shogun_libsvm.cpp

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/*
 * Copyright (c) 2000-2009 Chih-Chung Chang and Chih-Jen Lin
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 * notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 * notice, this list of conditions and the following disclaimer in the
 * documentation and/or other materials provided with the distribution.
 *
 * 3. Neither name of copyright holders nor the names of its contributors
 * may be used to endorse or promote products derived from this software
 * without specific prior written permission.
 *
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 * ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
 * A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR
 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
 * LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
 * NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
 * SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *
 * Shogun specific adjustments (w) 2006-2009 Soeren Sonnenburg
 */

#ifndef DOXYGEN_SHOULD_SKIP_THIS

#include <shogun/lib/memory.h>
#include <shogun/lib/external/shogun_libsvm.h>
#include <shogun/kernel/Kernel.h>
#include <shogun/io/SGIO.h>
#include <shogun/lib/Time.h>
#include <shogun/lib/Signal.h>
#include <shogun/lib/common.h>
#include <shogun/mathematics/Math.h>

#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
#include <float.h>
#include <string.h>
#include <stdarg.h>

#ifdef HAVE_PTHREAD
#include <pthread.h>
#endif

namespace shogun
{

typedef KERNELCACHE_ELEM Qfloat;
typedef float64_t schar;

template <class S, class T> inline void clone(T*& dst, S* src, int32_t n)
{
    dst = SG_MALLOC(T, n);
    memcpy((void *)dst,(void *)src,sizeof(T)*n);
}
#define INF HUGE_VAL
#define TAU 1e-12

class QMatrix;
class SVC_QMC;

//
// Kernel Cache
//
// l is the number of total data items
// size is the cache size limit in bytes
//
class Cache
{
public:
    Cache(int32_t l, int64_t size);
    ~Cache();

    // request data [0,len)
    // return some position p where [p,len) need to be filled
    // (p >= len if nothing needs to be filled)
    int32_t get_data(const int32_t index, Qfloat **data, int32_t len);
    void swap_index(int32_t i, int32_t j);  // future_option

private:
    int32_t l;
    int64_t size;
    struct head_t
    {
        head_t *prev, *next;    // a circular list
        Qfloat *data;
        int32_t len;        // data[0,len) is cached in this entry
    };

    head_t *head;
    head_t lru_head;
    void lru_delete(head_t *h);
    void lru_insert(head_t *h);
};

Cache::Cache(int32_t l_, int64_t size_):l(l_),size(size_)
{
    head = (head_t *)calloc(l,sizeof(head_t));  // initialized to 0
    size /= sizeof(Qfloat);
    size -= l * sizeof(head_t) / sizeof(Qfloat);
    size = CMath::max(size, (int64_t) 2*l); // cache must be large enough for two columns
    lru_head.next = lru_head.prev = &lru_head;
}

Cache::~Cache()
{
    for(head_t *h = lru_head.next; h != &lru_head; h=h->next)
        SG_FREE(h->data);
    SG_FREE(head);
}

void Cache::lru_delete(head_t *h)
{
    // delete from current location
    h->prev->next = h->next;
    h->next->prev = h->prev;
}

void Cache::lru_insert(head_t *h)
{
    // insert to last position
    h->next = &lru_head;
    h->prev = lru_head.prev;
    h->prev->next = h;
    h->next->prev = h;
}

int32_t Cache::get_data(const int32_t index, Qfloat **data, int32_t len)
{
    head_t *h = &head[index];
    if(h->len) lru_delete(h);
    int32_t more = len - h->len;

    if(more > 0)
    {
        // free old space
        while(size < more)
        {
            head_t *old = lru_head.next;
            lru_delete(old);
            SG_FREE(old->data);
            size += old->len;
            old->data = 0;
            old->len = 0;
        }

        // allocate new space
        h->data = SG_REALLOC(Qfloat, h->data, len);
        size -= more;
        CMath::swap(h->len,len);
    }

    lru_insert(h);
    *data = h->data;
    return len;
}

void Cache::swap_index(int32_t i, int32_t j)
{
    if(i==j) return;

    if(head[i].len) lru_delete(&head[i]);
    if(head[j].len) lru_delete(&head[j]);
    CMath::swap(head[i].data,head[j].data);
    CMath::swap(head[i].len,head[j].len);
    if(head[i].len) lru_insert(&head[i]);
    if(head[j].len) lru_insert(&head[j]);

    if(i>j) CMath::swap(i,j);
    for(head_t *h = lru_head.next; h!=&lru_head; h=h->next)
    {
        if(h->len > i)
        {
            if(h->len > j)
                CMath::swap(h->data[i],h->data[j]);
            else
            {
                // give up
                lru_delete(h);
                SG_FREE(h->data);
                size += h->len;
                h->data = 0;
                h->len = 0;
            }
        }
    }
}

//
// Kernel evaluation
//
// the static method k_function is for doing single kernel evaluation
// the constructor of Kernel prepares to calculate the l*l kernel matrix
// the member function get_Q is for getting one column from the Q Matrix
//
class QMatrix {
public:
    virtual Qfloat *get_Q(int32_t column, int32_t len) const = 0;
    virtual Qfloat *get_QD() const = 0;
    virtual void swap_index(int32_t i, int32_t j) const = 0;
    virtual ~QMatrix() {}

    float64_t max_train_time;
};

class LibSVMKernel;

// helper struct for threaded processing
struct Q_THREAD_PARAM
{
    int32_t i;
    int32_t start;
    int32_t end;
    Qfloat* data;
    float64_t* y;
    const LibSVMKernel* q;
};

extern Parallel* sg_parallel;

class LibSVMKernel: public QMatrix {
public:
    LibSVMKernel(int32_t l, svm_node * const * x, const svm_parameter& param);
    virtual ~LibSVMKernel();

    virtual Qfloat *get_Q(int32_t column, int32_t len) const = 0;
    virtual Qfloat *get_QD() const = 0;
    virtual void swap_index(int32_t i, int32_t j) const // no so const...
    {
        CMath::swap(x[i],x[j]);
        if(x_square) CMath::swap(x_square[i],x_square[j]);
    }

    static void* compute_Q_parallel_helper(void* p)
    {
        Q_THREAD_PARAM* params= (Q_THREAD_PARAM*) p;
        int32_t i=params->i;
        int32_t start=params->start;
        int32_t end=params->end;
        float64_t* y=params->y;
        Qfloat* data=params->data;
        const LibSVMKernel* q=params->q;

        if (y) // two class
        {
            for(int32_t j=start;j<end;j++)
                data[j] = (Qfloat) y[i]*y[j]*q->kernel_function(i,j);
        }
        else // one class, eps svr
        {
            for(int32_t j=start;j<end;j++)
                data[j] = (Qfloat) q->kernel_function(i,j);
        }

        return NULL;
    }

    void compute_Q_parallel(Qfloat* data, float64_t* lab, int32_t i, int32_t start, int32_t len) const
    {
        int32_t num_threads=sg_parallel->get_num_threads();
        if (num_threads < 2)
        {
            Q_THREAD_PARAM params;
            params.i=i;
            params.start=start;
            params.end=len;
            params.y=lab;
            params.data=data;
            params.q=this;
            compute_Q_parallel_helper((void*) &params);
        }
        else
        {
#ifdef HAVE_PTHREAD
            int32_t total_num=(len-start);
            pthread_t* threads = SG_MALLOC(pthread_t, num_threads-1);
            Q_THREAD_PARAM* params = SG_MALLOC(Q_THREAD_PARAM, num_threads);
            int32_t step= total_num/num_threads;

            int32_t t;

            num_threads--;
            for (t=0; t<num_threads; t++)
            {
                params[t].i=i;
                params[t].start=t*step;
                params[t].end=(t+1)*step;
                params[t].y=lab;
                params[t].data=data;
                params[t].q=this;

                int code=pthread_create(&threads[t], NULL,
                        compute_Q_parallel_helper, (void*)&params[t]);

                if (code != 0)
                {
                    SG_SWARNING("Thread creation failed (thread %d of %d) "
                            "with error:'%s'\n",t, num_threads, strerror(code));
                    num_threads=t;
                    break;
                }
            }

            params[t].i=i;
            params[t].start=t*step;
            params[t].end=len;
            params[t].y=lab;
            params[t].data=data;
            params[t].q=this;
            compute_Q_parallel_helper(&params[t]);

            for (t=0; t<num_threads; t++)
            {
                if (pthread_join(threads[t], NULL) != 0)
                    SG_SWARNING("pthread_join of thread %d/%d failed\n", t, num_threads);
            }

            SG_FREE(params);
            SG_FREE(threads);
#endif /* HAVE_PTHREAD */
        }
    }

    inline float64_t kernel_function(int32_t i, int32_t j) const
    {
        return kernel->kernel(x[i]->index,x[j]->index);
    }

private:
    CKernel* kernel;
    const svm_node **x;
    float64_t *x_square;

    // svm_parameter
    const int32_t kernel_type;
    const int32_t degree;
    const float64_t gamma;
    const float64_t coef0;
};

LibSVMKernel::LibSVMKernel(int32_t l, svm_node * const * x_, const svm_parameter& param)
: kernel_type(param.kernel_type), degree(param.degree),
 gamma(param.gamma), coef0(param.coef0)
{
    clone(x,x_,l);
    x_square = 0;
    kernel=param.kernel;
    max_train_time=param.max_train_time;
}

