graph.c

/*!
 * \file 
 *
 * \brief Various routines with dealing with sparse graphs 
 *
 * \author George Karypis
 * \version\verbatim $Id: graph.c 13328 2012-12-31 14:57:40Z karypis $ \endverbatim
 */

#include <GKlib.h>

#define OMPMINOPS       50000

/*************************************************************************/
/*! Allocate memory for a graph and initializes it 
    \returns the allocated graph. The various fields are set to NULL.
*/
/**************************************************************************/
gk_graph_t *gk_graph_Create()
{
  gk_graph_t *graph;

  graph = (gk_graph_t *)gk_malloc(sizeof(gk_graph_t), "gk_graph_Create: graph");

  gk_graph_Init(graph);

  return graph;
}


/*************************************************************************/
/*! Initializes the graph.
    \param graph is the graph to be initialized.
*/
/*************************************************************************/
void gk_graph_Init(gk_graph_t *graph)
{
  memset(graph, 0, sizeof(gk_graph_t));
  graph->nvtxs = -1;
}


/*************************************************************************/
/*! Frees all the memory allocated for a graph.
    \param graph is the graph to be freed.
*/
/*************************************************************************/
void gk_graph_Free(gk_graph_t **graph)
{
  if (*graph == NULL)
    return;
  gk_graph_FreeContents(*graph);
  gk_free((void **)graph, LTERM);
}


/*************************************************************************/
/*! Frees only the memory allocated for the graph's different fields and
    sets them to NULL.
    \param graph is the graph whose contents will be freed.
*/    
/*************************************************************************/
void gk_graph_FreeContents(gk_graph_t *graph)
{
  gk_free((void *)&graph->xadj, &graph->adjncy, 
          &graph->iadjwgt, &graph->fadjwgt,
          &graph->ivwgts, &graph->fvwgts,
          &graph->ivsizes, &graph->fvsizes,
          &graph->vlabels, 
          LTERM);
}


/**************************************************************************/
/*! Reads a sparse graph from the supplied file 
    \param filename is the file that stores the data.
    \param format is the graph format. The supported values are:
           GK_GRAPH_FMT_METIS.
    \param isfewgts is 1 if the edge-weights should be read as floats
    \param isfvwgts is 1 if the vertex-weights should be read as floats
    \param isfvsizes is 1 if the vertex-sizes should be read as floats
    \returns the graph that was read.
*/
/**************************************************************************/
gk_graph_t *gk_graph_Read(char *filename, int format, int isfewgts, 
                int isfvwgts, int isfvsizes)
{
  ssize_t i, k, l;
  size_t nfields, nvtxs, nedges, fmt, ncon, lnlen;
  int32_t ival;
  float fval;
  int readsizes=0, readwgts=0, readvals=0, numbering=0;
  char *line=NULL, *head, *tail, fmtstr[256];
  FILE *fpin=NULL;
  gk_graph_t *graph=NULL;


  if (!gk_fexists(filename)) 
    gk_errexit(SIGERR, "File %s does not exist!\n", filename);

  if (format == GK_GRAPH_FMT_METIS) {
    fpin = gk_fopen(filename, "r", "gk_graph_Read: fpin");
    do {
      if (gk_getline(&line, &lnlen, fpin) <= 0)
        gk_errexit(SIGERR, "Premature end of input file: file:%s\n", filename);
    } while (line[0] == '%');

    fmt = ncon = 0;
    nfields = sscanf(line, "%zu %zu %zu %zu", &nvtxs, &nedges, &fmt, &ncon);
    if (nfields < 2)
      gk_errexit(SIGERR, "Header line must contain at least 2 integers (#vtxs and #edges).\n");

    nedges *= 2;

    if (fmt > 111)
      gk_errexit(SIGERR, "Cannot read this type of file format [fmt=%zu]!\n", fmt);

    sprintf(fmtstr, "%03zu", fmt%1000);
    readsizes = (fmtstr[0] == '1');
    readwgts  = (fmtstr[1] == '1');
    readvals  = (fmtstr[2] == '1');
    numbering = 1;
    ncon      = (ncon == 0 ? 1 : ncon);
  }
  else {
    gk_errexit(SIGERR, "Unrecognized format: %d\n", format);
  }

  graph = gk_graph_Create();

  graph->nvtxs = nvtxs;

  graph->xadj   = gk_zmalloc(nvtxs+1, "gk_graph_Read: xadj");
  graph->adjncy = gk_i32malloc(nedges, "gk_graph_Read: adjncy");
  if (readvals) {
    if (isfewgts)
      graph->fadjwgt = gk_fmalloc(nedges, "gk_graph_Read: fadjwgt");
    else
      graph->iadjwgt = gk_i32malloc(nedges, "gk_graph_Read: iadjwgt");
  }

  if (readsizes) {
    if (isfvsizes)
      graph->fvsizes = gk_fmalloc(nvtxs, "gk_graph_Read: fvsizes");
    else
      graph->ivsizes = gk_i32malloc(nvtxs, "gk_graph_Read: ivsizes");
  }

  if (readwgts) {
    if (isfvwgts)
      graph->fvwgts = gk_fmalloc(nvtxs*ncon, "gk_graph_Read: fvwgts");
    else
      graph->ivwgts = gk_i32malloc(nvtxs*ncon, "gk_graph_Read: ivwgts");
  }


  /*----------------------------------------------------------------------
   * Read the sparse graph file
   *---------------------------------------------------------------------*/
  numbering = (numbering ? - 1 : 0);
  for (graph->xadj[0]=0, k=0, i=0; i<nvtxs; i++) {
    do {
      if (gk_getline(&line, &lnlen, fpin) == -1)
        gk_errexit(SIGERR, "Pregraphure end of input file: file while reading row %d\n", i);
    } while (line[0] == '%');

    head = line;
    tail = NULL;

    /* Read vertex sizes */
    if (readsizes) {
      if (isfvsizes) {
#ifdef __MSC__
        graph->fvsizes[i] = (float)strtod(head, &tail);
#else
        graph->fvsizes[i] = strtof(head, &tail);
#endif
        if (tail == head)
          gk_errexit(SIGERR, "The line for vertex %zd does not have size information\n", i+1);
        if (graph->fvsizes[i] < 0)
          gk_errexit(SIGERR, "The size for vertex %zd must be >= 0\n", i+1);
      }
      else {
        graph->ivsizes[i] = strtol(head, &tail, 0);
        if (tail == head)
          gk_errexit(SIGERR, "The line for vertex %zd does not have size information\n", i+1);
        if (graph->ivsizes[i] < 0)
          gk_errexit(SIGERR, "The size for vertex %zd must be >= 0\n", i+1);
      }
      head = tail;
    }

