space.c 64.5 KB
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/*******************************************************************************
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 * This file is part of SWIFT.
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 * Coypright (c) 2012 Pedro Gonnet (pedro.gonnet@durham.ac.uk)
 * 
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU Lesser General Public License as published
 * by the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 * 
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 * 
 * You should have received a copy of the GNU Lesser General Public License
 * along with this program.  If not, see <http://www.gnu.org/licenses/>.
 * 
 ******************************************************************************/

/* Config parameters. */
#include "../config.h"

/* Some standard headers. */
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <pthread.h>
#include <float.h>
#include <limits.h>
#include <math.h>

/* Local headers. */
#include "cycle.h"
#include "lock.h"
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#include "task.h"
#include "part.h"
#include "cell.h"
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#include "space.h"
#include "runner.h"

/* Error macro. */
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#define error(s) { fprintf( stderr , "%s:%s:%i: %s\n" , __FILE__ , __FUNCTION__ , __LINE__ , s ); abort(); }
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/* Split size. */
int space_splitsize = space_splitsize_default;
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int space_subsize = space_subsize_default;
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/* Map shift vector to sortlist. */
const int sortlistID[27] = {
    /* ( -1 , -1 , -1 ) */   0 ,
    /* ( -1 , -1 ,  0 ) */   1 , 
    /* ( -1 , -1 ,  1 ) */   2 ,
    /* ( -1 ,  0 , -1 ) */   3 ,
    /* ( -1 ,  0 ,  0 ) */   4 , 
    /* ( -1 ,  0 ,  1 ) */   5 ,
    /* ( -1 ,  1 , -1 ) */   6 ,
    /* ( -1 ,  1 ,  0 ) */   7 , 
    /* ( -1 ,  1 ,  1 ) */   8 ,
    /* (  0 , -1 , -1 ) */   9 ,
    /* (  0 , -1 ,  0 ) */   10 , 
    /* (  0 , -1 ,  1 ) */   11 ,
    /* (  0 ,  0 , -1 ) */   12 ,
    /* (  0 ,  0 ,  0 ) */   0 , 
    /* (  0 ,  0 ,  1 ) */   12 ,
    /* (  0 ,  1 , -1 ) */   11 ,
    /* (  0 ,  1 ,  0 ) */   10 , 
    /* (  0 ,  1 ,  1 ) */   9 ,
    /* (  1 , -1 , -1 ) */   8 ,
    /* (  1 , -1 ,  0 ) */   7 , 
    /* (  1 , -1 ,  1 ) */   6 ,
    /* (  1 ,  0 , -1 ) */   5 ,
    /* (  1 ,  0 ,  0 ) */   4 , 
    /* (  1 ,  0 ,  1 ) */   3 ,
    /* (  1 ,  1 , -1 ) */   2 ,
    /* (  1 ,  1 ,  0 ) */   1 , 
    /* (  1 ,  1 ,  1 ) */   0 
    };
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/**
 * @brief Mapping function to set dt_min and dt_max.
 */

void space_map_prepare ( struct cell *c , void *data ) {

    int k;
    float dt_min, dt_max, h_max, dx_max;
    struct part *p;
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    struct xpart *xp;
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    /* No children? */
    if ( !c->split ) {
    
        /* Init with first part. */
        p = &c->parts[0];
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        xp = p->xtras;
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        dt_min = p->dt;
        dt_max = p->dt;
        h_max = p->h;
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        dx_max = sqrtf( (p->x[0] - xp->x_old[0])*(p->x[0] - xp->x_old[0]) +
                        (p->x[1] - xp->x_old[1])*(p->x[1] - xp->x_old[1]) +
                        (p->x[2] - xp->x_old[2])*(p->x[2] - xp->x_old[2]) )*2 + p->h;
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        /* Loop over parts. */
        for ( k = 1 ; k < c->count ; k++ ) {
            p = &c->parts[k];
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            xp = p->xtras;
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            dt_min = fminf( dt_min , p->dt );
            dt_max = fmaxf( dt_max , p->dt );
            h_max = fmaxf( h_max , p->h );
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            dx_max = fmaxf( dx_max , sqrtf( (p->x[0] - xp->x_old[0])*(p->x[0] - xp->x_old[0]) +
                                            (p->x[1] - xp->x_old[1])*(p->x[1] - xp->x_old[1]) +
                                            (p->x[2] - xp->x_old[2])*(p->x[2] - xp->x_old[2]) )*2 + p->h );
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            }
            
        }
        
    /* Otherwise, agregate from children. */
    else {
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        /* Init with the first non-null child. */
        for ( k = 0 ; c->progeny[k] == 0 ; k++ );
        dt_min = c->progeny[k]->dt_min;
        dt_max = c->progeny[k]->dt_max;
        h_max = c->progeny[k]->h_max;
        dx_max = c->progeny[k]->dx_max;
        
        /* Loop over the remaining progeny. */
        for ( k += 1 ; k < 8 ; k++ )
            if ( c->progeny[k] != NULL ) {
                dt_min = fminf( dt_min , c->progeny[k]->dt_min );
                dt_max = fmaxf( dt_max , c->progeny[k]->dt_max );
                h_max = fmaxf( h_max , c->progeny[k]->h_max );
                dx_max = fmaxf( dx_max , c->progeny[k]->dx_max );
                }
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        }

    /* Store the values. */
    c->dt_min = dt_min;
    c->dt_max = dt_max;
    c->h_max = h_max;
    c->dx_max = dx_max;
    
    /* Clean out the task pointers. */
    c->sorts[0] = NULL;
    c->nr_tasks = 0;
    c->nr_density = 0;
    
    }


/**
 * @brief Check the integrity of the space and rebuild if necessary.
 *
 * @param s The #space.
 *
 * Runs through the tasks and marks those as "skip" which have no
 * effect for the current @c dt_max. Verifies the integrity of the
 * cell tree for those tasks and triggers a rebuild if necessary.
 */
 
void space_prepare ( struct space *s ) {

    int k;
    struct task *t;
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    float dt_step = s->dt_step, dx_max = 0.0f;
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    int counts[ task_type_count + 1 ];
    