LibSVMKernel::~LibSVMKernel()
{
    SG_FREE(x);
    SG_FREE(x_square);
}

// Generalized SMO+SVMlight algorithm
// Solves:
//
//  min 0.5(\alpha^T Q \alpha) + b^T \alpha
//
//      y^T \alpha = \delta
//      y_i = +1 or -1
//      0 <= alpha_i <= Cp for y_i = 1
//      0 <= alpha_i <= Cn for y_i = -1
//
// Given:
//
//  Q, p, y, Cp, Cn, and an initial feasible point \alpha
//  l is the size of vectors and matrices
//  eps is the stopping tolerance
//
// solution will be put in \alpha, objective value will be put in obj
//
class Solver {
public:
    Solver() {};
    virtual ~Solver() {};

    struct SolutionInfo {
        float64_t obj;
        float64_t rho;
        float64_t upper_bound_p;
        float64_t upper_bound_n;
        float64_t r;    // for Solver_NU
    };

    void Solve(
        int32_t l, const QMatrix& Q, const float64_t *p_, const schar *y_,
        float64_t *alpha_, float64_t Cp, float64_t Cn, float64_t eps,
        SolutionInfo* si, int32_t shrinking, bool use_bias);

protected:
    int32_t active_size;
    schar *y;
    float64_t *G;       // gradient of objective function
    enum { LOWER_BOUND, UPPER_BOUND, FREE };
    char *alpha_status; // LOWER_BOUND, UPPER_BOUND, FREE
    float64_t *alpha;
    const QMatrix *Q;
    const Qfloat *QD;
    float64_t eps;
    float64_t Cp,Cn;
    float64_t *p;
    int32_t *active_set;
    float64_t *G_bar;       // gradient, if we treat free variables as 0
    int32_t l;
    bool unshrink;  // XXX

    float64_t get_C(int32_t i)
    {
        return (y[i] > 0)? Cp : Cn;
    }
    void update_alpha_status(int32_t i)
    {
        if(alpha[i] >= get_C(i))
            alpha_status[i] = UPPER_BOUND;
        else if(alpha[i] <= 0)
            alpha_status[i] = LOWER_BOUND;
        else alpha_status[i] = FREE;
    }
    bool is_upper_bound(int32_t i) { return alpha_status[i] == UPPER_BOUND; }
    bool is_lower_bound(int32_t i) { return alpha_status[i] == LOWER_BOUND; }
    bool is_free(int32_t i) { return alpha_status[i] == FREE; }
    void swap_index(int32_t i, int32_t j);
    void reconstruct_gradient();
    virtual int32_t select_working_set(int32_t &i, int32_t &j, float64_t &gap);
    virtual float64_t calculate_rho();
    virtual void do_shrinking();
private:
    bool be_shrunk(int32_t i, float64_t Gmax1, float64_t Gmax2);
};

void Solver::swap_index(int32_t i, int32_t j)
{
    Q->swap_index(i,j);
    CMath::swap(y[i],y[j]);
    CMath::swap(G[i],G[j]);
    CMath::swap(alpha_status[i],alpha_status[j]);
    CMath::swap(alpha[i],alpha[j]);
    CMath::swap(p[i],p[j]);
    CMath::swap(active_set[i],active_set[j]);
    CMath::swap(G_bar[i],G_bar[j]);
}

void Solver::reconstruct_gradient()
{
    // reconstruct inactive elements of G from G_bar and free variables

    if(active_size == l) return;

    int32_t i,j;
    int32_t nr_free = 0;

    for(j=active_size;j<l;j++)
        G[j] = G_bar[j] + p[j];

    for(j=0;j<active_size;j++)
        if(is_free(j))
            nr_free++;

    if (nr_free*l > 2*active_size*(l-active_size))
    {
        for(i=active_size;i<l;i++)
        {
            const Qfloat *Q_i = Q->get_Q(i,active_size);
            for(j=0;j<active_size;j++)
                if(is_free(j))
                    G[i] += alpha[j] * Q_i[j];
        }
    }
    else
    {
        for(i=0;i<active_size;i++)
            if(is_free(i))
            {
                const Qfloat *Q_i = Q->get_Q(i,l);
                float64_t alpha_i = alpha[i];
                for(j=active_size;j<l;j++)
                    G[j] += alpha_i * Q_i[j];
            }
    }
}

void Solver::Solve(
    int32_t p_l, const QMatrix& p_Q, const float64_t *p_p,
    const schar *p_y, float64_t *p_alpha, float64_t p_Cp, float64_t p_Cn,
    float64_t p_eps, SolutionInfo* p_si, int32_t shrinking, bool use_bias)
{
    this->l = p_l;
    this->Q = &p_Q;
    QD=Q->get_QD();
    clone(p, p_p,l);
    clone(y, p_y,l);
    clone(alpha,p_alpha,l);
    this->Cp = p_Cp;
    this->Cn = p_Cn;
    this->eps = p_eps;
    unshrink = false;

    // initialize alpha_status
    {
        alpha_status = SG_MALLOC(char, l);
        for(int32_t i=0;i<l;i++)
            update_alpha_status(i);
    }

    // initialize active set (for shrinking)
    {
        active_set = SG_MALLOC(int32_t, l);
        for(int32_t i=0;i<l;i++)
            active_set[i] = i;
        active_size = l;
    }

    // initialize gradient
    CSignal::clear_cancel();
    CTime start_time;
    {
        G = SG_MALLOC(float64_t, l);
        G_bar = SG_MALLOC(float64_t, l);
        int32_t i;
        for(i=0;i<l;i++)
        {
            G[i] = p_p[i];
            G_bar[i] = 0;
        }
        SG_SINFO("Computing gradient for initial set of non-zero alphas\n");
        //CMath::display_vector(alpha, l, "alphas");
        for(i=0;i<l && !CSignal::cancel_computations(); i++)
        {
            if(!is_lower_bound(i))
            {
                const Qfloat *Q_i = Q->get_Q(i,l);
                float64_t alpha_i = alpha[i];
                int32_t j;
                for(j=0;j<l;j++)
                    G[j] += alpha_i*Q_i[j];
                if(is_upper_bound(i))
                    for(j=0;j<l;j++)
                        G_bar[j] += get_C(i) * Q_i[j];
            }
            SG_SPROGRESS(i, 0, l);
        }
        SG_SDONE();
    }

    // optimization step

    int32_t iter = 0;
    int32_t counter = CMath::min(l,1000)+1;

    while (!CSignal::cancel_computations())
    {
        if (Q->max_train_time > 0 && start_time.cur_time_diff() > Q->max_train_time)
          break;

        // show progress and do shrinking

        if(--counter == 0)
        {
            counter = CMath::min(l,1000);
            if(shrinking) do_shrinking();
            //SG_SINFO(".");
        }

        int32_t i,j;
        float64_t gap;
        if(select_working_set(i,j, gap)!=0)
        {
            // reconstruct the whole gradient
            reconstruct_gradient();
            // reset active set size and check
            active_size = l;
            //SG_SINFO("*");
            if(select_working_set(i,j, gap)!=0)
                break;
            else
                counter = 1;    // do shrinking next iteration
        }

        SG_SABS_PROGRESS(gap, -CMath::log10(gap), -CMath::log10(1), -CMath::log10(eps), 6);

        ++iter;

        // update alpha[i] and alpha[j], handle bounds carefully

        const Qfloat *Q_i = Q->get_Q(i,active_size);
        const Qfloat *Q_j = Q->get_Q(j,active_size);

        float64_t C_i = get_C(i);
        float64_t C_j = get_C(j);

        float64_t old_alpha_i = alpha[i];
        float64_t old_alpha_j = alpha[j];

        if (!use_bias)
        {
            double pi=G[i]-Q_i[i]*alpha[i]-Q_i[j]*alpha[j];
            double pj=G[j]-Q_i[j]*alpha[i]-Q_j[j]*alpha[j];
            double det=Q_i[i]*Q_j[j]-Q_i[j]*Q_i[j];
            double alpha_i=-(Q_j[j]*pi-Q_i[j]*pj)/det;
            alpha_i=CMath::min(C_i,CMath::max(0.0,alpha_i));
            double alpha_j=-(-Q_i[j]*pi+Q_i[i]*pj)/det;
            alpha_j=CMath::min(C_j,CMath::max(0.0,alpha_j));

            if (alpha_i==0 || alpha_i == C_i)
                alpha_j=CMath::min(C_j,CMath::max(0.0,-(pj+Q_i[j]*alpha_i)/Q_j[j]));
            if (alpha_j==0 || alpha_j == C_j)
                alpha_i=CMath::min(C_i,CMath::max(0.0,-(pi+Q_i[j]*alpha_j)/Q_i[i]));

            alpha[i]=alpha_i; alpha[j]=alpha_j;
        }
        else
        {
            if(y[i]!=y[j])
            {
                float64_t quad_coef = Q_i[i]+Q_j[j]+2*Q_i[j];
                if (quad_coef <= 0)
                    quad_coef = TAU;
                float64_t delta = (-G[i]-G[j])/quad_coef;
                float64_t diff = alpha[i] - alpha[j];
                alpha[i] += delta;
                alpha[j] += delta;

                if(diff > 0)
                {
                    if(alpha[j] < 0)
                    {
                        alpha[j] = 0;
                        alpha[i] = diff;
                    }
                }
                else
                {
                    if(alpha[i] < 0)
                    {
                        alpha[i] = 0;
                        alpha[j] = -diff;
                    }
                }
                if(diff > C_i - C_j)
                {
                    if(alpha[i] > C_i)
                    {
                        alpha[i] = C_i;
                        alpha[j] = C_i - diff;
                    }
                }
                else
                {
                    if(alpha[j] > C_j)
                    {
                        alpha[j] = C_j;
                        alpha[i] = C_j + diff;
                    }
                }
            }
            else
            {
                float64_t quad_coef = Q_i[i]+Q_j[j]-2*Q_i[j];
                if (quad_coef <= 0)
                    quad_coef = TAU;
                float64_t delta = (G[i]-G[j])/quad_coef;
                float64_t sum = alpha[i] + alpha[j];
                alpha[i] -= delta;
                alpha[j] += delta;

                if(sum > C_i)
                {
                    if(alpha[i] > C_i)
                    {
                        alpha[i] = C_i;
                        alpha[j] = sum - C_i;
                    }
                }
                else
                {
                    if(alpha[j] < 0)
                    {
                        alpha[j] = 0;
                        alpha[i] = sum;
                    }
                }
                if(sum > C_j)
                {
                    if(alpha[j] > C_j)
                    {
                        alpha[j] = C_j;
                        alpha[i] = sum - C_j;
                    }
                }
                else
                {
                    if(alpha[i] < 0)
                    {
                        alpha[i] = 0;
                        alpha[j] = sum;
                    }
                }
            }
        }

