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maximize-the-number-of-target-nodes-after-connecting-trees-i.cpp
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maximize-the-number-of-target-nodes-after-connecting-trees-i.cpp
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// Time: O(nlogn + mlogm)
// Space: O(n + m)
// dfs, centroid decomposition, prefix sum
class Solution {
public:
vector<int> maxTargetNodes(vector<vector<int>>& edges1, vector<vector<int>>& edges2, int k) {
const auto& centroid_decomposition = [](const auto& adj, int k) {
vector<int> result(size(adj));
vector<int> sizes(size(adj));
vector<bool> lookup(size(adj));
vector<vector<int>> prefix(size(adj));
vector<int> cnt;
vector<int> total, curr;
const function<void (int)> dfs = [&](int u) {
// https://usaco.guide/plat/centroid
const function<int (int, int)> find_subtree_size = [&](int u, int p) {
sizes[u] = 1;
for (const auto& v : adj[u]) {
if (v == p || lookup[v]) {
continue;
}
sizes[u] += find_subtree_size(v, u);
}
return sizes[u];
};
find_subtree_size(u, -1);
const int n = sizes[u];
const function<int (int, int)> find_centroid = [&](int u, int p) {
for (const auto& v : adj[u]) {
if (v == p || lookup[v]) {
continue;
}
if (sizes[v] * 2 > n) {
return find_centroid(v, u);
}
}
return u;
};
const function<void (int, int, int)> count = [&](int u, int p, int d) {
if (d > k) {
return;
}
if (d - 1 == size(cnt)) {
cnt.emplace_back(0);
}
++cnt[d - 1];
for (const auto& v : adj[u]) {
if (v == p || lookup[v]) {
continue;
}
count(v, u, d + 1);
}
};
const function<void (int, int, int)> update = [&](int u, int p, int d) {
if (d > k) {
return;
}
result[u] += total[min(k - d, static_cast<int>(size(total)) - 1)] - curr[min(k - d, static_cast<int>(size(curr)) - 1)];
for (const auto& v : adj[u]) {
if (v == p || lookup[v]) {
continue;
}
update(v, u, d + 1);
}
};
u = find_centroid(u, -1);
lookup[u] = true;
int max_d = 0;
for (const auto& v : adj[u]) {
if (lookup[v]) {
continue;
}
cnt.clear();
count(v, u, 0 + 1);
prefix[v].emplace_back(0);
for (int d = 0; d < size(cnt); ++d) {
prefix[v].emplace_back(prefix[v].back() + cnt[d]);
}
max_d = max(max_d, static_cast<int>(size(cnt)));
}
total.assign(max_d + 1, 1);
for (const auto& v : adj[u]) {
if (lookup[v]) {
continue;
}
for (int d = 0; d < size(total); ++d) {
total[d] += prefix[v][min(d, static_cast<int>(size(prefix[v])) - 1)];
}
}
result[u] += total[min(k, static_cast<int>(size(total)) - 1)];
for (const auto& v : adj[u]) {
if (lookup[v]) {
continue;
}
curr = move(prefix[v]);
update(v, u, 0 + 1);
}
for (const auto& v : adj[u]) {
if (lookup[v]) {
continue;
}
dfs(v);
}
};
if (k >= 0) {
dfs(0);
}
return result;
};
const auto& find_adj = [](const auto& edges) {
vector<vector<int>> adj(size(edges) + 1);
for (const auto& e : edges) {
adj[e[0]].emplace_back(e[1]);
adj[e[1]].emplace_back(e[0]);
}
return adj;
};
const auto& adj2 = find_adj(edges2);
int mx = ranges::max(centroid_decomposition(adj2, k - 1));
const auto& adj1 = find_adj(edges1);
vector<int> result(size(adj1));
const auto& ret = centroid_decomposition(adj1, k);
for (int u = 0; u < size(adj1); ++u) {
result[u] = mx + ret[u];
}
return result;
}
};
// Time: O((n + m) * k)
// Space: O((n + m) * k)
// dfs, tree dp
class Solution2 {
public:
vector<int> maxTargetNodes(vector<vector<int>>& edges1, vector<vector<int>>& edges2, int k) {
const auto& tree_dp = [](const auto& adj, int k) {
vector<int> result(size(adj));
k = min(k, static_cast<int>(size(adj)) - 1);
if (k == -1) {
return result;
}
vector<vector<int>> dp(size(adj), vector<int>(k + 1));
const function<void (int, int)> dfs1 = [&](int u, int p) {
for (const auto& v : adj[u]) {
if (v == p) {
continue;
}
dfs1(v, u);
}
++dp[u][0];
for (const auto& v : adj[u]) {
if (v == p) {
continue;
}
for (int d = 0; d < k; ++d) {
dp[u][d + 1] += dp[v][d];
}
}
};
const function<void(int, int, const vector<int>&)> dfs2 = [&](int u, int p, const auto& curr) {
const auto& update = [&](int v, int u, const auto& curr) {
vector<int> new_curr(size(curr));
for (int d = 0; d + 1 < size(new_curr); ++d) {
new_curr[d + 1] = curr[d] + (dp[u][d] - (d - 1 >= 0 ? dp[v][d - 1] : 0));
}
return new_curr;
};
for (const auto& v : adj[u]) {
if (v == p) {
continue;
}
dfs2(v, u, update(v, u, curr));
}
for (int d = 0; d < size(curr); ++d) {
result[u] += dp[u][d] + curr[d];
}
};
dfs1(0, -1);
dfs2(0, -1, vector<int>(k + 1));
return result;
};
const auto& find_adj = [](const auto& edges) {
vector<vector<int>> adj(size(edges) + 1);
for (const auto& e : edges) {
adj[e[0]].emplace_back(e[1]);
adj[e[1]].emplace_back(e[0]);
}
return adj;
};
const auto& adj2 = find_adj(edges2);
int mx = ranges::max(tree_dp(adj2, k - 1));
const auto& adj1 = find_adj(edges1);
vector<int> result(size(adj1));
const auto& ret = tree_dp(adj1, k);
for (int u = 0; u < size(adj1); ++u) {
result[u] = mx + ret[u];
}
return result;
}
};
// Time: O(n^2 + m^2)
// Space: O(n + m)
// brute force, bfs
class Solution3 {
public:
vector<int> maxTargetNodes(vector<vector<int>>& edges1, vector<vector<int>>& edges2, int k) {
const auto& bfs = [](int u, const auto& adj, int k) {
int result = 0;
vector<pair<int, int>> q = {{u, -1}};
while (!empty(q)) {
if (k == -1) {
break;
}
--k;
vector<pair<int, int>> new_q;
result += size(q);
for (const auto& [u, p] : q) {
for (const auto& v : adj[u]) {
if (v == p) {
continue;
}
new_q.emplace_back(v, u);
}
}
q = move(new_q);
}
return result;
};
const auto& find_adj = [](const auto& edges) {
vector<vector<int>> adj(size(edges) + 1);
for (const auto& e : edges) {
adj[e[0]].emplace_back(e[1]);
adj[e[1]].emplace_back(e[0]);
}
return adj;
};
const auto& adj2 = find_adj(edges2);
int mx = 0;
for (int u = 0; u < size(adj2); ++u) {
mx = max(mx, bfs(u, adj2, k - 1));
}
const auto& adj1 = find_adj(edges1);
vector<int> result(size(adj1));
for (int u = 0; u < size(adj1); ++u) {
result[u] = mx + bfs(u, adj1, k);
}
return result;
}
};