Unconstrained_FC.hpp

/*********************************************************************
 *
 * Linear Arrangement Library - A library that implements a collection
 * algorithms for linear arrangments of graphs.
 *
 * Copyright (C) 2019 - 2024
 *
 * This file is part of Linear Arrangement Library. The full code is available
 * at:
 *      https://github.com/LAL-project/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
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 * 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:
 *
 *     Juan Luis Esteban (esteban@cs.upc.edu)
 *         LOGPROG: Logics and Programming Research Group
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 *
 *     Lluís Alemany Puig (lluis.alemany.puig@upc.edu)
 *         LQMC (Quantitative, Mathematical, and Computational Linguisitcs)
 *         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/
 *
 ********************************************************************/
 
// C++ includes
#if defined DEBUG
#include <cassert>
#endif
 
#include <optional>
#include <limits>
#include <vector>
 
#include <lal/linear_arrangement.hpp>
#include <lal/detail/pairs_utils.hpp>
#include <lal/detail/graphs/traversal.hpp>
#include <lal/detail/properties/tree_centroid.hpp>
#include <lal/detail/graphs/size_subtrees.hpp>
#include <lal/detail/sorting/counting_sort.hpp>
#include <lal/detail/array.hpp>
#include <lal/detail/macros/basic_convert.hpp>
#include <lal/detail/linarr/D/Dopt_utils.hpp>
 
namespace lal {
namespace detail {
namespace Dmin {
namespace unconstrained {
 
namespace Chung {
 
typedef array<node_size> ordering;
 
/*
int calculate_q(uint64_t n, const ordering& ord) {
    const uint64_t k = to_uint64(ord.size()) - 1;
    const uint64_t t_0 = ord[0].size;
 
    // Maximum possible p_alpha
    int64_t q = to_int64(k)/2;
    uint64_t sum = 0;
    for (uint64_t i = 0; i <= 2*to_uint64(q); ++i) {
        sum += ord[i].size;
    }
 
    uint64_t z = n - sum;
    uint64_t tricky_formula = (t_0 + 2)/2 + (z + 2)/2;
    // t_0 >= t_1 >= ... >= t_k
    uint64_t t_2q = ord[2*q].size;
    while (q >= 0 and t_2q <= tricky_formula) {
        z += ord[2*q].size;
        if (q > 0) {
            z += ord[2*q - 1].size;
        }
        --q;
        tricky_formula = (t_0 + 2)/2 + (z + 2)/2;
        if (q >= 0) {
            t_2q = ord[2*q].size;
        }
    }
    return q;
}
*/
 
[[nodiscard]] std::optional<uint64_t> calculate_q
(const uint64_t n, const ordering& ord)
noexcept
{
#if defined DEBUG
    assert(ord.size() > 0);
#endif
 
    const uint64_t k = ord.size() - 1;
    const uint64_t t_0 = ord[0].size;
 
    // Maximum possible p_alpha
    uint64_t q = k/2;
    uint64_t sum = 0;
    for (uint64_t i = 0; i <= 2*q; ++i) {
        sum += ord[i].size;
    }
 
    uint64_t z = n - sum;
    uint64_t tricky_formula = (t_0 + 2)/2 + (z + 2)/2;
    // t_0 >= t_1 >= ... >= t_k
    uint64_t t_2q = ord[2*q].size;
 
    while (t_2q <= tricky_formula) {
        z += ord[2*q].size;
 
        if (q > 0) { z += ord[2*q - 1].size; }
        tricky_formula = (t_0 + 2)/2 + (z + 2)/2;
 
        if (q == 0) { return {}; }
        --q;
        t_2q = ord[2*q].size;
    }
    return q;
}
 
/*
int calculate_p(const uint64_t n, const ordering& ord) {
    if (ord.size() < 2) {
        return -1;
    }
 
