Hand Distribution Table
I am including this information in my web pages because I think it is
neat and that it is a method I think other writers of hand generators
might want to use.
It is also the only time I can think of when I have
used anything I learned in graduate school.
Creation of the table
My program, 'Deal', needed a fast way to determine whether a hand
was in a particular class. The goal was to stay out of the TCL interpreter
as much as possible. The answer was to implement a lookup table with an easy
indexing algorithm.
There are 560 hand shapes, where by "shape", I mean the ordered listing
of suit lengths: spades-hearts-diamonds-clubs.
The hand shapes are in easy 1-1 correspondence with the 3-subsets
of {0,...,15}.
In particular, the hand shape with s spades, h hearts,
d diamonds, and c clubs corresponds
to the 3-set {s, s+h+1,
s+h+d+2}.
There is a linear order on n-subsets, for fixed n, called the squashed
order [see Combinatorics on Finite Sets, Anderson, pp 112-119.]
The nice thing about this order is that it is easy to find the index of
an n-set in the order.
In the case of n=3, take a subset {x,y,z}, with 0<=x<y<z<=15.
The index of this set in the squashed order is
(z choose 3) + (y choose 2) + (x choose 1)
For the hand shape s-h-d-c, then, the corresponding index in the squashed
order is
(s+h+d+2 choose 3) + (s+h+1 choose 2) + (s choose 1)
For added speed, I pre-computed
(n+2 choose 3) and (n+1 choose 2) for n=0,...13, placing the values
in static arrays. The resulting C code looks like:
static int distTableIndex(s,h,d)
int s,h,d;
{
static choose2tab[]={0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91};
static choose3tab[]={0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455};
return choose3tab[s+h+d]+choose2tab[s+h]+s;
}
It is precisely because I can quickly compute this index that I have
chosen the squashed order. Perhaps other orders also happen to allow
for quick indexing, but this is the one I found in my personal math library.
Usage of the table
The table is useful for fast lookups. For example, lets say you were
looking for hands which were five-five or better in any two suits. You could,
of course, simply check each suit length, but the problem with this is
that in an interpreted language, like Tcl, that might be too slow. If,
however, you built a 560 element binary array, and do the computation
560 times up front, then when we start analyzing actual deals, we
can quickly determine whether a particular shape was in that class.
In reality, the table is not computed until the first request
is made of a shapeclass, which allows users to create large libraries
to be included but not instantiated until used.
For example:
shapecond Balanced {$s*$s+$h*$h+$d*$d+$c*$c<=47}
When this shapeclass is first used, a table of 560 entries is created,
and then the interpreter evaluates this expression only 560 times.
For rare hand classes, this is a significant improvement.
When Deal is confronted with a specific query about membership
in this class, all it has to do is use distTableIndex find the
index, then look it up in the table. This is significantly
faster than reinterpreting the Tcl code every time.
The concept of "shape class" led to the concept of "shape function".
These are functions which use the shape of the hand to determine the
value. For instance, I have a function which determines the opening
suit for a hand:
shapefunc opensuit {
if {$s>=5 && $s>=$h && $s>=$d && $s>=$c} {return spades}
if {$h>=5 && $h>=$d && $h>=$c} { return hearts }
if {$d>=4 && $d>=$c} {return diamonds}
if {$c<3} {return diamonds}
return clubs
}
[ The current implementation of shapefunc is a bit of a memory hog,
unfortunately, since it allocates strings for every element of the
table, even if the strings are duplicates. For instance, the function
above has only 4 return values, "spades", "hearts", "diamonds", and "clubs".
Still, shapefunc instantiates 560 strings. Smarter implementations
are certainly possible, and Tcl 8.0 alleviates the problem considerably
(because it allows deals with reference-counted strings.) ]
[ We could also instantiate the table an entry at a time, leaving null pointers
in the table until a value has been requested. This has the advantage that
we will often compute considerably fewer values from the table. The
disadvantage is that every time we need a value from the table, we will have
to do a check to see if a pointer value is null. It
is not clear to me this would be an improvement or not, but it would
add a complexity to the code that I am not willing to maintain. It might
seem that you would need to check a pointer anyway, to determine if the
shapeclass has been instantiated or not, but the implementation avoids
such a check.]
