diff --git a/Makefile b/Makefile
index 7b6c37c..b3eafce 100644
--- a/Makefile
+++ b/Makefile
@@ -1,5 +1,6 @@
 HEADERS=coxeter.h thickenings.h queue.h
 OPTIONS=-O3 -m64 -march=native -flto -funroll-loops -std=gnu99
+#OPTIONS=-O0 -g -std=gnu99
 
 all: generate process
 
diff --git a/coxeter.c b/coxeter.c
index 10a9e96..ca8ab50 100644
--- a/coxeter.c
+++ b/coxeter.c
@@ -176,6 +176,37 @@ int coxeter_hyperplanes(semisimple_type_t type)
   return hyperplanes;
 }
 
+unsigned long opposition_involution(semisimple_type_t type, unsigned long theta)
+{
+  int offset = 0;
+  unsigned long result = 0;
+  for(int i = 0; i < type.n; i++) {
+    unsigned long current = (theta >> offset) & ((1 << type.factors[i].rank) - 1);
+    unsigned long iota_current;
+
+    if(type.factors[i].series == 'B' || type.factors[i].series == 'C' || type.factors[i].series == 'F' || type.factors[i].series == 'G') {
+      iota_current = current;
+    } else if(type.factors[i].series == 'A') {
+      iota_current = 0;
+      for(int j = 0; j < type.factors[i].rank; j++)
+	iota_current += ((current >> j) & 1) << (type.factors[i].rank - 1 - j);
+    } else if(type.factors[i].series == 'D') {
+      if(type.factors[i].rank % 2 == 0) {
+	iota_current = current;
+      } else {
+	ERROR(1, "The opposition involution for type %c%d is not yet implemented!\n", type.factors[i].series, type.factors[i].rank);
+      }
+    } else if(type.factors[i].series == 'E') {
+      ERROR(1, "The opposition involution for En is not yet implemented!\n");
+    }
+
+    result += iota_current << offset;
+    offset += type.factors[i].rank;
+  }
+
+  return result;
+}
+
 static void generate_starting_vector(int rank, gsl_matrix *schlaefli, gsl_vector *result)
 {
   gsl_matrix *schlaefliLU = gsl_matrix_alloc(rank, rank);
diff --git a/coxeter.h b/coxeter.h
index babe2e9..b62278e 100644
--- a/coxeter.h
+++ b/coxeter.h
@@ -15,5 +15,6 @@ void generate_coxeter_graph(semisimple_type_t type, int *result);
 int coxeter_order(semisimple_type_t type);
 int coxeter_hyperplanes(semisimple_type_t type);
 int coxeter_rank(semisimple_type_t type);
+unsigned long opposition_involution(semisimple_type_t type, unsigned long theta);
 
 #endif
diff --git a/generate.c b/generate.c
index 6e4c87f..818ac16 100644
--- a/generate.c
+++ b/generate.c
@@ -8,56 +8,62 @@
 int main(int argc, const char *argv[])
 {
   semisimple_type_t type;
+  unsigned long right_invariance, left_invariance;
+  int rank, order, hyperplanes, cosets;
 
-  // heap stuff
   node_t *graph;
-  int *left, *right;
-  edgelist_t *edgelists;
-  int *words;
-
-  int rank, order;
 
   char string_buffer1[1000];
   const char *alphabet = "abcdefghijklmnopqrstuvwxyz";
 
+  // read arguments
+
   ERROR(argc < 2, "Too few arguments!\n");
 
-  type.n = argc - 1;
-  type.factors = (simple_type_t*)malloc((argc-1)*sizeof(simple_type_t));
+  type.n = 0;
   for(int i = 0; i < argc - 1; i++) {
+    if(argv[i+1][0] < 'A' || argv[i+1][0] > 'I')
+      break;
+    type.n++;
+  }
+
+  type.factors = (simple_type_t*)malloc(type.n*sizeof(simple_type_t));
+  for(int i = 0; i < type.n; i++) {
     type.factors[i].series = argv[i+1][0];
     type.factors[i].rank = argv[i+1][1] - '0';
     ERROR(argv[i+1][0] < 'A' || argv[i+1][0] > 'I' || argv[i+1][1] < '1' || argv[i+1][1] > '9', "Arguments must be Xn with X out of A-I and n out of 0-9\n");
   }
 