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    /* Traverse the cells and set their dt_min and dx_max. */
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    space_map_cells_post( s , 1 , &space_map_prepare , NULL );
    
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    /* Get the maximum displacement in the whole system. */
    for ( k = 0 ; k < s->nr_cells ; k++ )
        dx_max = fmaxf( dx_max , s->cells[k].dx_max );
    printf( "space_prepare: dx_max is %e.\n" , dx_max );
    
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    /* Run through the tasks and mark as skip or not. */
    for ( k = 0 ; k < s->nr_tasks ; k++ ) {
        t = &s->tasks[k];
        if ( t->type == task_type_sort ||
             t->type == task_type_self ||
             t->type == task_type_ghost ||
             ( t->type == task_type_sub && t->cj == NULL ) )
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            t->skip = ( t->ci->dt_min > dt_step );
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        else if ( t->type == task_type_pair || ( t->type == task_type_sub && t->cj != NULL ) ) {
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            t->skip = ( t->ci->dt_min > dt_step && t->cj->dt_min > dt_step );
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            if ( !t->skip && t->tight &&
                 ( t->ci->dx_max > t->ci->dmin || t->cj->dx_max > t->cj->dmin ) )
                break;
            }
        }
        
    /* Did this not go through? */
    if ( k < s->nr_tasks ) {
    
        /* Re-build the space. */
        space_rebuild( s , 0.0 );
    
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        /* Traverse the cells and set their dt_min and dx_max. */
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        space_map_cells_post( s , 1 , &space_map_prepare , NULL );
    
        }

    /* Store the condensed particle data. */         
    #pragma omp parallel for schedule(static)
    for ( k = 0 ; k < s->nr_parts ; k++ ) {
        s->cparts[k].x[0] = s->parts[k].x[0];
        s->cparts[k].x[1] = s->parts[k].x[1];
        s->cparts[k].x[2] = s->parts[k].x[2];
        s->cparts[k].h = s->parts[k].h;
        s->cparts[k].dt = s->parts[k].dt;
        }

    /* Now that we have the cell structre, re-build the tasks. */
    // tic = getticks();
    space_maketasks( s , 1 );
    // printf( "space_prepare: maketasks took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
    
    /* Count the number of each task type. */
    for ( k = 0 ; k <= task_type_count ; k++ )
        counts[k] = 0;
    for ( k = 0 ; k < s->nr_tasks ; k++ )
        if ( !s->tasks[k].skip )
            counts[ (int)s->tasks[k].type ] += 1;
        else
            counts[ task_type_count ] += 1;
    printf( "space_prepare: task counts are [ %s=%i" , taskID_names[0] , counts[0] );
    for ( k = 1 ; k < task_type_count ; k++ )
        printf( " %s=%i" , taskID_names[k] , counts[k] );
    printf( " skipped=%i ]\n" , counts[ task_type_count ] ); fflush(stdout);
    
    }
    
    
/** 
 * @brief Sort the tasks in topological order over all queues.
 *
 * @param s The #space.
 */
 
void space_ranktasks ( struct space *s ) {

    int i, j = 0, k, temp, left = 0, rank;
    struct task *t;
    int *tid = s->tasks_ind;
    
    /* Run throught the tasks and get all the waits right. */
    for ( k = 0 ; k < s->nr_tasks ; k++ ) {
        tid[k] = k;
        for ( j = 0 ; j < s->tasks[k].nr_unlock_tasks ; j++ )
            s->tasks[k].unlock_tasks[j]->wait += 1;
        }
        
    /* Main loop. */
    for ( j = 0 , rank = 0 ; left < s->nr_tasks ; rank++ ) {
        
        /* Load the tids of tasks with no waits. */
        for ( k = left ; k < s->nr_tasks ; k++ )
            if ( s->tasks[ tid[k] ].wait == 0 ) {
                temp = tid[j]; tid[j] = tid[k]; tid[k] = temp;
                j += 1;
                }
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        /* Did we get anything? */
        if ( j == left )
            error( "Unsatisfiable task dependencies detected." );
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        /* Traverse the task tree and add tasks with no weight. */
        for ( i = left ; i < j ; i++ ) {
            t = &s->tasks[ tid[i] ];
            t->rank = rank;
            s->tasks_ind[i] = t - s->tasks;
            /* printf( "engine_ranktasks: task %i of type %s has rank %i.\n" , i , 
                (t->type == task_type_self) ? "self" : (t->type == task_type_pair) ? "pair" : "sort" , rank ); */
            for ( k = 0 ; k < t->nr_unlock_tasks ; k++ )
                t->unlock_tasks[k]->wait -= 1;
            }
            
        /* The new left (no, not tony). */
        left = j;
            
        }
        
    }


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/**
 * @brief Get the shift-id of the given pair of cells, swapping them
 *      if need be.
 *
 * @param s The space
 * @param ci Pointer to first #cell.
 * @param cj Pointer second #cell.
 * @param shift Vector from ci to cj.
 *
 * @return The shift ID and set shift, may or may not swap ci and cj.
 */
 
int space_getsid ( struct space *s , struct cell **ci , struct cell **cj , double *shift ) {

    int k, sid = 0;
    struct cell *temp;
    double dx[3];

    /* Get the relative distance between the pairs, wrapping. */
    for ( k = 0 ; k < 3 ; k++ ) {
        dx[k] = (*cj)->loc[k] - (*ci)->loc[k];
        if ( dx[k] < -s->dim[k]/2 )
            shift[k] = s->dim[k];
        else if ( dx[k] > s->dim[k]/2 )
            shift[k] = -s->dim[k];
        else
            shift[k] = 0.0;
        dx[k] += shift[k];
        }
        
    /* Get the sorting index. */
    for ( k = 0 ; k < 3 ; k++ )
        sid = 3*sid + ( (dx[k] < 0.0) ? 0 : ( (dx[k] > 0.0) ? 2 : 1 ) );

