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enumerate-balanced-ideals
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Results seem sensible

This commit is contained in:
Florian Stecker 2016-11-23 20:58:05 +01:00
parent c4824abafd
commit 1e0959b7ce
6 changed files with 316 additions and 89 deletions
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4
Makefile
View File

@ -1,8 +1,8 @@
HEADERS=weyl.h thickenings.h queue.h bitvec.h
SPECIAL_OPTIONS=-O0 -g
#SPECIAL_OPTIONS=-O0 -g -D_DEBUG
#SPECIAL_OPTIONS=-O3 -pg -funroll-loops -fno-inline
#SPECIAL_OPTIONS=-O3 -flto -funroll-loops -Winline
SPECIAL_OPTIONS=-O3 -flto -funroll-loops -Winline
OPTIONS=-m64 -march=native -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS)

8
bitvec.h
View File

@ -107,6 +107,14 @@ static inline void bv_intersection(const bitvec_t *x, const bitvec_t *y, bitvec_
}
}
static inline void bv_difference(const bitvec_t *x, const bitvec_t *y, bitvec_t *result)
{
int i;
for (i=0; i < BV_QWORD_RANK; i++) {
result->v[i] = x->v[i] & ~y->v[i];
}
}
static inline int bv_disjoint(const bitvec_t *x, const bitvec_t *y)
{
for(int i = 0; i < BV_QWORD_RANK; i++)

