Rerooting DP
(graph/tree/rerooting_dp.hpp)

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Last update: 2026-08-13 01:41:40+09:00
Include: #include "graph/tree/rerooting_dp.hpp"

Overview

rerooting_dp is a generic all-roots tree DP helper. It computes one DP value for every possible root of an undirected tree, or for every vertex in each component of a forest.

The input uses m1une::graph::Graph<T> and should be built with add_edge. Inactive edges are ignored.

Function

template <class T, class DP, class Merge, class AddVertex, class AddEdge>
std::vector<DP> rerooting_dp(
    const m1une::graph::Graph<T>& g,
    DP id,
    Merge merge,
    AddVertex add_vertex,
    AddEdge add_edge
);

The callbacks mean:

Callback	Meaning
`id`	Identity DP value for `merge`.
`merge(a, b)`	Combines independent neighbor contributions. It should be associative.
`add_vertex(acc, v)`	Finalizes the merged contributions at vertex `v`.
`add_edge(dp, e)`	Converts a neighbor-side DP value through adjacency edge `e`.

For an edge contribution used at vertex v, e is the adjacency edge from v to that neighbor.

DP Meaning

Think of the tree as being cut at an edge. A DP value represents one connected side of the cut, rooted at the vertex closest to the edge.

For a vertex v, each neighbor to gives one independent contribution:

contribution_from_to = add_edge(dp_of_side_rooted_at_to, edge_v_to);

Then all neighbor contributions are merged, and v itself is added:

answer[v] = add_vertex(merge(all neighbor contributions), v);

The library computes this for every v. When it sends information from v to one child, it merges all contributions except that child, applies add_vertex(..., v), and passes that value across the edge later. This is the usual rerooting trick; prefix and suffix products make the “all except one” merge fast.

Most tree DPs use a commutative merge, such as +, max, or min. The implementation applies merge in adjacency-list order, so non-commutative operations are deterministic but depend on that order.

How to Design the Callbacks

Use this checklist:

Decide what DP means for a rooted connected piece.
Choose id, the value for an empty set of neighbor contributions.
Define merge(a, b), which combines two independent neighbor sides.
Define add_edge(dp, e), which changes distances or values when the piece is viewed from the other endpoint of edge e.
Define add_vertex(acc, v), which adds the current vertex after all neighbor sides have been merged.

If the final answer for root v is “the DP of the whole tree rooted at v”, then it is exactly result[v].

Complexity

rerooting_dp runs in $O(N)$ callback calls on a tree.

Example: Farthest Distance

The following computes the eccentricity of each vertex: the maximum number of edges from that vertex to any other vertex.

Here DP is one integer:

Meaning: maximum distance from the current root of this piece to a vertex in the piece.
id = 0: an empty set of children contributes distance 0.
merge = max: keep the farthest side.
add_edge(dp, e) = dp + 1: crossing one unweighted edge increases every distance by 1.
add_vertex(acc, v) = acc: the current vertex has distance 0, so it does not change the maximum.

#include "graph/graph.hpp"
#include "graph/tree/rerooting_dp.hpp"
#include <algorithm>
#include <iostream>

int main() {
    m1une::graph::Graph<int> g(4);
    g.add_edge(0, 1);
    g.add_edge(1, 2);
    g.add_edge(1, 3);

    auto ecc = m1une::tree::rerooting_dp(
        g,
        0,
        [](int a, int b) { return std::max(a, b); },
        [](int acc, int) { return acc; },
        [](int dp, const auto&) { return dp + 1; }
    );

    std::cout << ecc[0] << "\n"; // 2
}

Example: Sum of Distances

The following computes, for every vertex v, the sum of distances from v to all vertices.

Use a DP state with:

size: number of vertices in this piece.
sum: sum of distances from the current root of this piece to all vertices in the piece.

When crossing an edge of cost w, every distance increases by w, so sum += size * w.

#include "graph/graph.hpp"
#include "graph/tree/rerooting_dp.hpp"
#include <iostream>

struct DP {
    long long size;
    long long sum;
};

int main() {
    m1une::graph::Graph<long long> g(4);
    g.add_edge(0, 1, 1);
    g.add_edge(1, 2, 1);
    g.add_edge(1, 3, 1);

    auto res = m1une::tree::rerooting_dp(
        g,
        DP{0, 0},
        [](DP a, DP b) {
            return DP{a.size + b.size, a.sum + b.sum};
        },
        [](DP acc, int) {
            return DP{acc.size + 1, acc.sum};
        },
        [](DP dp, const auto& e) {
            return DP{dp.size, dp.sum + dp.size * e.cost};
        }
    );

    std::cout << res[0].sum << "\n"; // dist(0,0)+dist(0,1)+dist(0,2)+dist(0,3) = 5
}

Common Patterns

For counting vertices:

id = 0
merge(a, b) = a + b
add_vertex(acc, v) = acc + 1
add_edge(dp, e) = dp

For maximum weighted distance:

id = 0
merge(a, b) = std::max(a, b)
add_vertex(acc, v) = acc
add_edge(dp, e) = dp + e.cost

For DP states with several fields, merge usually adds or takes the best of each field, add_edge shifts the state across one edge, and add_vertex accounts for the current vertex.

