CGraphPartitioner.cpp

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          https://www.mrpt.org/                         |
   |                                                                        |
   | Copyright (c) 2005-2020, Individual contributors, see AUTHORS file     |
   | See: https://www.mrpt.org/Authors - All rights reserved.               |
   | Released under BSD License. See: https://www.mrpt.org/License          |
   +------------------------------------------------------------------------+ */

#include "graphs-precomp.h"  // Precompiled headers

#include <mrpt/graphs/CGraphPartitioner.h>
#include <mrpt/math/CMatrixD.h>
#include <mrpt/math/CMatrixF.h>
#include <mrpt/math/ops_matrices.h>  // laplacian()
#include <iostream>  // cout

using namespace mrpt;
using namespace mrpt::graphs;
using namespace std;

/*---------------------------------------------------------------
SpectralPartition
---------------------------------------------------------------*/
template <class GRAPH_MATRIX, typename num_t>
void CGraphPartitioner<GRAPH_MATRIX, num_t>::SpectralBisection(
    GRAPH_MATRIX& in_A, std::vector<uint32_t>& out_part1,
    std::vector<uint32_t>& out_part2, num_t& out_cut_value, bool forceSimetry)
{
    size_t nodeCount;  // Nodes count
    GRAPH_MATRIX Adj, eigenVectors;
    std::vector<typename GRAPH_MATRIX::Scalar> eigenValues;

    // Check matrix is square:
    if (in_A.cols() != int(nodeCount = in_A.rows()))
        THROW_EXCEPTION("Weights matrix is not square!!");

    // Shi & Malik's method
    //-----------------------------------------------------------------

    // forceSimetry?
    if (forceSimetry)
    {
        Adj.setSize(nodeCount, nodeCount);
        for (size_t i = 0; i < nodeCount; i++)
            for (size_t j = i; j < nodeCount; j++)
                Adj(i, j) = Adj(j, i) = 0.5f * (in_A(i, j) + in_A(j, i));
    }
    else
        Adj = in_A;

    // Compute eigen-vectors of laplacian:
    GRAPH_MATRIX LAPLACIAN;
    mrpt::math::laplacian(Adj, LAPLACIAN);

    LAPLACIAN.eig(eigenVectors, eigenValues);

    //  Execute the bisection
    // ------------------------------------
    // Second smallest eigen-vector
    double mean = 0;
    size_t colNo = 1;  // second smallest
    size_t nRows = eigenVectors.rows();

    // for (i=0;i<eigenVectors.cols();i++) mean+=eigenVectors(colNo,i);
    for (size_t i = 0; i < nRows; i++) mean += eigenVectors(i, colNo);
    mean /= nRows;

    out_part1.clear();
    out_part2.clear();

    for (size_t i = 0; i < nRows; i++)
    {
        if (eigenVectors(i, colNo) >= mean)
            out_part1.push_back(i);
        else
            out_part2.push_back(i);
    }

    // Special and strange case: Constant eigenvector: Split nodes in two
    //    equally sized parts arbitrarily:
    // ------------------------------------------------------------------
    if (!out_part1.size() || !out_part2.size())
    {
        out_part1.clear();
        out_part2.clear();
        // Assign 50%-50%:
        for (int i = 0; i < Adj.cols(); i++)
            if (i <= Adj.cols() / 2)
                out_part1.push_back(i);
            else
                out_part2.push_back(i);
    }

    // Compute the N-cut value
    out_cut_value = nCut(Adj, out_part1, out_part2);
}

/*---------------------------------------------------------------
RecursiveSpectralPartition
---------------------------------------------------------------*/
template <class GRAPH_MATRIX, typename num_t>
void CGraphPartitioner<GRAPH_MATRIX, num_t>::RecursiveSpectralPartition(
    GRAPH_MATRIX& in_A, std::vector<std::vector<uint32_t>>& out_parts,
    num_t threshold_Ncut, bool forceSimetry, bool useSpectralBisection,
    bool recursive, unsigned minSizeClusters, const bool verbose)
{
    MRPT_START

    size_t nodeCount;
    std::vector<uint32_t> p1, p2;
    num_t cut_value;
    size_t i, j;
    GRAPH_MATRIX Adj;

    out_parts.clear();