250
generate.c
View File

@ -5,15 +5,132 @@
#include <strings.h>
#include <stdio.h>
char stringbuffer[100];
char stringbuffer2[100];
typedef struct {
node_t *graph;
int cosets;
int rank;
int order;
int hyperplanes;
semisimple_type_t type;
unsigned long left_invariance;
unsigned long right_invariance;
const char *alphabet;
int *buffer;
} info_t;
int shorten(int i, unsigned long left, unsigned long right, node_t *graph, int rank)
{
int other, shorter = i;
do {
i = shorter;
for(int j = 0; j < rank; j++) {
other = graph[shorter].left[j];
if(left & (1 << j) &&
graph[other].wordlength < graph[shorter].wordlength)
shorter = other;
other = graph[shorter].right[j];
if(right & (1 << j) &&
graph[other].wordlength < graph[shorter].wordlength)
shorter = other;
}
} while(shorter != i);
return shorter;
}
void balanced_thickening_callback(const bitvec_t *pos, int size, void *data)
{
static long totcount = 0;
if(data) {
info_t *info = (info_t*)data;
unsigned long right_invariance = FIRSTBITS(info->rank);
unsigned long left_invariance = FIRSTBITS(info->rank);
int bit1, bit2left, bit2right, left, right;
for(int i = 0; i < size; i++) {
bit1 = i < size/2 ? bv_get_bit(pos, i) : !bv_get_bit(pos, size - 1 - i);
for(int j = 0; j < info->rank; j++) {
left = info->graph[i].left[j];
right = info->graph[i].right[j];
bit2left = left < size/2 ? bv_get_bit(pos, left) : !bv_get_bit(pos, size - 1 - left);
bit2right = right < size/2 ? bv_get_bit(pos, right) : !bv_get_bit(pos, size - 1 - right);
if(bit1 != bit2left)
left_invariance &= ~BIT(j);
if(bit1 != bit2right)
right_invariance &= ~BIT(j);
}
}
printf("left: ");
for(int j = 0; j < info->rank; j++)
printf("%c", left_invariance & (1 << j) ? info->alphabet[j] : ' ');
printf(" right: ");
for(int j = 0; j < info->rank; j++)
printf("%c", right_invariance & (1 << j) ? info->alphabet[j] : ' ');
if(info->buffer) {
printf(" generators:");
queue_t queue;
int current, left, right, shortest;
int *buffer = info->buffer;
for(int i = 0; i < size/2; i++) {
buffer[i] = bv_get_bit(pos, i);
buffer[size-1-i] = !buffer[i];
}
for(int i = size-1; i >= 0; i--) {
if(buffer[i]) {
int shortest = shorten(i, left_invariance, right_invariance, info->graph, info-> rank);
printf(" %s", alphabetize(info->graph[shortest].word,
info->graph[shortest].wordlength,
info->alphabet,
stringbuffer));
buffer[i] = 0;
queue_init(&queue);
queue_put(&queue, i);
while((current = queue_get(&queue)) != -1) {
for(edgelist_t *edge = info->graph[current].bruhat_lower; edge != (edgelist_t*)0; edge = edge->next) {
if(buffer[edge->to]) {
buffer[edge->to] = 0;
queue_put(&queue, edge->to);
}
}
for(int j = 0; j < info->rank; j++) {
left = info->graph[current].left[j];
if(left_invariance & (1 << j) &&
info->graph[left].wordlength < info->graph[current].wordlength &&
buffer[left]) {
buffer[left] = 0;
queue_put(&queue, left);
}
right = info->graph[current].left[j];
if(right_invariance & (1 << j) &&
info->graph[right].wordlength < info->graph[current].wordlength &&
buffer[right]) {
buffer[right] = 0;
queue_put(&queue, right);
}
}
}
}
}
}
printf("\n");
}
/*
if((++totcount) % 100000000 == 0) {
fprintf(stderr, "Found balanced ideal: ");
bv_print(stderr, pos, size/2);
fprintf(stderr, "\n");
}
} */
}
int main(int argc, const char *argv[])
@ -57,8 +174,6 @@ int main(int argc, const char *argv[])
right_invariance |= (1 << (argv[type.n + 2][i] - 'a'));
}
ERROR(strlen(alphabet) < rank, "The alphabet has too few letters\n");
// generate graph
graph = graph_alloc(type);
@ -68,48 +183,123 @@ int main(int argc, const char *argv[])
// print stuff
int output_level = 2;
if(getenv("OUTPUT_LEVEL"))
output_level = atoi(getenv("OUTPUT_LEVEL"));
rank = weyl_rank(type); // number of simple roots
order = weyl_order(type); // number of Weyl group elements
hyperplanes = weyl_hyperplanes(type); // number of positive roots
fprintf(stderr, "Rank: %d\tOrder: %d\tPositive Roots: %d\tCosets: %d\n", rank, order, hyperplanes, cosets);