-  rank = coxeter_rank(type);
-  order = coxeter_order(type);
+  left_invariance = right_invariance = 0;
+
+  if(argc - type.n >= 3) {
+    if(strcmp(argv[type.n + 1], "-") != 0)
+      for(int i = 0; i < strlen(argv[type.n + 1]); i++)
+	left_invariance |= (1 << (argv[type.n + 1][i] - 'a'));
+    if(strcmp(argv[type.n + 2], "-") != 0)
+      for(int i = 0; i < strlen(argv[type.n + 2]); i++)
+	right_invariance |= (1 << (argv[type.n + 2][i] - 'a'));
+  }
 
   ERROR(strlen(alphabet) < rank, "The alphabet has too few letters\n");
 
-  // initialize
-
-  graph = (node_t*)malloc(order*sizeof(node_t));
-  left = (int*)malloc(order*rank*sizeof(int));
-  right = (int*)malloc(order*rank*sizeof(int));
-
-  for(int i = 0; i < order; i++) {
-    graph[i].left = &left[rank*i];
-    graph[i].right = &right[rank*i];
-  }
-
   // generate graph
 
-  prepare_graph(type, graph, &edgelists, &words);
+  graph = graph_alloc(type);
+  cosets = prepare_simplified_graph(type, left_invariance, right_invariance, graph);
+
+  ERROR(cosets < 0, "The left invariance is not preserved by the opposition involution: %d %d!\n", left_invariance, opposition_involution(type, left_invariance));
 
   // print stuff
 
-  int hyperplane_count = coxeter_hyperplanes(type);
+  rank = coxeter_rank(type);                // number of simple roots
+  order = coxeter_order(type);              // number of Weyl group elements
+  hyperplanes = coxeter_hyperplanes(type);  // number of positive roots
 
-  fprintf(stderr, "Rank: %d\t\tOrder: %d\t\tHyperplanes: %d\n", rank, order, hyperplane_count);
+  fprintf(stderr, "Rank: %d\tOrder: %d\tPositive Roots: %d\tCosets: %d\n", rank, order, hyperplanes, cosets);
   fprintf(stderr, "\n");
-  fprintf(stderr, "Group elements: \n");
-  for(int i = 0, wl = 0; i < order; i++) {
+  fprintf(stderr, "Shortest coset representatives: \n");
+  for(int i = 0, wl = 0; i < cosets; i++) {
     if(i == 0) {
       fprintf(stderr, "1");
     } else if(graph[i].wordlength > wl) {
@@ -72,16 +78,15 @@ int main(int argc, const char *argv[])
 
   fwrite(&type.n, sizeof(int), 1, stdout);
   fwrite(type.factors, sizeof(simple_type_t), type.n, stdout);
-  long count = enumerate_balanced_thickenings(type, graph, order, alphabet, stdout);
+  fwrite(&left_invariance, sizeof(unsigned long), type.n, stdout);
+  fwrite(&right_invariance, sizeof(unsigned long), type.n, stdout);
+
+  long count = enumerate_balanced_thickenings(type, graph, cosets, alphabet, stdout);
 
   fprintf(stderr, "\n");
   fprintf(stderr, "Found %ld balanced thickenings\n\n", count);
 
-  free(graph);
-  free(left);
-  free(right);
-  free(edgelists);
-  free(words);
+  graph_free(type, graph);
   free(type.factors);
 