    // number of subtrees (T_0, T_1, ..., T_k)
    const uint64_t k = to_uint64(ord.size() - 1);
    const uint64_t t_0 = ord[0].size;
 
    int p = (k - 1)/2;
 
    uint64_t sum = 0;
    for (uint64_t i = 0; i <= 2*to_uint64(p) + 1; ++i) {
        sum += ord[i].size;
    }
 
    uint64_t y = n - sum;
    uint64_t tricky_formula = (t_0 + 2)/2 + (y + 2)/2;
    uint64_t t_2p_plus_1 = ord[2*p + 1].size;
 
    while (p >= 0 and t_2p_plus_1 <= tricky_formula) {
        y = y + ord[2*p + 1].size + ord[2*p].size;
        --p;
        tricky_formula = (t_0 + 2)/2 + (y + 2)/2;
        if (p >= 0) {
            t_2p_plus_1 = ord[2*p + 1].size;
        }
    }
    return p;
}
*/
 
[[nodiscard]] std::optional<uint64_t> calculate_p
(const uint64_t n, const ordering& ord)
noexcept
{
    if (ord.size() < 2) { return {}; }
 
    // number of subtrees (T_0, T_1, ..., T_k)
    const uint64_t k = ord.size() - 1;
    const uint64_t t_0 = ord[0].size;
 
    uint64_t p = (k - 1)/2;
 
    uint64_t sum = 0;
    for (uint64_t i = 0; i <= 2*p + 1; ++i) {
        sum += ord[i].size;
    }
 
    uint64_t y = n - sum;
    uint64_t tricky_formula = (t_0 + 2)/2 + (y + 2)/2;
    uint64_t t_2p_plus_1 = ord[2*p + 1].size;
 
    while (t_2p_plus_1 <= tricky_formula) {
        y = y + ord[2*p + 1].size + ord[2*p].size;
        tricky_formula = (t_0 + 2)/2 + (y + 2)/2;
 
        if (p == 0) { return {}; }
        --p;
        t_2p_plus_1 = ord[2*p + 1].size;
    }
    return p;
}
 
[[nodiscard]] array<uint64_t> get_P
(const uint64_t p, uint64_t i)
noexcept
{
    array<uint64_t> v(2*p + 1 + 1);
    uint64_t pos = v.size() - 1;
    uint64_t right_pos = pos;
    uint64_t left_pos = 1;
 
    uint64_t j = 0;
    while (j <= 2*p + 1) {
        if (j == i) {
            ++j;
        }
        else {
            v[pos] = j;
            if (pos > left_pos) {
                --right_pos;
                pos = left_pos;
            }
            else {
                ++left_pos;
                pos = right_pos;
            }
            ++j;
        }
    }
 
    return v;
}
 
[[nodiscard]] array<uint64_t> get_Q(const uint64_t q, const uint64_t i) noexcept {
    array<uint64_t> v(2*q + 1);
    uint64_t pos = v.size() - 1;
    uint64_t right_pos = pos;
    uint64_t left_pos = 1;
 
    uint64_t j = 0;
    while (j <= 2*q) {
        if (j == i) {
            ++j;
        }
        else {
            v[pos] = j;
            if (pos > left_pos) {
                --right_pos;
                pos = left_pos;
            }
            else {
                ++left_pos;
                pos = right_pos;
            }
            ++j;
        }
    }
    return v;
}
 
[[nodiscard]] ordering get_ordering(const graphs::free_tree& t, const node u)
noexcept
{
    // Let 'T_u' to be a tree rooted at vertex 'u'.
    // Order subtrees of 'T_u' by size.
    ordering ord(t.get_degree(u));
 
    // Retrieve size of every subtree. Let 'T_u[v]' be the subtree
    // of 'T_u' rooted at vertex 'v'. Now,
    //     s[v] := the size of the subtree 'T_u[v]'
    array<uint64_t> s(t.get_num_nodes());
    get_size_subtrees(t, u, s.begin());
 
    uint64_t M = 0; // maximum of the sizes (needed for the counting sort algorithm)
    const neighbourhood& u_neighs = t.get_neighbors(u);
    for (std::size_t i = 0; i < u_neighs.size(); ++i) {
        // i-th child of v_star
        const node ui = u_neighs[i];
        // size of subtree rooted at 'ui'
        const uint64_t s_ui = s[ui];
 
        ord[i].size = s_ui;
        ord[i].v = ui;
 
