enumerate-balanced-ideals/thickenings.c

#include <stdio.h>
#include <limits.h>
#include <stdlib.h>
#include <malloc.h>
#include <memory.h>

#include "thickenings.h"
#include "weyl.h"
#include "queue.h"

char *alphabetize(int *word, int len, const char *alphabet, char *buffer)
{
  if(len == 0) {
    buffer[0] = '1';
    buffer[1] = 0;
    return buffer;
  }

  int i = 0;
  for(i = 0; i < len; i++)
    buffer[i] = alphabet[word[i]];
  buffer[i] = 0;

  return buffer;
}

void print_thickening(int rank, int order, const signed char *thickening, int upto_level, const char *alphabet, FILE *f)
{
  for(int i = 0; i < order; i++) {
    if(thickening[i] == HEAD_MARKER)
      fprintf(f, "\e[41;37mx\e[0m");
    else if(thickening[i] < - upto_level || thickening[i] > upto_level)
      fprintf(f, " ");
    else if(thickening[i] < 0 && thickening[i] > -10)
      fprintf(f, "\e[47;30m%d\e[0m", -thickening[i]);
    else if(thickening[i] <= -10)
      fprintf(f, "\e[47;30m+\e[0m");
    else if(thickening[i] > 0 && thickening[i] < 10)
      fprintf(f, "\e[40;37m%d\e[0m", thickening[i]);
    else if(thickening[i] >= 10)
      fprintf(f, "\e[40;37m+\e[0m");
    else
      fprintf(f, " ");
  }

  fprintf(f, "\e[K");
}

static int compare_wordlength(const void *a, const void *b, void *gr)
{
  int i = *((int*)a);
  int j = *((int*)b);
  node_t *graph = (node_t*)gr;

  return graph[i].wordlength - graph[j].wordlength;
}

void prepare_graph(semisimple_type_t type, node_t *graph)
{
  queue_t queue;

  edgelist_t *edgelists_lower, *edgelists_higher;
  int rank, order, hyperplanes;
  edgelist_t *edge, *previous;
  int edgelist_count, hyperplane_count;
  int current;

  weylgroup_element_t *graph_data;
  node_t *graph_unsorted;
  int *ordering, *reverse_ordering, *seen;

  // initialize

  rank = weyl_rank(type);
  order = weyl_order(type);
  hyperplanes = weyl_hyperplanes(type);

  edgelists_higher = graph[0].bruhat_higher;
  edgelists_lower = &graph[0].bruhat_higher[order*hyperplanes/2];

  graph_data = weyl_alloc(type);
  graph_unsorted = (node_t*)malloc(order*sizeof(node_t));
  ordering = (int*)malloc(order*sizeof(int));
  reverse_ordering = (int*)malloc(order*sizeof(int));
  seen = (int*)malloc(order*sizeof(int));

  for(int i = 0; i < order; i++) {
    graph_unsorted[i].wordlength = INT_MAX;
    graph[i].bruhat_lower = 0;
    graph[i].bruhat_higher = 0;
    graph[i].is_hyperplane_reflection = 0;
  }

  LOG("Generate Weyl group.\n");

  weyl_generate(type, graph_data);

  for(int i = 0; i < order; i++)
    for(int j = 0; j < rank; j++) {
      graph_unsorted[i].left = graph_data[i].left;
      graph_unsorted[i].id = graph_data[i].id;
    }

  // find wordlengths

  LOG("Determine word lengths.\n");

  graph_unsorted[0].wordlength = 0;
  queue_init(&queue);
  queue_put(&queue, 0);
  while((current = queue_get(&queue)) != -1) {
    for(int i = 0; i < rank; i++) {
      int neighbor = graph_unsorted[current].left[i];
      if(graph_unsorted[neighbor].wordlength > graph_unsorted[current].wordlength + 1) {
	graph_unsorted[neighbor].wordlength = graph_unsorted[current].wordlength + 1;
	queue_put(&queue, neighbor);
      }
    }
  }

