File size: 3,531 Bytes
14c9c2b | 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 | #include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <cassert>
using namespace std;
typedef long long ll;
int n;
ll k;
vector<vector<ll>> val;
vector<vector<bool>> vis;
int query_count = 0;
ll query(int x, int y) {
if (!vis[x][y]) {
cout << "QUERY " << x << " " << y << endl;
cin >> val[x][y];
vis[x][y] = true;
++query_count;
}
return val[x][y];
}
// Count number of elements <= v using staircase walk
ll count(ll v) {
ll cnt = 0;
int i = 1, j = n;
while (i <= n && j >= 1) {
ll cur = query(i, j);
if (cur <= v) {
cnt += j;
++i;
} else {
--j;
}
}
return cnt;
}
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
cin >> n >> k;
val.assign(n+1, vector<ll>(n+1, 0));
vis.assign(n+1, vector<bool>(n+1, false));
// Quick checks for corner cases
if (k == 1) {
ll ans = query(1, 1);
cout << "DONE " << ans << endl;
return 0;
}
if (k == (ll)n * n) {
ll ans = query(n, n);
cout << "DONE " << ans << endl;
return 0;
}
// Sample a grid of points
int step = max(1, n / 20); // aim for about 20x20 = 400 samples
vector<int> rows, cols;
for (int i = 1; i <= n; i += step)
rows.push_back(i);
if (rows.back() != n)
rows.push_back(n);
for (int j = 1; j <= n; j += step)
cols.push_back(j);
if (cols.back() != n)
cols.push_back(n);
vector<ll> samples;
for (int r : rows) {
for (int c : cols) {
samples.push_back(query(r, c));
}
}
sort(samples.begin(), samples.end());
samples.erase(unique(samples.begin(), samples.end()), samples.end());
// Binary search on the sample values
int left = 0, right = (int)samples.size() - 1;
ll cnt_left = count(samples[left]);
if (cnt_left >= k) {
cout << "DONE " << samples[left] << endl;
return 0;
}
while (left < right) {
int mid = (left + right) / 2;
ll cnt_mid = count(samples[mid]);
if (cnt_mid >= k) {
right = mid;
} else {
left = mid + 1;
}
}
ll R = samples[left];
ll cnt_R = count(R); // already known if left==right from loop, but compute again to be safe
ll L = (left > 0) ? samples[left-1] : (R - 1);
ll cnt_L = (left > 0) ? count(L) : 0;
// Now apply secant method to narrow the interval [L, R]
ll x1 = L, y1 = cnt_L;
ll x2 = R, y2 = cnt_R;
const int MAX_SECANT = 5;
for (int iter = 0; iter < MAX_SECANT; ++iter) {
if (x2 - x1 <= 1)
break;
if (y2 == y1)
break;
// Use long double to avoid overflow in interpolation
long double fraction = (long double)(k - y1) / (y2 - y1);
long long delta = (long long)((long double)(x2 - x1) * fraction);
ll x3 = x1 + delta;
if (x3 <= x1) x3 = x1 + 1;
if (x3 >= x2) x3 = x2 - 1;
ll y3 = count(x3);
if (y3 < k) {
x1 = x3;
y1 = y3;
} else {
x2 = x3;
y2 = y3;
}
}
// Final binary search on [x1+1, x2]
ll lo = x1 + 1, hi = x2;
while (lo < hi) {
ll mid = (lo + hi) / 2;
ll cnt = count(mid);
if (cnt >= k) {
hi = mid;
} else {
lo = mid + 1;
}
}
cout << "DONE " << lo << endl;
return 0;
} |