    /* Read vertex weights */
    if (readwgts) {
      for (l=0; l<ncon; l++) {
        if (isfvwgts) {
#ifdef __MSC__
          graph->fvwgts[i*ncon+l] = (float)strtod(head, &tail);
#else
          graph->fvwgts[i*ncon+l] = strtof(head, &tail);
#endif
          if (tail == head)
            gk_errexit(SIGERR, "The line for vertex %zd does not have enough weights "
                    "for the %d constraints.\n", i+1, ncon);
          if (graph->fvwgts[i*ncon+l] < 0)
            gk_errexit(SIGERR, "The weight vertex %zd and constraint %zd must be >= 0\n", i+1, l);
        }
        else {
          graph->ivwgts[i*ncon+l] = strtol(head, &tail, 0);
          if (tail == head)
            gk_errexit(SIGERR, "The line for vertex %zd does not have enough weights "
                    "for the %d constraints.\n", i+1, ncon);
          if (graph->ivwgts[i*ncon+l] < 0)
            gk_errexit(SIGERR, "The weight vertex %zd and constraint %zd must be >= 0\n", i+1, l);
        }
        head = tail;
      }
    }

   
    /* Read the rest of the row */
    while (1) {
      ival = (int)strtol(head, &tail, 0);
      if (tail == head) 
        break;
      head = tail;
      
      if ((graph->adjncy[k] = ival + numbering) < 0)
        gk_errexit(SIGERR, "Error: Invalid column number %d at row %zd.\n", ival, i);

      if (readvals) {
        if (isfewgts) {
#ifdef __MSC__
          fval = (float)strtod(head, &tail);
#else
    	  fval = strtof(head, &tail);
#endif
          if (tail == head)
            gk_errexit(SIGERR, "Value could not be found for edge! Vertex:%zd, NNZ:%zd\n", i, k);

          graph->fadjwgt[k] = fval;
        }
        else {
    	  ival = strtol(head, &tail, 0);
          if (tail == head)
            gk_errexit(SIGERR, "Value could not be found for edge! Vertex:%zd, NNZ:%zd\n", i, k);

          graph->iadjwgt[k] = ival;
        }
        head = tail;
      }
      k++;
    }
    graph->xadj[i+1] = k;
  }

  if (k != nedges)
    gk_errexit(SIGERR, "gk_graph_Read: Something wrong with the number of edges in "
                       "the input file. nedges=%zd, Actualnedges=%zd.\n", nedges, k);

  gk_fclose(fpin);

  gk_free((void **)&line, LTERM);

  return graph;
}


/**************************************************************************/
/*! Writes a graph into a file.
    \param graph is the graph to be written,
    \param filename is the name of the output file.
    \param format is one of GK_GRAPH_FMT_METIS specifying
           the format of the output file.
*/
/**************************************************************************/
void gk_graph_Write(gk_graph_t *graph, char *filename, int format)
{
  ssize_t i, j;
  int hasvwgts, hasvsizes, hasewgts;
  FILE *fpout;

  if (format != GK_GRAPH_FMT_METIS)
    gk_errexit(SIGERR, "Unknown file format. %d\n", format);

  if (filename)
    fpout = gk_fopen(filename, "w", "gk_graph_Write: fpout");
  else
    fpout = stdout; 


  hasewgts  = (graph->iadjwgt || graph->fadjwgt);
  hasvwgts  = (graph->ivwgts || graph->fvwgts);
  hasvsizes = (graph->ivsizes || graph->fvsizes);

  /* write the header line */
  fprintf(fpout, "%d %zd", graph->nvtxs, graph->xadj[graph->nvtxs]/2);
  if (hasvwgts || hasvsizes || hasewgts) 
    fprintf(fpout, " %d%d%d", hasvsizes, hasvwgts, hasewgts);
  fprintf(fpout, "\n");


  for (i=0; i<graph->nvtxs; i++) {
    if (hasvsizes) {
      if (graph->ivsizes)
        fprintf(fpout, " %d", graph->ivsizes[i]);
      else
        fprintf(fpout, " %f", graph->fvsizes[i]);
    }

    if (hasvwgts) {
      if (graph->ivwgts)
        fprintf(fpout, " %d", graph->ivwgts[i]);
      else
        fprintf(fpout, " %f", graph->fvwgts[i]);
    }

    for (j=graph->xadj[i]; j<graph->xadj[i+1]; j++) {
      fprintf(fpout, " %d", graph->adjncy[j]+1);
      if (hasewgts) {
        if (graph->iadjwgt)
          fprintf(fpout, " %d", graph->iadjwgt[j]);
        else 
          fprintf(fpout, " %f", graph->fadjwgt[j]);
      }
    }
    fprintf(fpout, "\n");
  }
  if (filename)
    gk_fclose(fpout);
}


/*************************************************************************/
/*! Returns a copy of a graph.
    \param graph is the graph to be duplicated.
    \returns the newly created copy of the graph.
*/
/**************************************************************************/
gk_graph_t *gk_graph_Dup(gk_graph_t *graph)
{
  gk_graph_t *ngraph;

  ngraph = gk_graph_Create();

  ngraph->nvtxs  = graph->nvtxs;

  /* copy the adjacency structure */
  if (graph->xadj)
    ngraph->xadj = gk_zcopy(graph->nvtxs+1, graph->xadj, 
                            gk_zmalloc(graph->nvtxs+1, "gk_graph_Dup: xadj"));
  if (graph->ivwgts)
    ngraph->ivwgts = gk_i32copy(graph->nvtxs, graph->ivwgts, 
                            gk_i32malloc(graph->nvtxs, "gk_graph_Dup: ivwgts"));
  if (graph->ivsizes)
    ngraph->ivsizes = gk_i32copy(graph->nvtxs, graph->ivsizes, 
                            gk_i32malloc(graph->nvtxs, "gk_graph_Dup: ivsizes"));
  if (graph->vlabels)
    ngraph->vlabels = gk_i32copy(graph->nvtxs, graph->vlabels, 
                            gk_i32malloc(graph->nvtxs, "gk_graph_Dup: ivlabels"));
  if (graph->fvwgts)
    ngraph->fvwgts = gk_fcopy(graph->nvtxs, graph->fvwgts, 
                            gk_fmalloc(graph->nvtxs, "gk_graph_Dup: fvwgts"));
  if (graph->fvsizes)
    ngraph->fvsizes = gk_fcopy(graph->nvtxs, graph->fvsizes, 
                            gk_fmalloc(graph->nvtxs, "gk_graph_Dup: fvsizes"));


  if (graph->adjncy)
    ngraph->adjncy = gk_i32copy(graph->xadj[graph->nvtxs], graph->adjncy, 
                            gk_i32malloc(graph->xadj[graph->nvtxs], "gk_graph_Dup: adjncy"));
  if (graph->iadjwgt)
    ngraph->iadjwgt = gk_i32copy(graph->xadj[graph->nvtxs], graph->iadjwgt, 
                            gk_i32malloc(graph->xadj[graph->nvtxs], "gk_graph_Dup: iadjwgt"));
  if (graph->fadjwgt)
    ngraph->fadjwgt = gk_fcopy(graph->xadj[graph->nvtxs], graph->fadjwgt, 
                            gk_fmalloc(graph->xadj[graph->nvtxs], "gk_graph_Dup: fadjwgt"));

  return ngraph;
}


/*************************************************************************/
/*! Returns a subgraph containing a set of consecutive vertices.
    \param graph is the original graph.
    \param vstart is the starting vertex.
    \param nvtxs is the number of vertices from vstart to extract.
    \returns the newly created subgraph.
*/
/**************************************************************************/
gk_graph_t *gk_graph_ExtractSubgraph(gk_graph_t *graph, int vstart, int nvtxs)
{
  ssize_t i;
  gk_graph_t *ngraph;

  if (vstart+nvtxs > graph->nvtxs)
    return NULL;

  ngraph = gk_graph_Create();

  ngraph->nvtxs  = nvtxs;

