File size: 9,381 Bytes
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 | #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));
long long C = 1000000, D = 1000;
for (int i = 1; i <= n; i++)
for (int j = 1; j <= n; j++)
tc.A[i][j] = (long long)i * C + (long long)j * D + (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;
}
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 do_query(int r, int c) {
int key = r * 2001 + c;
if (memo[key] != -1) return memo[key];
query_count++;
memo[key] = tc.A[r][c];
return memo[key];
}
long long solve() {
long long k = tc.k;
long long N2 = (long long)n * n;
if (n == 1) return do_query(1, 1);
if (k == 1) return do_query(1, 1);
if (k == N2) return do_query(n, n);
long long heap_k = min(k, N2 - k + 1);
if (heap_k + n <= 24000) {
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(do_query(i, 1), i, 1);
long long result = -1;
for (long long t = 0; t < k; t++) {
auto [v, r, c] = pq.top(); pq.pop();
result = v;
if (c + 1 <= n) pq.emplace(do_query(r, c + 1), r, c + 1);
}
return result;
} else {
long long kk = N2 - k + 1;
priority_queue<tuple<long long, int, int>> pq;
for (int i = 1; i <= n; i++) pq.emplace(do_query(i, n), i, n);
long long result = -1;
for (long long t = 0; t < kk; t++) {
auto [v, r, c] = pq.top(); pq.pop();
result = v;
if (c - 1 >= 1) pq.emplace(do_query(r, c - 1), r, c - 1);
}
return result;
}
}
// Quickselect with improved pivot selection
vector<int> L(n + 1, 1), R(n + 1, n);
long long k_rem = k;
mt19937_64 rng(12345);
for (int iter = 0; iter < 100; iter++) {
vector<int> active;
long long total_cand = 0;
for (int i = 1; i <= n; i++) {
if (L[i] <= R[i]) {
active.push_back(i);
total_cand += R[i] - L[i] + 1;
}
}
int na = active.size();
if (total_cand == 0) break;
if (total_cand == 1) {
for (int i : active) return do_query(i, L[i]);
break;
}
long long budget = 49500 - query_count;
if (k_rem + na <= budget) {
priority_queue<tuple<long long, int, int>, vector<tuple<long long, int, int>>, greater<>> pq;
for (int i : active) pq.emplace(do_query(i, L[i]), i, L[i]);
for (long long t = 1; t < k_rem; t++) {
auto [v, r, c] = pq.top(); pq.pop();
if (c + 1 <= R[r]) pq.emplace(do_query(r, c + 1), r, c + 1);
}
return get<0>(pq.top());
}
long long rev_k = total_cand - k_rem + 1;
if (rev_k + na <= budget) {
priority_queue<tuple<long long, int, int>> pq;
for (int i : active) pq.emplace(do_query(i, R[i]), i, R[i]);
for (long long t = 1; t < rev_k; t++) {
auto [v, r, c] = pq.top(); pq.pop();
if (c - 1 >= L[r]) pq.emplace(do_query(r, c - 1), r, c - 1);
}
return get<0>(pq.top());
}
// Improved pivot selection: uniform random sampling from candidate area
// Build prefix sums of widths
vector<long long> pref(na + 1, 0);
for (int idx = 0; idx < na; idx++) {
int i = active[idx];
pref[idx + 1] = pref[idx] + (R[i] - L[i] + 1);
}
int num_samples = min(15, na);
vector<long long> samp;
for (int s = 0; s < num_samples; s++) {
long long pick = 1 + rng() % total_cand;
int idx = (int)(lower_bound(pref.begin() + 1, pref.end(), pick) - pref.begin()) - 1;
idx = max(0, min(na - 1, idx));
int i = active[idx];
int width = R[i] - L[i] + 1;
int col = L[i] + rng() % width;
samp.push_back(do_query(i, col));
}
sort(samp.begin(), samp.end());
// Pick at quantile matching k_rem/total_cand
double q = (double)(k_rem - 0.5) / total_cand;
int pidx = (int)(q * (double)(samp.size() - 1));
pidx = max(0, min((int)samp.size() - 1, pidx));
long long pivot = samp[pidx];
// Staircase walk (bottom-to-top)
vector<int> p_le(n + 1, 0);
{
int j = 0;
for (int idx = na - 1; idx >= 0; idx--) {
int i = active[idx];
j = max(j, L[i]);
while (j <= R[i] && do_query(i, j) <= pivot) j++;
p_le[i] = j - 1;
}
}
long long cle = 0;
for (int i : active) {
int rl = min(p_le[i], R[i]);
if (rl >= L[i]) cle += rl - L[i] + 1;
}
if (cle >= k_rem) {
for (int i : active) R[i] = min(R[i], p_le[i]);
} else {
k_rem -= cle;
for (int i : active) L[i] = max(L[i], p_le[i] + 1);
}
}
return -1;
}
};
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({"multiplicative 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({"multiplicative 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({"multiplicative 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({"multiplicative n=2000 k=1", []{ return gen_multiplicative(2000, 1); }});
tests.push_back({"multiplicative n=2000 k=4000000", []{ return gen_multiplicative(2000, 4000000); }});
tests.push_back({"multiplicative n=2000 k=100000", []{ return gen_multiplicative(2000, 100000); }});
tests.push_back({"multiplicative 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);
double score;
int used = s.query_count;
if (!correct) score = 0.0;
else if (used <= tc.n) score = 1.0;
else if (used >= 50000) score = 0.0;
else score = (50000.0 - used) / (50000.0 - tc.n);
printf("%-45s n=%4d k=%8lld queries=%6d correct=%s score=%.4f\n",
t.name.c_str(), tc.n, tc.k, used, correct ? "YES" : "NO", score);
}
return 0;
}
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