   return 0;
diff --git a/process-old.c b/process-old.c
new file mode 100644
index 0000000..0dd3fb4
--- /dev/null
+++ b/process-old.c
@@ -0,0 +1,193 @@
+#include <stdio.h>
+#include <string.h>
+#include <sys/stat.h>
+
+#include "thickenings.h"
+#include "coxeter.h"
+#include "queue.h"
+
+#define SWAP(t, a, b) do {t tmp = a; a = b; b = tmp;} while(0)
+
+char *stringify_SLn1_permutation(int *word, int wordlength, int rank, char *str)
+{
+  for(int i = 0; i <= rank; i++)
+    str[i] = '1' + i;
+  str[rank+1] = 0;
+
+  for(int i = 0; i < wordlength; i++) {
+    char tmp = str[word[i]];
+    str[word[i]] = str[word[i]+1];
+    str[word[i]+1] = tmp;
+  }
+  return str;
+}
+
+
+char *stringify_Onn1_permutation(int *word, int wordlength, int rank, char *str)
+{
+  for(int i = 0; i <= rank*2; i++)
+    str[i] = '1' + i;
+  str[2*rank+1] = 0;
+
+  for(int i = 0; i < wordlength; i++) {
+    if(word[i] == 0)
+      SWAP(char, str[rank-1], str[rank+1]);
+    else {
+      SWAP(char, str[rank-word[i]], str[rank-word[i]-1]);
+      SWAP(char, str[rank+word[i]], str[rank+word[i]+1]);
+    }
+  }
+  return str;
+}
+
+int main(int argc, const char *argv[])
+{
+  FILE *infile;
+  struct stat st;
+  int rank, order, hyperplanes;
+  semisimple_type_t type;
+  int n;
+  signed char *level;
+  node_t *graph;
+  int *left, *right;
+  int left_invariant, right_invariant;
+  int left_invariant_wanted = 0, right_invariant_wanted = 0;
+
+  unsigned long left_invariance, right_invariance;
+  edgelist_t *edgelists;
+  int *words;
+
+  queue_t queue;
+  int current;
+  int *seen;
+  int *generators;
+  int ngens;
+
+  char string_buffer1[1000];
+  const char *alphabet = "abcdefghijklmnopqrstuvwxyz";
+
+  // parse arguments
+
+  if(argc < 2)
+    infile = stdin;
+  else {
+    if(strcmp(argv[1], "-") == 0)
+      infile = stdin;
+    else
+      infile = fopen(argv[1], "rb");
+
+    if(argc >= 4) {
+      if(strcmp(argv[2], "-") != 0)
+	for(int i = 0; i < strlen(argv[2]); i++)
+	  left_invariant_wanted |= (1 << (argv[2][i] - 'a'));
+      if(strcmp(argv[3], "-") != 0)
+	for(int i = 0; i < strlen(argv[3]); i++)
+	  right_invariant_wanted |= (1 << (argv[3][i] - 'a'));
+    }
+  }
+
+  fread(&type.n, sizeof(int), 1, infile);  // we completely trust the input data
+  type.factors = malloc(type.n * sizeof(simple_type_t));
+  fread(type.factors, sizeof(simple_type_t), type.n, infile);
+  fread(&left_invariance, sizeof(simple_type_t), type.n, infile);
+  fread(&right_invariance, sizeof(simple_type_t), type.n, infile);
+
+  // get graph
+
+  rank = coxeter_rank(type);
+  order = coxeter_order(type);
+  hyperplanes = coxeter_hyperplanes(type);
+  ERROR(strlen(alphabet) < rank, "The alphabet has too few letters\n");
+  seen = (int*)malloc(order*sizeof(int));
+  generators = (int*)malloc(order*sizeof(int));
+  level = (signed char*)malloc(order*sizeof(int));
+
+  graph = graph_alloc(type);
+  prepare_graph(type, graph);
+
+  // finally do stuff
+
+  int counter = 0;
+
+  while(fread(level, sizeof(signed char), order, infile) == order) {
+
+    /*
+    if((counter++) % 100000 == 0)
+      print_thickening(rank, order, level, 0, 0, 0, alphabet, stdout);
+    continue;
+    */
+
+    left_invariant = right_invariant = -1; // all 1s
+    for(int j = 0; j < order; j++) {
+      for(int k = 0; k < rank; k++) {
+	if(level[j] > 0 && level[graph[j].left[k]] < 0 || level[j] < 0 && level[graph[j].left[k]] > 0) {
+	  left_invariant &= ~(1 << k);
+	}
+	if(level[j] > 0 && level[graph[j].right[k]] < 0 || level[j] < 0 && level[graph[j].right[k]] > 0) {
+	  right_invariant &= ~(1 << k);
+	}
+      }
+    }
+
+    if((~left_invariant & left_invariant_wanted) == 0 && (~right_invariant & right_invariant_wanted) == 0) {
+      ngens = 0;
+      memset(generators, 0, order*sizeof(int));
+      for(int j = 0; j < order; j++) {
+	if(level[j] == HEAD_MARKER && generators[j] == 0) { // ignore the generator, if it is equivalent to one already seen
+	  ngens++;
+	  queue_init(&queue);
+	  queue_put(&queue, j);
+	  while((current = queue_get(&queue)) != -1) {
+	    if(generators[current] == 0) { // visit everyone only once
+	      generators[current] = ngens;
+	      for(int k = 0; k < rank; k++) {
+		if(left_invariant & (1 << k))
+		queue_put(&queue, graph[current].left[k]);
+		if(right_invariant & (1 << k))
+		  queue_put(&queue, graph[current].right[k]);
+	      }
+	    }
+	  }
+	}
+      }
+
+      printf("left: ");
+      for(int j = 0; j < rank; j++)
+	printf("%c", left_invariant & (1 << j) ? alphabet[j] : ' ');
+      printf(" right: ");
+      for(int j = 0; j < rank; j++)
+	printf("%c", right_invariant & (1 << j) ? alphabet[j] : ' ');
+      printf(" generators: ");
+
+      memset(seen, 0, order*sizeof(int));
+      for(int i = 0; i < order; i++) {
+	if(generators[i] != 0 && seen[generators[i]-1] == 0) {
+	  seen[generators[i]-1] = 1;
+	  //	if(type.n == 1 && type.factors[0].series == 'A')
+	  //	  printf("%s ", stringify_SLn1_permutation(graph[i].word, graph[i].wordlength, rank, string_buffer1));
+	  //	else if(type.n == 1 && (type.factors[0].series == 'B' || type.factors[0].series == 'C'))
+	  //	  printf("%s ", stringify_Onn1_permutation(graph[i].word, graph[i].wordlength, rank, string_buffer1));
+	  //	else
+	  if(i == 0)
+	    printf("1 ");
+	  else
+	    printf("%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
+	}
+      }
+
+      printf("\n");
+    }
+  }
+
+  if(infile != stdin)
+    fclose(infile);
+
+  // cleanup
+
+  graph_free(type, graph);
+  free(seen);
+  free(generators);
+  free(type.factors);
+
+  return 0;
+}
diff --git a/process.c b/process.c
index c62cad2..0c9759a 100644
--- a/process.c
+++ b/process.c
@@ -6,105 +6,117 @@
 #include "coxeter.h"
 #include "queue.h"
 