  /* copy the adjancy structure */
  if (graph->xadj)
    ngraph->xadj = gk_zcopy(nvtxs+1, graph->xadj+vstart, 
                              gk_zmalloc(nvtxs+1, "gk_graph_ExtractSubgraph: xadj"));
  for (i=nvtxs; i>=0; i--)
    ngraph->xadj[i] -= ngraph->xadj[0];
  ASSERT(ngraph->xadj[0] == 0);

  if (graph->ivwgts)
    ngraph->ivwgts = gk_i32copy(nvtxs, graph->ivwgts+vstart, 
                            gk_i32malloc(nvtxs, "gk_graph_ExtractSubgraph: ivwgts"));
  if (graph->ivsizes)
    ngraph->ivsizes = gk_i32copy(nvtxs, graph->ivsizes+vstart, 
                            gk_i32malloc(nvtxs, "gk_graph_ExtractSubgraph: ivsizes"));
  if (graph->vlabels)
    ngraph->vlabels = gk_i32copy(nvtxs, graph->vlabels+vstart, 
                            gk_i32malloc(nvtxs, "gk_graph_ExtractSubgraph: vlabels"));

  if (graph->fvwgts)
    ngraph->fvwgts = gk_fcopy(nvtxs, graph->fvwgts+vstart, 
                            gk_fmalloc(nvtxs, "gk_graph_ExtractSubgraph: fvwgts"));
  if (graph->fvsizes)
    ngraph->fvsizes = gk_fcopy(nvtxs, graph->fvsizes+vstart, 
                            gk_fmalloc(nvtxs, "gk_graph_ExtractSubgraph: fvsizes"));


  ASSERT(ngraph->xadj[nvtxs] == graph->xadj[vstart+nvtxs]-graph->xadj[vstart]);
  if (graph->adjncy)
    ngraph->adjncy = gk_i32copy(graph->xadj[vstart+nvtxs]-graph->xadj[vstart], 
                            graph->adjncy+graph->xadj[vstart], 
                            gk_i32malloc(graph->xadj[vstart+nvtxs]-graph->xadj[vstart],
                                       "gk_graph_ExtractSubgraph: adjncy"));
  if (graph->iadjwgt)
    ngraph->iadjwgt = gk_i32copy(graph->xadj[vstart+nvtxs]-graph->xadj[vstart], 
                            graph->iadjwgt+graph->xadj[vstart], 
                            gk_i32malloc(graph->xadj[vstart+nvtxs]-graph->xadj[vstart],
                                       "gk_graph_ExtractSubgraph: iadjwgt"));
  if (graph->fadjwgt)
    ngraph->fadjwgt = gk_fcopy(graph->xadj[vstart+nvtxs]-graph->xadj[vstart], 
                            graph->fadjwgt+graph->xadj[vstart], 
                            gk_fmalloc(graph->xadj[vstart+nvtxs]-graph->xadj[vstart],
                                       "gk_graph_ExtractSubgraph: fadjwgt"));

  return ngraph;
}


/*************************************************************************/
/*! Returns a graph that has been reordered according to the permutation.
    \param[IN] graph is the graph to be re-ordered.
    \param[IN] perm is the new ordering of the graph's vertices
    \param[IN] iperm is the original ordering of the re-ordered graph's vertices
    \returns the newly created copy of the graph.

    \note Either perm or iperm can be NULL but not both.
*/
/**************************************************************************/
gk_graph_t *gk_graph_Reorder(gk_graph_t *graph, int32_t *perm, int32_t *iperm)
{
  ssize_t j, jj, *xadj;
  int i, k, u, v, nvtxs;
  int freeperm=0, freeiperm=0;
  int32_t *adjncy;
  gk_graph_t *ngraph;

  if (perm == NULL && iperm == NULL)
    return NULL;

  ngraph = gk_graph_Create();

  ngraph->nvtxs = nvtxs = graph->nvtxs;
  xadj   = graph->xadj;
  adjncy = graph->adjncy;

  /* allocate memory for the different structures that are present in graph */
  if (graph->xadj)
    ngraph->xadj = gk_zmalloc(nvtxs+1, "gk_graph_Reorder: xadj");

  if (graph->ivwgts)
    ngraph->ivwgts = gk_i32malloc(nvtxs, "gk_graph_Reorder: ivwgts");

  if (graph->ivsizes)
    ngraph->ivsizes = gk_i32malloc(nvtxs, "gk_graph_Reorder: ivsizes");

  if (graph->vlabels)
    ngraph->vlabels = gk_i32malloc(nvtxs, "gk_graph_Reorder: ivlabels");

  if (graph->fvwgts)
    ngraph->fvwgts = gk_fmalloc(nvtxs, "gk_graph_Reorder: fvwgts");

  if (graph->fvsizes)
    ngraph->fvsizes = gk_fmalloc(nvtxs, "gk_graph_Reorder: fvsizes");


  if (graph->adjncy)
    ngraph->adjncy = gk_i32malloc(graph->xadj[nvtxs], "gk_graph_Reorder: adjncy");

  if (graph->iadjwgt)
    ngraph->iadjwgt = gk_i32malloc(graph->xadj[nvtxs], "gk_graph_Reorder: iadjwgt");

  if (graph->fadjwgt)
    ngraph->fadjwgt = gk_fmalloc(graph->xadj[nvtxs], "gk_graph_Reorder: fadjwgt");


  /* create perm/iperm if not provided */
  if (perm == NULL) {
    freeperm = 1;
    perm = gk_i32malloc(nvtxs, "gk_graph_Reorder: perm"); 
    for (i=0; i<nvtxs; i++)
      perm[iperm[i]] = i;
  }
  if (iperm == NULL) {
    freeiperm = 1;
    iperm = gk_i32malloc(nvtxs, "gk_graph_Reorder: iperm"); 
    for (i=0; i<nvtxs; i++)
      iperm[perm[i]] = i;
  }

  /* fill-in the information of the re-ordered graph */
  ngraph->xadj[0] = jj = 0;
  for (v=0; v<nvtxs; v++) {
    u = iperm[v];
    for (j=xadj[u]; j<xadj[u+1]; j++, jj++) {
      ngraph->adjncy[jj] = perm[adjncy[j]];
      if (graph->iadjwgt)
        ngraph->iadjwgt[jj] = graph->iadjwgt[j];
      if (graph->fadjwgt)
        ngraph->fadjwgt[jj] = graph->fadjwgt[j];
    }
    if (graph->ivwgts)
      ngraph->ivwgts[v] = graph->ivwgts[u];
    if (graph->fvwgts)
      ngraph->fvwgts[v] = graph->fvwgts[u];
    if (graph->ivsizes)
      ngraph->ivsizes[v] = graph->ivsizes[u];
    if (graph->fvsizes)
      ngraph->fvsizes[v] = graph->fvsizes[u];
    if (graph->vlabels)
      ngraph->vlabels[v] = graph->vlabels[u];

    ngraph->xadj[v+1] = jj;
  }


  /* free memory */
  if (freeperm)
    gk_free((void **)&perm, LTERM);
  if (freeiperm)
    gk_free((void **)&iperm, LTERM);

  return ngraph;
}


/*************************************************************************/
/*! This function finds the connected components in a graph.