    // Check matrix is square:
    if (in_A.cols() != int(nodeCount = in_A.rows()))
        THROW_EXCEPTION("Weights matrix is not square!!");

    if (nodeCount == 1)
    {
        // Don't split, there is just a node!
        p1.push_back(0);
        out_parts.push_back(p1);
        return;
    }

    // forceSimetry?
    if (forceSimetry)
    {
        Adj.setSize(nodeCount, nodeCount);
        for (i = 0; i < nodeCount; i++)
            for (j = i; j < nodeCount; j++)
                Adj(i, j) = Adj(j, i) = 0.5f * (in_A(i, j) + in_A(j, i));
    }
    else
        Adj = in_A;

    // Make bisection
    if (useSpectralBisection)
        SpectralBisection(Adj, p1, p2, cut_value, false);
    else
        exactBisection(Adj, p1, p2, cut_value, false);

    if (verbose)
        std::cout << format(
            "Cut:%u=%u+%u,nCut=%.02f->", (unsigned int)nodeCount,
            (unsigned int)p1.size(), (unsigned int)p2.size(), cut_value);

    // Is it a useful partition?
    if (cut_value > threshold_Ncut || p1.size() < minSizeClusters ||
        p2.size() < minSizeClusters)
    {
        if (verbose) std::cout << "->NO!" << std::endl;

        // No:
        p1.clear();
        for (i = 0; i < nodeCount; i++) p1.push_back(i);
        out_parts.push_back(p1);
    }
    else
    {
        if (verbose) std::cout << "->YES!" << std::endl;

        // Yes:
        std::vector<std::vector<uint32_t>> p1_parts, p2_parts;

        if (recursive)
        {
            // Split "p1":
            // --------------------------------------------
            // sub-matrix:
            GRAPH_MATRIX A_1(p1.size(), p1.size());
            for (i = 0; i < p1.size(); i++)
                for (j = 0; j < p1.size(); j++) A_1(i, j) = in_A(p1[i], p1[j]);

            RecursiveSpectralPartition(
                A_1, p1_parts, threshold_Ncut, forceSimetry,
                useSpectralBisection, recursive, minSizeClusters);

            // Split "p2":
            // --------------------------------------------
            // sub-matrix:
            GRAPH_MATRIX A_2(p2.size(), p2.size());
            for (i = 0; i < p2.size(); i++)
                for (j = 0; j < p2.size(); j++) A_2(i, j) = in_A(p2[i], p2[j]);

            RecursiveSpectralPartition(
                A_2, p2_parts, threshold_Ncut, forceSimetry,
                useSpectralBisection, recursive, minSizeClusters);

            // Build "out_parts" from "p1_parts" + "p2_parts"
            //  taken care of indexes mapping!
            // -------------------------------------------------------------------------------------
            // remap p1 nodes:
            for (i = 0; i < p1_parts.size(); i++)
            {
                for (j = 0; j < p1_parts[i].size(); j++)
                    p1_parts[i][j] = p1[p1_parts[i][j]];
                out_parts.push_back(p1_parts[i]);
            }

            // remap p2 nodes:
            for (i = 0; i < p2_parts.size(); i++)
            {
                for (j = 0; j < p2_parts[i].size(); j++)
                    p2_parts[i][j] = p2[p2_parts[i][j]];

                out_parts.push_back(p2_parts[i]);
            }
        }
        else
        {
            // Force bisection only:
            out_parts.clear();
            out_parts.push_back(p1);
            out_parts.push_back(p2);
        }
    }

    MRPT_END
}

/*---------------------------------------------------------------
nCut
---------------------------------------------------------------*/
template <class GRAPH_MATRIX, typename num_t>
num_t CGraphPartitioner<GRAPH_MATRIX, num_t>::nCut(
    const GRAPH_MATRIX& in_A, const std::vector<uint32_t>& in_part1,
    const std::vector<uint32_t>& in_part2)
{
    size_t i, j;
    const size_t size1 = in_part1.size();
    const size_t size2 = in_part2.size();