        // update G

        float64_t delta_alpha_i = alpha[i] - old_alpha_i;
        float64_t delta_alpha_j = alpha[j] - old_alpha_j;

        for(int32_t k=0;k<active_size;k++)
        {
            G[k] += Q_i[k]*delta_alpha_i + Q_j[k]*delta_alpha_j;
        }

        // update alpha_status and G_bar

        {
            bool ui = is_upper_bound(i);
            bool uj = is_upper_bound(j);
            update_alpha_status(i);
            update_alpha_status(j);
            int32_t k;
            if(ui != is_upper_bound(i))
            {
                Q_i = Q->get_Q(i,l);
                if(ui)
                    for(k=0;k<l;k++)
                        G_bar[k] -= C_i * Q_i[k];
                else
                    for(k=0;k<l;k++)
                        G_bar[k] += C_i * Q_i[k];
            }

            if(uj != is_upper_bound(j))
            {
                Q_j = Q->get_Q(j,l);
                if(uj)
                    for(k=0;k<l;k++)
                        G_bar[k] -= C_j * Q_j[k];
                else
                    for(k=0;k<l;k++)
                        G_bar[k] += C_j * Q_j[k];
            }
        }

#ifdef MCSVM_DEBUG
        // calculate objective value
        {
            float64_t v = 0;
            for(i=0;i<l;i++)
                v += alpha[i] * (G[i] + p[i]);

            p_si->obj = v/2;

            float64_t primal=0;
            //float64_t gap=100000;
            SG_SPRINT("dual obj=%f primal obf=%f gap=%f\n", v/2, primal, gap);
        }
#endif
    }

    // calculate rho

    if (!use_bias)
        p_si->rho = 0;
    else
        p_si->rho = calculate_rho();

    // calculate objective value
    {
        float64_t v = 0;
        int32_t i;
        for(i=0;i<l;i++)
            v += alpha[i] * (G[i] + p[i]);

        p_si->obj = v/2;
    }

    // put back the solution
    {
        for(int32_t i=0;i<l;i++)
            p_alpha[active_set[i]] = alpha[i];
    }

    p_si->upper_bound_p = Cp;
    p_si->upper_bound_n = Cn;

    SG_SINFO("\noptimization finished, #iter = %d\n",iter);

    SG_FREE(p);
    SG_FREE(y);
    SG_FREE(alpha);
    SG_FREE(alpha_status);
    SG_FREE(active_set);
    SG_FREE(G);
    SG_FREE(G_bar);
}

// return 1 if already optimal, return 0 otherwise
int32_t Solver::select_working_set(
    int32_t &out_i, int32_t &out_j, float64_t &gap)
{
    // return i,j such that
    // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
    // j: minimizes the decrease of obj value
    //    (if quadratic coefficient <= 0, replace it with tau)
    //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)

    float64_t Gmax = -INF;
    float64_t Gmax2 = -INF;
    int32_t Gmax_idx = -1;
    int32_t Gmin_idx = -1;
    float64_t obj_diff_min = INF;

    for(int32_t t=0;t<active_size;t++)
        if(y[t]==+1)
        {
            if(!is_upper_bound(t))
                if(-G[t] >= Gmax)
                {
                    Gmax = -G[t];
                    Gmax_idx = t;
                }
        }
        else
        {
            if(!is_lower_bound(t))
                if(G[t] >= Gmax)
                {
                    Gmax = G[t];
                    Gmax_idx = t;
                }
        }

    int32_t i = Gmax_idx;
    const Qfloat *Q_i = NULL;
    if(i != -1) // NULL Q_i not accessed: Gmax=-INF if i=-1
        Q_i = Q->get_Q(i,active_size);

    for(int32_t j=0;j<active_size;j++)
    {
        if(y[j]==+1)
        {
            if (!is_lower_bound(j))
            {
                float64_t grad_diff=Gmax+G[j];
                if (G[j] >= Gmax2)
                    Gmax2 = G[j];
                if (grad_diff > 0)
                {
                    float64_t obj_diff;
                    float64_t quad_coef=Q_i[i]+QD[j]-2.0*y[i]*Q_i[j];
                    if (quad_coef > 0)
                        obj_diff = -(grad_diff*grad_diff)/quad_coef;
                    else
                        obj_diff = -(grad_diff*grad_diff)/TAU;

                    if (obj_diff <= obj_diff_min)
                    {
                        Gmin_idx=j;
                        obj_diff_min = obj_diff;
                    }
                }
            }
        }
        else
        {
            if (!is_upper_bound(j))
            {
                float64_t grad_diff= Gmax-G[j];
                if (-G[j] >= Gmax2)
                    Gmax2 = -G[j];
                if (grad_diff > 0)
                {
                    float64_t obj_diff;
                    float64_t quad_coef=Q_i[i]+QD[j]+2.0*y[i]*Q_i[j];
                    if (quad_coef > 0)
                        obj_diff = -(grad_diff*grad_diff)/quad_coef;
                    else
                        obj_diff = -(grad_diff*grad_diff)/TAU;

                    if (obj_diff <= obj_diff_min)
                    {
                        Gmin_idx=j;
                        obj_diff_min = obj_diff;
                    }
                }
            }
        }
    }

    gap=Gmax+Gmax2;
    if(gap < eps)
        return 1;

    out_i = Gmax_idx;
    out_j = Gmin_idx;
    return 0;
}

bool Solver::be_shrunk(int32_t i, float64_t Gmax1, float64_t Gmax2)
{
    if(is_upper_bound(i))
    {
        if(y[i]==+1)
            return(-G[i] > Gmax1);
        else
            return(-G[i] > Gmax2);
    }
    else if(is_lower_bound(i))
    {
        if(y[i]==+1)
            return(G[i] > Gmax2);
        else
            return(G[i] > Gmax1);
    }
    else
        return(false);
}


void Solver::do_shrinking()
{
    int32_t i;
    float64_t Gmax1 = -INF;     // max { -y_i * grad(f)_i | i in I_up(\alpha) }
    float64_t Gmax2 = -INF;     // max { y_i * grad(f)_i | i in I_low(\alpha) }

    // find maximal violating pair first
    for(i=0;i<active_size;i++)
    {
        if(y[i]==+1)
        {
            if(!is_upper_bound(i))
            {
                if(-G[i] >= Gmax1)
                    Gmax1 = -G[i];
            }
            if(!is_lower_bound(i))
            {
                if(G[i] >= Gmax2)
                    Gmax2 = G[i];
            }
        }
        else
        {
            if(!is_upper_bound(i))
            {
                if(-G[i] >= Gmax2)
                    Gmax2 = -G[i];
            }
            if(!is_lower_bound(i))
            {
                if(G[i] >= Gmax1)
                    Gmax1 = G[i];
            }
        }
    }

    if(unshrink == false && Gmax1 + Gmax2 <= eps*10)
    {
        unshrink = true;
        reconstruct_gradient();
        active_size = l;
    }

    for(i=0;i<active_size;i++)
        if (be_shrunk(i, Gmax1, Gmax2))
        {
            active_size--;
            while (active_size > i)
            {
                if (!be_shrunk(active_size, Gmax1, Gmax2))
                {
                    swap_index(i,active_size);
                    break;
                }
                active_size--;
            }
        }
}

float64_t Solver::calculate_rho()
{
    float64_t r;
    int32_t nr_free = 0;
    float64_t ub = INF, lb = -INF, sum_free = 0;
    for(int32_t i=0;i<active_size;i++)
    {
        float64_t yG = y[i]*G[i];

        if(is_upper_bound(i))
        {
            if(y[i]==-1)
                ub = CMath::min(ub,yG);
            else
                lb = CMath::max(lb,yG);
        }
        else if(is_lower_bound(i))
        {
            if(y[i]==+1)
                ub = CMath::min(ub,yG);
            else
                lb = CMath::max(lb,yG);
        }
        else
        {
            ++nr_free;
            sum_free += yG;
        }
    }

    if(nr_free>0)
        r = sum_free/nr_free;
    else
        r = (ub+lb)/2;

    return r;
}


//
//Solve with individually weighted examples
//
class WeightedSolver : public Solver
{

public:

    WeightedSolver(float64_t* cost_vec)
    {

        this->Cs = cost_vec;

    }

    virtual float64_t get_C(int32_t i)
    {

        return Cs[i];
    }

protected:

  float64_t* Cs;

};


//
// Solver for nu-svm classification and regression
//
// additional constraint: e^T \alpha = constant
//
class Solver_NU : public Solver
{
public:
    Solver_NU() {}
    void Solve(
        int32_t p_l, const QMatrix& p_Q, const float64_t *p_p,
        const schar *p_y, float64_t* p_alpha, float64_t p_Cp, float64_t p_Cn,
        float64_t p_eps, SolutionInfo* p_si, int32_t shrinking, bool use_bias)
    {
        this->si = p_si;
        Solver::Solve(p_l,p_Q,p_p,p_y,p_alpha,p_Cp,p_Cn,p_eps,p_si,
                shrinking,use_bias);
    }
private:
    SolutionInfo *si;
    int32_t select_working_set(int32_t &i, int32_t &j, float64_t &gap);
    float64_t calculate_rho();
    bool be_shrunk(
        int32_t i, float64_t Gmax1, float64_t Gmax2, float64_t Gmax3,
        float64_t Gmax4);
    void do_shrinking();
};

