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Hats: factor out the parent-choosing system.

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NFC, but I'm about to want to use it again elsewhere.
This commit is contained in:
Simon Tatham
2023-03-30 08:34:57 +01:00
parent 4720eeb1aa
commit 796d0f372f
1 changed files with 199 additions and 185 deletions
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384
hat.c
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@ -342,6 +342,166 @@ static inline size_t metamap_index(unsigned meta, unsigned meta2)
*/ */
#include "hat-tables.h" #include "hat-tables.h"
/*
* One set of tables that we write by hand: the permitted ways to
* extend the coordinate system outwards from a given metatile.
*
* One obvious approach would be to make a table of all the places
* each metatile can appear in the expansion of another (e.g. H can be
* subtile 0, 1 or 2 of another H, subtile 0 of a T, or 0 or 1 of a P
* or an F), and when we need to decide what our current topmost tile
* turns out to be a subtile of, choose equiprobably at random from
* those options.
*
* That's what I did originally, but a better approach is to skew the
* probabilities. We'd like to generate our patch of actual tiling
* uniformly at random, in the sense that if you selected uniformly
* from a very large region of the plane, the distribution of possible
* finite patches of tiling would converge to some limit as that
* region tended to infinity, and we'd be picking from that limiting
* distribution on finite patches.
*
* For this we have to refer back to the original paper, which
* indicates the subset of each metatile's expansion that can be
* considered to 'belong' to that metatile, such that every subtile
* belongs to exactly one parent metatile, and the overlaps are
* eliminated. Reading out the diagrams from their Figure 2.8:
*
* - H: we discard three of the outer F subtiles, in the symmetric
* positions index by our coordinates as 7, 10, 11. So we keep the
* remaining subtiles {0,1,2,3,4,5,6,8,9,12}, which consist of
* three H, one T, three P and three F.
*
* - T: only the central H expanded from a T is considered to belong
* to it, so we just keep {0}, a single H.
*
* - P: we discard everything intersected by a long edge of the
* parallelogram, leaving the central three tiles and the endmost
* pair of F. That is, we keep {0,1,4,5,10}, consisting of two H,
* one P and two F.
*
* - F: looks like P at one end, and we retain the corresponding set
* of tiles there, but at the other end we keep the two F on either
* side of the endmost one. So we keep {0,1,3,6,8,10}, consisting of
* two H, one P and _three_ F.
*
* Adding up the tile numbers gives us this matrix system:
*
* (H_1) (3 1 2 2)(H_0)
* (T_1) = (1 0 0 0)(T_0)
* (P_1) (3 0 1 1)(P_0)
* (F_1) (3 0 2 3)(F_0)
*
* which says that if you have a patch of metatiling consisting of H_0
* H tiles, T_0 T tiles etc, then this matrix shows the number H_1 of
* smaller H tiles, etc, expanded from it.
*
* If you expand _many_ times, that's equivalent to raising the matrix
* to a power:
*
* n
* (H_n) (3 1 2 2) (H_0)
* (T_n) = (1 0 0 0) (T_0)
* (P_n) (3 0 1 1) (P_0)
* (F_n) (3 0 2 3) (F_0)
*
* The limiting distribution of metatiles is obtained by looking at
* the four-way ratio between H_n, T_n, P_n and F_n as n tends to
* infinity. To calculate this, we find the eigenvalues and
* eigenvectors of the matrix, and extract the eigenvector
* corresponding to the eigenvalue of largest magnitude. (Things get
* more complicated in cases where there isn't a _unique_ eigenvalue
* of largest magnitude, but here, there is.)
*
* That eigenvector is
*
* [ 1 ] [ 1 ]
* [ (7 - 3 sqrt(5)) / 2 ] ~= [ 0.14589803375031545538 ]
* [ 3 sqrt(5) - 6 ] [ 0.70820393249936908922 ]
* [ (9 - 3 sqrt(5)) / 2 ] [ 1.14589803375031545538 ]
*
* So those are the limiting relative proportions of metatiles.
