enumerate-balanced-ideals/thickenings.c

#define _GNU_SOURCE

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

#include "thickenings.h"
#include "coxeter.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;

  int *graph_data;
  node_t *graph_unsorted;
  int *wordlength_order, *reverse_wordlength_order, *seen;

  // initialize

  rank = coxeter_rank(type);
  order = coxeter_order(type);
  hyperplanes = coxeter_hyperplanes(type);

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

  graph_data = (int*)malloc(order*rank*sizeof(int));
  graph_unsorted = (node_t*)malloc(order*sizeof(node_t));
  wordlength_order = (int*)malloc(order*sizeof(int));
  reverse_wordlength_order = (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;
  }

  // get coxeter graph

  generate_coxeter_graph(type, graph_data);

  for(int i = 0; i < order; i++)
    for(int j = 0; j < rank; j++)
      graph_unsorted[i].left = &graph_data[i*rank];

  // find wordlengths

  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);
      }
    }
  }

  // sort by wordlength

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

  // find words

  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);
      }
    }
  }

  // generate right edges

  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;
    }
  }

  // find opposites

  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;
  }

  // enumerate hyperplanes

  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++;
      }
    }
  }

  // generate folding order

  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");
	}
      }
    }
  }

  // remove redundant edges

  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;
	}
      }
    }
  }

  // reverse folding order

  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;
    }
  }

  free(graph_data);
  free(graph_unsorted);
  free(wordlength_order);
  free(reverse_wordlength_order);
  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;

  if(opposition_involution(type, left) != left)
    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);

  // initialize stuff

  rank = coxeter_rank(type);
  order = coxeter_order(type);
  hyperplanes = coxeter_hyperplanes(type);

  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;
  }

  // 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]);
      }
    }
  }

  // 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]];

  //  fprintf(stderr, "Number of double cosets: %d\n\n", ncosets);

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

  // 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]];
      }
    }

  // 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;
    }
  }

  // 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
	  //	  fprintf(stderr, "removing edge from %d to %d\n", i, edge->to);
	  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;
	}
      }
    }
  }

  // step 8: revert order
  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;
    }
  }

  // output as graphviz dot file
  /*
  fprintf(stdout, "difull_graph test123 {\n");
  for(int i = 0; i < ncosets; i++) {
    edge = simplified_graph[i].bruhat_lower;
    while(edge) {
      fprintf(stdout, "%s -> %s;\n",
      	     alphabetize(simplified_graph[i].word, simplified_graph[i].wordlength, alphabet, buffer),
	     alphabetize(simplified_graph[edge->to].word, simplified_graph[edge->to].wordlength, alphabet, buffer2));

      edge = edge->next;
    }
  }
  fprintf(stdout, "}\n"); */

  // some output
  /*  for(int i = 0; i < ncosets; i++)
      fprintf(stderr, "%s <=> %s\n", simplified_graph[i].wordlength == 0 ? "1" : alphabetize(simplified_graph[i].word, simplified_graph[i].wordlength, alphabet, buffer), simplified_graph[simplified_graph[i].opposite].wordlength == 0 ? "1" : alphabetize(simplified_graph[simplified_graph[i].opposite].word, simplified_graph[simplified_graph[i].opposite].wordlength, alphabet, buffer2)); */

  //  fprintf(stderr, "\nAdded %d edges.\n\n", edgelists_used);

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

  return ncosets;
}

node_t *graph_alloc(semisimple_type_t type)
{
  int rank = coxeter_rank(type);
  int order = coxeter_order(type);
  int hyperplanes = coxeter_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 = coxeter_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 rank;
  int order;
  int size;        // the size of the graph; this can vary from the order if we take quotients beforehand
  const node_t *graph;
  int printstep;
  const char *alphabet;
  FILE *outfile;
} enumeration_info_t;