  LOG("Sort by wordlength.\n");

  for(int i = 0; i < order; i++)
    ordering[i] = i;
  qsort_r(ordering, order, sizeof(int), compare_wordlength, graph_unsorted); // so ordering is a map new index -> old index
  for(int i = 0; i < order; i++)
    reverse_ordering[ordering[i]] = i; // reverse_ordering is a map old index -> new index
  for(int i = 0; i < order; i++) {
    // we have only set left, wordlength and id so far, so just copy these
    graph[i].wordlength = graph_unsorted[ordering[i]].wordlength;
    graph[i].id = graph_unsorted[ordering[i]].id;
    for(int j = 0; j < rank; j++)
      graph[i].left[j] = reverse_ordering[graph_unsorted[ordering[i]].left[j]]; // rewrite references
  }

  LOG("Find shortest words.\n");

  for(int i = 0; i < order; i++)
    memset(graph[i].word, 0, hyperplanes*sizeof(int));
  queue_init(&queue);
  queue_put(&queue, 0);
  while((current = queue_get(&queue)) != -1) {
    for(int i = 0; i < rank; i++) {
      int neighbor = graph[current].left[i];
      if(graph[neighbor].wordlength == graph[current].wordlength + 1 && graph[neighbor].word[0] == 0) {
	memcpy(&graph[neighbor].word[1], &graph[current].word[0], graph[current].wordlength*sizeof(int));
	graph[neighbor].word[0] = i;
	queue_put(&queue, neighbor);
      }
    }
  }

  LOG("Generate right edges.\n");

  for(int i = 0; i < order; i++) {
    for(int j = 0; j < rank; j++) {
      current = graph[0].left[j];
      for(int k = graph[i].wordlength - 1; k >= 0; k--) { // apply group element from right to left
	current = graph[current].left[graph[i].word[k]];
      }
      graph[i].right[j] = current;
    }
  }

  LOG("Find opposites.\n");

  node_t *longest = &graph[order-1];
  for(int i = 0; i < order; i++) {
    current = i;
    for(int k = longest->wordlength - 1; k >= 0; k--)
      current = graph[current].left[longest->word[k]];
    graph[i].opposite = current;
  }

  LOG("Enumerate hyperplanes.\n");

  hyperplane_count = 0;
  for(int i = 0; i < order; i++) {
    for(int j = 0; j < rank; j++) {
      current = 0;
      int *word1 = graph[i].word;
      int word1len = graph[i].wordlength;
      int *word2 = graph[graph[i].right[j]].word; // want to calculate word2 * word1^{-1}
      int word2len = graph[graph[i].right[j]].wordlength;
      for(int k = 0; k < word1len; k++) // apply inverse, i.e. go from left to right
	current = graph[current].left[word1[k]];
      for(int k = word2len - 1; k >= 0; k--) // now from right to left
	current = graph[current].left[word2[k]];
      if(graph[current].is_hyperplane_reflection == 0) {
	graph[current].is_hyperplane_reflection = 1;
	hyperplane_count++;
      }
    }
  }

  LOG("Determine Bruhat order.\n");

  edgelist_count = 0;
  for(int i = 0; i < order; i++) {
    if(graph[i].is_hyperplane_reflection) {
      for(int j = 0; j < order; j++) {

	current = j;
	for(int k = graph[i].wordlength - 1; k >= 0; k--) // apply hyperplane reflection
	  current = graph[current].left[graph[i].word[k]];

	if(graph[j].wordlength < graph[current].wordlength) { // current has higher bruhat order than j
	  edgelists_lower[edgelist_count].to = j;
	  edgelists_lower[edgelist_count].next = graph[current].bruhat_lower;
	  graph[current].bruhat_lower = &edgelists_lower[edgelist_count];
	  edgelist_count++;
	} else if(graph[j].wordlength > graph[current].wordlength) { // j has higher bruhat order than current; these are already included from the other side
	} else {
	  ERROR(1, "Chambers of equal word lengths should not be folded on each other!\n");
	}
      }
    }
  }