-#define SWAP(t, a, b) do {t tmp = a; a = b; b = tmp;} while(0)
-
-char *stringify_SLn1_permutation(int *word, int wordlength, int rank, char *str)
-{
-  for(int i = 0; i <= rank; i++)
-    str[i] = '1' + i;
-  str[rank+1] = 0;
-
-  for(int i = 0; i < wordlength; i++) {
-    char tmp = str[word[i]];
-    str[word[i]] = str[word[i]+1];
-    str[word[i]+1] = tmp;
-  }
-  return str;
-}
-
-
-char *stringify_Onn1_permutation(int *word, int wordlength, int rank, char *str)
-{
-  for(int i = 0; i <= rank*2; i++)
-    str[i] = '1' + i;
-  str[2*rank+1] = 0;
-
-  for(int i = 0; i < wordlength; i++) {
-    if(word[i] == 0)
-      SWAP(char, str[rank-1], str[rank+1]);
-    else {
-      SWAP(char, str[rank-word[i]], str[rank-word[i]-1]);
-      SWAP(char, str[rank+word[i]], str[rank+word[i]+1]);
-    }
-  }
-  return str;
-}
-
 int main(int argc, const char *argv[])
 {
   FILE *infile;
-  struct stat st;
-  int rank, order;
   semisimple_type_t type;
-  int n;
-  int *thickenings;
-  signed char *level;
+  unsigned long left_invariance, right_invariance; // these are the invariances we have already modded out
+  unsigned long left_invariant, right_invariant; // these are the invariances of the thickening under consideration
+  int rank, cosets;
   node_t *graph;
-  int *left, *right;
-  int left_invariant, right_invariant;
-  int left_invariant_wanted = 0, right_invariant_wanted = 0;
-  edgelist_t *edgelists;
-  int *words;
-
+  signed char *thickening;
+  int *seen, *generators;
   queue_t queue;
+  int ngenerators;
   int current;
-  int *seen;
-  int *generators;
-  int ngens;
 
   char string_buffer1[1000];
   const char *alphabet = "abcdefghijklmnopqrstuvwxyz";
 
-  // parse arguments
-
   if(argc < 2)
     infile = stdin;
-  else {
-    if(strcmp(argv[1], "-") == 0)
-      infile = stdin;
-    else
-      infile = fopen(argv[1], "rb");
+  else
+    infile = fopen(argv[1], "rb");
 
