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KokiYmgch's blog

By KokiYmgch, history, 9 months ago, In English,

I wrote about the easy implementation of centroid decomposition on a tree.

Japanese here: http://www.learning-algorithms.com/entry/2018/01/20/031005

First of all, the implementation of centroid decomposition tends to be complicated, and you might have seen someone's code which has too many functions named 'dfs n' (n = 1, 2, 3, ...). I, for one, don't want to code something like that!

So, let me introduce my implementation of centroid decompositon. I hope you get something new from it.

Firstly, we need to know one of the centroids of the tree. Be careful not to forget that some vertices are going to die while repeating the decompositon.

The function which returns the centroid is easily implemented in the following way:

int OneCentroid(int root, const vector<vector<int>> &g, const vector<bool> &dead) {
        static vector<int> sz(g.size());
        function<void (int, int)> get_sz = [&](int u, int prev) {
                sz[u] = 1;
                for (auto v : g[u]) if (v != prev && !dead[v]) {
                        get_sz(v, u);
                        sz[u] += sz[v];
                }
        };
        get_sz(root, -1);
        int n = sz[root];
        function<int (int, int)> dfs = [&](int u, int prev) {
                for (auto v : g[u]) if (v != prev && !dead[v]) {
                        if (sz[v] > n / 2) {
                                return dfs(v, u);
                        }
                }
                return u;
        };
        return dfs(root, -1);
}

Then, using this centroid, you can implement centroid decomposition like this.

void CentroidDecomposition(const vector<vector<int>> &g) {
        int n = (int) g.size();
        vector<bool> dead(n, false);
        function<void (int)> rec = [&](int start) {
                int c = OneCentroid(start, g, dead);           //2
                dead[c] = true;                                //2
                for (auto u : g[c]) if (!dead[u]) {
                        rec(u);                                //3
                }
                /*
                        compute something with the centroid    //4
                */
                dead[c] = false;                               //5
        };
        rec(0);                                                //1
}

This works following way:

  1. Calculate on the entire tree. All the vertices are alive now.

  2. Find the centroid of the current tree, and make it die.

  3. Calculate on the subtree which doesn't include the centroid. Go to 2 with this subtree.

  4. Calculate something required which includes the centroid.

  5. Make the centroid alive again, because this is DFS.

Simply enough, when you use this, you just need to change the part 4. All the other parts are the same, which means you can use it generally.

Let me show you an example.

https://beta.atcoder.jp/contests/yahoo-procon2018-final-open/tasks/yahoo_procon2018_final_c

(I guess this statement is available only in Japanese. Sorry for inconvenience!)

Summary: You are given a tree with N vertices. Answer the Q queries below.

Query v k : Find the number of the vertices, such that the distance from v is exactly k.

N, Q ≤ 105

The obvious solution to this problem is, for each query v, k, make v-rooted tree and count the number of the vertices whose depth is equal to k. This solution, however, requires time O(NQ).

When you want to count something on a tree, especially when it's related to a path, centroid decomposition is one of the good directions you are heading for.

First of all, let all the queries on the tree, and deal with them all at once. It's easy to see that these queries are actually asking the number of the paths whose length is k and the end point is v.

If you decompose the tree, as I mentioned above, you only need to count the paths which include the centroid.

More specifically, just calculate the the number of the distances from the centroid, and make the paths whose length is exacly k, and count them. Again, I didn't change almost anything but the part 4 of the implementation above.

#include <cstdio>
#include <vector>
#include <algorithm>
#include <functional>
#include <map>
#include <cassert>
#include <cmath>
using namespace std;

int OneCentroid(int root, const vector<vector<int>> &g, const vector<bool> &dead) {
        static vector<int> sz(g.size());
        function<void (int, int)> get_sz = [&](int u, int prev) {
                sz[u] = 1;
                for (auto v : g[u]) if (v != prev && !dead[v]) {
                        get_sz(v, u);
                        sz[u] += sz[v];
                }
        };
        get_sz(root, -1);
        int n = sz[root];
        function<int (int, int)> dfs = [&](int u, int prev) {
                for (auto v : g[u]) if (v != prev && !dead[v]) {
                        if (sz[v] > n / 2) {
                                return dfs(v, u);
                        }
                }
                return u;
        };
        return dfs(root, -1);
}

vector<int> CentroidDecomposition(const vector<vector<int>> &g, const vector<vector<pair<int, int>>> &l, int q) {
        int n = (int) g.size();
        vector<int> ans(q, 0);
        vector<bool> dead(n, false);
        function<void (int)> rec = [&](int start) {
                int c = OneCentroid(start, g, dead);
                dead[c] = true;
                for (auto u : g[c]) if (!dead[u]) {
                        rec(u);
                }