  LOG("Perform transitive reduction.\n");

  for(int i = 0; i < order; i++) {
    memset(seen, 0, order*sizeof(int));
    queue_init(&queue);

    for(int len = 1; len <= graph[i].wordlength; len++) {
      // remove all edges originating from i of length len which connect to something already seen using shorter edges
      edge = graph[i].bruhat_lower;
      previous = (edgelist_t*)0;

      while(edge) {
	if(graph[i].wordlength - graph[edge->to].wordlength != len) {
	  previous = edge;
	} else if(seen[edge->to]) {
	  if(previous)
	    previous->next = edge->next;
	  else
	    graph[i].bruhat_lower = edge->next;
	} else {
	  previous = edge;
	  seen[edge->to] = 1;
	  queue_put(&queue, edge->to);
	}
	edge = edge->next;
      }

      // see which nodes we can reach using only edges up to length len, mark them as seen
      while((current = queue_get(&queue)) != -1) {
	edge = graph[current].bruhat_lower;
	while(edge) {
	  if(!seen[edge->to]) {
	    seen[edge->to] = 1;
	    queue_put(&queue, edge->to);
	  }
	  edge = edge->next;
	}
      }
    }
  }

  LOG("Revert Bruhat order.\n");

  edgelist_count = 0;
  for(int i = 0; i < order; i++) {
    edge = graph[i].bruhat_lower;
    while(edge) {
      edgelists_higher[edgelist_count].to = i;
      edgelists_higher[edgelist_count].next = graph[edge->to].bruhat_higher;
      graph[edge->to].bruhat_higher = &edgelists_higher[edgelist_count];
      edgelist_count++;
      edge = edge->next;
    }
  }

  LOG("Sort opposites.\n");

  // additional sorting step to force opposite property (opposite of j is at n - j - 1)

  for(int i = 0; i < order; i++)
    reverse_ordering[i] = -1;
  for(int i = 0, j = 0; i < order; i++) {  // i = old index, j = new index
    if(reverse_ordering[i] == -1) {
      reverse_ordering[i] = j;
      ordering[j] = i;
      reverse_ordering[graph[i].opposite] = order - j - 1;
      ordering[order - j - 1] = graph[i].opposite;
      j++;
    }
  }
  memcpy(graph_unsorted, graph, order*sizeof(node_t));
  for(int i = 0; i < order; i++) {
    graph[i] = graph_unsorted[ordering[i]];
    graph[i].opposite = reverse_ordering[graph[i].opposite];
    for(int j = 0; j < rank; j++) {
      graph[i].left[j] = reverse_ordering[graph[i].left[j]];
      graph[i].right[j] = reverse_ordering[graph[i].right[j]];
    }
    for(edge = graph[i].bruhat_lower; edge; edge = edge->next)
      edge->to = reverse_ordering[edge->to];
    for(edge = graph[i].bruhat_higher; edge; edge = edge->next)
      edge->to = reverse_ordering[edge->to];
  }

  weyl_free(graph_data);
  free(graph_unsorted);
  free(ordering);
  free(reverse_ordering);
  free(seen);
}

static int edgelist_contains(edgelist_t *list, int x) {
  while(list) {
    if(list->to == x)
      return 1;
    list = list->next;
  }
  return 0;
}

static edgelist_t *edgelist_add(edgelist_t *list, int new, edgelist_t *storage, int *storage_index)
{
  edgelist_t *new_link = &storage[*storage_index];
  new_link->next = list;
  new_link->to = new;
  (*storage_index)++;
  return new_link;
}

int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigned long right, node_t *simplified_graph)
{
  node_t *full_graph;
  int edgelists_used;
  int rank, order, hyperplanes;
  int *reduced, *group, *simplified;
  int *seen;
  int current;
  edgelist_t *edge, *previous;
  queue_t queue;
  int ncosets;