-    if(argc >= 4) {
-      if(strcmp(argv[2], "-") != 0)
-	for(int i = 0; i < strlen(argv[2]); i++)
-	  left_invariant_wanted |= (1 << (argv[2][i] - 'a'));
-      if(strcmp(argv[3], "-") != 0)
-	for(int i = 0; i < strlen(argv[3]); i++)
-	  right_invariant_wanted |= (1 << (argv[3][i] - 'a'));
-    }
-  }
-
-  fread(&type.n, sizeof(int), 1, infile);  // we completely trust the input data
+  // we completely trust the input data
+  ERROR(fread(&type.n, sizeof(int), 1, infile) == 0, "The input file seems to be empty!\n");
   type.factors = malloc(type.n * sizeof(simple_type_t));
   fread(type.factors, sizeof(simple_type_t), type.n, infile);
-
-  // get graph
+  fread(&left_invariance, sizeof(simple_type_t), type.n, infile);
+  fread(&right_invariance, sizeof(simple_type_t), type.n, infile);
 
   rank = coxeter_rank(type);
-  order = coxeter_order(type);
-  ERROR(strlen(alphabet) < rank, "The alphabet has too few letters\n");
-  graph = (node_t*)malloc(order*sizeof(node_t));
-  left = (int*)malloc(order*rank*sizeof(int));
-  right = (int*)malloc(order*rank*sizeof(int));
+  graph = graph_alloc(type);
+  cosets = prepare_simplified_graph(type, left_invariance, right_invariance, graph);
+
+  thickening = (signed char*)malloc(cosets*sizeof(signed char));
+  generators = (int*)malloc(cosets*sizeof(int));
+  seen = (int*)malloc(cosets*sizeof(int));
+
+  while(fread(thickening, sizeof(signed char), cosets, infile) == cosets) {
+
+    // determine invariances of this thickening
+    left_invariant = right_invariant = -1; // set all bits to 1
+    for(int j = 0; j < cosets; j++) {
+      for(int k = 0; k < rank; k++) {
+	if(thickening[j] > 0 && thickening[graph[j].left[k]] < 0 ||
+	   thickening[j] < 0 && thickening[graph[j].left[k]] > 0)
+	  left_invariant &= ~(1 << k);
+	if(thickening[j] > 0 && thickening[graph[j].right[k]] < 0 ||
+	   thickening[j] < 0 && thickening[graph[j].right[k]] > 0)
+	  right_invariant &= ~(1 << k);
+      }
+    }
+
+    // print this stuff
+    printf("left: ");
+    for(int j = 0; j < rank; j++)
+      printf("%c", left_invariant & (1 << j) ? alphabet[j] : ' ');
+    printf(" right: ");
+    for(int j = 0; j < rank; j++)
+      printf("%c", right_invariant & (1 << j) ? alphabet[j] : ' ');
+    printf(" generators: ");
+
+    // find a minimal set of weyl group elements such that the union of the ideals generated by their cosets wrt the invariances determined above gives the thickening
+    // in the first step, mark everything which is equivalent to a "head" by a generator id
+    ngenerators = 0;
+    memset(generators, 0, cosets*sizeof(int));
+    for(int j = 0; j < cosets; j++) {
+      if(thickening[j] == HEAD_MARKER && generators[j] == 0) { // ignore the generator, if it is equivalent to one already seen
+	ngenerators++;
+	queue_init(&queue);
+	queue_put(&queue, j);
+	while((current = queue_get(&queue)) != -1) {
+	  if(generators[current] == 0) { // visit everyone only once
+	    generators[current] = ngenerators;
+	    for(int k = 0; k < rank; k++) {
+	      if(left_invariant & (1 << k))
+		queue_put(&queue, graph[current].left[k]);
+	      if(right_invariant & (1 << k))
+		queue_put(&queue, graph[current].right[k]);
+	    }
+	  }
+	}
+      }
+    }
+
+    // in the second step, go through the list in ascending word length order and print the first appearance of each generator id
+    memset(seen, 0, cosets*sizeof(int));
+    for(int i = 0; i < cosets; i++) {
+      if(generators[i] != 0 && seen[generators[i]-1] == 0) {
+	seen[generators[i]-1] = 1;
+	printf("%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
+      }
+    }
+
+    printf("\n");
+  }
+
+  if(infile != stdin)
+    fclose(infile);
+
+  graph_free(type, graph);
+  free(type.factors);
+  free(thickening);
+}
+  /*******************************************************************************************
+
   seen = (int*)malloc(order*sizeof(int));
   generators = (int*)malloc(order*sizeof(int));
   level = (signed char*)malloc(order*sizeof(int));
-  for(int i = 0; i < order; i++) {
-    graph[i].left = &left[i*rank];
-    graph[i].right = &right[i*rank];
-  }
-  prepare_graph(type, graph, &edgelists, &words);
+
+  graph = graph_alloc(type);
+  prepare_graph(type, graph);
 