        M = std::max(M, s_ui);
    }
    sorting::counting_sort
    <node_size, sorting::sort_type::non_increasing>
    (
        ord.begin(), ord.end(), M, ord.size(),
        [](const node_size& p) { return p.size; }
    );
 
    return ord;
}
 
template <char root, bool make_arrangement>
void calculate_mla
(
    graphs::free_tree& t,
    const node one_node,
    const position start,
    linear_arrangement& mla,
    uint64_t& cost
)
noexcept
{
    array<node> reachable(t.get_num_nodes_component(one_node));
    {
    auto it = reachable.begin();
    BFS bfs(t);
    bfs.set_process_current([&](const auto&, node u) { *it++ = u; });
    bfs.start_at(one_node);
    }
    const uint64_t size_tree = t.get_num_nodes_component(one_node);
 
    static_assert(
        root == Dopt_utils::NO_ANCHOR or
        root == Dopt_utils::RIGHT_ANCHOR or
        root == Dopt_utils::LEFT_ANCHOR
    );
 
#if defined DEBUG
    assert(size_tree > 0);
#endif
 
    // Base case
    if (size_tree == 1) {
#if defined DEBUG
        assert(one_node == reachable[0]);
        assert(start <= t.get_num_nodes());
#endif
        if constexpr (make_arrangement) {
        mla.assign(one_node, start);
        }
        cost = 0;
        return;
    }
 
    if constexpr (root == Dopt_utils::NO_ANCHOR) {
        const node u = retrieve_centroid(t, one_node).first;
 
        const ordering ord = get_ordering(t, u);
 
        const auto q = calculate_q(size_tree, ord);
        if (not q) {
            const uint64_t n_0 = ord[0].size;
            const node t_0 = ord[0].v;
 
            t.remove_edge(u, t_0, false, false);
 
            uint64_t c1 = 0;
            calculate_mla<Dopt_utils::RIGHT_ANCHOR, make_arrangement>
                (t, t_0, start, mla, c1);
 
            uint64_t c2 = 0;
            calculate_mla<Dopt_utils::LEFT_ANCHOR, make_arrangement>
                (t, u, start + n_0, mla, c2);
 
            cost = c1 + c2 + 1;
 
            t.add_edge(u, t_0, false, false);
        }
        else {
            // uq: unsigned 'q'
            const uint64_t uq = *q;
            cost = std::numeric_limits<uint64_t>::max();
 
            std::vector<edge> edges(2*uq + 1);
            for (uint64_t i = 0; i <= 2*uq; ++i) {
                edges[i] = {u, ord[i].v};
            }
 
            // Transform g into Y
            t.remove_edges(edges, false, false);
 
            // Central tree size
            uint64_t size_rest_of_trees = 0;
            for (uint64_t i = 2*uq + 1; i < ord.size(); ++i) {
                size_rest_of_trees += ord[i].size;
            }
 
            for (uint64_t i = 0; i <= 2*uq; ++i) {
                const auto Q_i = get_Q(uq, i);
 
                t.add_edge(u, ord[i].v, false, false);
 
                uint64_t c_i = 0;
                linear_arrangement arr_aux = mla;
                uint64_t start_aux = start;
 
                // Left part of the arrangement
                for (uint64_t j = 1; j <= uq; ++j) {
                    const position pos_in_ord = Q_i[j];
 
                    uint64_t c_i_j = 0;
                    calculate_mla<Dopt_utils::RIGHT_ANCHOR, make_arrangement>
                    (
                        t,
                        ord[pos_in_ord].v, start_aux,
                        arr_aux, c_i_j
                    );
                    start_aux += ord[pos_in_ord].size;
                    c_i += c_i_j;
                }
 
                // Central part of the arrangement
                uint64_t c_i_j = 0;
                calculate_mla<Dopt_utils::NO_ANCHOR, make_arrangement>(t, u, start_aux, arr_aux, c_i_j);
                c_i += c_i_j;
 
                // Right part of the arrangement
                start_aux += ord[i].size + 1 + size_rest_of_trees;
                for (uint64_t j = uq + 1; j <= 2*uq; ++j) {
                    const position pos_in_ord = Q_i[j];
 
                    uint64_t c_i_j_in = 0;
                    calculate_mla<Dopt_utils::LEFT_ANCHOR, make_arrangement>
                    (
                        t,
                        ord[pos_in_ord].v, start_aux,
                        arr_aux, c_i_j_in
                    );
                    start_aux += ord[pos_in_ord].size;
                    c_i += c_i_j_in;
                }
 