    \param graph is the graph structure
    \param cptr is the ptr structure of the CSR representation of the 
           components. The length of this vector must be graph->nvtxs+1.
    \param cind is the indices structure of the CSR representation of 
           the components. The length of this vector must be graph->nvtxs.

    \returns the number of components that it found.

    \note The cptr and cind parameters can be NULL, in which case only the
          number of connected components is returned.
*/
/*************************************************************************/
int gk_graph_FindComponents(gk_graph_t *graph, int32_t *cptr, int32_t *cind)
{
  ssize_t i, ii, j, jj, k, nvtxs, first, last, ntodo, ncmps;
  ssize_t *xadj;
  int32_t *adjncy, *pos, *todo;
  int32_t mustfree_ccsr=0, mustfree_where=0;

  nvtxs  = graph->nvtxs;
  xadj   = graph->xadj;
  adjncy = graph->adjncy;

  /* Deal with NULL supplied cptr/cind vectors */
  if (cptr == NULL) {
    cptr = gk_i32malloc(nvtxs+1, "gk_graph_FindComponents: cptr");
    cind = gk_i32malloc(nvtxs, "gk_graph_FindComponents: cind");
    mustfree_ccsr = 1;
  }

  /* The list of vertices that have not been touched yet. 
     The valid entries are from [0..ntodo). */
  todo = gk_i32incset(nvtxs, 0, gk_i32malloc(nvtxs, "gk_graph_FindComponents: todo"));

  /* For a vertex that has not been visited, pos[i] is the position in the
     todo list that this vertex is stored. 
     If a vertex has been visited, pos[i] = -1. */
  pos = gk_i32incset(nvtxs, 0, gk_i32malloc(nvtxs, "gk_graph_FindComponents: pos"));


  /* Find the connected componends */
  ncmps = -1;
  ntodo = nvtxs;     /* All vertices have not been visited */
  first = last = 0;  /* Point to the first and last vertices that have been touched
                        but not explored. 
                        These vertices are stored in cind[first]...cind[last-1]. */
  while (ntodo > 0) {
    if (first == last) { /* Find another starting vertex */
      cptr[++ncmps] = first;  /* Mark the end of the current CC */

      ASSERT(pos[todo[0]] != -1);
      i = todo[0];

      cind[last++] = i;
      pos[i] = -1;
    }

    i = cind[first++];  /* Get the first visited but unexplored vertex */

    /* Remove i from the todo list and put the last item in the todo 
       list at the position that i was so that the todo list will be
       consequtive. The pos[] array is updated accordingly to keep track
       the location of the vertices in the todo[] list. */
    k = pos[i];
    j = todo[k] = todo[--ntodo];
    pos[j] = k;

    for (j=xadj[i]; j<xadj[i+1]; j++) {
      k = adjncy[j];
      if (pos[k] != -1) {
        cind[last++] = k;
        pos[k] = -1;
      }
    }
  }
  cptr[++ncmps] = first;

  if (mustfree_ccsr)
    gk_free((void **)&cptr, &cind, LTERM);

  gk_free((void **)&pos, &todo, LTERM);

  return (int) ncmps;
}


/*************************************************************************/
/*! This function computes a permutation of the vertices based on a
    breadth-first-traversal. It can be used for re-ordering the graph
    to reduce its bandwidth for better cache locality.
    The algorithm used is a simplified version of the method used to find
    the connected components.

    \param[IN]  graph is the graph structure
    \param[IN]  v is the starting vertex of the BFS
    \param[OUT] perm[i] stores the ID of vertex i in the re-ordered graph.
    \param[OUT] iperm[i] stores the ID of the vertex that corresponds to 
                the ith vertex in the re-ordered graph.

    \note The perm or iperm (but not both) can be NULL, at which point, 
          the corresponding arrays are not returned. Though the program
          works fine when both are NULL, doing that is not smart.
          The returned arrays should be freed with gk_free().
*/
/*************************************************************************/
void gk_graph_ComputeBFSOrdering(gk_graph_t *graph, int v, int32_t **r_perm,
          int32_t **r_iperm)
{
  ssize_t j, *xadj;
  int i, k, nvtxs, first, last;
  int32_t *adjncy, *cot, *pos;

  if (graph->nvtxs <= 0)
    return;

  nvtxs  = graph->nvtxs;
  xadj   = graph->xadj;
  adjncy = graph->adjncy;

  /* This array will function like pos + touched of the CC method */
  pos = gk_i32incset(nvtxs, 0, gk_i32malloc(nvtxs, "gk_graph_ComputeBFSOrdering: pos"));

  /* This array ([C]losed[O]pen[T]odo => cot) serves three purposes. 
     Positions from [0...first) is the current iperm[] vector of the explored vertices; 
     Positions from [first...last) is the OPEN list (i.e., visited vertices);
     Positions from [last...nvtxs) is the todo list. */
  cot = gk_i32incset(nvtxs, 0, gk_i32malloc(nvtxs, "gk_graph_ComputeBFSOrdering: cot"));


  /* put v at the front of the todo list */
  pos[0] = cot[0] = v;
  pos[v] = cot[v] = 0;

  /* Find the connected componends induced by the partition */
  first = last = 0;
  while (first < nvtxs) {
    if (first == last) { /* Find another starting vertex */
      k = cot[last];
      ASSERT(pos[k] != -1);
      pos[k] = -1; /* mark node as being visited */
      last++;
    }

    i = cot[first++];  /* the ++ advances the explored vertices */
    for (j=xadj[i]; j<xadj[i+1]; j++) {
      k = adjncy[j];
      /* if a node has already been visited, its perm[] will be -1 */
      if (pos[k] != -1) {
        /* pos[k] is the location within iperm of where k resides (it is in the 'todo' part); 
           It is placed in that location cot[last] (end of OPEN list) that we 
           are about to overwrite and update pos[cot[last]] to reflect that. */
        cot[pos[k]]    = cot[last]; /* put the head of the todo list to 
                                       where k was in the todo list */
        pos[cot[last]] = pos[k];    /* update perm to reflect the move */

        cot[last++] = k;  /* put node at the end of the OPEN list */
        pos[k]      = -1; /* mark node as being visited */
      }
    }
  }

  /* time to decide what to return */
  if (r_perm != NULL) {
    /* use the 'pos' array to build the perm array */
    for (i=0; i<nvtxs; i++)
      pos[cot[i]] = i;

    *r_perm = pos;
    pos = NULL;
  }

  if (r_iperm != NULL) {
    *r_iperm = cot;
    cot = NULL;
  }


  /* cleanup memory */
  gk_free((void **)&pos, &cot, LTERM);

}


/*************************************************************************/
/*! This function computes a permutation of the vertices based on a
    best-first-traversal. It can be used for re-ordering the graph
    to reduce its bandwidth for better cache locality.

    \param[IN]  graph is the graph structure.
    \param[IN]  v is the starting vertex of the best-first traversal.
    \param[IN]  type indicates the criteria to use to measure the 'bestness'
                of a vertex.
    \param[OUT] perm[i] stores the ID of vertex i in the re-ordered graph.
    \param[OUT] iperm[i] stores the ID of the vertex that corresponds to 
                the ith vertex in the re-ordered graph.