    /* Switch the cells around? */
    if ( runner_flip[sid] ) {
        temp = *ci; *ci = *cj; *cj = temp;
        for ( k = 0 ; k < 3 ; k++ )
            shift[k] = -shift[k];
        }
    sid = sortlistID[sid];
    
    /* Return the sort ID. */
    return sid;

    }


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/**
 * @breif Recursively dismantle a cell tree.
 *
 */
 
void space_rebuild_recycle ( struct space *s , struct cell *c ) {
    
    int k;
    
    if ( c->split )
        for ( k = 0 ; k < 8 ; k++ )
            if ( c->progeny[k] != NULL ) {
                space_rebuild_recycle( s , c->progeny[k] );
                space_recycle( s , c->progeny[k] );
                c->progeny[k] = NULL;
                }
    
    }

/**
 * @breif Recursively rebuild a cell tree.
 *
 */
 
int space_rebuild_recurse ( struct space *s , struct cell *c ) {
    
    int k, count, changes = 0, wasmt[8];
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    float h, h_limit, h_max = 0.0f, dt_min = c->parts[0].dt, dt_max = dt_min;
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    struct cell *temp;
    
    /* If the cell is already split, check that the split is still ok. */
    if ( c->split ) {
    
        /* Check the depth. */
        if ( c->depth > s->maxdepth )
            s->maxdepth = c->depth;

        /* Set the minimum cutoff. */
        h_limit = fmin( c->h[0] , fmin( c->h[1] , c->h[2] ) ) / 2;

        /* Count the particles below that. */
        for ( count = 0 , k = 0 ; k < c->count ; k++ ) {
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            h = c->parts[k].h;
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            if ( h <= h_limit )
                count += 1;
            if ( h > h_max )
                h_max = h;
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            if ( c->parts[k].dt < dt_min )
                dt_min = c->parts[k].dt;
            if ( c->parts[k].dt > dt_max )
                dt_max = c->parts[k].dt;
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            }
        c->h_max = h_max;
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        c->dt_min = dt_min;
        c->dt_max = dt_max;
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        /* Un-split? */
        if ( count < c->count*space_splitratio || c->count < space_splitsize ) {
        
            /* Get rid of the progeny. */
            space_rebuild_recycle( s , c );
            
            /* Re-set the split flag. */
            c->split = 0;
        
            }
        
        /* Otherwise, recurse on the kids. */
        else {
        
            /* Populate all progeny. */
            for ( k = 0 ; k < 8 ; k++ )
                if ( ( wasmt[k] = ( c->progeny[k] == NULL ) ) ) {
                    temp = space_getcell( s );
                    temp->count = 0;
                    temp->loc[0] = c->loc[0];
                    temp->loc[1] = c->loc[1];
                    temp->loc[2] = c->loc[2];
                    temp->h[0] = c->h[0]/2;
                    temp->h[1] = c->h[1]/2;
                    temp->h[2] = c->h[2]/2;
                    if ( k & 4 )
                        temp->loc[0] += temp->h[0];
                    if ( k & 2 )
                        temp->loc[1] += temp->h[1];
                    if ( k & 1 )
                        temp->loc[2] += temp->h[2];
                    temp->depth = c->depth + 1;
                    temp->split = 0;
                    temp->h_max = 0.0;
                    temp->parent = c;
                    c->progeny[k] = temp;
                    }
        
            /* Make sure each part is in its place. */
            cell_split( c );
            
            /* Remove empty progeny. */
            for ( k = 0 ; k < 8 ; k++ )
                if ( c->progeny[k]->count == 0 ) {
                    changes += !wasmt[k];
                    space_recycle( s , c->progeny[k] );
                    c->progeny[k] = NULL;
                    }
                else
                    changes += wasmt[k];
        
            /* Recurse. */
            for ( k = 0 ; k < 8 ; k++ )
                if ( c->progeny[k] != NULL )
                    changes += space_rebuild_recurse( s , c->progeny[k] );
                    
            }
    
        }
        
    /* Otherwise, try to split it anyway. */
    else {
        space_split( s , c );
        changes += c->split;
        }
        
    /* Return the grand total. */
    return changes;
    
    }

/**
 * @breif Re-build the cells as well as the tasks.
 *
 * @param s The #space in which to update the cells.
 * @param force Flag to force re-building the cells and tasks.
 *
 */
 
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void space_rebuild ( struct space *s , double cell_max ) {
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    float h_max = s->cell_min, h_min = s->parts[0].h, dmin;
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    int i, j, k, cdim[3];
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    struct cell *restrict c;
    struct part *restrict finger, *restrict p;
    struct cpart *restrict cfinger;
    int *ind;
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    // ticks tic;
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    /* Be verbose about this. */
    printf( "space_rebuild: (re)building space...\n" ); fflush(stdout);
    
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    /* Run through the parts and get the current h_max. */
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    // tic = getticks();
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    for ( k = 0 ; k < s->nr_parts ; k++ ) {
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        if ( s->parts[k].h > h_max )
            h_max = s->parts[k].h;
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        else if ( s->parts[k].h < h_min )
            h_min = s->parts[k].h;
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        }
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    s->h_min = h_min;
    s->h_max = h_max;
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    printf( "space_rebuild: h_min/h_max is %.3e/%.3e.\n" , h_min , h_max );
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    // printf( "space_rebuild: getting h_min and h_max took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    /* Get the new putative cell dimensions. */
    for ( k = 0 ; k < 3 ; k++ )
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        cdim[k] = floor( s->dim[k] / fmax( h_max*space_stretch , cell_max ) );
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    /* Do we need to re-build the upper-level cells? */
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    // tic = getticks();
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    if ( s->cells == NULL ||
         cdim[0] < s->cdim[0] || cdim[1] < s->cdim[1] || cdim[2] < s->cdim[2] ) {
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        /* Free the old cells, if they were allocated. */
        if ( s->cells != NULL ) {
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            for ( k = 0 ; k < s->nr_cells ; k++ ) {
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                space_rebuild_recycle( s , &s->cells[k] );
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                if ( s->cells[k].sort != NULL )
                    free( s->cells[k].sort );
                }
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            free( s->cells );
            s->maxdepth = 0;
            }
            