*
* So if we have a particular metatile, how likely is it for its
* parent to be one of those? We have to adjust by the number of
* metatiles of each type that each tile has as its children. For
* example, the P and F tiles have one P child each, but the H has
* three P children. So if we have a P, the proportion of H in its
* potential ancestry is three times what's shown here. (And T can't
* occur at all as a parent.)
*
* In other words, we should choose _each coordinate_ with probability
* corresponding to one of those numbers (scaled down so they all sum
* to 1). Continuing to use P as an example, it will be:
*
* - child 4 of H with relative probability 1
* - child 5 of H with relative probability 1
* - child 6 of H with relative probability 1
* - child 4 of P with relative probability 0.70820393249936908922
* - child 3 of F with relative probability 1.14589803375031545538
*
* and then we obtain the true probabilities by scaling those values
* down so that they sum to 1.
*
* The tables below give a reasonable approximation in 32-bit
* integers to these proportions.
*/
typedef struct MetatilePossibleParent {
TileType type;
unsigned index;
unsigned long probability;
} MetatilePossibleParent;
/* The above probabilities scaled up by 10000000 */
#define PROB_H 10000000
#define PROB_T 1458980
#define PROB_P 7082039
#define PROB_F 11458980
static const MetatilePossibleParent parents_H[] = {
{ TT_H, 0, PROB_H },
{ TT_H, 1, PROB_H },
{ TT_H, 2, PROB_H },
{ TT_T, 0, PROB_T },
{ TT_P, 0, PROB_P },
{ TT_P, 1, PROB_P },
{ TT_F, 0, PROB_F },
{ TT_F, 1, PROB_F },
};
static const MetatilePossibleParent parents_T[] = {
{ TT_H, 3, PROB_H },
};
static const MetatilePossibleParent parents_P[] = {
{ TT_H, 4, PROB_H },
{ TT_H, 5, PROB_H },
{ TT_H, 6, PROB_H },
{ TT_P, 4, PROB_P },
{ TT_F, 3, PROB_F },
};
static const MetatilePossibleParent parents_F[] = {
{ TT_H, 8, PROB_H },
{ TT_H, 9, PROB_H },
{ TT_H, 12, PROB_H },
{ TT_P, 5, PROB_P },
{ TT_P, 10, PROB_P },
{ TT_F, 6, PROB_F },
{ TT_F, 8, PROB_F },
{ TT_F, 10, PROB_F },
};
static const MetatilePossibleParent *const possible_parents[] = {
parents_H, parents_T, parents_P, parents_F,
};
static const size_t n_possible_parents[] = {
lenof(parents_H), lenof(parents_T), lenof(parents_P), lenof(parents_F),
};
#undef PROB_H
#undef PROB_T
#undef PROB_P
#undef PROB_F
/* /*
* Coordinate system for tracking kites within a randomly selected * Coordinate system for tracking kites within a randomly selected
* part of the recursively expanded hat tiling. * part of the recursively expanded hat tiling.
@ -405,6 +565,35 @@ static HatCoords *hc_copy(HatCoords *hc_in)
return hc_out; return hc_out;
} }
static const MetatilePossibleParent *choose_mpp(
random_state *rs, const MetatilePossibleParent *parents, size_t nparents)
{
/*
* If we needed to do this _efficiently_, we'd rewrite all those
* tables above as cumulative frequency tables and use binary
* search. But this happens about log n times in a grid of area n,
* so it hardly matters, and it's easier to keep the tables
* legible.
*/
unsigned long limit = 0, value;
size_t i;
for (i = 0; i < nparents; i++)
limit += parents[i].probability;
value = random_upto(rs, limit);
for (i = 0; i+1 < nparents; i++) {
if (value < parents[i].probability)
return &parents[i];
value -= parents[i].probability;
}
assert(i == nparents - 1);
assert(value < parents[i].probability);
return &parents[i];
}
/* /*
* HatCoordContext is the shared context of a whole run of the * HatCoordContext is the shared context of a whole run of the
* algorithm. Its 'prototype' HatCoords object represents the * algorithm. Its 'prototype' HatCoords object represents the
@ -500,197 +689,22 @@ static HatCoords *initial_coords(HatCoordContext *ctx)
*/ */
static void ensure_coords(HatCoordContext *ctx, HatCoords *hc, size_t n) static void ensure_coords(HatCoordContext *ctx, HatCoords *hc, size_t n)
{ {
/*
* One table that we write by hand: the permitted ways to extend
* the coordinate system outwards from a given metatile.