// return 1 if already optimal, return 0 otherwise
int32_t Solver_NU::select_working_set(
    int32_t &out_i, int32_t &out_j, float64_t &gap)
{
    // return i,j such that y_i = y_j and
    // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
    // j: minimizes the decrease of obj value
    //    (if quadratic coefficient <= 0, replace it with tau)
    //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)

    float64_t Gmaxp = -INF;
    float64_t Gmaxp2 = -INF;
    int32_t Gmaxp_idx = -1;

    float64_t Gmaxn = -INF;
    float64_t Gmaxn2 = -INF;
    int32_t Gmaxn_idx = -1;

    int32_t Gmin_idx = -1;
    float64_t obj_diff_min = INF;

    for(int32_t t=0;t<active_size;t++)
        if(y[t]==+1)
        {
            if(!is_upper_bound(t))
                if(-G[t] >= Gmaxp)
                {
                    Gmaxp = -G[t];
                    Gmaxp_idx = t;
                }
        }
        else
        {
            if(!is_lower_bound(t))
                if(G[t] >= Gmaxn)
                {
                    Gmaxn = G[t];
                    Gmaxn_idx = t;
                }
        }

    int32_t ip = Gmaxp_idx;
    int32_t in = Gmaxn_idx;
    const Qfloat *Q_ip = NULL;
    const Qfloat *Q_in = NULL;
    if(ip != -1) // NULL Q_ip not accessed: Gmaxp=-INF if ip=-1
        Q_ip = Q->get_Q(ip,active_size);
    if(in != -1)
        Q_in = Q->get_Q(in,active_size);

    for(int32_t j=0;j<active_size;j++)
    {
        if(y[j]==+1)
        {
            if (!is_lower_bound(j))
            {
                float64_t grad_diff=Gmaxp+G[j];
                if (G[j] >= Gmaxp2)
                    Gmaxp2 = G[j];
                if (grad_diff > 0)
                {
                    float64_t obj_diff;
                    float64_t quad_coef = Q_ip[ip]+QD[j]-2*Q_ip[j];
                    if (quad_coef > 0)
                        obj_diff = -(grad_diff*grad_diff)/quad_coef;
                    else
                        obj_diff = -(grad_diff*grad_diff)/TAU;

                    if (obj_diff <= obj_diff_min)
                    {
                        Gmin_idx=j;
                        obj_diff_min = obj_diff;
                    }
                }
            }
        }
        else
        {
            if (!is_upper_bound(j))
            {
                float64_t grad_diff=Gmaxn-G[j];
                if (-G[j] >= Gmaxn2)
                    Gmaxn2 = -G[j];
                if (grad_diff > 0)
                {
                    float64_t obj_diff;
                    float64_t quad_coef = Q_in[in]+QD[j]-2*Q_in[j];
                    if (quad_coef > 0)
                        obj_diff = -(grad_diff*grad_diff)/quad_coef;
                    else
                        obj_diff = -(grad_diff*grad_diff)/TAU;

                    if (obj_diff <= obj_diff_min)
                    {
                        Gmin_idx=j;
                        obj_diff_min = obj_diff;
                    }
                }
            }
        }
    }

    gap=CMath::max(Gmaxp+Gmaxp2,Gmaxn+Gmaxn2);
    if(gap < eps)
        return 1;

    if (y[Gmin_idx] == +1)
        out_i = Gmaxp_idx;
    else
        out_i = Gmaxn_idx;
    out_j = Gmin_idx;

    return 0;
}

bool Solver_NU::be_shrunk(
    int32_t i, float64_t Gmax1, float64_t Gmax2, float64_t Gmax3,
    float64_t Gmax4)
{
    if(is_upper_bound(i))
    {
        if(y[i]==+1)
            return(-G[i] > Gmax1);
        else
            return(-G[i] > Gmax4);
    }
    else if(is_lower_bound(i))
    {
        if(y[i]==+1)
            return(G[i] > Gmax2);
        else
            return(G[i] > Gmax3);
    }
    else
        return(false);
}

void Solver_NU::do_shrinking()
{
    float64_t Gmax1 = -INF; // max { -y_i * grad(f)_i | y_i = +1, i in I_up(\alpha) }
    float64_t Gmax2 = -INF; // max { y_i * grad(f)_i | y_i = +1, i in I_low(\alpha) }
    float64_t Gmax3 = -INF; // max { -y_i * grad(f)_i | y_i = -1, i in I_up(\alpha) }
    float64_t Gmax4 = -INF; // max { y_i * grad(f)_i | y_i = -1, i in I_low(\alpha) }

    // find maximal violating pair first
    int32_t i;
    for(i=0;i<active_size;i++)
    {
        if(!is_upper_bound(i))
        {
            if(y[i]==+1)
            {
                if(-G[i] > Gmax1) Gmax1 = -G[i];
            }
            else    if(-G[i] > Gmax4) Gmax4 = -G[i];
        }
        if(!is_lower_bound(i))
        {
            if(y[i]==+1)
            {
                if(G[i] > Gmax2) Gmax2 = G[i];
            }
            else    if(G[i] > Gmax3) Gmax3 = G[i];
        }
    }

    if(unshrink == false && CMath::max(Gmax1+Gmax2,Gmax3+Gmax4) <= eps*10)
    {
        unshrink = true;
        reconstruct_gradient();
        active_size = l;
    }

    for(i=0;i<active_size;i++)
        if (be_shrunk(i, Gmax1, Gmax2, Gmax3, Gmax4))
        {
            active_size--;
            while (active_size > i)
            {
                if (!be_shrunk(active_size, Gmax1, Gmax2, Gmax3, Gmax4))
                {
                    swap_index(i,active_size);
                    break;
                }
                active_size--;
            }
        }
}

float64_t Solver_NU::calculate_rho()
{
    int32_t nr_free1 = 0,nr_free2 = 0;
    float64_t ub1 = INF, ub2 = INF;
    float64_t lb1 = -INF, lb2 = -INF;
    float64_t sum_free1 = 0, sum_free2 = 0;

    for(int32_t i=0; i<active_size; i++)
    {
        if(y[i]==+1)
        {
            if(is_upper_bound(i))
                lb1 = CMath::max(lb1,G[i]);
            else if(is_lower_bound(i))
                ub1 = CMath::min(ub1,G[i]);
            else
            {
                ++nr_free1;
                sum_free1 += G[i];
            }
        }
        else
        {
            if(is_upper_bound(i))
                lb2 = CMath::max(lb2,G[i]);
            else if(is_lower_bound(i))
                ub2 = CMath::min(ub2,G[i]);
            else
            {
                ++nr_free2;
                sum_free2 += G[i];
            }
        }
    }

    float64_t r1,r2;
    if(nr_free1 > 0)
        r1 = sum_free1/nr_free1;
    else
        r1 = (ub1+lb1)/2;

    if(nr_free2 > 0)
        r2 = sum_free2/nr_free2;
    else
        r2 = (ub2+lb2)/2;

    si->r = (r1+r2)/2;
    return (r1-r2)/2;
}

class SVC_QMC: public LibSVMKernel
{
public:
    SVC_QMC(const svm_problem& prob, const svm_parameter& param, const schar *y_, int32_t n_class, float64_t fac)
    :LibSVMKernel(prob.l, prob.x, param)
    {
        nr_class=n_class;
        factor=fac;
        clone(y,y_,prob.l);
        cache = new Cache(prob.l,(int64_t)(param.cache_size*(1l<<20)));
        QD = SG_MALLOC(Qfloat, prob.l);
        for(int32_t i=0;i<prob.l;i++)
        {
            QD[i]= factor*(nr_class-1)*kernel_function(i,i);
        }
    }

    Qfloat *get_Q(int32_t i, int32_t len) const
    {
        Qfloat *data;
        int32_t start;
        if((start = cache->get_data(i,&data,len)) < len)
        {
            compute_Q_parallel(data, NULL, i, start, len);

            for(int32_t j=start;j<len;j++)
            {
                if (y[i]==y[j])
                    data[j] *= (factor*(nr_class-1));
                else
                    data[j] *= (-factor);
            }
        }
        return data;
    }

    inline Qfloat get_orig_Qij(Qfloat Q, int32_t i, int32_t j)
    {
        if (y[i]==y[j])
            return Q/(nr_class-1);
        else
            return -Q;
    }

    Qfloat *get_QD() const
    {
        return QD;
    }

    void swap_index(int32_t i, int32_t j) const
    {
        cache->swap_index(i,j);
        LibSVMKernel::swap_index(i,j);
        CMath::swap(y[i],y[j]);
        CMath::swap(QD[i],QD[j]);
    }

    ~SVC_QMC()
    {
        SG_FREE(y);
        delete cache;
        SG_FREE(QD);
    }
private:
    float64_t factor;
    float64_t nr_class;
    schar *y;
    Cache *cache;
    Qfloat *QD;
};

//
// Solver for nu-svm classification and regression
//
// additional constraint: e^T \alpha = constant
//
class Solver_NUMC : public Solver
{
public:
    Solver_NUMC(int32_t n_class, float64_t svm_nu)
    {
        nr_class=n_class;
        nu=svm_nu;
    }

    void Solve(
        int32_t p_l, const QMatrix& p_Q, const float64_t *p_p,
        const schar *p_y, float64_t* p_alpha, float64_t p_Cp, float64_t p_Cn,
        float64_t p_eps, SolutionInfo* p_si, int32_t shrinking, bool use_bias)
    {
        this->si = p_si;
        Solver::Solve(p_l,p_Q,p_p,p_y,p_alpha,p_Cp,p_Cn,p_eps,p_si,shrinking, use_bias);
    }
    float64_t compute_primal(const schar* p_y, float64_t* p_alpha, float64_t* biases,float64_t* normwcw);

private:
    SolutionInfo *si;
    int32_t select_working_set(int32_t &i, int32_t &j, float64_t &gap);
    float64_t calculate_rho();
    bool be_shrunk(
        int32_t i, float64_t Gmax1, float64_t Gmax2, float64_t Gmax3,
        float64_t Gmax4);
    void do_shrinking();

private:
    int32_t nr_class;
    float64_t  nu;
};

float64_t Solver_NUMC::compute_primal(const schar* p_y, float64_t* p_alpha, float64_t* biases, float64_t* normwcw)
{
    clone(y, p_y,l);
    clone(alpha,p_alpha,l);

    alpha_status = SG_MALLOC(char, l);
    for(int32_t i=0;i<l;i++)
        update_alpha_status(i);

    float64_t* class_count = SG_MALLOC(float64_t, nr_class);
    float64_t* outputs = SG_MALLOC(float64_t, l);

    for (int32_t i=0; i<nr_class; i++)
    {
        class_count[i]=0;
        biases[i+1]=0;
    }

    for (int32_t i=0; i<active_size; i++)
    {
        update_alpha_status(i);
        if(!is_upper_bound(i) && !is_lower_bound(i))
            class_count[(int32_t) y[i]]++;
    }