fprintf(stderr, "\n");
if(output_level >= 1) {
printf("Poset: ");
if(left_invariance) {
printf("<");
for(int j = 0; j < rank; j++)
if(left_invariance & BIT(j))
fputc(alphabet[j], stdout);
printf("> \\ ");
}
/*
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) {
fprintf(stderr, "\n%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
wl = graph[i].wordlength;
} else
fprintf(stderr, "%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
for(int i = 0; i < type.n; i++)
printf("%s%c%d", i == 0 ? "" : " x ", type.factors[i].series, type.factors[i].rank);
if(right_invariance) {
printf(" / <");
for(int j = 0; j < rank; j++)
if(right_invariance & BIT(j))
fputc(alphabet[j], stdout);
printf(">");
}
fprintf(stdout, "\n");
fprintf(stdout, "Rank: %d\tOrder: %d\tPositive Roots: %d\tCosets: %d\n\n", rank, order, hyperplanes, cosets);
}
if(output_level >= 3) {
fprintf(stdout, "Shortest coset representatives: \n");
for(int i = 0, wl = 0; i < cosets; i++) {
if(i == 0) {
fprintf(stdout, "1");
} else if(graph[i].wordlength > wl) {
fprintf(stdout, "\n%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
wl = graph[i].wordlength;
} else
fprintf(stdout, "%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
}
fprintf(stdout, "\n\n");
}
if(output_level >= 4) {
edgelist_t *edge;
fprintf(stdout, "Bruhat order in graphviz format:\n");
fprintf(stdout, "digraph test123 {\n");
for(int i = 0; i < cosets; i++) {
edge = graph[i].bruhat_lower;
while(edge) {
fprintf(stdout, "%s -> %s;\n",
alphabetize(graph[i].word, graph[i].wordlength, alphabet, stringbuffer),
alphabetize(graph[edge->to].word, graph[edge->to].wordlength, alphabet, stringbuffer2));
edge = edge->next;
}
}
fprintf(stdout, "}\n\n");
}
if(output_level >= 4) {
fprintf(stdout, "Opposites:\n");
for(int i = 0; i < cosets; i++) {
fprintf(stdout, "%s <-> %s\n",
alphabetize(graph[i].word, graph[i].wordlength, alphabet, stringbuffer),
alphabetize(graph[graph[i].opposite].word, graph[graph[i].opposite].wordlength, alphabet, stringbuffer2));
}
fprintf(stdout, "\n");
}
fprintf(stderr, "\n\n");
*/
fixpoints = 0;
for(int i = 0; i < cosets; i++)
if(graph[i].opposite == i) {
if(fixpoints == 0)
fprintf(stderr, "No thickenings since the longest element fixes the following cosets: %s", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
else
fprintf(stderr, " %s", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
if(output_level >= 1) {
if(fixpoints == 0)
fprintf(stdout, "No thickenings since the longest element fixes the following cosets: %s", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
else
fprintf(stdout, " %s", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
}
fixpoints++;
}
if(output_level >= 1 && fixpoints)
fprintf(stdout, "\n\n");
if(fixpoints > 0) {
fprintf(stderr, "\n\n");
} else {
fwrite(&type.n, sizeof(int), 1, stdout);
fwrite(type.factors, sizeof(simple_type_t), type.n, stdout);
fwrite(&left_invariance, sizeof(unsigned long), type.n, stdout);
fwrite(&right_invariance, sizeof(unsigned long), type.n, stdout);
long count = enumerate_balanced_thickenings(graph, cosets, balanced_thickening_callback, (void*)0);
if(!fixpoints) {
int *buffer = (int*)malloc(cosets*sizeof(int));
fprintf(stderr, "Found %ld balanced thickenings\n\n", count);
info_t info;
info.graph = graph;
info.cosets = cosets;
info.rank = rank;
info.order = order;
info.hyperplanes = hyperplanes;
info.type = type;
info.left_invariance = left_invariance;
info.right_invariance = right_invariance;
info.alphabet = alphabet;
info.buffer = buffer;
long count;
if(output_level >= 2) {
fprintf(stdout, "Balanced ideals:\n", count);
count = enumerate_balanced_thickenings(graph, cosets, balanced_thickening_callback, &info);
fprintf(stdout, "\n", count);
}
else
count = enumerate_balanced_thickenings(graph, cosets, 0, 0);
if(output_level >= 1)
fprintf(stdout, "Found %ld balanced ideal%s\n", count, count == 1 ? "" : "s");
}
graph_free(type, graph);