                /* 
                        changed from here
                */
                map<int, int> cnt;
                function<void (int, int, int, bool)> add_cnt = [&](int u, int prev, int d, bool add) {
                        cnt[d] += (add ? 1 : -1);
                        for (auto v : g[u]) if (v != prev && !dead[v]) {
                                add_cnt(v, u, d + 1, add);
                        }
                };
                function<void (int, int, int)> calc = [&](int u, int prev, int d) {
                        for (auto it : l[u]) {
                                int dd, idx;
                                tie(dd, idx) = it;
                                if (dd - d >= 0 && cnt.count(dd - d)) {
                                        ans[idx] += cnt[dd - d];
                                }
                        }
                        for (auto v : g[u]) if (v != prev && !dead[v]) {
                                calc(v, u, d + 1);
                        }
                };
                add_cnt(c, -1, 0, true);
                for (auto it : l[c]) {
                        int dd, idx;
                        tie(dd, idx) = it;
                        ans[idx] += cnt[dd];
                }
                for (auto u : g[c]) if (!dead[u]) {
                        add_cnt(u, c, 1, false);
                        calc(u, c, 1);
                        add_cnt(u, c, 1, true);
                }
                //

                dead[c] = false;
        };
        rec(0); 
        return ans;
}

int main() {
        int n, q;
        scanf("%d %d", &n, &q);
        vector<vector<int>> g(n);
        for (int i = 0; i < n - 1; i ++) {
                int a, b;
                scanf("%d %d", &a, &b);
                a --, b --;
                g[a].push_back(b);
                g[b].push_back(a);
        }
        vector<vector<pair<int, int>>> l(n); //dist, query idx
        for (int i = 0; i < q; i ++) {
                int v, k;
                scanf("%d %d", &v, &k);
                v --;
                l[v].emplace_back(k, i);
        }
        auto ans = CentroidDecomposition(g, l, q);
        for (int i = 0; i < q; i ++) {
                printf("%d\n", ans[i]);
        }
        return 0;
}

You can practice centroid decomposition on these problems too! Try them if you would like!

http://codeforces.com/contest/914/problem/E

https://csacademy.com/contest/round-58/task/path-inversions

These problems ask you to count the number of the specific paths too.

Thank you for your reading!

 
 
 
 
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9 months ago, # |
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Thank you very much for this beautiful implementation. I have a request, would you please explain to me how CentroidDecomposition considers all n2 cases?

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    9 months ago, # ^ |
      Vote: I like it 0 Vote: I do not like it

    I'm sorry, I don't think I understand what exactly you're trying to say, but in the example I showed above, the complexity just happened to be . Therefore, if you rewrite the part 4 as it needs the time O(n2), then the entire complexity is going to be accordingly.

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      9 months ago, # ^ |
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      I mean, how does CentroidDecomposition consider all n2 cases in O(nlogn)

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        9 months ago, # ^ |
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        It's based on a divide and conquer algorithm. When you divide the tree with n vertices on the centroid, each subtree's size will not be more than (according to the definition of the centroid), which means you can finish the decomposition in times.

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          9 months ago, # ^ |
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          Yeah, I've got that asymptotics'll be O(nlogn) and why, and I also have written CentroidDecomposition several times, but I still didn't get one thing, why it takes all possible n2 or n * (n - 1) / 2 cases?

          And if your answer's going to be DivideAndConquer again, would you please explain why it is correct?

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            9 months ago, # ^ |
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            Sorry I'm still not sure what you don't understand. In that example, every time you take the centroid of the current tree, you have O(n2) possible cases (paths) of course, but it's made faster by kind of DP (the array cnt) and consequently O(n), and I think it is not the point here.

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9 months ago, # |
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http://www.usaco.org/index.php?page=viewproblem2&cpid=286

Also think this problem is notable because it is used in the algorithms live! episode on centroid decomposition (I believe it's episode 12)