  rank = weyl_rank(type);
  order = weyl_order(type);
  hyperplanes = weyl_hyperplanes(type);

  for(int i = 0; i < rank; i++) {
    int oppi = weyl_opposition(type, i);
    if(left & BIT(i) && !(left & BIT(oppi)) ||
       left & BIT(oppi) && !(left & BIT(i)))
      return -1;
  }

  edgelist_t *edgelists_higher = &simplified_graph[0].bruhat_higher[0];
  edgelist_t *edgelists_lower = &simplified_graph[0].bruhat_higher[order*hyperplanes/2];

  // get full graph

  full_graph = graph_alloc(type);
  prepare_graph(type, full_graph);

  LOG("Full graph generated.\n");

  // initialize stuff

  reduced = (int*)malloc(order*sizeof(int));
  group = (int*)malloc(order*sizeof(int));
  simplified = (int*)malloc(order*sizeof(int));
  for(int i = 0; i < order; i++) {
    group[i] = -1;
    reduced[i] = i;
  }

  LOG("Group by double coset.\n");

  // step 1: group
  for(int i = 0; i < order; i++) {
    if(group[i] != -1)
      continue;

    queue_init(&queue);
    queue_put(&queue, i);
    while((current = queue_get(&queue)) != -1) {
      if(group[current] != -1)
        continue;
      group[current] = i;

      for(int j = 0; j < rank; j++) {
        if(left & (1 << j))
          queue_put(&queue, full_graph[current].left[j]);
        if(right & (1 << j))
          queue_put(&queue, full_graph[current].right[j]);
      }
    }
  }

  LOG("Find minimal length elements.\n");

  // step 2: find minimum
  for(int i = 0; i < order; i++)
    if(full_graph[i].wordlength < full_graph[reduced[group[i]]].wordlength)
      reduced[group[i]] = i;

  // step 3: assign minimum to all
  for(int i = 0; i < order; i++)
    reduced[i] = reduced[group[i]];

  // step 4: assign indices to cosets
  ncosets = 0;
  for(int i = 0; i < order; i++)
    if(reduced[i] == i)
      simplified[i] = ncosets++;

  for(int i = 0; i < order; i++)
    simplified[i] = simplified[reduced[i]];

  seen = (int*) malloc(ncosets*sizeof(int));
  edgelists_used = 0;

  LOG("Copy minimal elements.\n");

  // step 5: set up nodes from minima
  current = 0;
  for(int i = 0; i < order; i++)
    if(reduced[i] == i) { // is minimum
      memcpy(simplified_graph[simplified[i]].word, full_graph[i].word, full_graph[i].wordlength*sizeof(int));
      simplified_graph[simplified[i]].wordlength = full_graph[i].wordlength;
      simplified_graph[simplified[i]].opposite = simplified[full_graph[i].opposite];
      simplified_graph[simplified[i]].bruhat_lower = (edgelist_t*)0;
      simplified_graph[simplified[i]].bruhat_higher = (edgelist_t*)0;
      for(int j = 0; j < rank; j++) {
	simplified_graph[simplified[i]].left[j] = simplified[full_graph[i].left[j]];
	simplified_graph[simplified[i]].right[j] = simplified[full_graph[i].right[j]];
      }
    }

  LOG("Find induced order.\n");

  // step 6: find order relations
  for(int i = 0; i < order; i++) {
    edge = full_graph[i].bruhat_lower;
    while(edge) {
      int this = simplified[i];
      int that = simplified[edge->to];
      if(this != that) {
	// found something
	if(!edgelist_contains(simplified_graph[this].bruhat_lower, that))
	  simplified_graph[this].bruhat_lower = edgelist_add(simplified_graph[this].bruhat_lower, that, edgelists_lower, &edgelists_used);
	ERROR(simplified_graph[this].wordlength <= simplified_graph[that].wordlength, "The order assumption is being violated!\n");
      }
      edge = edge->next;
    }
  }