*
* One obvious approach would be to make a table of all the places
* each metatile can appear in the expansion of another (e.g. H
* can be subtile 0, 1 or 2 of another H, subtile 0 of a T, or 0
* or 1 of a P or an F), and when we need to decide what our
* current topmost tile turns out to be a subtile of, choose
* equiprobably at random from those options.
*
* That's what I did originally, but a better approach is to skew
* the probabilities. We'd like to generate our patch of actual
* tiling uniformly at random, in the sense that if you selected
* uniformly from a very large region of the plane, the
* distribution of possible finite patches of tiling would
* converge to some limit as that region tended to infinity, and
* we'd be picking from that limiting distribution on finite
* patches.
*
* For this we have to refer back to the original paper, which
* indicates the subset of each metatile's expansion that can be
* considered to 'belong' to that metatile, such that every
* subtile belongs to exactly one parent metatile, and the
* overlaps are eliminated. Reading out the diagrams from their
* Figure 2.8:
*
* - H: we discard three of the outer F subtiles, in the symmetric
* positions index by our coordinates as 7, 10, 11. So we keep
* the remaining subtiles {0,1,2,3,4,5,6,8,9,12}, which consist
* of three H, one T, three P and three F.
*
* - T: only the central H expanded from a T is considered to
* belong to it, so we just keep {0}, a single H.
*
* - P: we discard everything intersected by a long edge of the
* parallelogram, leaving the central three tiles and the
* endmost pair of F. That is, we keep {0,1,4,5,10}, consisting
* of two H, one P and two F.
*
* - F: looks like P at one end, and we retain the corresponding
* set of tiles there, but at the other end we keep the two F on
* either side of the endmost one. So we keep {0,1,3,6,8,10},
* consisting of two H, one P and _three_ F.
*
* Adding up the tile numbers gives us this matrix system:
*
* (H_1) (3 1 2 2)(H_0)
* (T_1) = (1 0 0 0)(T_0)
* (P_1) (3 0 1 1)(P_0)
* (F_1) (3 0 2 3)(F_0)
*
* which says that if you have a patch of metatiling consisting of
* H_0 H tiles, T_0 T tiles etc, then this matrix shows the number
* H_1 of smaller H tiles, etc, expanded from it.
*
* If you expand _many_ times, that's equivalent to raising the
* matrix to a power:
*
* n
* (H_n) (3 1 2 2) (H_0)
* (T_n) = (1 0 0 0) (T_0)
* (P_n) (3 0 1 1) (P_0)
* (F_n) (3 0 2 3) (F_0)
*
* The limiting distribution of metatiles is obtained by looking
* at the four-way ratio between H_n, T_n, P_n and F_n as n tends
* to infinity. To calculate this, we find the eigenvalues and
* eigenvectors of the matrix, and extract the eigenvector
* corresponding to the eigenvalue of largest magnitude. (Things
* get more complicated in cases where that's not unique, but
* here, it is.)
*
* That eigenvector is
*
* [ 1 ] [ 1 ]
* [ (7 - 3 sqrt(5)) / 2 ] ~= [ 0.14589803375031545538 ]
* [ 3 sqrt(5) - 6 ] [ 0.70820393249936908922 ]
* [ (9 - 3 sqrt(5)) / 2 ] [ 1.14589803375031545538 ]
*
* So those are the limiting relative proportions of metatiles.
*
* So if we have a particular metatile, how likely is it for its
* parent to be one of those? We have to adjust by the number of
* metatiles of each type that each tile has as its children. For
* example, the P and F tiles have one P child each, but the H has
* three P children. So if we have a P, the proportion of H in its
* potential ancestry is three times what's shown here. (And T
* can't occur at all as a parent.)