    \note The perm or iperm (but not both) can be NULL, at which point, 
          the corresponding arrays are not returned. Though the program
          works fine when both are NULL, doing that is not smart.
          The returned arrays should be freed with gk_free().
*/
/*************************************************************************/
void gk_graph_ComputeBestFOrdering0(gk_graph_t *graph, int v, int type, 
          int32_t **r_perm, int32_t **r_iperm)
{
  ssize_t j, jj, *xadj;
  int i, k, u, nvtxs;
  int32_t *adjncy, *perm, *degrees, *minIDs, *open;
  gk_i32pq_t *queue;

  if (graph->nvtxs <= 0)
    return;

  nvtxs  = graph->nvtxs;
  xadj   = graph->xadj;
  adjncy = graph->adjncy;

  /* the degree of the vertices in the closed list */
  degrees = gk_i32smalloc(nvtxs, 0, "gk_graph_ComputeBestFOrdering: degrees");

  /* the minimum vertex ID of an open vertex to the closed list */ 
  minIDs  = gk_i32smalloc(nvtxs, nvtxs+1, "gk_graph_ComputeBestFOrdering: minIDs");

  /* the open list */ 
  open  = gk_i32malloc(nvtxs, "gk_graph_ComputeBestFOrdering: open");

  /* if perm[i] >= 0, then perm[i] is the order of vertex i; 
     otherwise perm[i] == -1.
  */
  perm = gk_i32smalloc(nvtxs, -1, "gk_graph_ComputeBestFOrdering: perm");

  /* create the queue and put everything in it */
  queue = gk_i32pqCreate(nvtxs);
  for (i=0; i<nvtxs; i++)
    gk_i32pqInsert(queue, i, 0);
  gk_i32pqUpdate(queue, v, 1);

  open[0] = v;

  /* start processing the nodes */
  for (i=0; i<nvtxs; i++) {
    if ((v = gk_i32pqGetTop(queue)) == -1) 
      gk_errexit(SIGERR, "The priority queue got empty ahead of time [i=%d].\n", i);
    if (perm[v] != -1)
      gk_errexit(SIGERR, "The perm[%d] has already been set.\n", v);
    perm[v] = i;


    for (j=xadj[v]; j<xadj[v+1]; j++) {
      u = adjncy[j];
      if (perm[u] == -1) {
        degrees[u]++;
        minIDs[u] = (i < minIDs[u] ? i : minIDs[u]);

        switch (type) {
          case 1: /* DFS */
            gk_i32pqUpdate(queue, u, 1);
            break;
          case 2: /* Max in closed degree */
            gk_i32pqUpdate(queue, u, degrees[u]);
            break;
          case 3: /* Sum of orders in closed list */
            for (k=0, jj=xadj[u]; jj<xadj[u+1]; jj++) {
              if (perm[adjncy[jj]] != -1)
                k += perm[adjncy[jj]];
            }
            gk_i32pqUpdate(queue, u, k);
            break;
          case 4: /* Sum of order-differences (w.r.t. current number) in closed 
                     list (updated once in a while) */
            for (k=0, jj=xadj[u]; jj<xadj[u+1]; jj++) {
              if (perm[adjncy[jj]] != -1)
                k += (i-perm[adjncy[jj]]);
            }
            gk_i32pqUpdate(queue, u, k);
            break;
          default:
            ;
        }
      }
    }
  }


  /* time to decide what to return */
  if (r_perm != NULL) {
    *r_perm = perm;
    perm = NULL;
  }

  if (r_iperm != NULL) {
    /* use the 'degrees' array to build the iperm array */
    for (i=0; i<nvtxs; i++)
      degrees[perm[i]] = i;

    *r_iperm = degrees;
    degrees = NULL;
  }



  /* cleanup memory */
  gk_i32pqDestroy(queue);
  gk_free((void **)&perm, &degrees, &minIDs, &open, LTERM);

}


/*************************************************************************/
/*! This function computes a permutation of the vertices based on a
    best-first-traversal. It can be used for re-ordering the graph
    to reduce its bandwidth for better cache locality.

    \param[IN]  graph is the graph structure.
    \param[IN]  v is the starting vertex of the best-first traversal.
    \param[IN]  type indicates the criteria to use to measure the 'bestness'
                of a vertex.
    \param[OUT] perm[i] stores the ID of vertex i in the re-ordered graph.
    \param[OUT] iperm[i] stores the ID of the vertex that corresponds to 
                the ith vertex in the re-ordered graph.

    \note The perm or iperm (but not both) can be NULL, at which point, 
          the corresponding arrays are not returned. Though the program
          works fine when both are NULL, doing that is not smart.
          The returned arrays should be freed with gk_free().
*/
/*************************************************************************/
void gk_graph_ComputeBestFOrdering(gk_graph_t *graph, int v, int type, 
          int32_t **r_perm, int32_t **r_iperm)
{
  ssize_t j, jj, *xadj;
  int i, k, u, nvtxs, nopen, ntodo;
  int32_t *adjncy, *perm, *degrees, *wdegrees, *sod, *level, *ot, *pos;
  gk_i32pq_t *queue;

  if (graph->nvtxs <= 0)
    return;

  nvtxs  = graph->nvtxs;
  xadj   = graph->xadj;
  adjncy = graph->adjncy;

  /* the degree of the vertices in the closed list */
  degrees = gk_i32smalloc(nvtxs, 0, "gk_graph_ComputeBestFOrdering: degrees");

  /* the weighted degree of the vertices in the closed list for type==3 */
  wdegrees = gk_i32smalloc(nvtxs, 0, "gk_graph_ComputeBestFOrdering: wdegrees");

  /* the sum of differences for type==4 */
  sod = gk_i32smalloc(nvtxs, 0, "gk_graph_ComputeBestFOrdering: sod");

  /* the encountering level of a vertex type==5 */
  level = gk_i32smalloc(nvtxs, 0, "gk_graph_ComputeBestFOrdering: level");

  /* The open+todo list of vertices. 
     The vertices from [0..nopen] are the open vertices.
     The vertices from [nopen..ntodo) are the todo vertices.
     */
  ot = gk_i32incset(nvtxs, 0, gk_i32malloc(nvtxs, "gk_graph_FindComponents: ot"));

  /* For a vertex that has not been explored, pos[i] is the position in the ot list. */
  pos = gk_i32incset(nvtxs, 0, gk_i32malloc(nvtxs, "gk_graph_FindComponents: pos"));

  /* if perm[i] >= 0, then perm[i] is the order of vertex i; otherwise perm[i] == -1. */
  perm = gk_i32smalloc(nvtxs, -1, "gk_graph_ComputeBestFOrdering: perm");

  /* create the queue and put the starting vertex in it */
  queue = gk_i32pqCreate(nvtxs);
  gk_i32pqInsert(queue, v, 1);

  /* put v at the front of the open list */
  pos[0] = ot[0] = v;
  pos[v] = ot[v] = 0;
  nopen = 1;
  ntodo = nvtxs;