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        /* Set the new cell dimensions only if smaller. */
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        for ( k = 0 ; k < 3 ; k++ ) {
            s->cdim[k] = cdim[k];
            s->h[k] = s->dim[k] / cdim[k];
            s->ih[k] = 1.0 / s->h[k];
            }
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        dmin = fminf( s->h[0] , fminf( s->h[1] , s->h[2] ) );
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        /* Allocate the highest level of cells. */
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        s->tot_cells = s->nr_cells = cdim[0] * cdim[1] * cdim[2];
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        if ( posix_memalign( (void *)&s->cells , 64 , s->nr_cells * sizeof(struct cell) ) != 0 )
            error( "Failed to allocate cells." );
        bzero( s->cells , s->nr_cells * sizeof(struct cell) );
        for ( k = 0 ; k < s->nr_cells ; k++ )
            if ( lock_init( &s->cells[k].lock ) != 0 )
                error( "Failed to init spinlock." );

        /* Set the cell location and sizes. */
        for ( i = 0 ; i < cdim[0] ; i++ )
            for ( j = 0 ; j < cdim[1] ; j++ )
                for ( k = 0 ; k < cdim[2] ; k++ ) {
                    c = &s->cells[ cell_getid( cdim , i , j , k ) ];
                    c->loc[0] = i*s->h[0]; c->loc[1] = j*s->h[1]; c->loc[2] = k*s->h[2];
                    c->h[0] = s->h[0]; c->h[1] = s->h[1]; c->h[2] = s->h[2];
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                    c->dmin = dmin;
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                    c->depth = 0;
                    }
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        /* Be verbose about the change. */         
        printf( "space_rebuild: set cell dimensions to [ %i %i %i ].\n" , cdim[0] , cdim[1] , cdim[2] ); fflush(stdout);
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        } /* re-build upper-level cells? */
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    // printf( "space_rebuild: rebuilding upper-level cells took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    /* Run through the particles and get their cell index. */
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    // tic = getticks();
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    if ( ( ind = (int *)malloc( sizeof(int) * s->nr_parts ) ) == NULL )
        error( "Failed to allocate temporary particle indices." );
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    for ( k = 0 ; k < s->nr_cells ; k++ )
        s->cells[ k ].count = 0;
    for ( k = 0 ; k < s->nr_parts ; k++ )  {
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        p = &s->parts[k];
        for ( j = 0 ; j < 3 ; j++ )
            if ( p->x[j] < 0.0 )
                p->x[j] += s->dim[j];
            else if ( p->x[j] >= s->dim[j] )
                p->x[j] -= s->dim[j];
        ind[k] = cell_getid( s->cdim , p->x[0]*s->ih[0] , p->x[1]*s->ih[1] , p->x[2]*s->ih[2] );
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        s->cells[ ind[k] ].count += 1;
        }
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    // printf( "space_rebuild: getting particle indices took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    /* Sort the parts according to their cells. */
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    // tic = getticks();
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    parts_sort( s->parts , ind , s->nr_parts , 0 , s->nr_cells );
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    // printf( "space_rebuild: parts_sort took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    /* We no longer need the indices as of here. */
    free( ind );    
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    /* Store the current positions. */         
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    // tic = getticks();
    #pragma omp parallel for schedule(static)
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    for ( k = 0 ; k < s->nr_parts ; k++ ) {
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        s->parts[k].xtras->x_old[0] = s->parts[k].x[0];
        s->parts[k].xtras->x_old[1] = s->parts[k].x[1];
        s->parts[k].xtras->x_old[2] = s->parts[k].x[2];
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        }
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    // printf( "space_rebuild: storing old positions took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    /* Hook the cells up to the parts. */
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    // tic = getticks();
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    finger = s->parts;
    cfinger = s->cparts;
    for ( k = 0 ; k < s->nr_cells ; k++ ) {
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        c = &s->cells[ k ];
        c->parts = finger;
        c->cparts = cfinger;
        finger = &finger[ c->count ];
        cfinger = &cfinger[ c->count ];
        }
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    // printf( "space_rebuild: hooking up cells took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    /* At this point, we have the upper-level cells, old or new. Now make
       sure that the parts in each cell are ok. */
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    // tic = getticks();
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    #pragma omp parallel for schedule(dynamic,1) shared(s)
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    for ( k = 0 ; k < s->nr_cells ; k++ )
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        space_rebuild_recurse( s , &s->cells[k] );
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    // printf( "space_rebuild: space_rebuild_recurse took %.3f ms.\n" , (double)(getticks() - tic) / CPU_TPS * 1000 );
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    }


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/**
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 * @brief Sort the particles and condensed particles according to the given indices.
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 *
 * @param parts The list of #part
 * @param ind The indices with respect to which the parts are sorted.
 * @param N The number of parts
 * @param min Lowest index.
 * @param max highest index.
 *
 * This function calls itself recursively.
 */
 
void parts_sort ( struct part *parts , int *ind , int N , int min , int max ) {

    int pivot = (min + max) / 2;
    int i = 0, j = N-1;
    int temp_i;
    struct part temp_p;
    
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    /* If N is small enough, just do insert sort. */
    if ( N < 16 ) {
    
        for ( i = 1 ; i < N ; i++ )
            if ( ind[i] < ind[i-1] ) {
                temp_i = ind[i];
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                temp_p = parts[i];
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                for ( j = i ; j > 0 && ind[j-1] > temp_i ; j-- ) {
                    ind[j] = ind[j-1];
                    parts[j] = parts[j-1];
                    }
                ind[j] = temp_i;
                parts[j] = temp_p;
                }
    