*
* In other words, we should choose _each coordinate_ with
* probability corresponding to one of those numbers (scaled down
* so they all sum to 1). Continuing to use P as an example, it
* will be:
*
* - child 4 of H with relative probability 1
* - child 5 of H with relative probability 1
* - child 6 of H with relative probability 1
* - child 4 of P with relative probability 0.70820393249936908922
* - child 3 of F with relative probability 1.14589803375031545538
*
* and then we obtain the true probabilities by scaling those
* values down so that they sum to 1.
*
* The tables below give a reasonable approximation in 32-bit
* integers to these proportions.
*/
typedef struct MetatilePossibleParent {
TileType type;
unsigned index;
unsigned long probability;
} MetatilePossibleParent;
/* The above probabilities scaled up by 10000000 */
#define PROB_H 10000000
#define PROB_T 1458980
#define PROB_P 7082039
#define PROB_F 11458980
static const MetatilePossibleParent parents_H[] = {
{ TT_H, 0, PROB_H },
{ TT_H, 1, PROB_H },
{ TT_H, 2, PROB_H },
{ TT_T, 0, PROB_T },
{ TT_P, 0, PROB_P },
{ TT_P, 1, PROB_P },
{ TT_F, 0, PROB_F },
{ TT_F, 1, PROB_F },
};
static const MetatilePossibleParent parents_T[] = {
{ TT_H, 3, PROB_H },
};
static const MetatilePossibleParent parents_P[] = {
{ TT_H, 4, PROB_H },
{ TT_H, 5, PROB_H },
{ TT_H, 6, PROB_H },
{ TT_P, 4, PROB_P },
{ TT_F, 3, PROB_F },
};
static const MetatilePossibleParent parents_F[] = {
{ TT_H, 8, PROB_H },
{ TT_H, 9, PROB_H },
{ TT_H, 12, PROB_H },
{ TT_P, 5, PROB_P },
{ TT_P, 10, PROB_P },
{ TT_F, 6, PROB_F },
{ TT_F, 8, PROB_F },
{ TT_F, 10, PROB_F },
};
#undef PROB_H
#undef PROB_T
#undef PROB_P
#undef PROB_F
static const MetatilePossibleParent *const possible_parents[] = {
parents_H, parents_T, parents_P, parents_F,
};
static const size_t n_possible_parents[] = {
lenof(parents_H), lenof(parents_T), lenof(parents_P), lenof(parents_F),
};
if (ctx->prototype->nc < n) { if (ctx->prototype->nc < n) {
hc_make_space(ctx->prototype, n); hc_make_space(ctx->prototype, n);
while (ctx->prototype->nc < n) { while (ctx->prototype->nc < n) {
TileType type = ctx->prototype->c[ctx->prototype->nc - 1].type; TileType type = ctx->prototype->c[ctx->prototype->nc - 1].type;
assert(ctx->prototype->c[ctx->prototype->nc - 1].index == -1); assert(ctx->prototype->c[ctx->prototype->nc - 1].index == -1);
const MetatilePossibleParent *parents = possible_parents[type]; const MetatilePossibleParent *parent;
size_t parent_index;
if (ctx->rs) { if (ctx->rs)
unsigned long limit = 0, value; parent = choose_mpp(ctx->rs, possible_parents[type],
size_t nparents = n_possible_parents[type], i; n_possible_parents[type]);
for (i = 0; i < nparents; i++) else
limit += parents[i].probability; parent = possible_parents[type];
value = random_upto(ctx->rs, limit);
for (i = 0; i < nparents; i++) { ctx->prototype->c[ctx->prototype->nc - 1].index = parent->index;
if (value < parents[i].probability)
break;
value -= parents[i].probability;
}
assert(i < nparents);
parent_index = i;
} else {
parent_index = 0;
}
ctx->prototype->c[ctx->prototype->nc - 1].index =
parents[parent_index].index;
ctx->prototype->c[ctx->prototype->nc].index = -1; ctx->prototype->c[ctx->prototype->nc].index = -1;
ctx->prototype->c[ctx->prototype->nc].type = ctx->prototype->c[ctx->prototype->nc].type = parent->type;
parents[parent_index].type;
ctx->prototype->nc++; ctx->prototype->nc++;
} }
} }