                // Adding parts of the anchors over trees nearer to the central tree
                c_i += size_tree*uq;
 
                uint64_t subs = 0;
                for (uint64_t j = 1; j <= uq; ++j) {
                    subs += (uq - j + 1)*(ord[Q_i[j]].size + ord[Q_i[2*uq - j + 1]].size);
                }
                c_i -= subs;
                c_i += uq; // NOT IN CHUNG'S PAPER
 
                if (c_i < cost) {
                    cost = c_i;
                    if constexpr (make_arrangement) {
                    mla = std::move(arr_aux);
                    }
                }
#if defined DEBUG
                assert(u != ord[i].v);
#endif
                t.remove_edge(u, ord[i].v, false, false);
            }
 
            // Transform g into its previous form
            t.add_edges(edges, false, false);
        }
    }
    else { // root == ANCHOR
        const ordering ord = get_ordering(t, one_node);
 
        const auto p = calculate_p(size_tree, ord);
        if (not p) {
            const uint64_t n_0 = ord[0].size;
            const node t_0 = ord[0].v;
#if defined DEBUG
            assert(one_node != t_0);
#endif
 
            t.remove_edge(one_node, t_0, false, false);
 
            uint64_t c1 = 0;
            uint64_t c2 = 0;
 
            if constexpr (root == Dopt_utils::LEFT_ANCHOR) {
                calculate_mla<Dopt_utils::NO_ANCHOR, make_arrangement>
                    (t, one_node, start, mla, c1);
 
                calculate_mla<Dopt_utils::LEFT_ANCHOR, make_arrangement>
                    (t, t_0, start + size_tree - ord[0].size, mla, c2);
            }
            else {
                calculate_mla<Dopt_utils::RIGHT_ANCHOR, make_arrangement>
                    (t, t_0, start, mla, c1);
 
                calculate_mla<Dopt_utils::NO_ANCHOR, make_arrangement>
                    (t, one_node, start + n_0, mla, c2);
            }
 
            cost = c1 + c2 + size_tree - ord[0].size;
            t.add_edge(one_node, t_0, false, false);
 
            /*
            if constexpr (root == LEFT_ANCHOR) {
                //if (2*mla[one_node - 1] - 2*start > size_tree - 1) {
 
                    // Left anchor and the root is too much to the right
                    for (uint64_t i = 0; i < size_tree; ++i) {
                        const uint64_t aux = start + size_tree - 1 - mla[reachable[i]] + start;
                        mla[reachable[i]] = aux;
                    }
                //}
            }
            */
        }
        else {
            // up: unsigned 'p'
            const uint64_t up = *p;
            cost = std::numeric_limits<uint64_t>::max();
 
            std::vector<edge> edges(2*up + 2);
            for (uint64_t i = 0; i <= 2*up + 1; ++i) {
                edges[i] = {one_node, ord[i].v};
            }
 
            // Transform g into Y
            t.remove_edges(edges, false, false);
 
            // Central tree size
            uint64_t size_rest_of_trees= 0;
            for (uint64_t i = 2*up + 2; i < ord.size() ;++i) {
                size_rest_of_trees += ord[i].size;
            }
 
            for (uint64_t i = 0; i <= 2*up + 1; ++i) {
                const auto P_i = get_P(up, i);
                t.add_edge(one_node, ord[i].v, false, false);
 
                uint64_t c_i = 0;
                linear_arrangement arr_aux = mla;
                uint64_t start_aux = start;
 
                if constexpr (root == Dopt_utils::LEFT_ANCHOR) {
                    // Left part of the arrangement
                    for (uint64_t j = 1; j <= up; ++j) {
                        const position pos_in_ord = P_i[j];
 
                        uint64_t c_i_j_in = 0;
                        calculate_mla<Dopt_utils::RIGHT_ANCHOR, make_arrangement>
                        (
                            t,
                            ord[pos_in_ord].v, start_aux,
                            arr_aux, c_i_j_in
                        );
                        start_aux += ord[pos_in_ord].size;
                        c_i += c_i_j_in;
                    }
 