// calculate transitive closure; that is, fill current_level in every spot which must be marked with the current head (but was not already marked before), and -current_level in every opposite spot (including opposite to the head)
static int transitive_closure(const enumeration_info_t *info, signed char *level, int head, int current_level)
{
  int is_slim = 1;
  queue_t queue;
  int current;
  edgelist_t *edge;

  queue_init(&queue);
  level[info->graph[head].opposite] = -current_level;
  queue_put(&queue, head);

  for(int i = head + 1; level[i] != HEAD_MARKER && i < info->size; i++) { // everything which is right to the head and empty will not get marked in this or higher levels, so we can mark its opposite
    if(level[i] == current_level) {
      is_slim = 0;
      break;
    } if(level[i] == 0) {
      level[i] = -current_level;
      level[info->graph[i].opposite] = current_level;
      queue_put(&queue, info->graph[i].opposite);
    }
  }

  if(is_slim) {
    while((current = queue_get(&queue)) != -1) {
      edge = info->graph[current].bruhat_lower;
      while(edge) {
	if(level[edge->to] < 0) {
	  is_slim = 0;
	  break;
	} else if(level[edge->to] == 0) {
	  level[edge->to] = current_level;
	  level[info->graph[edge->to].opposite] = -current_level;
	  queue_put(&queue, edge->to);
	}
	edge = edge->next;
      }
    }
  }

  return is_slim;
}

static inline void output_thickening(const enumeration_info_t *info, signed char *level, int current_level, int is_slim, int is_fat, long count)
{
  // if printstep is set accordingly, write state to stderr
  if(is_slim && is_fat && info->printstep > 0 && (count + 1) % info->printstep == 0) {
    print_thickening(info->rank, info->size, level, current_level, info->alphabet, stderr);
    fprintf(stderr, "\n");
  }
  else if(info->printstep < 0) {
    print_thickening(info->rank, info->size, level, current_level - !is_slim, info->alphabet, stderr);
    fprintf(stderr, " ");
    if(is_slim) {
      fprintf(stderr, "S");
      if(is_fat)
	fprintf(stderr, "F");
    }
    fprintf(stderr, "\n");
  }
}

static long enumerate_tree(const enumeration_info_t *info, signed char *level, int current_level, int head)
{
  ERROR(current_level >= HEAD_MARKER, "HEAD_MARKER too small!\n");

  level[head] = HEAD_MARKER;

  int is_slim = transitive_closure(info, level, head, current_level);
  int is_balanced = 0;
  int count = 0;

  // we have a candidate, check if it is a balanced thickening; if so, write to stdout
  if(is_slim) {
    is_balanced = 1;
    for(int i = head - 1; i >= 0; i--)
      if(level[i] == 0)
	is_balanced = 0;
  }

  // comment this out (or just put it inside the if block) to save 1/3 of the runtime
  output_thickening(info, level, current_level, is_slim, is_balanced, count);

  if(is_slim) {
    if(is_balanced) {
      count++;
      fwrite(level, sizeof(signed char), info->size, info->outfile);
    } else {
      for(int i = head - 1; i >= 0; i--)
	if(level[i] == 0)
	  count += enumerate_tree(info, level, current_level + 1, i);
    }
  }

  // clean up
  level[head] = 0;
  for(int i = 0; i < info->size; i++)
    if(level[i] >= current_level && level[i] != HEAD_MARKER || level[i] <= -current_level)
      level[i] = 0;

  return count;
}

long enumerate_balanced_thickenings(semisimple_type_t type, node_t *graph, int size, const char *alphabet, FILE *outfile)
{
  signed char *level;
  long count = 0;
  enumeration_info_t info;
  queue_t queue;
  int current;

  info.rank = coxeter_rank(type);
  info.order = coxeter_order(type);
  info.size = size;
  info.graph = graph;
  info.alphabet = (char*)alphabet;
  info.outfile = outfile;

  info.printstep = 0;
  if(getenv("PRINTSTEP"))
    info.printstep = atoi(getenv("PRINTSTEP"));

  // 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;

  level = (signed char*)malloc(info.size*sizeof(int));
  memset(level, 0, info.size*sizeof(int));

  for(int i = info.size - 1; i >= 0; i--)
    count += enumerate_tree(&info, level, 1, i);

  free(level);

  return count;
}