5
queue.h
View File

@ -7,6 +7,11 @@
#define QUEUE_SIZE 5000
#define ERROR(condition, msg, ...) if(condition){fprintf(stderr, msg, ##__VA_ARGS__); exit(1);}
#ifdef _DEBUG
#define LOG(msg, ...) fprintf(stderr, msg, ##__VA_ARGS__)
#else
#define LOG(msg, ...)
#endif
typedef struct {
unsigned int start;

128
thickenings.c
View File

@ -91,12 +91,10 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
graph[i].is_hyperplane_reflection = 0;
}
// get coxeter graph
LOG("Generate Weyl group.\n");
weyl_generate(type, graph_data);
fprintf(stderr, "Weyl group generated.\n");
for(int i = 0; i < order; i++)
for(int j = 0; j < rank; j++) {
graph_unsorted[i].left = graph_data[i].left;
@ -105,6 +103,8 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
// find wordlengths
LOG("Determine word lengths.\n");
graph_unsorted[0].wordlength = 0;
queue_init(&queue);
queue_put(&queue, 0);
@ -118,9 +118,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Wordlengths calculated.\n");
// sort by wordlength
LOG("Sort by wordlength.\n");
for(int i = 0; i < order; i++)
ordering[i] = i;
@ -135,9 +133,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
graph[i].left[j] = reverse_ordering[graph_unsorted[ordering[i]].left[j]]; // rewrite references
}
fprintf(stderr, "Sorted by wordlength.\n");
// find words
LOG("Find shortest words.\n");
for(int i = 0; i < order; i++)
memset(graph[i].word, 0, hyperplanes*sizeof(int));
@ -154,9 +150,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Shortest words found.\n");
// generate right edges
LOG("Generate right edges.\n");
for(int i = 0; i < order; i++) {
for(int j = 0; j < rank; j++) {
@ -168,9 +162,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Right edges generated.\n");
// find opposites
LOG("Find opposites.\n");
node_t *longest = &graph[order-1];
for(int i = 0; i < order; i++) {
@ -180,9 +172,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
graph[i].opposite = current;
}
fprintf(stderr, "Opposites found.\n");
// enumerate hyperplanes
LOG("Enumerate hyperplanes.\n");
hyperplane_count = 0;
for(int i = 0; i < order; i++) {
@ -203,9 +193,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Hyperplanes enumerated.\n");
// generate folding order
LOG("Determine Bruhat order.\n");
edgelist_count = 0;
for(int i = 0; i < order; i++) {
@ -229,9 +217,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Bruhat order generated.\n");
// remove redundant edges
LOG("Perform transitive reduction.\n");
for(int i = 0; i < order; i++) {
memset(seen, 0, order*sizeof(int));
@ -272,9 +258,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Redundant edges removed.\n");
// reverse folding order
LOG("Revert Bruhat order.\n");
edgelist_count = 0;
for(int i = 0; i < order; i++) {
@ -288,7 +272,7 @@ void prepare_graph(semisimple_type_t type, node_t *graph)
}
}
fprintf(stderr, "Bruhat order reversed.\n");
LOG("Sort opposites.\n");
// additional sorting step to force opposite property (opposite of j is at n - j - 1)
@ -373,7 +357,7 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
full_graph = graph_alloc(type);
prepare_graph(type, full_graph);
fprintf(stderr, "Full graph generated.\n");
LOG("Full graph generated.\n");
// initialize stuff
@ -385,6 +369,8 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
reduced[i] = i;
}
LOG("Group by double coset.\n");
// step 1: group
for(int i = 0; i < order; i++) {
if(group[i] != -1)
@ -406,6 +392,8 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
}
}
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)
@ -424,12 +412,11 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
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;
LOG("Copy minimal elements.\n");
// step 5: set up nodes from minima
current = 0;
for(int i = 0; i < order; i++)
@ -445,6 +432,8 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
}
}
LOG("Find induced order.\n");
// step 6: find order relations
for(int i = 0; i < order; i++) {
edge = full_graph[i].bruhat_lower;
@ -461,6 +450,8 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
}
}
LOG("Perform transitive reduction.\n");
// step 7: remove redundant edges
for(int i = 0; i < ncosets; i++) {
memset(seen, 0, ncosets*sizeof(int));
@ -477,7 +468,6 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
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
@ -505,6 +495,8 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
}
}
LOG("Revert order.\n");
// step 8: revert order
edgelists_used = 0;
for(int i = 0; i < ncosets; i++) {
@ -517,33 +509,57 @@ int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigne
}
}
// 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));
LOG("Sort opposites.\n");
edge = edge->next;
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++;
}
}
fprintf(stdout, "}\n"); */
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++;
}
// 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);
// 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;
}
@ -637,6 +653,11 @@ static long enumerate_tree(const enumeration_info_t *info, const bitvec_t *pos,
// 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;
@ -728,12 +749,15 @@ long enumerate_balanced_thickenings(node_t *graph, int size, void (*callback) (c
}
free(principal);
/*
// output principal ideals
#ifdef _DEBUG
for(int i = 0; i < info.size; i++) {
fprintf(stderr, "%d: ", 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;

10
weyl.c
View File

@ -283,10 +283,10 @@ void weyl_cartan_matrix(semisimple_type_t type, int *m)
}
break;
case 'B': // not sure at all about the order of B and C
case 'B': // not sure at all about the order of B and C
if(type.factors[k].rank >= 2) {
A[0][1] = -1;
A[1][0] = -2;
A[0][1] = -2;
A[1][0] = -1;
}
for(int i = 2; i < type.factors[k].rank; i++) {
A[i][i-1] = -1;
@ -296,8 +296,8 @@ void weyl_cartan_matrix(semisimple_type_t type, int *m)
case 'C':
if(type.factors[k].rank >= 2) {
A[0][1] = -2;
A[1][0] = -1;
A[0][1] = -1;
A[1][0] = -2;
}
for(int i = 2; i < type.factors[k].rank; i++) {
A[i][i-1] = -1;