  LOG("Perform transitive reduction.\n");

  // step 7: remove redundant edges
  for(int i = 0; i < ncosets; i++) {
    memset(seen, 0, ncosets*sizeof(int));
    queue_init(&queue);

    for(int len = 1; len <= simplified_graph[i].wordlength; len++) {
      edge = simplified_graph[i].bruhat_lower;
      previous = (edgelist_t*)0;

      while(edge) {
	// only look at edges of this length now
	if(simplified_graph[i].wordlength - simplified_graph[edge->to].wordlength != len) {
	  // we only consider edges of length len in this pass
	  previous = edge;
	} else if(seen[edge->to]) {
	  // this edge is redundant, remove it
	  if(previous)
	    previous->next = edge->next;
	  else
	    simplified_graph[i].bruhat_lower = edge->next;
	} else {
	  // this edge was not redundant, add to seen
	  previous = edge;
	  seen[edge->to] = 1;
	  queue_put(&queue, edge->to);
	}
	edge = edge->next;
      }

      // calculate transitive closure of seen nodes
      while((current = queue_get(&queue)) != -1) {
	edge = simplified_graph[current].bruhat_lower;
	while(edge) {
	  if(!seen[edge->to]) {
	    seen[edge->to] = 1;
	    queue_put(&queue, edge->to);
	  }
	  edge = edge->next;
	}
      }
    }
  }

  LOG("Revert order.\n");

  // step 8: revert order
  edgelists_used = 0;
  for(int i = 0; i < ncosets; i++) {
    edge = simplified_graph[i].bruhat_lower;
    while(edge) {
      simplified_graph[edge->to].bruhat_higher =
	edgelist_add(simplified_graph[edge->to].bruhat_higher,
		     i, edgelists_higher, &edgelists_used);
      edge = edge->next;
    }
  }

  LOG("Sort opposites.\n");

  int *ordering = (int*)malloc(ncosets*sizeof(int));
  int *reverse_ordering = (int*)malloc(ncosets*sizeof(int));
  node_t *unsorted = (node_t*)malloc(ncosets*sizeof(node_t));
  int opp, pos;

  pos = 0;
  for(int i = 0; i < ncosets; i++) {  // first all the pairs
    opp = simplified_graph[i].opposite;
    if(opp > i) { // first occurrence of this pair
      ordering[pos] = i;
      ordering[ncosets-1-pos] = opp;
      reverse_ordering[i] = pos;
      reverse_ordering[opp] = ncosets-1-pos;
      pos++;
    }
  }
  for(int i = 0; i < ncosets; i++)  // and finally the self-opposites
    if(simplified_graph[i].opposite == i) {
      ordering[pos] = i;
      reverse_ordering[i] = pos;
      pos++;
    }

  // now really do it
  memcpy(unsorted, simplified_graph, ncosets*sizeof(node_t));
  for(int i = 0; i < ncosets; i++) {
    simplified_graph[i] = unsorted[ordering[i]];
    simplified_graph[i].opposite = reverse_ordering[simplified_graph[i].opposite];
    for(edgelist_t *edge = simplified_graph[i].bruhat_lower; edge != (edgelist_t*)0; edge = edge->next)
      edge->to = reverse_ordering[edge->to];
    for(edgelist_t *edge = simplified_graph[i].bruhat_higher; edge != (edgelist_t*)0; edge = edge->next)
      edge->to = reverse_ordering[edge->to];
    for(int j = 0; j < rank; j++) {
      simplified_graph[i].left[j] = reverse_ordering[simplified_graph[i].left[j]];
      simplified_graph[i].right[j] = reverse_ordering[simplified_graph[i].right[j]];
    }
  }

  free(ordering);
  free(reverse_ordering);
  free(unsorted);
  free(seen);
  free(reduced);
  free(group);
  free(simplified);
  graph_free(type, full_graph);

