tree_centroid.hpp

/*********************************************************************
 *
 *  Linear Arrangement Library - A library that implements a collection
 *  algorithms for linear arrangments of graphs.
 *
 *  Copyright (C) 2019 - 2021
 *
 *  This file is part of Linear Arrangement Library. To see the full code
 *  visit the webpage:
 *      https://github.com/lluisalemanypuig/linear-arrangement-library.git
 *
 *  Linear Arrangement Library is free software: you can redistribute it
 *  and/or modify it under the terms of the GNU Affero General Public License
 *  as published by the Free Software Foundation, either version 3 of the
 *  License, or (at your option) any later version.
 *
 *  Linear Arrangement Library is distributed in the hope that it will be
 *  useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 *  GNU Affero General Public License for more details.
 *
 *  You should have received a copy of the GNU Affero General Public License
 *  along with Linear Arrangement Library.  If not, see <http://www.gnu.org/licenses/>.
 *
 *  Contact:
 *
 *      Lluís Alemany Puig (lalemany@cs.upc.edu)
 *          LARCA (Laboratory for Relational Algorithmics, Complexity and Learning)
 *          CQL (Complexity and Quantitative Linguistics Lab)
 *          Jordi Girona St 1-3, Campus Nord UPC, 08034 Barcelona.   CATALONIA, SPAIN
 *          Webpage: https://cqllab.upc.edu/people/lalemany/
 *
 *      Ramon Ferrer i Cancho (rferrericancho@cs.upc.edu)
 *          LARCA (Laboratory for Relational Algorithmics, Complexity and Learning)
 *          CQL (Complexity and Quantitative Linguistics Lab)
 *          Office S124, Omega building
 *          Jordi Girona St 1-3, Campus Nord UPC, 08034 Barcelona.   CATALONIA, SPAIN
 *          Webpage: https://cqllab.upc.edu/people/rferrericancho/
 *
 ********************************************************************/
 
#pragma once
 
// C++ includes
#if defined DEBUG
#include <cassert>
#endif
#include <vector>
 
// lal includes
#include <lal/definitions.hpp>
#include <lal/graphs/rooted_tree.hpp>
#include <lal/internal/graphs/size_subtrees.hpp>
#include <lal/internal/sorting/counting_sort.hpp>
 
namespace lal {
namespace internal {
 
namespace __lal {
 
// N: actual number of vertices of the tree
// n: number of vertices of the connected component of x
// x: start at node x
template<
    class tree_type,
    std::enable_if_t<
        std::is_base_of_v<graphs::free_tree, tree_type> ||
        std::is_base_of_v<graphs::rooted_tree, tree_type>,
    bool> = true
>
std::pair<node, node> retrieve_centroid(
    const tree_type& t,
    const uint32_t N, const uint32_t n, const node x,
    std::vector<std::vector<std::pair<node,uint32_t>>>& L,
    std::vector<std::pair<edge, uint32_t>>& sizes_edge
)
{
#if defined DEBUG
    assert(n > 0);
#endif
 
    typedef std::pair<edge,uint32_t> edge_size;
    typedef std::vector<edge_size>::iterator Iterator_Type;
 
    // calculate s(u,v) with H&S algorithm (lemma 8)
    {
    sizes_edge.resize(2*(n - 1));
    auto it = sizes_edge.begin();
    internal::calculate_bidirectional_sizes
    <tree_type, Iterator_Type>(t, n, x, it);
 
    // sort all tuples in sizes_edge using the sizes
    internal::counting_sort<edge_size, Iterator_Type, countingsort::decreasing_t>
    (
        sizes_edge.begin(), sizes_edge.end(), n, sizes_edge.size(),
        [](const edge_size& edge_pair) -> size_t { return edge_pair.second; }
    );
    }
 
    // put the s(u,v) into an adjacency list
    // M[u] : adjacency list of vertex u sorted decreasingly according
    // to the sizes of the subtrees.
    L.resize(N);
    for (const auto& [uv, suv] : sizes_edge) {
        const auto [u, v] = uv;
        L[u].push_back(std::make_pair(v,suv));
    }
 
    // find the first centroidal vertex
    node c1 = N + 1;
    bool found = false;
    node u = x;
    while (not found) {
        const auto& [v, suv] = L[u][0];
        u = (suv > n/2 ? v : u);
        found = suv <= n/2;
    }
    c1 = u;
 
#if defined DEBUG
    assert(c1 < N);
#endif
 
    // find the second centroidal vertex among the
    // neighbours of the first centroidal vertex
    node c2 = N + 1;
    for (const auto& p : L[c1]) {
        const node v = p.first;
        if (L[v][0].second <= n/2) {
            c2 = v;
            break;
        }
    }
 