    //CMath::display_vector(class_count, nr_class, "class_count");

    float64_t mu=((float64_t) nr_class)/(nu*l);
    //SG_SPRINT("nr_class=%d, l=%d, active_size=%d, nu=%f, mu=%f\n", nr_class, l, active_size, nu, mu);

    float64_t rho=0;
    float64_t quad=0;
    float64_t* zero_counts  = SG_MALLOC(float64_t, nr_class);
    float64_t normwc_const = 0;

    for (int32_t i=0; i<nr_class; i++)
    {
        zero_counts[i]=-INF;
        normwcw[i]=0;
    }

    for (int32_t i=0; i<active_size; i++)
    {
        float64_t sum_free=0;
        float64_t sum_atbound=0;
        float64_t sum_zero_count=0;

        Qfloat* Q_i = Q->get_Q(i,active_size);
        outputs[i]=0;

        for (int j=0; j<active_size; j++)
        {
            quad+= alpha[i]*alpha[j]*Q_i[j];
            float64_t tmp= alpha[j]*Q_i[j]/mu;

            if(!is_upper_bound(i) && !is_lower_bound(i))
                sum_free+=tmp;
            else
                sum_atbound+=tmp;

            if (class_count[(int32_t) y[i]] == 0 && y[j]==y[i])
                sum_zero_count+= tmp;

            SVC_QMC* QMC=(SVC_QMC*) Q;
            float64_t norm_tmp=alpha[i]*alpha[j]*QMC->get_orig_Qij(Q_i[j], i, j);
            if (y[i]==y[j])
                normwcw[(int32_t) y[i]]+=norm_tmp;

            normwcw[(int32_t) y[i]]-=2.0/nr_class*norm_tmp;
            normwc_const+=norm_tmp;
        }

        if (class_count[(int32_t) y[i]] == 0)
        {
            if (zero_counts[(int32_t) y[i]]<sum_zero_count)
                zero_counts[(int32_t) y[i]]=sum_zero_count;
        }

        biases[(int32_t) y[i]+1]-=sum_free;
        if (class_count[(int32_t) y[i]] != 0.0)
            rho+=sum_free/class_count[(int32_t) y[i]];
        outputs[i]+=sum_free+sum_atbound;
    }

    for (int32_t i=0; i<nr_class; i++)
    {
        if (class_count[i] == 0.0)
            rho+=zero_counts[i];

        normwcw[i]+=normwc_const/CMath::sq(nr_class);
        normwcw[i]=CMath::sqrt(normwcw[i]);
    }

    SGVector<float64_t>::display_vector(normwcw, nr_class, "normwcw");

    rho/=nr_class;

    SG_SPRINT("rho=%f\n", rho);

    float64_t sumb=0;
    for (int32_t i=0; i<nr_class; i++)
    {
        if (class_count[i] != 0.0)
            biases[i+1]=biases[i+1]/class_count[i]+rho;
        else
            biases[i+1]+=rho-zero_counts[i];

        SG_SPRINT("biases=%f\n", biases[i+1]);

        sumb+=biases[i+1];
    }
    SG_SPRINT("sumb=%f\n", sumb);

    SG_FREE(zero_counts);

    for (int32_t i=0; i<l; i++)
        outputs[i]+=biases[(int32_t) y[i]+1];

    biases[0]=rho;

    //CMath::display_vector(outputs, l, "outputs");


    float64_t xi=0;
    for (int32_t i=0; i<active_size; i++)
    {
        if (is_lower_bound(i))
            continue;
        xi+=rho-outputs[i];
    }

    //SG_SPRINT("xi=%f\n", xi);

    //SG_SPRINT("quad=%f Cp=%f xi*mu=%f\n", quad, nr_class*rho, xi*mu);

    float64_t primal=0.5*quad- nr_class*rho+xi*mu;

    //SG_SPRINT("primal=%10.10f\n", primal);

    SG_FREE(y);
    SG_FREE(alpha);
    SG_FREE(alpha_status);

    return primal;
}


// return 1 if already optimal, return 0 otherwise
int32_t Solver_NUMC::select_working_set(
    int32_t &out_i, int32_t &out_j, float64_t &gap)
{
    // return i,j such that y_i = y_j and
    // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
    // j: minimizes the decrease of obj value
    //    (if quadratic coefficient <= 0, replace it with tau)
    //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)

    int32_t retval=0;
    float64_t best_gap=0;
    int32_t best_out_i=-1;
    int32_t best_out_j=-1;

    float64_t* Gmaxp = SG_MALLOC(float64_t, nr_class);
    float64_t* Gmaxp2 = SG_MALLOC(float64_t, nr_class);
    int32_t* Gmaxp_idx = SG_MALLOC(int32_t, nr_class);

    int32_t* Gmin_idx = SG_MALLOC(int32_t, nr_class);
    float64_t* obj_diff_min = SG_MALLOC(float64_t, nr_class);

    for (int32_t i=0; i<nr_class; i++)
    {
        Gmaxp[i]=-INF;
        Gmaxp2[i]=-INF;
        Gmaxp_idx[i]=-1;
        Gmin_idx[i]=-1;
        obj_diff_min[i]=INF;
    }

    for(int32_t t=0;t<active_size;t++)
    {
        int32_t cidx=y[t];
        if(!is_upper_bound(t))
        {
            if(-G[t] >= Gmaxp[cidx])
            {
                Gmaxp[cidx] = -G[t];
                Gmaxp_idx[cidx] = t;
            }
        }
    }

    for(int32_t j=0;j<active_size;j++)
    {
        int32_t cidx=y[j];
        int32_t ip = Gmaxp_idx[cidx];
        const Qfloat *Q_ip = NULL;
        if(ip != -1) // NULL Q_ip not accessed: Gmaxp=-INF if ip=-1
            Q_ip = Q->get_Q(ip,active_size);

        if (!is_lower_bound(j))
        {
            float64_t grad_diff=Gmaxp[cidx]+G[j];
            if (G[j] >= Gmaxp2[cidx])
                Gmaxp2[cidx] = G[j];
            if (grad_diff > 0)
            {
                float64_t obj_diff;
                float64_t quad_coef = Q_ip[ip]+QD[j]-2*Q_ip[j];
                if (quad_coef > 0)
                    obj_diff = -(grad_diff*grad_diff)/quad_coef;
                else
                    obj_diff = -(grad_diff*grad_diff)/TAU;

                if (obj_diff <= obj_diff_min[cidx])
                {
                    Gmin_idx[cidx]=j;
                    obj_diff_min[cidx] = obj_diff;
                }
            }
        }

        gap=Gmaxp[cidx]+Gmaxp2[cidx];
        if (gap>=best_gap && Gmin_idx[cidx]>=0 &&
                Gmaxp_idx[cidx]>=0 && Gmin_idx[cidx]<active_size)
        {
            out_i = Gmaxp_idx[cidx];
            out_j = Gmin_idx[cidx];

            best_gap=gap;
            best_out_i=out_i;
            best_out_j=out_j;
        }
    }

    gap=best_gap;
    out_i=best_out_i;
    out_j=best_out_j;

    SG_SDEBUG("i=%d j=%d best_gap=%f y_i=%f y_j=%f\n", out_i, out_j, gap, y[out_i], y[out_j]);


    if(gap < eps)
        retval=1;

    SG_FREE(Gmaxp);
    SG_FREE(Gmaxp2);
    SG_FREE(Gmaxp_idx);
    SG_FREE(Gmin_idx);
    SG_FREE(obj_diff_min);

    return retval;
}

bool Solver_NUMC::be_shrunk(
    int32_t i, float64_t Gmax1, float64_t Gmax2, float64_t Gmax3,
    float64_t Gmax4)
{
    return false;
}

void Solver_NUMC::do_shrinking()
{
}

float64_t Solver_NUMC::calculate_rho()
{
    return 0;
}


//
// Q matrices for various formulations
//
class SVC_Q: public LibSVMKernel
{
public:
    SVC_Q(const svm_problem& prob, const svm_parameter& param, const schar *y_)
    :LibSVMKernel(prob.l, prob.x, param)
    {
        clone(y,y_,prob.l);
        cache = new Cache(prob.l,(int64_t)(param.cache_size*(1l<<20)));
        QD = SG_MALLOC(Qfloat, prob.l);
        for(int32_t i=0;i<prob.l;i++)
            QD[i]= (Qfloat)kernel_function(i,i);
    }

    Qfloat *get_Q(int32_t i, int32_t len) const
    {
        Qfloat *data;
        int32_t start;
        if((start = cache->get_data(i,&data,len)) < len)
            compute_Q_parallel(data, y, i, start, len);

        return data;
    }

    Qfloat *get_QD() const
    {
        return QD;
    }

    void swap_index(int32_t i, int32_t j) const
    {
        cache->swap_index(i,j);
        LibSVMKernel::swap_index(i,j);
        CMath::swap(y[i],y[j]);
        CMath::swap(QD[i],QD[j]);
    }