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        }
        
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    /* Otherwise, recurse with Quicksort. */
    else {
    
        /* One pass of quicksort. */
        while ( i < j ) {
            while ( i < N && ind[i] <= pivot )
                i++;
            while ( j >= 0 && ind[j] > pivot )
                j--;
            if ( i < j ) {
                temp_i = ind[i]; ind[i] = ind[j]; ind[j] = temp_i;
                temp_p = parts[i]; parts[i] = parts[j]; parts[j] = temp_p;
                }
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            }
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        /* Verify sort. */
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        /* for ( int k = 0 ; k <= j ; k++ )
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            if ( ind[k] > pivot ) {
                printf( "parts_sort: sorting failed at k=%i, ind[k]=%i, pivot=%i, i=%i, j=%i, N=%i.\n" , k , ind[k] , pivot , i , j , N );
                error( "Sorting failed (<=pivot)." );
                }
        for ( int k = j+1 ; k < N ; k++ )
            if ( ind[k] <= pivot ) {
                printf( "parts_sort: sorting failed at k=%i, ind[k]=%i, pivot=%i, i=%i, j=%i, N=%i.\n" , k , ind[k] , pivot , i , j , N );
                error( "Sorting failed (>pivot)." );
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                } */
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        /* Recurse on the left? */
        if ( j > 0  && pivot > min )
            parts_sort( parts , ind , j+1 , min , pivot );
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        /* Recurse on the right? */
        if ( i < N && pivot+1 < max )
            parts_sort( &parts[i], &ind[i], N-i , pivot+1 , max );
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        }
    
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    }


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/**
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 * @brief Mapping function to free the sorted indices buffers.
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 */

void space_map_clearsort ( struct cell *c , void *data ) {

    if ( c->sort != NULL ) {
        free( c->sort );
        c->sort = NULL;
        }

    }


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/**
 * @brief Mapping function to append a ghost task to each cell.
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 *
 * Looks for the super cell, e.g. the highest-level cell above each
 * cell for which a pair is defined. All ghosts below this cell will
 * depend on the ghost of their parents (sounds spooky, but it isn't).
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 */

void space_map_mkghosts ( struct cell *c , void *data ) {

    struct space *s = (struct space *)data;
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    struct cell *finger;
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    /* Find the super cell, i.e. the highest cell hierarchically above
       this one to still have at least one task associated with it. */
    c->super = c;
    for ( finger = c->parent ; finger != NULL ; finger = finger->parent )
        if ( finger->nr_tasks > 0 )
            c->super = finger;
            
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    /* Make the ghost task */
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    if ( c->super != c || c->nr_tasks > 0 )
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        c->ghost = space_addtask( s , task_type_ghost , task_subtype_none , 0 , 0 , c , NULL , 0 );
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    /* If we are not the super cell ourselves, make our ghost depend
       on our parent cell. */
    if ( c->super != c )
        task_addunlock( c->parent->ghost , c->ghost );
    
    }


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/**
 * @brief Map a function to all particles in a aspace.
 *
 * @param s The #space we are working in.
 */
 
void space_map_parts ( struct space *s , void (*fun)( struct part *p , struct cell *c , void *data ) , void *data ) {

    int i;

    void rec_map ( struct cell *c ) {
    
        int k;
        
        /* No progeny? */
        if ( !c->split )
            for ( k = 0 ; k < c->count ; k++ )
                fun( &c->parts[k] , c , data );
                
        /* Otherwise, recurse. */
        else
            for ( k = 0 ; k < 8 ; k++ )
                if ( c->progeny[k] != NULL )
                    rec_map( c->progeny[k] );
                
        }
        
    /* Call the recursive function on all higher-level cells. */
    for ( i = 0 ; i < s->nr_cells ; i++ )
        rec_map( &s->cells[i] );

    }


/**
 * @brief Map a function to all particles in a aspace.
 *
 * @param s The #space we are working in.
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 * @param full Map to all cells, including cells with sub-cells.
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 */
 
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void space_map_cells_post ( struct space *s , int full , void (*fun)( struct cell *c , void *data ) , void *data ) {

    int i;

    void rec_map ( struct cell *c ) {
    
        int k;
        
        /* Recurse. */
        if ( c->split )
            for ( k = 0 ; k < 8 ; k++ )
                if ( c->progeny[k] != NULL )
                    rec_map( c->progeny[k] );
                
        /* No progeny? */
        if ( full || !c->split )
            fun( c , data );
                
        }
        
    /* Call the recursive function on all higher-level cells. */
    for ( i = 0 ; i < s->nr_cells ; i++ )
        rec_map( &s->cells[i] );

    }


void space_map_cells_pre ( struct space *s , int full , void (*fun)( struct cell *c , void *data ) , void *data ) {
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    int i;

    void rec_map ( struct cell *c ) {
    
        int k;
        
        /* No progeny? */
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        if ( full || !c->split )
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            fun( c , data );
                
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        /* Recurse. */
        if ( c->split )
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            for ( k = 0 ; k < 8 ; k++ )
                if ( c->progeny[k] != NULL )
                    rec_map( c->progeny[k] );
                
        }
        
    /* Call the recursive function on all higher-level cells. */
    for ( i = 0 ; i < s->nr_cells ; i++ )
        rec_map( &s->cells[i] );

    }


/**
 * @brief Add a #task to the #space.
 *
 * @param s The #space we are working in.
 */
 
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struct task *space_addtask ( struct space *s , int type , int subtype , int flags , int wait , struct cell *ci , struct cell *cj , int tight ) {
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    struct task *t;
    
    /* Lock the space. */
    lock_lock( &s->lock );
    
    /* Get the next free task. */
    t = &s->tasks[ s->nr_tasks ];
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    /* Copy the data. */
    t->type = type;
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    t->subtype = subtype;
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    t->flags = flags;
    t->wait = wait;
    t->ci = ci;
    t->cj = cj;
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    t->skip = 0;
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    t->tight = tight;
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    t->nr_unlock_tasks = 0;
    t->nr_unlock_cells = 0;
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    /* Increase the task counter. */
    s->nr_tasks += 1;
    