  /* start processing the nodes */
  for (i=0; i<nvtxs; i++) {
    if (nopen == 0) { /* deal with non-connected graphs */
      gk_i32pqInsert(queue, ot[0], 1);  
      nopen++;
    }

    if ((v = gk_i32pqGetTop(queue)) == -1)
      gk_errexit(SIGERR, "The priority queue got empty ahead of time [i=%d].\n", i);

    if (perm[v] != -1)
      gk_errexit(SIGERR, "The perm[%d] has already been set.\n", v);
    perm[v] = i;

    if (ot[pos[v]] != v)
      gk_errexit(SIGERR, "Something went wrong [ot[pos[%d]]!=%d.\n", v, v);
    if (pos[v] >= nopen)
      gk_errexit(SIGERR, "The position of v is not in open list. pos[%d]=%d is >=%d.\n", v, pos[v], nopen);

    /* remove v from the open list and re-arrange the todo part of the list */
    ot[pos[v]]       = ot[nopen-1];
    pos[ot[nopen-1]] = pos[v];
    if (ntodo > nopen) {
      ot[nopen-1]      = ot[ntodo-1];
      pos[ot[ntodo-1]] = nopen-1;
    }
    nopen--;
    ntodo--;

    for (j=xadj[v]; j<xadj[v+1]; j++) {
      u = adjncy[j];
      if (perm[u] == -1) {
        /* update ot list, if u is not in the open list by putting it at the end
           of the open list. */
        if (degrees[u] == 0) {
          ot[pos[u]]     = ot[nopen];
          pos[ot[nopen]] = pos[u];
          ot[nopen]      = u;
          pos[u]         = nopen;
          nopen++;

          level[u] = level[v]+1;
          gk_i32pqInsert(queue, u, 0);  
        }


        /* update the in-closed degree */
        degrees[u]++;

        /* update the queues based on the type */
        switch (type) {
          case 1: /* DFS */
            gk_i32pqUpdate(queue, u, 1000*(i+1)+degrees[u]);
            break;

          case 2: /* Max in closed degree */
            gk_i32pqUpdate(queue, u, degrees[u]);
            break;

          case 3: /* Sum of orders in closed list */
            wdegrees[u] += i;
            gk_i32pqUpdate(queue, u, wdegrees[u]);
            break;

          case 4: /* Sum of order-differences */
            /* this is handled at the end of the loop */
            ;
            break;

          case 5: /* BFS with in degree priority */
            gk_i32pqUpdate(queue, u, -(1000*level[u] - degrees[u]));
            break;

          case 6: /* Hybrid of 1+2 */
            gk_i32pqUpdate(queue, u, (i+1)*degrees[u]);
            break;

          default:
            ;
        }
      }
    }

    if (type == 4) { /* update all the vertices in the open list */
      for (j=0; j<nopen; j++) {
        u = ot[j];
        if (perm[u] != -1)
          gk_errexit(SIGERR, "For i=%d, the open list contains a closed vertex: ot[%zd]=%d, perm[%d]=%d.\n", i, j, u, u, perm[u]);
        sod[u] += degrees[u];
        if (i<1000 || i%25==0)
          gk_i32pqUpdate(queue, u, sod[u]);
      }
    }

    /*
    for (j=0; j<ntodo; j++) {
      if (pos[ot[j]] != j)
        gk_errexit(SIGERR, "pos[ot[%zd]] != %zd.\n", j, j);
    }
    */

  }


  /* time to decide what to return */
  if (r_perm != NULL) {
    *r_perm = perm;
    perm = NULL;
  }

  if (r_iperm != NULL) {
    /* use the 'degrees' array to build the iperm array */
    for (i=0; i<nvtxs; i++)
      degrees[perm[i]] = i;

    *r_iperm = degrees;
    degrees = NULL;
  }



  /* cleanup memory */
  gk_i32pqDestroy(queue);
  gk_free((void **)&perm, &degrees, &wdegrees, &sod, &ot, &pos, &level, LTERM);

}


/*************************************************************************/
/*! This function computes the single-source shortest path lengths from the
    root node to all the other nodes in the graph. If the graph is not 
    connected then, the sortest part to the vertices in the other components 
    is -1.

    \param[IN]  graph is the graph structure.
    \param[IN]  v is the root of the single-source shortest path computations.
    \param[IN]  type indicates the criteria to use to measure the 'bestness'
                of a vertex.
    \param[OUT] sps[i] stores the length of the shortest path from v to vertex i.
                If no such path exists, then it is -1. Note that the returned
                array will be either an array of int32_t or an array of floats.
                The specific type is determined by the existance of non NULL
                iadjwgt and fadjwgt arrays. If both of these arrays exist, then
                priority is given to iadjwgt.

    \note The returned array should be freed with gk_free().
*/
/*************************************************************************/
void gk_graph_SingleSourceShortestPaths(gk_graph_t *graph, int v, void **r_sps)
{
  ssize_t *xadj;
  int i, u, nvtxs;
  int32_t *adjncy, *inqueue;

  if (graph->nvtxs <= 0)
    return;

  nvtxs  = graph->nvtxs;
  xadj   = graph->xadj;
  adjncy = graph->adjncy;

  inqueue = gk_i32smalloc(nvtxs, 0, "gk_graph_SingleSourceShortestPaths: inqueue");

  /* determine if you will be computing using int32_t or float and proceed from there */
  if (graph->iadjwgt != NULL) {
    gk_i32pq_t *queue;
    int32_t *adjwgt;
    int32_t *sps;

    adjwgt = graph->iadjwgt;

    queue = gk_i32pqCreate(nvtxs);
    gk_i32pqInsert(queue, v, 0);
    inqueue[v] = 1;

    sps = gk_i32smalloc(nvtxs, -1, "gk_graph_SingleSourceShortestPaths: sps");
    sps[v] = 0;

    /* start processing the nodes */
    while ((v = gk_i32pqGetTop(queue)) != -1) {
      inqueue[v] = 2;

      /* relax the adjacent edges */
      for (i=xadj[v]; i<xadj[v+1]; i++) {
        u = adjncy[i];
        if (inqueue[u] == 2)
          continue;

        if (sps[u] < 0 || sps[v]+adjwgt[i] < sps[u]) {
          sps[u] = sps[v]+adjwgt[i];

          if (inqueue[u])
            gk_i32pqUpdate(queue, u, -sps[u]);
          else {
            gk_i32pqInsert(queue, u, -sps[u]);
            inqueue[u] = 1;
          }
        }
      }
    }

    *r_sps = (void *)sps;

    gk_i32pqDestroy(queue);
  }
  else {
    gk_fpq_t *queue;
    float *adjwgt;
    float *sps;

    adjwgt = graph->fadjwgt;

    queue = gk_fpqCreate(nvtxs);
    gk_fpqInsert(queue, v, 0);
    inqueue[v] = 1;

    sps = gk_fsmalloc(nvtxs, -1, "gk_graph_SingleSourceShortestPaths: sps");
    sps[v] = 0;

    /* start processing the nodes */
    while ((v = gk_fpqGetTop(queue)) != -1) {
      inqueue[v] = 2;

      /* relax the adjacent edges */
      for (i=xadj[v]; i<xadj[v+1]; i++) {
        u = adjncy[i];
        if (inqueue[u] == 2)
          continue;

        if (sps[u] < 0 || sps[v]+adjwgt[i] < sps[u]) {
          sps[u] = sps[v]+adjwgt[i];

          if (inqueue[u])
            gk_fpqUpdate(queue, u, -sps[u]);
          else {
            gk_fpqInsert(queue, u, -sps[u]);
            inqueue[u] = 1;
          }
        }
      }
    }