    return (c1 < c2 ? std::make_pair(c1, c2) : std::make_pair(c2, c1));
}
 
} // -- namespace __lal
 
// -----------------------------------------------------------------------------
 
/*
 * @brief Calculate the centroid of the connected component that has node @e x.
 *
 * Here, "centroid" should NOT be confused with "centre". The centre is the set
 * of (at most) two vertices that have minimum eccentricity. The centroid is the
 * set of (at most) two vertices that have minimum weight, where the weight is
 * the maximum size of the subtrees rooted at that vertex. In both case, if
 * the set has two vertices then they are adjacent in the tree. See \cite Harary1969a
 * for further details.
 *
 * A graph of type @ref lal::graphs::tree may lack some edges tree so it can have
 * several connected components. Vertex @e x belongs to one of these connected
 * components. So, this method finds the centroidal nodes of the connected
 * component node @e x belongs to.
 *
 * This function uses @ref lal::internal::calculate_suvs, an algorithm described
 * in \cite Hochberg2003a (see function's documentation for details).
 * @param t Input tree.
 * @param x Input node.
 * @param[out] L A sorted and enriched adjacency list where @e M[u] is a list of
 * pairs \f$(v,sv)\f$ where @e v is a neighbour of @e u and @e sv is the size of
 * the subtree rooted at @e v with parent @e u. The list is sorted decreasingly.
 * @param[out] sizes_edge See documentation of method internal::calculate_suvs.
 * @returns A tuple of two values: the nodes in the centroid. If the
 * tree has a single centroidal node, only the first node is valid and the second
 * is assigned an invalid vertex index. It is guaranteed that the first vertex
 * has smaller index value than the second.
 */
template<
    class T,
    std::enable_if_t<
        std::is_base_of_v<graphs::free_tree, T> ||
        std::is_base_of_v<graphs::rooted_tree, T>,
    bool> = true
>
std::pair<node, node> retrieve_centroid(
    const T& t, const node x,
    std::vector<std::vector<std::pair<node,uint32_t>>>& L,
    std::vector<std::pair<edge, uint32_t>>& sizes_edge
)
{
    // actual number of vertices of the tree
    const uint32_t component_size = t.get_num_nodes();
    // calculate the size of the connected component
    const uint32_t n = t.get_num_nodes_component(x);
    // easy case
    if (n == 1) {
        return std::make_pair(x, component_size+1);
    }
    // general case
    return __lal::retrieve_centroid(t, component_size, n, x, L, sizes_edge);
}
 
/*
 * @brief Calculate the centroid of the connected component that has node @e x.
 *
 * For details on the parameters and return value see documentation of the
 * function above.
 */
template<
    class T,
    std::enable_if_t<
        std::is_base_of_v<graphs::free_tree, T> ||
        std::is_base_of_v<graphs::rooted_tree, T>,
    bool> = true
>
std::pair<node, node> retrieve_centroid(const T& t, const node x) {
    std::vector<std::vector<std::pair<node,uint32_t>>> M;
    std::vector<std::pair<edge, uint32_t>> sizes_edge;
    return retrieve_centroid(t, x, M, sizes_edge);
}
 
// -----------------------------------------------------------------------------
 
/*
 * @brief Calculate the centroid of the tree @e t.
 *
 * Here, "centroid" should NOT be confused with "centre". The centre is the set
 * of (at most) two vertices that have minimum eccentricity. The centroid is the
 * set of (at most) two vertices that have minimum weight, where the weight is
 * the maximum size of the subtrees rooted at that vertex. In both case, if
 * the set has two vertices then they are adjacent in the tree. See \cite Harary1969a
 * for further details.
 *
 * This function uses @ref lal::internal::calculate_suvs, an algorithm described
 * in \cite Hochberg2003a (see function's documentation for details).
 * @param t Input tree.
 * @param[out] M A sorted and enriched adjacency list where @e M[u] is a list of
 * pairs (v,sv) where @e v is a neighbour of @e u and @e sv is the size of the
 * subtree rooted at @e v with parent @e u. The list is sorted decreasingly.
 * @param[out] sizes_edge See documentation of method internal::calculate_suvs.
 * @returns A tuple of two values: the nodes in the centroid. If the
 * tree has a single centroidal node, only the first node is valid and the second
 * is assigned an invalid vertex index. It is guaranteed that the first vertex
 * has smaller index value than the second.
 * @pre The tree @e t is a valid tree (see @ref lal::graphs::tree::is_tree).
 */
template<
    class T,
    std::enable_if_t<
        std::is_base_of_v<graphs::free_tree, T> ||
        std::is_base_of_v<graphs::rooted_tree, T>,
    bool> = true
>
std::pair<node, node> retrieve_centroid(
    const T& t,
    std::vector<std::vector<std::pair<node,uint32_t>>>& L,
    std::vector<std::pair<edge, uint32_t>>& sizes_edge
)
{
    // actual number of vertices of the tree
    const uint32_t n = t.get_num_nodes();
    // easy case
    if (n == 1) {
        return std::make_pair(0, n+1);
    }
    // general case
    return __lal::retrieve_centroid(t, n, n, 0, L, sizes_edge);
}
 
/*
 * @brief Calculate the centroid of the tree @e t.
 *
 * For details on the parameters and return value see documentation of the
 * function above.
 */
template<
    class T,
    std::enable_if_t<
        std::is_base_of_v<graphs::free_tree, T> ||
        std::is_base_of_v<graphs::rooted_tree, T>,
    bool> = true
>
std::pair<node, node> retrieve_centroid(const T& t) {
    std::vector<std::vector<std::pair<node,uint32_t>>> L;
    std::vector<std::pair<edge, uint32_t>> sizes_edge;
    return retrieve_centroid(t, L, sizes_edge);
}
 
} // -- namespace internal
} // -- namespace lal