Note
The above definition of Balanced is an interesting oddity,
which I leave it up to the reader to try to understand.
Think about it for a moment before you look
here.
Table
Here is the raw data of the table. Not very interesting, but it does
help to make it clear how the squashed ordering works, and why the index
values are computed as they are.
<![CDATA[
Index| S H D C
==================
0 | 0 0 0 13
1 | 0 0 1 12
2 | 0 1 0 12
3 | 1 0 0 12
4 | 0 0 2 11
5 | 0 1 1 11
6 | 1 0 1 11
7 | 0 2 0 11
8 | 1 1 0 11
9 | 2 0 0 11
10 | 0 0 3 10
11 | 0 1 2 10
12 | 1 0 2 10
13 | 0 2 1 10
14 | 1 1 1 10
15 | 2 0 1 10
16 | 0 3 0 10
17 | 1 2 0 10
18 | 2 1 0 10
19 | 3 0 0 10
20 | 0 0 4 9
21 | 0 1 3 9
22 | 1 0 3 9
23 | 0 2 2 9
24 | 1 1 2 9
25 | 2 0 2 9
26 | 0 3 1 9
27 | 1 2 1 9
28 | 2 1 1 9
29 | 3 0 1 9
30 | 0 4 0 9
31 | 1 3 0 9
32 | 2 2 0 9
33 | 3 1 0 9
34 | 4 0 0 9
35 | 0 0 5 8
36 | 0 1 4 8
37 | 1 0 4 8
38 | 0 2 3 8
39 | 1 1 3 8
40 | 2 0 3 8
41 | 0 3 2 8
42 | 1 2 2 8
43 | 2 1 2 8
44 | 3 0 2 8
45 | 0 4 1 8
46 | 1 3 1 8
47 | 2 2 1 8
48 | 3 1 1 8
49 | 4 0 1 8
50 | 0 5 0 8
51 | 1 4 0 8
52 | 2 3 0 8
53 | 3 2 0 8
54 | 4 1 0 8
55 | 5 0 0 8
56 | 0 0 6 7
57 | 0 1 5 7
58 | 1 0 5 7
59 | 0 2 4 7
60 | 1 1 4 7
61 | 2 0 4 7
62 | 0 3 3 7
63 | 1 2 3 7
64 | 2 1 3 7
65 | 3 0 3 7
66 | 0 4 2 7
67 | 1 3 2 7
68 | 2 2 2 7
69 | 3 1 2 7
70 | 4 0 2 7
71 | 0 5 1 7
72 | 1 4 1 7
73 | 2 3 1 7
74 | 3 2 1 7
75 | 4 1 1 7
76 | 5 0 1 7
77 | 0 6 0 7
78 | 1 5 0 7
79 | 2 4 0 7
80 | 3 3 0 7
81 | 4 2 0 7
82 | 5 1 0 7
83 | 6 0 0 7
84 | 0 0 7 6
85 | 0 1 6 6
86 | 1 0 6 6
87 | 0 2 5 6
88 | 1 1 5 6
89 | 2 0 5 6
90 | 0 3 4 6
91 | 1 2 4 6
92 | 2 1 4 6
93 | 3 0 4 6
94 | 0 4 3 6
95 | 1 3 3 6
96 | 2 2 3 6
97 | 3 1 3 6
98 | 4 0 3 6
99 | 0 5 2 6
100 | 1 4 2 6
101 | 2 3 2 6
102 | 3 2 2 6
103 | 4 1 2 6
104 | 5 0 2 6
105 | 0 6 1 6
106 | 1 5 1 6
107 | 2 4 1 6