    *r_sps = (void *)sps;

    gk_fpqDestroy(queue);
  }

  gk_free((void **)&inqueue, LTERM);

}



#ifdef XXX

/*************************************************************************/
/*! Sorts the adjacency lists in increasing vertex order
    \param graph the graph itself,
*/
/**************************************************************************/
void gk_graph_SortAdjacencies(gk_graph_t *graph)
{
  int n, nn=0;
  ssize_t *ptr;
  int *ind;
  float *val;

  switch (what) {
    case GK_CSR_ROW:
      if (!graph->rowptr)
        gk_errexit(SIGERR, "Row-based view of the graphrix does not exists.\n");

      n   = graph->nrows;
      ptr = graph->rowptr;
      ind = graph->rowind;
      val = graph->rowval;
      break;

    case GK_CSR_COL:
      if (!graph->colptr)
        gk_errexit(SIGERR, "Column-based view of the graphrix does not exists.\n");

      n   = graph->ncols;
      ptr = graph->colptr;
      ind = graph->colind;
      val = graph->colval;
      break;

    default:
      gk_errexit(SIGERR, "Invalid index type of %d.\n", what);
      return;
  }

  #pragma omp parallel if (n > 100)
  {
    ssize_t i, j, k;
    gk_ikv_t *cand;
    float *tval;

    #pragma omp single
    for (i=0; i<n; i++) 
      nn = gk_max(nn, ptr[i+1]-ptr[i]);
  
    cand = gk_ikvmalloc(nn, "gk_graph_SortIndices: cand");
    tval = gk_fmalloc(nn, "gk_graph_SortIndices: tval");
  
    #pragma omp for schedule(static)
    for (i=0; i<n; i++) {
      for (k=0, j=ptr[i]; j<ptr[i+1]; j++) {
        if (j > ptr[i] && ind[j] < ind[j-1])
          k = 1; /* an inversion */
        cand[j-ptr[i]].val = j-ptr[i];
        cand[j-ptr[i]].key = ind[j];
        tval[j-ptr[i]]     = val[j];
      }
      if (k) {
        gk_ikvsorti(ptr[i+1]-ptr[i], cand);
        for (j=ptr[i]; j<ptr[i+1]; j++) {
          ind[j] = cand[j-ptr[i]].key;
          val[j] = tval[cand[j-ptr[i]].val];
        }
      }
    }

    gk_free((void **)&cand, &tval, LTERM);
  }

}


/*************************************************************************/
/*! Returns a subgraphrix containing a certain set of rows.
    \param graph is the original graphrix.
    \param nrows is the number of rows to extract.
    \param rind is the set of row numbers to extract.
    \returns the row structure of the newly created subgraphrix.
*/
/**************************************************************************/
gk_graph_t *gk_graph_ExtractRows(gk_graph_t *graph, int nrows, int *rind)
{
  ssize_t i, ii, j, nnz;
  gk_graph_t *ngraph;

  ngraph = gk_graph_Create();

  ngraph->nrows = nrows;
  ngraph->ncols = graph->ncols;

  for (nnz=0, i=0; i<nrows; i++)  
    nnz += graph->rowptr[rind[i]+1]-graph->rowptr[rind[i]];

  ngraph->rowptr = gk_zmalloc(ngraph->nrows+1, "gk_graph_ExtractPartition: rowptr");
  ngraph->rowind = gk_imalloc(nnz, "gk_graph_ExtractPartition: rowind");
  ngraph->rowval = gk_fmalloc(nnz, "gk_graph_ExtractPartition: rowval");

  ngraph->rowptr[0] = 0;
  for (nnz=0, j=0, ii=0; ii<nrows; ii++) {
    i = rind[ii];
    gk_icopy(graph->rowptr[i+1]-graph->rowptr[i], graph->rowind+graph->rowptr[i], ngraph->rowind+nnz);
    gk_fcopy(graph->rowptr[i+1]-graph->rowptr[i], graph->rowval+graph->rowptr[i], ngraph->rowval+nnz);
    nnz += graph->rowptr[i+1]-graph->rowptr[i];
    ngraph->rowptr[++j] = nnz;
  }
  ASSERT(j == ngraph->nrows);

  return ngraph;
}


/*************************************************************************/
/*! Returns a subgraphrix corresponding to a specified partitioning of rows.
    \param graph is the original graphrix.
    \param part is the partitioning vector of the rows.
    \param pid is the partition ID that will be extracted.
    \returns the row structure of the newly created subgraphrix.
*/
/**************************************************************************/
gk_graph_t *gk_graph_ExtractPartition(gk_graph_t *graph, int *part, int pid)
{
  ssize_t i, j, nnz;
  gk_graph_t *ngraph;

  ngraph = gk_graph_Create();

  ngraph->nrows = 0;
  ngraph->ncols = graph->ncols;

  for (nnz=0, i=0; i<graph->nrows; i++) {
    if (part[i] == pid) {
      ngraph->nrows++;
      nnz += graph->rowptr[i+1]-graph->rowptr[i];
    }
  }

  ngraph->rowptr = gk_zmalloc(ngraph->nrows+1, "gk_graph_ExtractPartition: rowptr");
  ngraph->rowind = gk_imalloc(nnz, "gk_graph_ExtractPartition: rowind");
  ngraph->rowval = gk_fmalloc(nnz, "gk_graph_ExtractPartition: rowval");

  ngraph->rowptr[0] = 0;
  for (nnz=0, j=0, i=0; i<graph->nrows; i++) {
    if (part[i] == pid) {
      gk_icopy(graph->rowptr[i+1]-graph->rowptr[i], graph->rowind+graph->rowptr[i], ngraph->rowind+nnz);
      gk_fcopy(graph->rowptr[i+1]-graph->rowptr[i], graph->rowval+graph->rowptr[i], ngraph->rowval+nnz);
      nnz += graph->rowptr[i+1]-graph->rowptr[i];
      ngraph->rowptr[++j] = nnz;
    }
  }
  ASSERT(j == ngraph->nrows);

  return ngraph;
}


/*************************************************************************/
/*! Splits the graphrix into multiple sub-graphrices based on the provided
    color array.
    \param graph is the original graphrix.
    \param color is an array of size equal to the number of non-zeros
           in the graphrix (row-wise structure). The graphrix is split into
           as many parts as the number of colors. For meaningfull results,
           the colors should be numbered consecutively starting from 0.
    \returns an array of graphrices for each supplied color number.
*/
/**************************************************************************/
gk_graph_t **gk_graph_Split(gk_graph_t *graph, int *color)
{
  ssize_t i, j;
  int nrows, ncolors;
  ssize_t *rowptr;
  int *rowind;
  float *rowval;
  gk_graph_t **sgraphs;

  nrows  = graph->nrows;
  rowptr = graph->rowptr;
  rowind = graph->rowind;
  rowval = graph->rowval;

  ncolors = gk_imax(rowptr[nrows], color)+1;