  LOG("Simplified graph generated.\n");

  return ncosets;
}

node_t *graph_alloc(semisimple_type_t type)
{
  int rank = weyl_rank(type);
  int order = weyl_order(type);
  int hyperplanes = weyl_hyperplanes(type);

  node_t *graph = (node_t*)malloc(order*sizeof(node_t));
  int *left = (int*)malloc(order*rank*sizeof(int));
  int *right = (int*)malloc(order*rank*sizeof(int));
  edgelist_t *edgelists = (edgelist_t*)malloc(order*hyperplanes*sizeof(edgelist_t));
  int *words = (int*)malloc(order*hyperplanes*sizeof(int));

  for(int i = 0; i < order; i++) {
    graph[i].left = &left[rank*i];
    graph[i].right = &right[rank*i];
    graph[i].word = &words[hyperplanes*i];
  }

  graph[0].bruhat_higher = edgelists;

  return graph;
}

void graph_free(semisimple_type_t type, node_t *graph)
{
  free(graph[0].left);
  free(graph[0].right);
  free(graph[0].word);

  int order = weyl_order(type);

  // find the head of all edgelists by just taking the one having the lowest address
  edgelist_t *edgelists = graph[0].bruhat_lower;
  for(int i = 0; i < order; i++) {
    if(graph[i].bruhat_lower < edgelists && graph[i].bruhat_lower != 0)
      edgelists = graph[i].bruhat_lower;
    if(graph[i].bruhat_higher < edgelists && graph[i].bruhat_higher != 0)
      edgelists = graph[i].bruhat_higher;
  }
  free(edgelists);
}

/*********************************** THE ACTUAL ENUMERATION ****************************************/

typedef struct {
  int size;        // the size of the weyl group. We store however only the first size/2 elements
  bitvec_t *principal_pos;
  bitvec_t *principal_neg;
  int *principal_is_slim;
  void (*callback)(const bitvec_t *, int, void*);
  void *callback_data;
} enumeration_info_t;

/*
  This function enumerates all balanced ideals satisfying certain constraints, given by its arguments pos, neg and next_neg

  - info: constant information which just gets passed on to recursive calls, mainly contains the principal ideals
  - pos: a set of elements which have to be positive (that is, in the ideal)
  - neg: a set of elements which have to be negative (not in the ideal)
  - next_neg: this element has to be the first negative one not already contained in neg; if next_neg is info.size/2, then everything not in neg has to be positive
  - already_known: not a constraint, but just a hint to speed things up; tells the function that the first already_known elements are set either in neg or in pos; must be less or equal to next_neg
  - returns number of balanced ideals found

  uses the bitvector functions bv_union, bv_copy, bv_set_range_except, bv_disjoint, bv_next_zero

 */

static long enumerate_tree(const enumeration_info_t *info, const bitvec_t *pos, const bitvec_t *neg, int next_neg, int already_known)
{
  static long totcount = 0;
  bitvec_t newpos, newneg, known;
  int next_next_neg;
  long count = 0;

  // the omission of next_neg means inclusion of info->size - 1 - next_neg
  // add its principal ideal to pos and the opposite to neg
  if(next_neg != info->size/2) {
    // if the principal ideal we want to add is not slim by itself, we don't even have to try; but there is not really a performance benefit, it rather makes it slower
    //    if(!info->principal_is_slim[info->size - 1 - next_neg])
    //      return 0;
    bv_union(&info->principal_pos[info->size - 1 - next_neg], pos, &newpos);
    bv_union(&info->principal_neg[info->size - 1 - next_neg], neg, &newneg);
  } else { // or, if there is no next_neg, just copy
    bv_copy(pos, &newpos);
    bv_copy(neg, &newneg);
  }

  // everything before next_neg which was unknown should be set to positive; to speed this up, we can start with already_known
  bv_set_range_except(&newpos, neg, already_known, next_neg);

  #ifdef _DEBUG
  bv_print_nice(stderr, &newpos, &newneg, -1, info->size/2);
  fprintf(stderr, "\n");
  #endif