   // finally do stuff
 
@@ -112,24 +124,6 @@ int main(int argc, const char *argv[])
 
   while(fread(level, sizeof(signed char), order, infile) == order) {
 
-    /*
-    if((counter++) % 100000 == 0)
-      print_thickening(rank, order, level, 0, 0, 0, alphabet, stdout);
-    continue;
-    */
-
-    left_invariant = right_invariant = -1; // all 1s
-    for(int j = 0; j < order; j++) {
-      for(int k = 0; k < rank; k++) {
-	if(level[j] > 0 && level[graph[j].left[k]] < 0 || level[j] < 0 && level[graph[j].left[k]] > 0) {
-	  left_invariant &= ~(1 << k);
-	}
-	if(level[j] > 0 && level[graph[j].right[k]] < 0 || level[j] < 0 && level[graph[j].right[k]] > 0) {
-	  right_invariant &= ~(1 << k);
-	}
-      }
-    }
-
     if((~left_invariant & left_invariant_wanted) == 0 && (~right_invariant & right_invariant_wanted) == 0) {
       ngens = 0;
       memset(generators, 0, order*sizeof(int));
@@ -185,15 +179,11 @@ int main(int argc, const char *argv[])
 
   // cleanup
 
-  free(thickenings);
-  free(edgelists);
-  free(words);
-  free(graph);
-  free(left);
-  free(right);
+  graph_free(type, graph);
   free(seen);
   free(generators);
   free(type.factors);
 
   return 0;
 }
+*/
diff --git a/queue.h b/queue.h
index aab9539..a19cc31 100644
--- a/queue.h
+++ b/queue.h
@@ -4,7 +4,7 @@
 #include <stdio.h>
 #include <stdlib.h>
 
-#define QUEUE_SIZE 2000
+#define QUEUE_SIZE 5000
 
 #define ERROR(condition, msg, ...) if(condition){fprintf(stderr, msg, ##__VA_ARGS__); exit(1);}
 
diff --git a/test.c b/test.c
index e9163c4..7d86a67 100644
--- a/test.c
+++ b/test.c
@@ -24,6 +24,60 @@ edgelist_t *edgelist_add(edgelist_t *list, int new, edgelist_t *storage, int *st
   return new_link;
 }
 
+int main(int argc, const char *argv[])
+{
+  unsigned long left_invariance = atoi(argv[2]);
+  unsigned long right_invariance = atoi(argv[3]);
+  semisimple_type_t type;
+  type.n = 1;
+  type.factors = (simple_type_t*)malloc(type.n*sizeof(simple_type_t));
+  type.factors[0].series = 'B';
+  type.factors[0].rank = atoi(argv[1]);
+
+  char alphabet[] = "abcdefghijklmnopqrstuvwxyz";
+  char buffer[1024], buffer2[1024];
+
+  node_t *graph;
+  int *leftbuf, *rightbuf;
+  edgelist_t *edgelists;
+  int *words;
+  int rank, order, hyperplanes, cosets;
+
+  // initialize
+
+  rank = coxeter_rank(type);
+  order = coxeter_order(type);
+  hyperplanes = coxeter_hyperplanes(type);
+
+  graph = (node_t*)malloc(order*sizeof(node_t));
+  leftbuf = (int*)malloc(order*rank*sizeof(int));
+  rightbuf = (int*)malloc(order*rank*sizeof(int));
+  edgelists = (edgelist_t*)malloc(4*order*hyperplanes*sizeof(edgelist_t));
+  words = (int*)malloc(order*hyperplanes*sizeof(int));
+
+  for(int i = 0; i < order; i++) {
+    graph[i].left = &leftbuf[rank*i];
+    graph[i].right = &rightbuf[rank*i];
+  }
+
+  // generate graph
+
+  cosets = prepare_simplified_graph(type, left_invariance, right_invariance, graph, edgelists, words);
+
+  // do something
+
+  fprintf(stderr, "There are %d double cosets.\n", cosets);
+
+  // cleanup
+
+  free(leftbuf);
+  free(rightbuf);
+  free(graph);
+  free(edgelists);
+  free(words);
+}
+
+/*
 int main(int argc, const char *argv[])
 {
   semisimple_type_t type;
@@ -245,3 +299,4 @@ int main(int argc, const char *argv[])
 