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    /* Unock the space. */
    lock_unlock_blind( &s->lock );
    
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    /* Return a pointer to the new task. */
    return t;

    }



/**
 * @brief Split tasks that may be too large.
 *
 * @param s The #space we are working in.
 */
 
void space_splittasks ( struct space *s ) {

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    int j, k, sid, tid;
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    struct cell *ci, *cj;
    double hi, hj, shift[3];
    struct task *t;
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    float dt_step = s->dt_step;
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    int pts[7][8] = { { -1 , 12 , 10 ,  9 ,  4 ,  3 ,  1 ,  0 } ,
                      { -1 , -1 , 11 , 10 ,  5 ,  4 ,  2 ,  1 } ,
                      { -1 , -1 , -1 , 12 ,  7 ,  6 ,  4 ,  3 } , 
                      { -1 , -1 , -1 , -1 ,  8 ,  7 ,  5 ,  4 } ,
                      { -1 , -1 , -1 , -1 , -1 , 12 , 10 ,  9 } ,
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                      { -1 , -1 , -1 , -1 , -1 , -1 , 11 , 10 } ,
                      { -1 , -1 , -1 , -1 , -1 , -1 , -1 , 12 } };
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    /* Loop through the tasks... */
    for ( tid = 0 ; tid < s->nr_tasks ; tid++ ) {
    
        /* Get a pointer on the task. */
        t = &s->tasks[tid];
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        /* Self-interaction? */
        if ( t->type == task_type_self ) {
        
            /* Get a handle on the cell involved. */
            ci = t->ci;
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            /* Ingore this task? */
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            if ( ci->dt_min > dt_step ) {
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                t->skip = 1;
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                continue;
                }
            
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            /* Is this cell even split? */
            if ( !ci->split )
                continue;
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            /* Make a sub? */
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            if ( space_dosub && ci->count < space_subsize ) {
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                /* convert to a self-subtask. */
                t->type = task_type_sub;
                
                /* Wait for this tasks sorts, as we will now have pairwise
                   components in this sub. */
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                space_addsorts( s , t , ci , NULL , -1 );
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                }
                
            /* Otherwise, make tasks explicitly. */
            else {
            
                /* Take a step back (we're going to recycle the current task)... */
                tid -= 1;

                /* Add the self taks. */
                for ( k = 0 ; ci->progeny[k] == NULL ; k++ );
                t->ci = ci->progeny[k];
                for ( k += 1 ; k < 8 ; k++ )
                    if ( ci->progeny[k] != NULL )
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                        space_addtask( s , task_type_self , task_subtype_density , 0 , 0 , ci->progeny[k] , NULL , 0 );
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                /* Make a task for each pair of progeny. */
                for ( j = 0 ; j < 8 ; j++ )
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                    if ( ci->progeny[j] != NULL )
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                        for ( k = j + 1 ; k < 8 ; k++ )
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                            if ( ci->progeny[k] != NULL )
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                                space_addtask( s , task_type_pair , task_subtype_density , pts[j][k] , 0 , ci->progeny[j] , ci->progeny[k] , 0 );
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                }
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            }
    
        /* Pair interaction? */
        else if ( t->type == task_type_pair ) {
            
            /* Get a handle on the cells involved. */
            ci = t->ci;
            cj = t->cj;
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            hi = fmin( ci->h[0] , fmin( ci->h[1] , ci->h[2] ) );
            hj = fmin( cj->h[0] , fmin( cj->h[1] , cj->h[2] ) );
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            /* Ingore this task? */
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            if ( ci->dt_min > dt_step && cj->dt_min > dt_step ) {
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                t->skip = 1;
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                continue;
                }
            
            /* Get the sort ID, use space_getsid and not t->flags
               to make sure we get ci and cj swapped if needed. */
            sid = space_getsid( s , &ci , &cj , shift );
                
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            /* Should this task be split-up? */
            if ( ci->split && cj->split &&
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                 ci->h_max < hi/2 && cj->h_max < hj/2 ) {
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                /* Replace by a single sub-task? */
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                if ( space_dosub &&
                     ci->count < space_subsize && cj->count < space_subsize &&
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                     sid != 0 && sid != 2 && sid != 6 && sid != 8 ) {
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                    /* Make this task a sub task. */
                    t->type = task_type_sub;
                    t->flags = sid;
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                    t->ci = ci; t->cj = cj;
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                    /* Create the sorts recursively. */
                    space_addsorts( s , t , ci , cj , sid );
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                    /* Don't go any further. */
                    continue;
                
                    }

                /* Take a step back (we're going to recycle the current task)... */
                tid -= 1;

                /* For each different sorting type... */
                switch ( sid ) {

                    case 0: /* (  1 ,  1 ,  1 ) */
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                        t->ci = ci->progeny[7]; t->cj = cj->progeny[0]; t->flags = 0;
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                        break;

                    case 1: /* (  1 ,  1 ,  0 ) */
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                        t->ci = ci->progeny[6]; t->cj = cj->progeny[0]; t->flags = 1; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 1 , 0 , ci->progeny[7] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 0 , 0 , ci->progeny[6] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 2 , 0 , ci->progeny[7] , cj->progeny[0] , 1 );
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                        break;

                    case 2: /* (  1 ,  1 , -1 ) */
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                        t->ci = ci->progeny[6]; t->cj = cj->progeny[1]; t->flags = 2; t->tight = 1;
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                        break;

                    case 3: /* (  1 ,  0 ,  1 ) */
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                        t->ci = ci->progeny[5]; t->cj = cj->progeny[0]; t->flags = 3; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 3 , 0 , ci->progeny[7] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 0 , 0 , ci->progeny[5] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 6 , 0 , ci->progeny[7] , cj->progeny[0] , 1 );
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                        break;