  // check if this leads to any conflicts (equivalently, violates slimness)
  if(!bv_disjoint(&newpos, &newneg))
    return 0;

  // what do we know so far?
  bv_union(&newpos, &newneg, &known);

  next_next_neg = bv_next_zero(&known, next_neg + 1);

  if(next_next_neg >= info->size/2) {
    // there is no unknown left, so we found a balanced ideal
    if(info->callback)
      info->callback(&newpos, info->size, info->callback_data);
    return 1;
  }

  do {
    count += enumerate_tree(info, &newpos, &newneg, next_next_neg, next_neg + 1);
    next_next_neg = bv_next_zero(&known, next_next_neg + 1);
  } while(next_next_neg <= info->size/2);

  return count;
}

/*
   enumerates all balanced ideals

   - graph: hasse diagram of the bruhat order (of double cosets) with opposition pairing
   - size: number of nodes in graph
   - callback to call when a balanced ideal was found
   - arbitrary data for callback function

   returns the number of balanced ideals
*/

long enumerate_balanced_thickenings(node_t *graph, int size, void (*callback) (const bitvec_t *, int, void*), void *callback_data)
{
  signed char *level;
  long count = 0;
  enumeration_info_t info;
  queue_t queue;
  int current;
  edgelist_t *edge;

  info.size = size;
  info.callback = callback;
  info.callback_data = callback_data;
  info.principal_pos = (bitvec_t*)malloc(info.size*sizeof(bitvec_t));
  info.principal_neg = (bitvec_t*)malloc(info.size*sizeof(bitvec_t));
  info.principal_is_slim = (int*)malloc(info.size*sizeof(int));

  // the algorithm only works if the opposition pairing does not stabilize any element
  // if this happens, there can be no balanced thickenings
  for(int i = 0; i < info.size; i++)
    if(graph[i].opposite == i)
      return 0;

  // we can only handle bitvectors up to 64*BV_QWORD_RANK bits, but we only store half of the weyl group
  if(info.size > 128*BV_QWORD_RANK)
    return -1;

  // generate principal ideals
  int *principal = (int*)malloc(info.size*sizeof(int));
  for(int i = 0; i < info.size; i++) {
    memset(principal, 0, info.size*sizeof(int));
    principal[i] = 1;
    queue_init(&queue);
    queue_put(&queue, i);
    while((current = queue_get(&queue)) != -1)
      for(edge = graph[current].bruhat_lower; edge; edge = edge->next)
	if(!principal[edge->to]) {
	  principal[edge->to] = 1;
	  queue_put(&queue, edge->to);
	}

    // copy the first half into bitvectors
    bv_clear(&info.principal_pos[i]);
    bv_clear(&info.principal_neg[i]);
    info.principal_is_slim[i] = 1;
    for(int j = 0; j < info.size/2; j++)
      if(principal[j])
	bv_set_bit(&info.principal_pos[i], j);
    for(int j = 0; j < info.size/2; j++)
      if(principal[info.size - 1 - j]) {
	bv_set_bit(&info.principal_neg[i], j);
	if(bv_get_bit(&info.principal_pos[i], j))
	  info.principal_is_slim[i] = 0;
      }
  }
  free(principal);

  // output principal ideals
#ifdef _DEBUG
  for(int i = 0; i < info.size; i++) {
    fprintf(stderr, "%2d: ", i);
    bv_print_nice(stderr, &info.principal_pos[i], &info.principal_neg[i], -1, info.size/2);
    fprintf(stderr, "\n");
  }
  fprintf(stderr,"\n");
#endif

  // enumerate balanced ideals
  bitvec_t pos, neg;
  bv_clear(&pos);
  bv_clear(&neg);
  for(int i = 0; i <= info.size/2; i++)
    count += enumerate_tree(&info, &pos, &neg, i, 0);

  free(info.principal_is_slim);
  free(info.principal_pos);
  free(info.principal_neg);

  return count;
}