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_TREE_REROOTING_DP_HPP
#define M1UNE_TREE_REROOTING_DP_HPP 1

#include <vector>

#include "../graph.hpp"

namespace m1une {
namespace tree {

template <class T, class DP, class Merge, class AddVertex, class AddEdge>
std::vector<DP> rerooting_dp(const m1une::graph::Graph<T>& g, DP id, Merge merge, AddVertex add_vertex,
                             AddEdge add_edge) {
    int n = g.size();
    std::vector<int> parent(n, -2), parent_edge(n, -1), order;
    order.reserve(n);
    for (int root = 0; root < n; root++) {
        if (parent[root] != -2) continue;
        parent[root] = -1;
        std::vector<int> stack = {root};
        while (!stack.empty()) {
            int v = stack.back();
            stack.pop_back();
            order.push_back(v);
            for (const auto& e : g[v]) {
                if (!e.alive) continue;
                if (parent[e.to] != -2) continue;
                parent[e.to] = v;
                parent_edge[e.to] = e.id;
                stack.push_back(e.to);
            }
        }
    }

    std::vector<DP> down(n, id), outside(n, id), answer(n, id);
    for (int i = n - 1; i >= 0; i--) {
        int v = order[i];
        DP acc = id;
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            if (parent[e.to] != v) continue;
            acc = merge(acc, add_edge(down[e.to], e));
        }
        down[v] = add_vertex(acc, v);
    }

    for (int v : order) {
        int d = int(g[v].size());
        std::vector<DP> contrib(d, id);
        for (int i = 0; i < d; i++) {
            const auto& e = g[v][i];
            if (!e.alive) continue;
            if (parent[e.to] == v) {
                contrib[i] = add_edge(down[e.to], e);
            } else if (parent[v] == e.to && parent_edge[v] == e.id) {
                contrib[i] = add_edge(outside[v], e);
            }
        }

        std::vector<DP> pref(d + 1, id), suff(d + 1, id);
        for (int i = 0; i < d; i++) pref[i + 1] = merge(pref[i], contrib[i]);
        for (int i = d - 1; i >= 0; i--) suff[i] = merge(contrib[i], suff[i + 1]);
        answer[v] = add_vertex(pref[d], v);

        for (int i = 0; i < d; i++) {
            const auto& e = g[v][i];
            if (!e.alive) continue;
            if (parent[e.to] != v) continue;
            outside[e.to] = add_vertex(merge(pref[i], suff[i + 1]), v);
        }
    }

    return answer;
}

}  // namespace tree
}  // namespace m1une

#endif  // M1UNE_TREE_REROOTING_DP_HPP

#line 1 "graph/tree/rerooting_dp.hpp"



#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#include <cassert>
#include <utility>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 7 "graph/tree/rerooting_dp.hpp"

namespace m1une {
namespace tree {

template <class T, class DP, class Merge, class AddVertex, class AddEdge>
std::vector<DP> rerooting_dp(const m1une::graph::Graph<T>& g, DP id, Merge merge, AddVertex add_vertex,
                             AddEdge add_edge) {
    int n = g.size();
    std::vector<int> parent(n, -2), parent_edge(n, -1), order;
    order.reserve(n);
    for (int root = 0; root < n; root++) {
        if (parent[root] != -2) continue;
        parent[root] = -1;
        std::vector<int> stack = {root};
        while (!stack.empty()) {
            int v = stack.back();
            stack.pop_back();
            order.push_back(v);
            for (const auto& e : g[v]) {
                if (!e.alive) continue;
                if (parent[e.to] != -2) continue;
                parent[e.to] = v;
                parent_edge[e.to] = e.id;
                stack.push_back(e.to);
            }
        }
    }

    std::vector<DP> down(n, id), outside(n, id), answer(n, id);
    for (int i = n - 1; i >= 0; i--) {
        int v = order[i];
        DP acc = id;
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            if (parent[e.to] != v) continue;
            acc = merge(acc, add_edge(down[e.to], e));
        }
        down[v] = add_vertex(acc, v);
    }

    for (int v : order) {
        int d = int(g[v].size());
        std::vector<DP> contrib(d, id);
        for (int i = 0; i < d; i++) {
            const auto& e = g[v][i];
            if (!e.alive) continue;
            if (parent[e.to] == v) {
                contrib[i] = add_edge(down[e.to], e);
            } else if (parent[v] == e.to && parent_edge[v] == e.id) {
                contrib[i] = add_edge(outside[v], e);
            }
        }

        std::vector<DP> pref(d + 1, id), suff(d + 1, id);
        for (int i = 0; i < d; i++) pref[i + 1] = merge(pref[i], contrib[i]);
        for (int i = d - 1; i >= 0; i--) suff[i] = merge(contrib[i], suff[i + 1]);
        answer[v] = add_vertex(pref[d], v);

        for (int i = 0; i < d; i++) {
            const auto& e = g[v][i];
            if (!e.alive) continue;
            if (parent[e.to] != v) continue;
            outside[e.to] = add_vertex(merge(pref[i], suff[i + 1]), v);
        }
    }

    return answer;
}

}  // namespace tree
}  // namespace m1une

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

Rerooting DP
(graph/tree/rerooting_dp.hpp)

Overview

Function

DP Meaning

How to Design the Callbacks

Complexity

Example: Farthest Distance

Example: Sum of Distances

Common Patterns

Depends on

Required by

Verified with

Code

Rerooting DP (graph/tree/rerooting_dp.hpp)

Overview

Function

DP Meaning

How to Design the Callbacks

Complexity

Example: Farthest Distance

Example: Sum of Distances

Common Patterns

Depends on

Required by

Verified with

Code

Rerooting DP
(graph/tree/rerooting_dp.hpp)