    ~SVC_Q()
    {
        SG_FREE(y);
        delete cache;
        SG_FREE(QD);
    }
private:
    schar *y;
    Cache *cache;
    Qfloat *QD;
};


class ONE_CLASS_Q: public LibSVMKernel
{
public:
    ONE_CLASS_Q(const svm_problem& prob, const svm_parameter& param)
    :LibSVMKernel(prob.l, prob.x, param)
    {
        cache = new Cache(prob.l,(int64_t)(param.cache_size*(1l<<20)));
        QD = SG_MALLOC(Qfloat, prob.l);
        for(int32_t i=0;i<prob.l;i++)
            QD[i]= (Qfloat)kernel_function(i,i);
    }

    Qfloat *get_Q(int32_t i, int32_t len) const
    {
        Qfloat *data;
        int32_t start;
        if((start = cache->get_data(i,&data,len)) < len)
            compute_Q_parallel(data, NULL, i, start, len);

        return data;
    }

    Qfloat *get_QD() const
    {
        return QD;
    }

    void swap_index(int32_t i, int32_t j) const
    {
        cache->swap_index(i,j);
        LibSVMKernel::swap_index(i,j);
        CMath::swap(QD[i],QD[j]);
    }

    ~ONE_CLASS_Q()
    {
        delete cache;
        SG_FREE(QD);
    }
private:
    Cache *cache;
    Qfloat *QD;
};

class SVR_Q: public LibSVMKernel
{
public:
    SVR_Q(const svm_problem& prob, const svm_parameter& param)
    :LibSVMKernel(prob.l, prob.x, param)
    {
        l = prob.l;
        cache = new Cache(l,(int64_t)(param.cache_size*(1l<<20)));
        QD = SG_MALLOC(Qfloat, 2*l);
        sign = SG_MALLOC(schar, 2*l);
        index = SG_MALLOC(int32_t, 2*l);
        for(int32_t k=0;k<l;k++)
        {
            sign[k] = 1;
            sign[k+l] = -1;
            index[k] = k;
            index[k+l] = k;
            QD[k]= (Qfloat)kernel_function(k,k);
            QD[k+l]=QD[k];
        }
        buffer[0] = SG_MALLOC(Qfloat, 2*l);
        buffer[1] = SG_MALLOC(Qfloat, 2*l);
        next_buffer = 0;
    }

    void swap_index(int32_t i, int32_t j) const
    {
        CMath::swap(sign[i],sign[j]);
        CMath::swap(index[i],index[j]);
        CMath::swap(QD[i],QD[j]);
    }

    Qfloat *get_Q(int32_t i, int32_t len) const
    {
        Qfloat *data;
        int32_t real_i = index[i];
        if(cache->get_data(real_i,&data,l) < l)
            compute_Q_parallel(data, NULL, real_i, 0, l);

        // reorder and copy
        Qfloat *buf = buffer[next_buffer];
        next_buffer = 1 - next_buffer;
        schar si = sign[i];
        for(int32_t j=0;j<len;j++)
            buf[j] = si * sign[j] * data[index[j]];
        return buf;
    }

    Qfloat *get_QD() const
    {
        return QD;
    }

    ~SVR_Q()
    {
        delete cache;
        SG_FREE(sign);
        SG_FREE(index);
        SG_FREE(buffer[0]);
        SG_FREE(buffer[1]);
        SG_FREE(QD);
    }

private:
    int32_t l;
    Cache *cache;
    schar *sign;
    int32_t *index;
    mutable int32_t next_buffer;
    Qfloat *buffer[2];
    Qfloat *QD;
};

//
// construct and solve various formulations
//
static void solve_c_svc(
    const svm_problem *prob, const svm_parameter* param,
    float64_t *alpha, Solver::SolutionInfo* si, float64_t Cp, float64_t Cn)
{
    int32_t l = prob->l;
    schar *y = SG_MALLOC(schar, l);

    int32_t i;

    for(i=0;i<l;i++)
    {
        alpha[i] = 0;
        if(prob->y[i] > 0) y[i] = +1; else y[i]=-1;
    }

    Solver s;
    s.Solve(l, SVC_Q(*prob,*param,y), prob->pv, y,
        alpha, Cp, Cn, param->eps, si, param->shrinking, param->use_bias);

    float64_t sum_alpha=0;
    for(i=0;i<l;i++)
        sum_alpha += alpha[i];

    if (Cp==Cn)
        SG_SINFO("nu = %f\n", sum_alpha/(param->C*prob->l));

    for(i=0;i<l;i++)
        alpha[i] *= y[i];

    SG_FREE(y);
}


//two weighted datasets
void solve_c_svc_weighted(
    const svm_problem *prob, const svm_parameter* param,
    float64_t *alpha, Solver::SolutionInfo* si, float64_t Cp, float64_t Cn)
{
    int l = prob->l;
    float64_t *minus_ones = SG_MALLOC(float64_t, l);
    schar *y = SG_MALLOC(schar, l);

    int i;

    for(i=0;i<l;i++)
    {
        alpha[i] = 0;
        minus_ones[i] = -1;
        if(prob->y[i] > 0) y[i] = +1; else y[i]=-1;
    }

    WeightedSolver s = WeightedSolver(prob->C);
    s.Solve(l, SVC_Q(*prob,*param,y), minus_ones, y,
        alpha, Cp, Cn, param->eps, si, param->shrinking, param->use_bias);

    float64_t sum_alpha=0;
    for(i=0;i<l;i++)
        sum_alpha += alpha[i];

    //if (Cp==Cn)
    //  SG_SINFO("nu = %f\n", sum_alpha/(prob->C*prob->l));

    for(i=0;i<l;i++)
        alpha[i] *= y[i];

    SG_FREE(minus_ones);
    SG_FREE(y);
}

static void solve_nu_svc(
    const svm_problem *prob, const svm_parameter *param,
    float64_t *alpha, Solver::SolutionInfo* si)
{
    int32_t i;
    int32_t l = prob->l;
    float64_t nu = param->nu;

    schar *y = SG_MALLOC(schar, l);

    for(i=0;i<l;i++)
        if(prob->y[i]>0)
            y[i] = +1;
        else
            y[i] = -1;

    float64_t sum_pos = nu*l/2;
    float64_t sum_neg = nu*l/2;

    for(i=0;i<l;i++)
        if(y[i] == +1)
        {
            alpha[i] = CMath::min(1.0,sum_pos);
            sum_pos -= alpha[i];
        }
        else
        {
            alpha[i] = CMath::min(1.0,sum_neg);
            sum_neg -= alpha[i];
        }

    float64_t *zeros = SG_MALLOC(float64_t, l);

    for(i=0;i<l;i++)
        zeros[i] = 0;

    Solver_NU s;
    s.Solve(l, SVC_Q(*prob,*param,y), zeros, y,
        alpha, 1.0, 1.0, param->eps, si,  param->shrinking, param->use_bias);
    float64_t r = si->r;

    SG_SINFO("C = %f\n",1/r);

    for(i=0;i<l;i++)
        alpha[i] *= y[i]/r;

    si->rho /= r;
    si->obj /= (r*r);
    si->upper_bound_p = 1/r;
    si->upper_bound_n = 1/r;

    SG_FREE(y);
    SG_FREE(zeros);
}

static void solve_nu_multiclass_svc(const svm_problem *prob,
        const svm_parameter *param, Solver::SolutionInfo* si, svm_model* model)
{
    int32_t l = prob->l;
    float64_t nu = param->nu;

    float64_t *alpha = SG_MALLOC(float64_t, prob->l);
    schar *y = SG_MALLOC(schar, l);

    for(int32_t i=0;i<l;i++)
    {
        alpha[i] = 0;
        y[i]=prob->y[i];
    }

    int32_t nr_class=param->nr_class;
    float64_t* sum_class = SG_MALLOC(float64_t, nr_class);

    for (int32_t j=0; j<nr_class; j++)
        sum_class[j] = nu*l/nr_class;

    for(int32_t i=0;i<l;i++)
    {
        alpha[i] = CMath::min(1.0,sum_class[int32_t(y[i])]);
        sum_class[int32_t(y[i])] -= alpha[i];
    }
    SG_FREE(sum_class);


    float64_t *zeros = SG_MALLOC(float64_t, l);

    for (int32_t i=0;i<l;i++)
        zeros[i] = 0;

    Solver_NUMC s(nr_class, nu);
    SVC_QMC Q(*prob,*param,y, nr_class, ((float64_t) nr_class)/CMath::sq(nu*l));

    s.Solve(l, Q, zeros, y,
        alpha, 1.0, 1.0, param->eps, si,  param->shrinking, param->use_bias);


    int32_t* class_sv_count=SG_MALLOC(int32_t, nr_class);

    for (int32_t i=0; i<nr_class; i++)
        class_sv_count[i]=0;

    for (int32_t i=0; i<l; i++)
    {
        if (CMath::abs(alpha[i]) > 0)
            class_sv_count[(int32_t) y[i]]++;
    }

    model->l=l;
    // rho[0]= rho in mcsvm paper, rho[1]...rho[nr_class] is bias in mcsvm paper
    model->rho = SG_MALLOC(float64_t, nr_class+1);
    model->nr_class = nr_class;
    model->label = NULL;
    model->SV = SG_MALLOC(svm_node*,nr_class);
    model->nSV = SG_MALLOC(int32_t, nr_class);
    model->sv_coef = SG_MALLOC(float64_t *,nr_class);
    model->normwcw = SG_MALLOC(float64_t,nr_class);

    for (int32_t i=0; i<nr_class+1; i++)
        model->rho[i]=0;

    model->objective = si->obj;

    if (param->use_bias)
    {
        SG_SDEBUG("Computing biases and primal objective\n");
        float64_t primal = s.compute_primal(y, alpha, model->rho, model->normwcw);
        SG_SINFO("Primal = %10.10f\n", primal);
    }

    for (int32_t i=0; i<nr_class; i++)
    {
        model->nSV[i]=class_sv_count[i];
        model->SV[i] = SG_MALLOC(svm_node,class_sv_count[i]);
        model->sv_coef[i] = SG_MALLOC(float64_t,class_sv_count[i]);
        class_sv_count[i]=0;
    }

    for (int32_t i=0;i<prob->l;i++)
    {
        if(fabs(alpha[i]) > 0)
        {
            model->SV[(int32_t) y[i]][class_sv_count[(int32_t) y[i]]].index = prob->x[i]->index;
            model->sv_coef[(int32_t) y[i]][class_sv_count[(int32_t) y[i]]] = alpha[i];
            class_sv_count[(int32_t) y[i]]++;
        }
    }