                    case 4: /* (  1 ,  0 ,  0 ) */
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                        t->ci = ci->progeny[4]; t->cj = cj->progeny[0]; t->flags = 4; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 5 , 0 , ci->progeny[5] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 7 , 0 , ci->progeny[6] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 8 , 0 , ci->progeny[7] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 3 , 0 , ci->progeny[4] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 4 , 0 , ci->progeny[5] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 6 , 0 , ci->progeny[6] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 7 , 0 , ci->progeny[7] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 1 , 0 , ci->progeny[4] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 2 , 0 , ci->progeny[5] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 4 , 0 , ci->progeny[6] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 5 , 0 , ci->progeny[7] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 0 , 0 , ci->progeny[4] , cj->progeny[3] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 1 , 0 , ci->progeny[5] , cj->progeny[3] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 3 , 0 , ci->progeny[6] , cj->progeny[3] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 4 , 0 , ci->progeny[7] , cj->progeny[3] , 1 );
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                        break;

                    case 5: /* (  1 ,  0 , -1 ) */
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                        t->ci = ci->progeny[4]; t->cj = cj->progeny[1]; t->flags = 5; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 5 , 0 , ci->progeny[6] , cj->progeny[3] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 2 , 0 , ci->progeny[4] , cj->progeny[3] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 8 , 0 , ci->progeny[6] , cj->progeny[1] , 1 );
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                        break;

                    case 6: /* (  1 , -1 ,  1 ) */
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                        t->ci = ci->progeny[5]; t->cj = cj->progeny[2]; t->flags = 6; t->tight = 1;
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                        break;

                    case 7: /* (  1 , -1 ,  0 ) */
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                        t->ci = ci->progeny[4]; t->cj = cj->progeny[3]; t->flags = 6; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 8 , 0 , ci->progeny[5] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 7 , 0 , ci->progeny[4] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 7 , 0 , ci->progeny[5] , cj->progeny[3] , 1 );
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                        break;

                    case 8: /* (  1 , -1 , -1 ) */
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                        t->ci = ci->progeny[4]; t->cj = cj->progeny[3]; t->flags = 8; t->tight = 1;
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                        break;

                    case 9: /* (  0 ,  1 ,  1 ) */
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                        t->ci = ci->progeny[3]; t->cj = cj->progeny[0]; t->flags = 9; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 9 , 0 , ci->progeny[7] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 0 , 0 , ci->progeny[3] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 8 , 0 , ci->progeny[7] , cj->progeny[0] , 1 );
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                        break;

                    case 10: /* (  0 ,  1 ,  0 ) */
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                        t->ci = ci->progeny[2]; t->cj = cj->progeny[0]; t->flags = 10; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 11 , 0 , ci->progeny[3] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 7 , 0 , ci->progeny[6] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 6 , 0 , ci->progeny[7] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 9 , 0 , ci->progeny[2] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 10 , 0 , ci->progeny[3] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 8 , 0 , ci->progeny[6] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 7 , 0 , ci->progeny[7] , cj->progeny[1] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 1 , 0 , ci->progeny[2] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 2 , 0 , ci->progeny[3] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 10 , 0 , ci->progeny[6] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 11 , 0 , ci->progeny[7] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 0 , 0 , ci->progeny[2] , cj->progeny[5] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 1 , 0 , ci->progeny[3] , cj->progeny[5] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 9 , 0 , ci->progeny[6] , cj->progeny[5] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 10 , 0 , ci->progeny[7] , cj->progeny[5] , 1 );
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                        break;

                    case 11: /* (  0 ,  1 , -1 ) */
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                        t->ci = ci->progeny[2]; t->cj = cj->progeny[1]; t->flags = 11; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 11 , 0 , ci->progeny[6] , cj->progeny[5] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 2 , 0 , ci->progeny[2] , cj->progeny[5] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 6 , 0 , ci->progeny[6] , cj->progeny[1] , 1 );
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                        break;

                    case 12: /* (  0 ,  0 ,  1 ) */
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                        t->ci = ci->progeny[1]; t->cj = cj->progeny[0]; t->flags = 12; t->tight = 1;
                        t = space_addtask( s , task_type_pair , t->subtype , 11 , 0 , ci->progeny[3] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 5 , 0 , ci->progeny[5] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 2 , 0 , ci->progeny[7] , cj->progeny[0] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 9 , 0 , ci->progeny[1] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 12 , 0 , ci->progeny[3] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 8 , 0 , ci->progeny[5] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 5 , 0 , ci->progeny[7] , cj->progeny[2] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 3 , 0 , ci->progeny[1] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 6 , 0 , ci->progeny[3] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 12 , 0 , ci->progeny[5] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 11 , 0 , ci->progeny[7] , cj->progeny[4] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 0 , 0 , ci->progeny[1] , cj->progeny[6] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 3 , 0 , ci->progeny[3] , cj->progeny[6] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 9 , 0 , ci->progeny[5] , cj->progeny[6] , 1 );
                        t = space_addtask( s , task_type_pair , t->subtype , 12 , 0 , ci->progeny[7] , cj->progeny[6] , 1 );
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                        break;

                    }

                } /* split this task? */
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            /* Otherwise, if not spilt, stitch-up the sorting. */
            else {
            
                /* Create the sort for ci. */
                if ( ci->sorts[0] == NULL )
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                    ci->sorts[0] = space_addtask( s , task_type_sort , 0 , 1 << sid , 0 , ci , NULL , 0 );
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                ci->sorts[0]->flags |= (1 << sid);
                task_addunlock( ci->sorts[0] , t );
                
                /* Create the sort for cj. */
                if ( cj->sorts[0] == NULL )
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                    cj->sorts[0] = space_addtask( s , task_type_sort , 0 , 1 << sid , 0 , cj , NULL , 0 );
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                cj->sorts[0]->flags |= (1 << sid);
                task_addunlock( cj->sorts[0] , t );
                
                }
                
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            } /* pair interaction? */
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        } /* loop over all tasks. */
        
    }
    
    
/**
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 * @brief Generate the sorts for a sub recursively.
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 *
 * @param s The #space we are working in.
 */
 