  sgraphs = (gk_graph_t **)gk_malloc(sizeof(gk_graph_t *)*ncolors, "gk_graph_Split: sgraphs");
  for (i=0; i<ncolors; i++) {
    sgraphs[i] = gk_graph_Create();
    sgraphs[i]->nrows  = graph->nrows;
    sgraphs[i]->ncols  = graph->ncols;
    sgraphs[i]->rowptr = gk_zsmalloc(nrows+1, 0, "gk_graph_Split: sgraphs[i]->rowptr"); 
  }

  for (i=0; i<nrows; i++) {
    for (j=rowptr[i]; j<rowptr[i+1]; j++) 
      sgraphs[color[j]]->rowptr[i]++;
  }
  for (i=0; i<ncolors; i++) 
    MAKECSR(j, nrows, sgraphs[i]->rowptr);

  for (i=0; i<ncolors; i++) {
    sgraphs[i]->rowind = gk_imalloc(sgraphs[i]->rowptr[nrows], "gk_graph_Split: sgraphs[i]->rowind"); 
    sgraphs[i]->rowval = gk_fmalloc(sgraphs[i]->rowptr[nrows], "gk_graph_Split: sgraphs[i]->rowval"); 
  }

  for (i=0; i<nrows; i++) {
    for (j=rowptr[i]; j<rowptr[i+1]; j++) {
      sgraphs[color[j]]->rowind[sgraphs[color[j]]->rowptr[i]] = rowind[j];
      sgraphs[color[j]]->rowval[sgraphs[color[j]]->rowptr[i]] = rowval[j];
      sgraphs[color[j]]->rowptr[i]++;
    }
  }

  for (i=0; i<ncolors; i++) 
    SHIFTCSR(j, nrows, sgraphs[i]->rowptr);

  return sgraphs;
}


/*************************************************************************/
/*! Prunes certain rows/columns of the graphrix. The prunning takes place 
    by analyzing the row structure of the graphrix. The prunning takes place
    by removing rows/columns but it does not affect the numbering of the
    remaining rows/columns.
   
    \param graph the graphrix to be prunned,
    \param what indicates if the rows (GK_CSR_ROW) or the columns (GK_CSR_COL)
           of the graphrix will be prunned,
    \param minf is the minimum number of rows (columns) that a column (row) must
           be present in order to be kept,
    \param maxf is the maximum number of rows (columns) that a column (row) must
          be present at in order to be kept.
    \returns the prunned graphrix consisting only of its row-based structure. 
          The input graphrix is not modified. 
*/
/**************************************************************************/
gk_graph_t *gk_graph_Prune(gk_graph_t *graph, int what, int minf, int maxf)
{
  ssize_t i, j, nnz;
  int nrows, ncols;
  ssize_t *rowptr, *nrowptr;
  int *rowind, *nrowind, *collen;
  float *rowval, *nrowval;
  gk_graph_t *ngraph;

  ngraph = gk_graph_Create();
  
  nrows = ngraph->nrows = graph->nrows;
  ncols = ngraph->ncols = graph->ncols;

  rowptr = graph->rowptr;
  rowind = graph->rowind;
  rowval = graph->rowval;

  nrowptr = ngraph->rowptr = gk_zmalloc(nrows+1, "gk_graph_Prune: nrowptr");
  nrowind = ngraph->rowind = gk_imalloc(rowptr[nrows], "gk_graph_Prune: nrowind");
  nrowval = ngraph->rowval = gk_fmalloc(rowptr[nrows], "gk_graph_Prune: nrowval");


  switch (what) {
    case GK_CSR_COL:
      collen = gk_ismalloc(ncols, 0, "gk_graph_Prune: collen");

      for (i=0; i<nrows; i++) {
        for (j=rowptr[i]; j<rowptr[i+1]; j++) {
          ASSERT(rowind[j] < ncols);
          collen[rowind[j]]++;
        }
      }
      for (i=0; i<ncols; i++)
        collen[i] = (collen[i] >= minf && collen[i] <= maxf ? 1 : 0);

      nrowptr[0] = 0;
      for (nnz=0, i=0; i<nrows; i++) {
        for (j=rowptr[i]; j<rowptr[i+1]; j++) {
          if (collen[rowind[j]]) {
            nrowind[nnz] = rowind[j];
            nrowval[nnz] = rowval[j];
            nnz++;
          }
        }
        nrowptr[i+1] = nnz;
      }
      gk_free((void **)&collen, LTERM);
      break;

    case GK_CSR_ROW:
      nrowptr[0] = 0;
      for (nnz=0, i=0; i<nrows; i++) {
        if (rowptr[i+1]-rowptr[i] >= minf && rowptr[i+1]-rowptr[i] <= maxf) {
          for (j=rowptr[i]; j<rowptr[i+1]; j++, nnz++) {
            nrowind[nnz] = rowind[j];
            nrowval[nnz] = rowval[j];
          }
        }
        nrowptr[i+1] = nnz;
      }
      break;

    default:
      gk_graph_Free(&ngraph);
      gk_errexit(SIGERR, "Unknown prunning type of %d\n", what);
      return NULL;
  }

  return ngraph;
}



/*************************************************************************/
/*! Normalizes the rows/columns of the graphrix to be unit 
    length.
    \param graph the graphrix itself,
    \param what indicates what will be normalized and is obtained by
           specifying GK_CSR_ROW, GK_CSR_COL, GK_CSR_ROW|GK_CSR_COL. 
    \param norm indicates what norm is to normalize to, 1: 1-norm, 2: 2-norm
*/
/**************************************************************************/
void gk_graph_Normalize(gk_graph_t *graph, int what, int norm)
{
  ssize_t i, j;
  int n;
  ssize_t *ptr;
  float *val, sum;

  if (what&GK_CSR_ROW && graph->rowval) {
    n   = graph->nrows;
    ptr = graph->rowptr;
    val = graph->rowval;

    #pragma omp parallel if (ptr[n] > OMPMINOPS) 
    {
      #pragma omp for private(j,sum) schedule(static)
      for (i=0; i<n; i++) {
        for (sum=0.0, j=ptr[i]; j<ptr[i+1]; j++){
  	if (norm == 2)
  	  sum += val[j]*val[j];
  	else if (norm == 1)
  	  sum += val[j]; /* assume val[j] > 0 */ 
        }
        if (sum > 0) {
  	if (norm == 2)
  	  sum=1.0/sqrt(sum); 
  	else if (norm == 1)
  	  sum=1.0/sum; 
          for (j=ptr[i]; j<ptr[i+1]; j++)
            val[j] *= sum;
  	
        }
      }
    }
  }

  if (what&GK_CSR_COL && graph->colval) {
    n   = graph->ncols;
    ptr = graph->colptr;
    val = graph->colval;

    #pragma omp parallel if (ptr[n] > OMPMINOPS)
    {
    #pragma omp for private(j,sum) schedule(static)
      for (i=0; i<n; i++) {
        for (sum=0.0, j=ptr[i]; j<ptr[i+1]; j++)
  	if (norm == 2)
  	  sum += val[j]*val[j];
  	else if (norm == 1)
  	  sum += val[j]; 
        if (sum > 0) {
  	if (norm == 2)
  	  sum=1.0/sqrt(sum); 
  	else if (norm == 1)
  	  sum=1.0/sum; 
          for (j=ptr[i]; j<ptr[i+1]; j++)
            val[j] *= sum;
        }
      }
    }
  }
}


#endif