File size: 5,754 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 | #include <bits/stdc++.h>
using namespace std;
/*
* Heavy-path tree reconstruction.
* Step 1: Query d(1, v) for all v. (n-1 queries)
* Step 2: Sort nodes by distance, process in order.
* Step 3: For each node, find parent by querying heavy-path leaf of current tree.
* Uses O(log n) queries per node for random trees.
* Total: ~n + n*O(log n) queries, well under 5n for random trees.
*/
int main() {
ios_base::sync_with_stdio(false);
cin.tie(NULL);
int T;
cin >> T;
while (T--) {
int n;
cin >> n;
if (n == 1) {
cout << "!" << endl;
continue;
}
if (n == 2) {
cout << "? 1 2" << endl;
long long d;
cin >> d;
cout << "! 1 2 " << d << endl;
continue;
}
// Step 1: Query distances from root 1 to all others
vector<long long> d1(n + 1, 0);
for (int v = 2; v <= n; v++) {
cout << "? 1 " << v << "\n";
cout.flush();
cin >> d1[v];
}
// Sort nodes by distance from root
vector<int> order(n);
iota(order.begin(), order.end(), 1);
sort(order.begin(), order.end(), [&](int a, int b) {
return d1[a] < d1[b];
});
// Tree structure
vector<int> par(n + 1, 0);
vector<vector<int>> children(n + 1);
vector<int> sub_size(n + 1, 1);
vector<int> heavy(n + 1, 0); // heavy child (0 = leaf)
auto get_heavy_leaf = [&](int u) -> int {
while (heavy[u] != 0) u = heavy[u];
return u;
};
auto update_sizes = [&](int u) {
int cur = par[u];
int child = u;
while (cur != 0) {
sub_size[cur]++;
if (heavy[cur] == 0 || sub_size[child] > sub_size[heavy[cur]]) {
heavy[cur] = child;
}
child = cur;
cur = par[cur];
}
};
vector<tuple<int,int,long long>> edges;
for (int idx = 1; idx < n; idx++) {
int v = order[idx];
long long dv = d1[v];
int cur = 1;
int next_target = -1;
long long next_d = -1;
while (true) {
int leaf;
long long d_v_leaf;
if (next_target != -1) {
leaf = next_target;
d_v_leaf = next_d;
next_target = -1;
} else {
leaf = get_heavy_leaf(cur);
if (leaf == cur) {
// cur is a leaf in current tree, v attaches here
par[v] = cur;
children[cur].push_back(v);
edges.push_back({cur, v, (int)(dv - d1[cur])});
update_sizes(v);
break;
}
cout << "? " << v << " " << leaf << "\n";
cout.flush();
cin >> d_v_leaf;
}
// Compute LCA depth of v and leaf
long long lca_d = (dv + d1[leaf] - d_v_leaf) / 2;
// Walk up from leaf to find the LCA node
int lca_node = leaf;
int child_towards_leaf = 0;
while (d1[lca_node] > lca_d) {
child_towards_leaf = lca_node;
lca_node = par[lca_node];
}
if (lca_node == leaf) {
par[v] = leaf;
children[leaf].push_back(v);
edges.push_back({leaf, v, (int)(dv - d1[leaf])});
update_sizes(v);
break;
}
// Check light children of lca_node (excluding the one toward leaf)
vector<pair<int,int>> light_children;
for (int c : children[lca_node]) {
if (c != child_towards_leaf) {
light_children.push_back({sub_size[c], c});
}
}
if (light_children.empty()) {
par[v] = lca_node;
children[lca_node].push_back(v);
edges.push_back({lca_node, v, (int)(dv - d1[lca_node])});
update_sizes(v);
break;
}
// Sort by size descending
sort(light_children.rbegin(), light_children.rend());
bool found = false;
for (auto [sz, c] : light_children) {
int c_leaf = get_heavy_leaf(c);
cout << "? " << v << " " << c_leaf << "\n";
cout.flush();
long long d_v_cleaf;
cin >> d_v_cleaf;
long long lca2_d = (dv + d1[c_leaf] - d_v_cleaf) / 2;
if (lca2_d > d1[lca_node]) {
cur = c;
next_target = c_leaf;
next_d = d_v_cleaf;
found = true;
break;
}
}
if (!found) {
par[v] = lca_node;
children[lca_node].push_back(v);
edges.push_back({lca_node, v, (int)(dv - d1[lca_node])});
update_sizes(v);
break;
}
}
}
cout << "!";
for (auto& [u, v, w] : edges) {
cout << " " << u << " " << v << " " << w;
}
cout << "\n";
cout.flush();
}
return 0;
}
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