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, edgelist_t **edgelists_pointer, int **words_pointer) // the edgelists_pointer and words_pointer arguments are just for freeing afterwards
{
  queue_t queue;

  int rank, order;
  edgelist_t *edge, *previous;
  int edgelist_count, max_wordlength, hyperplane_count;
  int current;

  int *graph_data;
  node_t *graph_unsorted;
  int *wordlength_order, *reverse_wordlength_order, *seen, *words;
  edgelist_t *edgelists;

  // initialize

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

  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].left = graph[i].left;
    graph_unsorted[i].right = graph[i].right;
    graph_unsorted[i].word = 0;
    graph_unsorted[i].wordlength = INT_MAX;
    graph_unsorted[i].bruhat_lower = 0;
    graph_unsorted[i].bruhat_higher = 0;
    graph_unsorted[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[j] = graph_data[i*rank + j];

  // 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);
      }
    }
  }

  max_wordlength = 0;
  for(int i = 0; i < order; i++)
    if(graph_unsorted[i].wordlength > max_wordlength)
      max_wordlength = graph_unsorted[i].wordlength;

  // 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++) {
    graph[i] = graph_unsorted[wordlength_order[i]]; // copy the whole thing
    for(int j = 0; j < rank; j++)
      graph[i].left[j] = reverse_wordlength_order[graph[i].left[j]]; // rewrite references
  }

  // find words

  words = (int*)malloc(order*max_wordlength*sizeof(int));
  memset(words, 0, order*max_wordlength*sizeof(int));
  graph[0].word = &words[0];
  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) {
	graph[neighbor].word = &words[neighbor*max_wordlength];
	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

  edgelists = (edgelist_t*)malloc(order*hyperplane_count*sizeof(edgelist_t));
  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[edgelist_count].to = j;
	  edgelists[edgelist_count].next = graph[current].bruhat_lower;
	  graph[current].bruhat_lower = &edgelists[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

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

  *edgelists_pointer = edgelists;
  *words_pointer = words;

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

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

typedef struct {
  int rank;
  int order;
  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->order; 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->order, level, current_level, info->alphabet, stderr);
    fprintf(stderr, "\n");
  }
  else if(info->printstep < 0) {
    print_thickening(info->rank, info->order, 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_fat;
  int count = 0;

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

    output_thickening(info, level, current_level, is_slim, is_fat, count);

    if(is_fat) {
      count++;
      fwrite(level, sizeof(signed char), info->order, 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->order; 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, const char *alphabet, FILE *outfile)
{
  //  int rank, order;
  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.graph = graph;
  info.alphabet = (char*)alphabet;
  info.outfile = outfile;

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

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

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

  free(level);

  return count;
}