= n; j++) for (int i = 2; i <= n; i++) tc.A[i][j] = max(tc.A[i][j], tc.A[i-1][j]);
vector<long long> all;
for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) all.push_back(tc.A[i][j]);
sort(all.begin(), all.end()); tc.answer = all[k-1]; return tc;
}
// Simulating gpt5.2_2 approach exactly
struct Solver {
const TestCase& tc;
int query_count;
vector<long long> memo;
int n;
Solver(const TestCase& t) : tc(t), query_count(0), n(t.n) { memo.assign(2002 * 2002, -1); }
long long get(int x, int y) {
int key = x * 2001 + y;
if (memo[key] != -1) return memo[key];
query_count++;
memo[key] = tc.A[x][y];
return memo[key];
}
pair<long long, vector<int>> count_leq(long long x) {
vector<int> pos(n + 1, 0);
long long cnt = 0;
int j = n;
for (int i = 1; i <= n; i++) {
while (j >= 1) {
if (get(i, j) <= x) break;
j--;
}
pos[i] = j;
cnt += j;
if (j == 0) break;
}
return {cnt, pos};
}
pair<long long, vector<int>> count_lt(long long x) {
vector<int> pos(n + 1, 0);
long long cnt = 0;
int j = n;
for (int i = 1; i <= n; i++) {
while (j >= 1) {
if (get(i, j) < x) break;
j--;
}
pos[i] = j;
cnt += j;
if (j == 0) break;
}
return {cnt, pos};
}
long long solve() {
long long k = tc.k;
long long N2 = (long long)n * n;
if (n == 1) return get(1, 1);
if (k <= 1) return get(1, 1);
if (k >= N2) return get(n, n);
// Heap for small k
long long tSmall = min(k, N2 - k + 1);
if (tSmall + n <= 49000) {
if (k <= N2 - k + 1) {
priority_queue<tuple<long long, int, int>, vector<tuple<long long, int, int>>, greater<>> pq;
for (int i = 1; i <= n; i++) pq.emplace(get(i, 1), i, 1);
long long ans = 0;
for (long long s = 1; s <= k; s++) {
auto [v, r, c] = pq.top(); pq.pop();
ans = v;
if (s == k) break;
if (c < n) pq.emplace(get(r, c + 1), r, c + 1);
}
return ans;
} else {
priority_queue<tuple<long long, int, int>> pq;
for (int i = 1; i <= n; i++) pq.emplace(get(i, n), i, n);
long long ans = 0;
long long t = N2 - k + 1;
for (long long s = 1; s <= t; s++) {
auto [v, r, c] = pq.top(); pq.pop();
ans = v;
if (s == t) break;
if (c > 1) pq.emplace(get(r, c - 1), r, c - 1);
}
return ans;
}
}
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
// Initial upper bound
int rBound = max(1, min(n, (int)((k + n - 1) / n)));
long long highTh = get(rBound, n);
auto [cntHigh, posHigh] = count_leq(highTh);
if (cntHigh < k) {
highTh = get(n, n);
tie(cntHigh, posHigh) = count_leq(highTh);
}
vector<int> posLow(n + 1, 0);
long long cntLow = 0;
long long lowTh = LLONG_MIN;
const int SAMPLES = 9;
while (true) {
if (cntHigh < k) {
highTh = get(n, n);
tie(cntHigh, posHigh) = count_leq(highTh);
if (cntHigh < k) break;
}
long long candTotal = cntHigh - cntLow;
if (candTotal <= 0) return highTh;
long long needSmall = k - cntLow;
long long needLarge = cntHigh - k + 1;
vector<long long> pref(n + 1, 0);
int nonempty = 0;
for (int i = 1; i <= n; i++) {
int len = posHigh[i] - posLow[i];
if (len > 0) nonempty++;
pref[i] = pref[i - 1] + max(0, len);
}
candTotal = pref[n];
if (candTotal <= 0) return highTh;
long long remaining = 49500 - query_count;
long long enumCost = min(needSmall, needLarge) + nonempty + 10;
if (enumCost <= remaining) {
// Enumerate
if (needSmall <= needLarge) {
priority_queue<tuple<long long, int, int>, vector<tuple<long long, int, int>>, greater<>> pq;
for (int i = 1; i <= n; i++) {
int L = posLow[i] + 1, R = posHigh[i];
if (L >= 1 && L <= n && L <= R) pq.emplace(get(i, L), i, L);
}
long long ans = 0;
for (long long t = 1; t <= needSmall; t++) {
auto [v, r, c] = pq.top(); pq.pop();
ans = v;
if (t == needSmall) break;
if (c < posHigh[r]) pq.emplace(get(r, c + 1), r, c + 1);
}
return ans;
} else {
priority_queue<tuple<long long, int, int>> pq;
for (int i = 1; i <= n; i++) {
int L = posLow[i] + 1, R = posHigh[i];
if (R >= 1 && R <= n && L <= R) pq.emplace(get(i, R), i, R);