    SG_FREE(y);
    SG_FREE(zeros);
    SG_FREE(alpha);
}

static void solve_one_class(
    const svm_problem *prob, const svm_parameter *param,
    float64_t *alpha, Solver::SolutionInfo* si)
{
    int32_t l = prob->l;
    float64_t *zeros = SG_MALLOC(float64_t, l);
    schar *ones = SG_MALLOC(schar, l);
    int32_t i;

    int32_t n = (int32_t)(param->nu*prob->l);   // # of alpha's at upper bound

    for(i=0;i<n;i++)
        alpha[i] = 1;
    if(n<prob->l)
        alpha[n] = param->nu * prob->l - n;
    for(i=n+1;i<l;i++)
        alpha[i] = 0;

    for(i=0;i<l;i++)
    {
        zeros[i] = 0;
        ones[i] = 1;
    }

    Solver s;
    s.Solve(l, ONE_CLASS_Q(*prob,*param), zeros, ones,
        alpha, 1.0, 1.0, param->eps, si, param->shrinking, param->use_bias);

    SG_FREE(zeros);
    SG_FREE(ones);
}

static void solve_epsilon_svr(
    const svm_problem *prob, const svm_parameter *param,
    float64_t *alpha, Solver::SolutionInfo* si)
{
    int32_t l = prob->l;
    float64_t *alpha2 = SG_MALLOC(float64_t, 2*l);
    float64_t *linear_term = SG_MALLOC(float64_t, 2*l);
    schar *y = SG_MALLOC(schar, 2*l);
    int32_t i;

    for(i=0;i<l;i++)
    {
        alpha2[i] = 0;
        linear_term[i] = param->p - prob->y[i];
        y[i] = 1;

        alpha2[i+l] = 0;
        linear_term[i+l] = param->p + prob->y[i];
        y[i+l] = -1;
    }

    Solver s;
    s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y,
        alpha2, param->C, param->C, param->eps, si, param->shrinking, param->use_bias);

    float64_t sum_alpha = 0;
    for(i=0;i<l;i++)
    {
        alpha[i] = alpha2[i] - alpha2[i+l];
        sum_alpha += fabs(alpha[i]);
    }
    SG_SINFO("nu = %f\n",sum_alpha/(param->C*l));

    SG_FREE(alpha2);
    SG_FREE(linear_term);
    SG_FREE(y);
}

static void solve_nu_svr(
    const svm_problem *prob, const svm_parameter *param,
    float64_t *alpha, Solver::SolutionInfo* si)
{
    int32_t l = prob->l;
    float64_t C = param->C;
    float64_t *alpha2 = SG_MALLOC(float64_t, 2*l);
    float64_t *linear_term = SG_MALLOC(float64_t, 2*l);
    schar *y = SG_MALLOC(schar, 2*l);
    int32_t i;

    float64_t sum = C * param->nu * l / 2;
    for(i=0;i<l;i++)
    {
        alpha2[i] = alpha2[i+l] = CMath::min(sum,C);
        sum -= alpha2[i];

        linear_term[i] = - prob->y[i];
        y[i] = 1;

        linear_term[i+l] = prob->y[i];
        y[i+l] = -1;
    }

    Solver_NU s;
    s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y,
        alpha2, C, C, param->eps, si, param->shrinking, param->use_bias);

    SG_SINFO("epsilon = %f\n",-si->r);

    for(i=0;i<l;i++)
        alpha[i] = alpha2[i] - alpha2[i+l];

    SG_FREE(alpha2);
    SG_FREE(linear_term);
    SG_FREE(y);
}

//
// decision_function
//
struct decision_function
{
    float64_t *alpha;
    float64_t rho;
    float64_t objective;
};

decision_function svm_train_one(
    const svm_problem *prob, const svm_parameter *param,
    float64_t Cp, float64_t Cn)
{
    float64_t *alpha = SG_MALLOC(float64_t, prob->l);
    Solver::SolutionInfo si;
    switch(param->svm_type)
    {
        case C_SVC:
            solve_c_svc(prob,param,alpha,&si,Cp,Cn);
            break;
        case NU_SVC:
            solve_nu_svc(prob,param,alpha,&si);
            break;
        case ONE_CLASS:
            solve_one_class(prob,param,alpha,&si);
            break;
        case EPSILON_SVR:
            solve_epsilon_svr(prob,param,alpha,&si);
            break;
        case NU_SVR:
            solve_nu_svr(prob,param,alpha,&si);
            break;
    }

    SG_SINFO("obj = %.16f, rho = %.16f\n",si.obj,si.rho);

    // output SVs
    if (param->svm_type != ONE_CLASS)
    {
        int32_t nSV = 0;
        int32_t nBSV = 0;
        for(int32_t i=0;i<prob->l;i++)
        {
            if(fabs(alpha[i]) > 0)
            {
                ++nSV;
                if(prob->y[i] > 0)
                {
                    if(fabs(alpha[i]) >= si.upper_bound_p)
                        ++nBSV;
                }
                else
                {
                    if(fabs(alpha[i]) >= si.upper_bound_n)
                        ++nBSV;
                }
            }
        }
        SG_SINFO("nSV = %d, nBSV = %d\n",nSV,nBSV);
    }

    decision_function f;
    f.alpha = alpha;
    f.rho = si.rho;
    f.objective=si.obj;
    return f;
}

//
// label: label name, start: begin of each class, count: #data of classes, perm: indices to the original data
// perm, length l, must be allocated before calling this subroutine
void svm_group_classes(
    const svm_problem *prob, int32_t *nr_class_ret, int32_t **label_ret,
    int32_t **start_ret, int32_t **count_ret, int32_t *perm)
{
    int32_t l = prob->l;
    int32_t max_nr_class = 16;
    int32_t nr_class = 0;
    int32_t *label = SG_MALLOC(int32_t, max_nr_class);
    int32_t *count = SG_MALLOC(int32_t, max_nr_class);
    int32_t *data_label = SG_MALLOC(int32_t, l);
    int32_t i;

    for(i=0;i<l;i++)
    {
        int32_t this_label=(int32_t) prob->y[i];
        int32_t j;
        for(j=0;j<nr_class;j++)
        {
            if(this_label == label[j])
            {
                ++count[j];
                break;
            }
        }
        data_label[i] = j;
        if(j == nr_class)
        {
            if(nr_class == max_nr_class)
            {
                max_nr_class *= 2;
                label=SG_REALLOC(int32_t, label,max_nr_class);
                count=SG_REALLOC(int32_t, count,max_nr_class);
            }
            label[nr_class] = this_label;
            count[nr_class] = 1;
            ++nr_class;
        }
    }

    int32_t *start = SG_MALLOC(int32_t, nr_class);
    start[0] = 0;
    for(i=1;i<nr_class;i++)
        start[i] = start[i-1]+count[i-1];
    for(i=0;i<l;i++)
    {
        perm[start[data_label[i]]] = i;
        ++start[data_label[i]];
    }
    start[0] = 0;
    for(i=1;i<nr_class;i++)
        start[i] = start[i-1]+count[i-1];

    *nr_class_ret = nr_class;
    *label_ret = label;
    *start_ret = start;
    *count_ret = count;
    SG_FREE(data_label);
}

//
// Interface functions
//
svm_model *svm_train(const svm_problem *prob, const svm_parameter *param)
{
    svm_model *model = SG_MALLOC(svm_model,1);
    model->param = *param;
    model->free_sv = 0; // XXX

    if(param->svm_type == ONE_CLASS ||
       param->svm_type == EPSILON_SVR ||
       param->svm_type == NU_SVR)
    {
        SG_SINFO("training one class svm or doing epsilon sv regression\n");

        // regression or one-class-svm
        model->nr_class = 2;
        model->label = NULL;
        model->nSV = NULL;
        model->sv_coef = SG_MALLOC(float64_t *,1);
        decision_function f = svm_train_one(prob,param,0,0);
        model->rho = SG_MALLOC(float64_t, 1);
        model->rho[0] = f.rho;
        model->objective = f.objective;

        int32_t nSV = 0;
        int32_t i;
        for(i=0;i<prob->l;i++)
            if(fabs(f.alpha[i]) > 0) ++nSV;
        model->l = nSV;
        model->SV = SG_MALLOC(svm_node *,nSV);
        model->sv_coef[0] = SG_MALLOC(float64_t, nSV);
        int32_t j = 0;
        for(i=0;i<prob->l;i++)
            if(fabs(f.alpha[i]) > 0)
            {
                model->SV[j] = prob->x[i];
                model->sv_coef[0][j] = f.alpha[i];
                ++j;
            }

        SG_FREE(f.alpha);
    }
    else if(param->svm_type == NU_MULTICLASS_SVC)
    {
        Solver::SolutionInfo si;
        solve_nu_multiclass_svc(prob,param,&si,model);
        SG_SINFO("obj = %.16f, rho = %.16f\n",si.obj,si.rho);
    }
    else
    {
        // classification
        int32_t l = prob->l;
        int32_t nr_class;
        int32_t *label = NULL;
        int32_t *start = NULL;
        int32_t *count = NULL;
        int32_t *perm = SG_MALLOC(int32_t, l);