                    // Central part of the arrangement
                    uint64_t c_i_j = 0;
                    calculate_mla<Dopt_utils::NO_ANCHOR, make_arrangement>
                        (t, one_node, start_aux, arr_aux, c_i_j);
 
                    start_aux += ord[i].size + 1 + size_rest_of_trees;
                    c_i += c_i_j;
 
                    // Right part of the arrangement
                    for (uint64_t j = up + 1; j <= 2*up + 1; ++j) {
                        const position pos_in_ord = P_i[j];
 
                        uint64_t c_i_j_in = 0;
                        calculate_mla<Dopt_utils::LEFT_ANCHOR, make_arrangement>
                        (
                            t,
                            ord[pos_in_ord].v, start_aux,
                            arr_aux, c_i_j_in
                        );
                        start_aux += ord[pos_in_ord].size;
                        c_i += c_i_j_in;
                    }
                }
                else { // root == RIGHT_ANCHOR
 
                    // Right part of the arrangement
                    for (uint64_t j = 2*up + 1; j >= up + 1; --j) {
                        const position pos_in_ord = P_i[j];
 
                        uint64_t c_i_j_in = 0;
                        calculate_mla<Dopt_utils::RIGHT_ANCHOR, make_arrangement>
                        (
                            t,
                            ord[pos_in_ord].v, start_aux,
                            arr_aux, c_i_j_in
                        );
                        start_aux += ord[pos_in_ord].size;
                        c_i += c_i_j_in;
                    }
 
                    // Central part of the arrangement
                    uint64_t c_i_j = 0;
                    calculate_mla<Dopt_utils::NO_ANCHOR, make_arrangement>
                        (t, one_node, start_aux, arr_aux, c_i_j);
 
                    start_aux += ord[i].size + 1 + size_rest_of_trees;
                    c_i += c_i_j;
 
                    // Left part of the arrangement
                    for (uint64_t j = up; j >= 1; --j) {
                        const position pos_in_ord = P_i[j];
 
                        uint64_t c_i_j_in = 0;
                        calculate_mla<Dopt_utils::LEFT_ANCHOR, make_arrangement>
                        (
                            t,
                            ord[pos_in_ord].v, start_aux,
                            arr_aux, c_i_j_in
                        );
                        start_aux += ord[pos_in_ord].size;
                        c_i += c_i_j_in;
                    }
                }
 
                // Adding parts of the anchors over trees nearer to the central tree
                c_i += size_tree*(up + 1);
                c_i -= (up + 1)*ord[P_i[P_i.size() - 1]].size;
 
                uint64_t subs = 0;
                for (uint64_t j = 1; j <= up; ++j) {
                    subs += (up - j + 1)*(ord[P_i[j]].size + ord[P_i[2*up - j + 1]].size);
                }
                c_i -= subs;
                c_i += up; // NOT IN CHUNG'S PAPER
 
                if (c_i < cost) {
                    cost = c_i;
                    mla = arr_aux;
                }
#if defined DEBUG
                assert(one_node != ord[i].v);
#endif
                t.remove_edge(one_node, ord[i].v, false, false);
            }
 
            // Transform g into its previous form
            t.add_edges(edges, false, false);
 
        }
    }
}
 
} // -- namespace dmin_Chung
 
template <bool make_arrangement>
[[nodiscard]] std::conditional_t<
    make_arrangement,
    std::pair<uint64_t, linear_arrangement>,
    uint64_t
>
FanChung_2(const graphs::free_tree& t) noexcept
{
#if defined DEBUG
    assert(t.is_tree());
#endif
 
    graphs::free_tree T = t;
 
    uint64_t Dmin = 0;
    linear_arrangement arr(make_arrangement ? t.get_num_nodes() : 0);
    Chung::calculate_mla<Dopt_utils::NO_ANCHOR, make_arrangement>(T, 0, 0, arr, Dmin);
 
    if constexpr (make_arrangement) {
        return {Dmin, std::move(arr)};
    }
    else {
        return Dmin;
    }
}
 
} // -- namespace unconstrained
} // -- namespace Dmin
} // -- namespace detail
} // -- namespace lal