    // Compute the N-cut value
    // -----------------------------------------------
    num_t cut_AB = 0;
    for (i = 0; i < size1; i++)
        for (j = 0; j < size2; j++) cut_AB += in_A(in_part1[i], in_part2[j]);

    num_t assoc_AA = 0;
    for (i = 0; i < size1; i++)
        for (j = i; j < size1; j++)
            if (i != j) assoc_AA += in_A(in_part1[i], in_part1[j]);

    num_t assoc_BB = 0;
    for (i = 0; i < size2; i++)
        for (j = i; j < size2; j++)
            if (i != j) assoc_BB += in_A(in_part2[i], in_part2[j]);

    num_t assoc_AV = assoc_AA + cut_AB;
    num_t assoc_BV = assoc_BB + cut_AB;

    if (!cut_AB)
        return 0;
    else
        return cut_AB / assoc_AV + cut_AB / assoc_BV;
}

/*---------------------------------------------------------------
exactBisection
---------------------------------------------------------------*/
template <class GRAPH_MATRIX, typename num_t>
void CGraphPartitioner<GRAPH_MATRIX, num_t>::exactBisection(
    GRAPH_MATRIX& in_A, std::vector<uint32_t>& out_part1,
    std::vector<uint32_t>& out_part2, num_t& out_cut_value, bool forceSimetry)
{
    size_t nodeCount;  // Nodes count
    size_t i, j;
    GRAPH_MATRIX Adj;
    std::vector<bool> partition, bestPartition;
    std::vector<uint32_t> part1, part2;
    num_t partCutValue, bestCutValue = std::numeric_limits<num_t>::max();
    bool end = false;

    // Check matrix is square:
    if (in_A.cols() != int(nodeCount = in_A.rows()))
        THROW_EXCEPTION("Weights matrix is not square!!");

    ASSERT_(nodeCount >= 2);

    // forceSimetry?
    if (forceSimetry)
    {
        Adj.setSize(nodeCount, nodeCount);
        for (i = 0; i < nodeCount; i++)
            for (j = i; j < nodeCount; j++)
                Adj(i, j) = Adj(j, i) = 0.5f * (in_A(i, j) + in_A(j, i));
    }
    else
        Adj = in_A;

    // Brute force: compute all posible partitions:
    //-----------------------------------------------------------------

    // First combination: 1000...0000
    // Last combination:  1111...1110
    partition.clear();
    partition.resize(nodeCount, false);
    partition[0] = true;

    while (!end)
    {
        // Build partitions from binary vector:
        part1.clear();
        part2.clear();

        for (i = 0; i < nodeCount; i++)
        {
            if (partition[i])
                part2.push_back(i);
            else
                part1.push_back(i);
        }

        // Compute the n-cut:
        partCutValue = nCut(Adj, part1, part2);

        if (partCutValue < bestCutValue)
        {
            bestCutValue = partCutValue;
            bestPartition = partition;
        }

        // Next combo in the binary vector:
        i = 0;
        bool carry = false;
        do
        {
            carry = partition[i];
            partition[i] = !partition[i];
            i++;
        } while (carry && i < nodeCount);

        // End criterion:
        end = true;
        for (i = 0; end && i < nodeCount; i++) end = end && partition[i];

    };  // End of while

    // Return the best partition:
    out_cut_value = bestCutValue;

    out_part1.clear();
    out_part2.clear();

    for (i = 0; i < nodeCount; i++)
    {
        if (bestPartition[i])
            out_part2.push_back(i);
        else
            out_part1.push_back(i);
    }
}

// Explicit instantiations:
namespace mrpt::graphs
{
// Explicit instantiations:
template class CGraphPartitioner<mrpt::math::CMatrixDouble, double>;
template class CGraphPartitioner<mrpt::math::CMatrixFloat, float>;
template class CGraphPartitioner<mrpt::math::CMatrixD, double>;
template class CGraphPartitioner<mrpt::math::CMatrixF, float>;
}  // namespace mrpt::graphs