   return 0;
 }
+*/
diff --git a/thickenings.c b/thickenings.c
index 4855d62..594171a 100644
--- a/thickenings.c
+++ b/thickenings.c
@@ -57,24 +57,28 @@ static int compare_wordlength(const void *a, const void *b, void *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
+void prepare_graph(semisimple_type_t type, node_t *graph)
 {
   queue_t queue;
 
-  int rank, order;
+  edgelist_t *edgelists_lower, *edgelists_higher;
+  int rank, order, hyperplanes;
   edgelist_t *edge, *previous;
-  int edgelist_count, max_wordlength, hyperplane_count;
+  int edgelist_count, hyperplane_count;
   int current;
 
   int *graph_data;
   node_t *graph_unsorted;
-  int *wordlength_order, *reverse_wordlength_order, *seen, *words;
-  edgelist_t *edgelists;
+  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));
@@ -83,13 +87,10 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
   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;
+    graph[i].bruhat_lower = 0;
+    graph[i].bruhat_higher = 0;
+    graph[i].is_hyperplane_reflection = 0;
   }
 
   // get coxeter graph
@@ -98,7 +99,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
 
   for(int i = 0; i < order; i++)
     for(int j = 0; j < rank; j++)
-      graph_unsorted[i].left[j] = graph_data[i*rank + j];
+      graph_unsorted[i].left = &graph_data[i*rank];
 
   // find wordlengths
 
@@ -115,11 +116,6 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
     }
   }
 
-  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++)
@@ -128,23 +124,22 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
   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
+    // 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[i].left[j]]; // rewrite references
+      graph[i].left[j] = reverse_wordlength_order[graph_unsorted[wordlength_order[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];
+  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) {
-	graph[neighbor].word = &words[neighbor*max_wordlength];
+      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);
@@ -197,7 +192,6 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
 
   // 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) {
@@ -208,9 +202,9 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
 	  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];
+	  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 {
@@ -263,20 +257,18 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
 
   // reverse folding order
 
+  edgelist_count = 0;
   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];
+      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;
     }
   }
 
-  *edgelists_pointer = edgelists;
-  *words_pointer = words;
-
   free(graph_data);
   free(graph_unsorted);
   free(wordlength_order);
@@ -284,6 +276,264 @@ void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists
   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 {
@@ -401,7 +651,6 @@ static long enumerate_tree(const enumeration_info_t *info, signed char *level, i
 
 long enumerate_balanced_thickenings(semisimple_type_t type, node_t *graph, int size, const char *alphabet, FILE *outfile)
 {
-  //  int rank, order;
   signed char *level;
   long count = 0;
   enumeration_info_t info;
diff --git a/thickenings.h b/thickenings.h
index f2f813a..7c6b5d7 100644
--- a/thickenings.h
+++ b/thickenings.h
@@ -25,10 +25,22 @@ typedef struct {
   int is_hyperplane_reflection; // boolean value
 } node_t;
 
+// printing functions
 char *alphabetize(int *word, int len, const char *alphabet, char *buffer);
 void print_thickening(int rank, int order, const signed char *thickening, int level, const char *alphabet, FILE *f);
-static int compare_wordlength(const void *a, const void *b, void *gr);
-void prepare_graph(semisimple_type_t type, node_t *graph, edgelist_t **edgelists_pointer, int **words_pointer);
+
+// generating the graph of the bruhat order
+void prepare_graph(semisimple_type_t type, node_t *graph);
+int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigned long right, node_t *simplified_graph);
+
+// enumerate balanced thickenings
 long enumerate_balanced_thickenings(semisimple_type_t type, node_t *graph, int size, const char *alphabet, FILE *outfile);
+node_t *graph_alloc(semisimple_type_t type);
+void graph_free(semisimple_type_t type, node_t *graph);
+
+// various helper functions
+static int compare_wordlength(const void *a, const void *b, void *gr);
+static int edgelist_contains(edgelist_t *list, int x);
+static edgelist_t *edgelist_add(edgelist_t *list, int new, edgelist_t *storage, int *storage_index);
 
 #endif