Solving 894D for arbitrary trees in O(nlog^2n)

Revision en3, by f2lk6wf90d, 2017-11-21 14:45:14

This article is about a solution to a variation of 894D - Ralph And His Tour in Binary Country, with (almost) complete binary trees replaced by arbitrary trees. The solution turned out to be more complicated than I thought, so I decided to post it as a separate blog.
The main idea is to fix the LCA, just like the original problem.
Note: We are calculating the sum of all Hi - L, such that

where x is any destination vertex and dv is the distance from the root to the v-th vertex.
Therefore, we could calculate the answer using the following functions:

// dist[v] = sorted list of d_x, where x is any node in v's subtree
// prefsum[v] = prefix sum array of dist[v]
typedef long long i64;
i64 helper(int v, i64 val) {
	if(v == 0) return 0;
	auto x = std::upper_bound(dist[v].begin(), dist[v].end(), val);
	if(x == dist[v].begin()) return 0;
	i64 cnt = std::distance(dist[v].begin(), x);
	return cnt * val - prefsum[v][cnt - 1];
i64 solve(int v, int c, i64 happiness, int start) { // par[root] = 0
	if(v == 0) return 0;
	i64 val = happiness - d[start] + 2 * d[v];
	return helper(v, val) - helper(c, val) + solve(par[v], v, happiness, start);
i64 query(int v, i64 happiness) {
	return solve(v, 0, happiness, v);

We could use this code to calculate the answer for an arbitrary tree, however queries would take time.
We can speed this algorithm up by using centroid decomposition. We will create a "centroid tree", and create a vector for each node that stores the distances from each node to the nodes in its subtree.
Let d(x, y) be the distance from node x to node y. Then, we need to calculate the sum of all Hi - L such that , where v is an ancestor of Ai in the centroid tree, and x is a node in v's subtree, but not in the subtree containing Ai.
Note that we can use the exact same helper function to calculate the answer for a single vertex, however we also need to find a way to subtract the answer for v's children. We can reduce subtree queries to range queries on an array using the Euler tour technique.
We also need a data structure that supports the following operations:
1. count(l, r, x): count the number of elements in [l, r] such that ai ≤ x.
2. sum(l, r, x): calculate the sum of elements in [l, r] such that ai ≤ x.
We can use a wavelet tree/persistent segment tree/simple segment tree (offline).
In the following code, let idx[v] be the index of vertex v in the DFS order, and sz[v] be the size of v's subtree in the centroid tree. Then, the code can be adapted for the centroid tree like this:

// we build a separate data structure for every vertex
data_structure DS[MAXN];
i64 helper(int v, int l, int r, int val) {
	if(l > r) return 0;
	return DS[v].count(l, r, val) * val - DS[v].sum(l, r, val);
i64 solve(int v, int c, int happiness, int start) {
	if(v == 0) return 0;
	int x = idx[c] - idx[v] + 1;
	int y = x + sz[c] - 1;
	i64 val = happiness - d[v][idx[start] - idx[v] + 1];
	return (c == 0 ? helper(v, 1, sz[v], val) : helper(v, 1, x - 1, val) + helper(v, y + 1, sz[v], val)) + solve(par[v], v, happiness, start);
i64 query(int v, i64 happiness) {
	return solve(v, 0, happiness, v);

This solution takes time per query. (solve is called times per query, and every helper query takes time.)
Disclaimer: This is the first time I have used centroid decomposition. Feel free to point out any flaws in this approach. I will try to post a full solution later. Update: The initial version stated that this was a solution for arbitrary binary trees. It is actually a solution for any tree.


  Rev. Lang. By When Δ Comment
en3 English f2lk6wf90d 2017-11-21 14:45:14 22 Tiny change: 'ss - d[v][x];\n retur' -> 'ss - d[v][idx[start] - idx[v] + 1];\n retur'
en2 English f2lk6wf90d 2017-11-21 00:16:10 138
en1 English f2lk6wf90d 2017-11-20 22:40:43 3906 Initial revision (published)