        // group training data of the same class
        svm_group_classes(prob,&nr_class,&label,&start,&count,perm);
        svm_node **x = SG_MALLOC(svm_node *,l);
        float64_t *C = SG_MALLOC(float64_t,l);
        float64_t *pv = SG_MALLOC(float64_t,l);


        int32_t i;
        for(i=0;i<l;i++) {
            x[i] = prob->x[perm[i]];
            C[i] = prob->C[perm[i]];

            if (prob->pv)
            {
                pv[i] = prob->pv[perm[i]];
            }
            else
            {
                //no custom linear term is set
                pv[i] = -1.0;
            }

        }


        // calculate weighted C
        float64_t *weighted_C = SG_MALLOC(float64_t, nr_class);
        for(i=0;i<nr_class;i++)
            weighted_C[i] = param->C;
        for(i=0;i<param->nr_weight;i++)
        {
            int32_t j;
            for(j=0;j<nr_class;j++)
                if(param->weight_label[i] == label[j])
                    break;
            if(j == nr_class)
                SG_SWARNING("warning: class label %d specified in weight is not found\n", param->weight_label[i]);
            else
                weighted_C[j] *= param->weight[i];
        }

        // train k*(k-1)/2 models

        bool *nonzero = SG_MALLOC(bool,l);
        for(i=0;i<l;i++)
            nonzero[i] = false;
        decision_function *f = SG_MALLOC(decision_function,nr_class*(nr_class-1)/2);

        int32_t p = 0;
        for(i=0;i<nr_class;i++)
            for(int32_t j=i+1;j<nr_class;j++)
            {
                svm_problem sub_prob;
                int32_t si = start[i], sj = start[j];
                int32_t ci = count[i], cj = count[j];
                sub_prob.l = ci+cj;
                sub_prob.x = SG_MALLOC(svm_node *,sub_prob.l);
                sub_prob.y = SG_MALLOC(float64_t,sub_prob.l+1); //dirty hack to surpress valgrind err
                sub_prob.C = SG_MALLOC(float64_t,sub_prob.l+1);
                sub_prob.pv = SG_MALLOC(float64_t,sub_prob.l+1);

                int32_t k;
                for(k=0;k<ci;k++)
                {
                    sub_prob.x[k] = x[si+k];
                    sub_prob.y[k] = +1;
                    sub_prob.C[k] = C[si+k];
                    sub_prob.pv[k] = pv[si+k];

                }
                for(k=0;k<cj;k++)
                {
                    sub_prob.x[ci+k] = x[sj+k];
                    sub_prob.y[ci+k] = -1;
                    sub_prob.C[ci+k] = C[sj+k];
                    sub_prob.pv[ci+k] = pv[sj+k];
                }
                sub_prob.y[sub_prob.l]=-1; //dirty hack to surpress valgrind err
                sub_prob.C[sub_prob.l]=-1;
                sub_prob.pv[sub_prob.l]=-1;

                f[p] = svm_train_one(&sub_prob,param,weighted_C[i],weighted_C[j]);
                for(k=0;k<ci;k++)
                    if(!nonzero[si+k] && fabs(f[p].alpha[k]) > 0)
                        nonzero[si+k] = true;
                for(k=0;k<cj;k++)
                    if(!nonzero[sj+k] && fabs(f[p].alpha[ci+k]) > 0)
                        nonzero[sj+k] = true;
                SG_FREE(sub_prob.x);
                SG_FREE(sub_prob.y);
                SG_FREE(sub_prob.C);
                SG_FREE(sub_prob.pv);
                ++p;
            }

        // build output

        model->objective = f[0].objective;
        model->nr_class = nr_class;

        model->label = SG_MALLOC(int32_t, nr_class);
        for(i=0;i<nr_class;i++)
            model->label[i] = label[i];

        model->rho = SG_MALLOC(float64_t, nr_class*(nr_class-1)/2);
        for(i=0;i<nr_class*(nr_class-1)/2;i++)
            model->rho[i] = f[i].rho;

        int32_t total_sv = 0;
        int32_t *nz_count = SG_MALLOC(int32_t, nr_class);
        model->nSV = SG_MALLOC(int32_t, nr_class);
        for(i=0;i<nr_class;i++)
        {
            int32_t nSV = 0;
            for(int32_t j=0;j<count[i];j++)
                if(nonzero[start[i]+j])
                {
                    ++nSV;
                    ++total_sv;
                }
            model->nSV[i] = nSV;
            nz_count[i] = nSV;
        }

        SG_SINFO("Total nSV = %d\n",total_sv);

        model->l = total_sv;
        model->SV = SG_MALLOC(svm_node *,total_sv);
        p = 0;
        for(i=0;i<l;i++)
            if(nonzero[i]) model->SV[p++] = x[i];

        int32_t *nz_start = SG_MALLOC(int32_t, nr_class);
        nz_start[0] = 0;
        for(i=1;i<nr_class;i++)
            nz_start[i] = nz_start[i-1]+nz_count[i-1];

        model->sv_coef = SG_MALLOC(float64_t *,nr_class-1);
        for(i=0;i<nr_class-1;i++)
            model->sv_coef[i] = SG_MALLOC(float64_t, total_sv);

        p = 0;
        for(i=0;i<nr_class;i++)
            for(int32_t j=i+1;j<nr_class;j++)
            {
                // classifier (i,j): coefficients with
                // i are in sv_coef[j-1][nz_start[i]...],
                // j are in sv_coef[i][nz_start[j]...]

                int32_t si = start[i];
                int32_t sj = start[j];
                int32_t ci = count[i];
                int32_t cj = count[j];

                int32_t q = nz_start[i];
                int32_t k;
                for(k=0;k<ci;k++)
                    if(nonzero[si+k])
                        model->sv_coef[j-1][q++] = f[p].alpha[k];
                q = nz_start[j];
                for(k=0;k<cj;k++)
                    if(nonzero[sj+k])
                        model->sv_coef[i][q++] = f[p].alpha[ci+k];
                ++p;
            }

        SG_FREE(label);
        SG_FREE(count);
        SG_FREE(perm);
        SG_FREE(start);
        SG_FREE(x);
        SG_FREE(C);
        SG_FREE(pv);
        SG_FREE(weighted_C);
        SG_FREE(nonzero);
        for(i=0;i<nr_class*(nr_class-1)/2;i++)
            SG_FREE(f[i].alpha);
        SG_FREE(f);
        SG_FREE(nz_count);
        SG_FREE(nz_start);
    }
    return model;
}

void svm_destroy_model(svm_model* model)
{
    if(model->free_sv && model->l > 0)
        SG_FREE((void *)(model->SV[0]));
    for(int32_t i=0;i<model->nr_class-1;i++)
        SG_FREE(model->sv_coef[i]);
    SG_FREE(model->SV);
    SG_FREE(model->sv_coef);
    SG_FREE(model->rho);
    SG_FREE(model->label);
    SG_FREE(model->nSV);
    SG_FREE(model);
}

void svm_destroy_param(svm_parameter* param)
{
    SG_FREE(param->weight_label);
    SG_FREE(param->weight);
}

const char *svm_check_parameter(
    const svm_problem *prob, const svm_parameter *param)
{
    // svm_type

    int32_t svm_type = param->svm_type;
    if(svm_type != C_SVC &&
       svm_type != NU_SVC &&
       svm_type != ONE_CLASS &&
       svm_type != EPSILON_SVR &&
       svm_type != NU_SVR &&
       svm_type != NU_MULTICLASS_SVC)
        return "unknown svm type";

    // kernel_type, degree

    int32_t kernel_type = param->kernel_type;
    if(kernel_type != LINEAR &&
       kernel_type != POLY &&
       kernel_type != RBF &&
       kernel_type != SIGMOID &&
       kernel_type != PRECOMPUTED)
        return "unknown kernel type";

    if(param->degree < 0)
        return "degree of polynomial kernel < 0";

    // cache_size,eps,C,nu,p,shrinking

    if(param->cache_size <= 0)
        return "cache_size <= 0";

    if(param->eps <= 0)
        return "eps <= 0";

    if(svm_type == C_SVC ||
       svm_type == EPSILON_SVR ||
       svm_type == NU_SVR)
        if(param->C <= 0)
            return "C <= 0";

    if(svm_type == NU_SVC ||
       svm_type == ONE_CLASS ||
       svm_type == NU_SVR)
        if(param->nu <= 0 || param->nu > 1)
            return "nu <= 0 or nu > 1";

    if(svm_type == EPSILON_SVR)
        if(param->p < 0)
            return "p < 0";

    if(param->shrinking != 0 &&
       param->shrinking != 1)
        return "shrinking != 0 and shrinking != 1";


    // check whether nu-svc is feasible

    if(svm_type == NU_SVC)
    {
        int32_t l = prob->l;
        int32_t max_nr_class = 16;
        int32_t nr_class = 0;
        int32_t *label = SG_MALLOC(int32_t, max_nr_class);
        int32_t *count = SG_MALLOC(int32_t, max_nr_class);

        int32_t i;
        for(i=0;i<l;i++)
        {
            int32_t this_label = (int32_t) prob->y[i];
            int32_t j;
            for(j=0;j<nr_class;j++)
                if(this_label == label[j])
                {
                    ++count[j];
                    break;
                }
            if(j == nr_class)
            {
                if(nr_class == max_nr_class)
                {
                    max_nr_class *= 2;
                    label=SG_REALLOC(int32_t, label, max_nr_class);
                    count=SG_REALLOC(int32_t, count, max_nr_class);
                }
                label[nr_class] = this_label;
                count[nr_class] = 1;
                ++nr_class;
            }
        }

        for(i=0;i<nr_class;i++)
        {
            int32_t n1 = count[i];
            for(int32_t j=i+1;j<nr_class;j++)
            {
                int32_t n2 = count[j];
                if(param->nu*(n1+n2)/2 > CMath::min(n1,n2))
                {
                    SG_FREE(label);
                    SG_FREE(count);
                    return "specified nu is infeasible";
                }
            }
        }
        SG_FREE(label);
        SG_FREE(count);
    }

    return NULL;
}
}
#endif // DOXYGEN_SHOULD_SKIP_THIS