}
long long ans = 0;
for (long long t = 1; t <= needLarge; t++) {
auto [v, r, c] = pq.top(); pq.pop();
ans = v;
if (t == needLarge) break;
if (c > posLow[r] + 1) pq.emplace(get(r, c - 1), r, c - 1);
}
return ans;
}
}
if (remaining < 2 * n + SAMPLES + 200) {
if (nonempty + min(needSmall, needLarge) + 10 <= remaining) {
// same enum
if (needSmall <= needLarge) {
priority_queue<tuple<long long, int, int>, vector<tuple<long long, int, int>>, greater<>> pq;
for (int i = 1; i <= n; i++) {
int L = posLow[i] + 1, R = posHigh[i];
if (L >= 1 && L <= n && L <= R) pq.emplace(get(i, L), i, L);
}
long long ans = 0;
for (long long t = 1; t <= needSmall; t++) {
auto [v, r, c] = pq.top(); pq.pop(); ans = v;
if (t == needSmall) break;
if (c < posHigh[r]) pq.emplace(get(r, c + 1), r, c + 1);
}
return ans;
} else {
priority_queue<tuple<long long, int, int>> pq;
for (int i = 1; i <= n; i++) {
int L = posLow[i] + 1, R = posHigh[i];
if (R >= 1 && R <= n && L <= R) pq.emplace(get(i, R), i, R);
}
long long ans = 0;
for (long long t = 1; t <= needLarge; t++) {
auto [v, r, c] = pq.top(); pq.pop(); ans = v;
if (t == needLarge) break;
if (c > posLow[r] + 1) pq.emplace(get(r, c - 1), r, c - 1);
}
return ans;
}
}
return highTh;
}
// Sample pivots
vector<long long> samp;
for (int s = 0; s < SAMPLES; s++) {
long long pick = 1 + rng() % candTotal;
int row = (int)(lower_bound(pref.begin() + 1, pref.end(), pick) - pref.begin());
if (row < 1 || row > n) continue;
int len = posHigh[row] - posLow[row];
if (len <= 0) continue;
int col = posLow[row] + 1 + rng() % len;
col = max(1, min(n, col));
samp.push_back(get(row, col));
}
if (samp.empty()) return highTh;
sort(samp.begin(), samp.end());
double q = (double)needSmall / (double)candTotal;
int idx = (int)(q * (double)(samp.size() - 1));
idx = max(0, min((int)samp.size() - 1, idx));
long long pivot = samp[idx];
auto [cntPivot, posPivot] = count_leq(pivot);
if (cntPivot >= k) {
if (pivot == highTh && cntPivot == cntHigh) {
auto [cntLess, posLess] = count_lt(pivot);
if (cntLess < k) return pivot;
highTh = pivot - 1;
posHigh = posLess;
cntHigh = cntLess;
} else {
highTh = pivot;
posHigh = posPivot;
cntHigh = cntPivot;
}
} else {
lowTh = pivot;
posLow = posPivot;
cntLow = cntPivot;
}
}
return 0;
}
};
int main() {
struct TestDef { string name; function<TestCase()> gen; };
vector<TestDef> tests;
tests.push_back({"additive n=100 k=5000", []{ return gen_additive(100, 5000); }});
tests.push_back({"mult n=100 k=5000", []{ return gen_multiplicative(100, 5000); }});
tests.push_back({"additive n=500 k=125000", []{ return gen_additive(500, 125000); }});
tests.push_back({"mult n=500 k=125000", []{ return gen_multiplicative(500, 125000); }});
tests.push_back({"random n=500 k=125000", []{ return gen_random_sorted(500, 125000); }});
tests.push_back({"additive n=2000 k=2000000", []{ return gen_additive(2000, 2000000); }});
tests.push_back({"mult n=2000 k=2000000", []{ return gen_multiplicative(2000, 2000000); }});
tests.push_back({"shifted n=2000 k=2000000", []{ return gen_shifted(2000, 2000000); }});
tests.push_back({"random n=2000 k=2000000", []{ return gen_random_sorted(2000, 2000000); }});
tests.push_back({"mult n=2000 k=100000", []{ return gen_multiplicative(2000, 100000); }});
tests.push_back({"mult n=2000 k=3900000", []{ return gen_multiplicative(2000, 3900000); }});
for (auto& t : tests) {
auto tc = t.gen();
Solver s(tc);
long long result = s.solve();
bool correct = (result == tc.answer);
int used = s.query_count;
double score = !correct ? 0.0 : (used <= tc.n ? 1.0 : (used >= 50000 ? 0.0 : (50000.0 - used) / (50000.0 - tc.n)));
printf("%-45s q=%6d %s score=%.4f\n", t.name.c_str(), used, correct ? "OK" : "WRONG", score);
}
}
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