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void space_addsorts ( struct space *s , struct task *t , struct cell *ci , struct cell *cj , int sid ) {
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    float h;
    double shift[3];
    int j, k;
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    /* Get the cell dimensions. */
    h = fmin( ci->h[0] , fmin( ci->h[1] , ci->h[2] ) );
    
    /* Single-cell sub? */
    if ( cj == NULL ) {
    
        /* If there is further splitting, add the pairs recursively. */
        if ( ci->split ) {
        
            /* Recurse for each progeny. */
            for ( j = 0 ; j < 8 ; j++ )
                if ( ci->progeny[j] != NULL )
                    space_addsorts( s , t , ci->progeny[j] , NULL , -1 );

            /* Recurse for each pair of progeny. */
            for ( j = 0 ; j < 8 ; j++ )
                if ( ci->progeny[j] != NULL )
                    for ( k = j + 1 ; k < 8 ; k++ )
                        if ( ci->progeny[k] != NULL )
                            space_addsorts( s , t , ci->progeny[j] , ci->progeny[k] , -1 );
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            }
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        }
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    /* Otherwise, it's a pair. */
    else {
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        /* Get the sort ID if not specified. */
        // if ( sid < 0 )
            sid = space_getsid( s , &ci , &cj , shift );
        
        /* If there is no further splitting, add the sorts. */
        if ( !ci->split || !cj->split ||
             ci->h_max*2 >= h || cj->h_max*2 >= h ) {
            
            /* Create and add the sort for ci. */
            if ( ci->sorts[0] == NULL )
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                ci->sorts[0] = space_addtask( s , task_type_sort , 0 , 1 << sid , 0 , ci , NULL , 0 );
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            ci->sorts[0]->flags |= (1 << sid);
            task_addunlock( ci->sorts[0] , t );
            
            /* Create and add the sort for cj. */
            if ( cj->sorts[0] == NULL )
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                cj->sorts[0] = space_addtask( s , task_type_sort , 0 , 1 << sid , 0 , cj , NULL , 0 );
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            cj->sorts[0]->flags |= (1 << sid);
            task_addunlock( cj->sorts[0] , t );
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            }
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        /* Otherwise, recurse. */
        else {
                
            /* For each different sorting type... */
            switch ( sid ) {

                case 0: /* (  1 ,  1 ,  1 ) */
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 0 );
                    break;

                case 1: /* (  1 ,  1 ,  0 ) */
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[0] , 1 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[1] , 1 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[1] , 0 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 2 );
                    break;

                case 2: /* (  1 ,  1 , -1 ) */
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[1] , 2 );
                    break;

                case 3: /* (  1 ,  0 ,  1 ) */
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[0] , 3 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[2] , 3 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[2] , 0 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 6 );
                    break;

                case 4: /* (  1 ,  0 ,  0 ) */
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[0] , 4 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[0] , 5 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[0] , 7 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 8 );
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[1] , 3 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[1] , 4 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[1] , 6 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[1] , 7 );
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[2] , 1 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[2] , 2 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[2] , 4 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[2] , 5 );
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[3] , 0 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[3] , 1 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[3] , 3 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[3] , 4 );
                    break;

                case 5: /* (  1 ,  0 , -1 ) */
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[1] , 5 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[3] , 5 );
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[3] , 2 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[1] , 8 );
                    break;

                case 6: /* (  1 , -1 ,  1 ) */
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[2] , 6 );
                    break;

                case 7: /* (  1 , -1 ,  0 ) */
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[3] , 6 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[2] , 8 );
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[2] , 7 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[3] , 7 );
                    break;

                case 8: /* (  1 , -1 , -1 ) */
                    space_addsorts( s , t , ci->progeny[4] , cj->progeny[3] , 8 );
                    break;

                case 9: /* (  0 ,  1 ,  1 ) */
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[0] , 9 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[4] , 9 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[4] , 0 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 8 );
                    break;

                case 10: /* (  0 ,  1 ,  0 ) */
                    space_addsorts( s , t , ci->progeny[2] , cj->progeny[0] , 10 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[0] , 11 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[0] , 7 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 6 );
                    space_addsorts( s , t , ci->progeny[2] , cj->progeny[1] , 9 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[1] , 10 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[1] , 8 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[1] , 7 );
                    space_addsorts( s , t , ci->progeny[2] , cj->progeny[4] , 1 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[4] , 2 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[4] , 10 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[4] , 11 );
                    space_addsorts( s , t , ci->progeny[2] , cj->progeny[5] , 0 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[5] , 1 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[5] , 9 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[5] , 10 );
                    break;

                case 11: /* (  0 ,  1 , -1 ) */
                    space_addsorts( s , t , ci->progeny[2] , cj->progeny[1] , 11 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[5] , 11 );
                    space_addsorts( s , t , ci->progeny[2] , cj->progeny[5] , 2 );
                    space_addsorts( s , t , ci->progeny[6] , cj->progeny[1] , 6 );
                    break;

                case 12: /* (  0 ,  0 ,  1 ) */
                    space_addsorts( s , t , ci->progeny[1] , cj->progeny[0] , 12 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[0] , 11 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[0] , 5 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[0] , 2 );
                    space_addsorts( s , t , ci->progeny[1] , cj->progeny[2] , 9 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[2] , 12 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[2] , 8 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[2] , 5 );
                    space_addsorts( s , t , ci->progeny[1] , cj->progeny[4] , 3 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[4] , 6 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[4] , 12 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[4] , 11 );
                    space_addsorts( s , t , ci->progeny[1] , cj->progeny[6] , 0 );
                    space_addsorts( s , t , ci->progeny[3] , cj->progeny[6] , 3 );
                    space_addsorts( s , t , ci->progeny[5] , cj->progeny[6] , 9 );
                    space_addsorts( s , t , ci->progeny[7] , cj->progeny[6] , 12 );
                    break;

                } /* switch. */

            } /* recurse. */
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