108 | 3 3 1 6
109 | 4 2 1 6
110 | 5 1 1 6
111 | 6 0 1 6
112 | 0 7 0 6
113 | 1 6 0 6
114 | 2 5 0 6
115 | 3 4 0 6
116 | 4 3 0 6
117 | 5 2 0 6
118 | 6 1 0 6
119 | 7 0 0 6
120 | 0 0 8 5
121 | 0 1 7 5
122 | 1 0 7 5
123 | 0 2 6 5
124 | 1 1 6 5
125 | 2 0 6 5
126 | 0 3 5 5
127 | 1 2 5 5
128 | 2 1 5 5
129 | 3 0 5 5
130 | 0 4 4 5
131 | 1 3 4 5
132 | 2 2 4 5
133 | 3 1 4 5
134 | 4 0 4 5
135 | 0 5 3 5
136 | 1 4 3 5
137 | 2 3 3 5
138 | 3 2 3 5
139 | 4 1 3 5
140 | 5 0 3 5
141 | 0 6 2 5
142 | 1 5 2 5
143 | 2 4 2 5
144 | 3 3 2 5
145 | 4 2 2 5
146 | 5 1 2 5
147 | 6 0 2 5
148 | 0 7 1 5
149 | 1 6 1 5
150 | 2 5 1 5
151 | 3 4 1 5
152 | 4 3 1 5
153 | 5 2 1 5
154 | 6 1 1 5
155 | 7 0 1 5
156 | 0 8 0 5
157 | 1 7 0 5
158 | 2 6 0 5
159 | 3 5 0 5
160 | 4 4 0 5
161 | 5 3 0 5
162 | 6 2 0 5
163 | 7 1 0 5
164 | 8 0 0 5
165 | 0 0 9 4
166 | 0 1 8 4
167 | 1 0 8 4
168 | 0 2 7 4
169 | 1 1 7 4
170 | 2 0 7 4
171 | 0 3 6 4
172 | 1 2 6 4
173 | 2 1 6 4
174 | 3 0 6 4
175 | 0 4 5 4
176 | 1 3 5 4
177 | 2 2 5 4
178 | 3 1 5 4
179 | 4 0 5 4
180 | 0 5 4 4
181 | 1 4 4 4
182 | 2 3 4 4
183 | 3 2 4 4
184 | 4 1 4 4
185 | 5 0 4 4
186 | 0 6 3 4
187 | 1 5 3 4
188 | 2 4 3 4
189 | 3 3 3 4
190 | 4 2 3 4
191 | 5 1 3 4
192 | 6 0 3 4
193 | 0 7 2 4
194 | 1 6 2 4
195 | 2 5 2 4
196 | 3 4 2 4
197 | 4 3 2 4
198 | 5 2 2 4
199 | 6 1 2 4
200 | 7 0 2 4
201 | 0 8 1 4
202 | 1 7 1 4
203 | 2 6 1 4
204 | 3 5 1 4
205 | 4 4 1 4
206 | 5 3 1 4
207 | 6 2 1 4
208 | 7 1 1 4
209 | 8 0 1 4
210 | 0 9 0 4
211 | 1 8 0 4
212 | 2 7 0 4
213 | 3 6 0 4
214 | 4 5 0 4
215 | 5 4 0 4
216 | 6 3 0 4
217 | 7 2 0 4
218 | 8 1 0 4
219 | 9 0 0 4
220 | 0 0 10 3
221 | 0 1 9 3
222 | 1 0 9 3
223 | 0 2 8 3
224 | 1 1 8 3
225 | 2 0 8 3
226 | 0 3 7 3
227 | 1 2 7 3
228 | 2 1 7 3
229 | 3 0 7 3
230 | 0 4 6 3
231 | 1 3 6 3
232 | 2 2 6 3
233 | 3 1 6 3
234 | 4 0 6 3
235 | 0 5 5 3
236 | 1 4 5 3
237 | 2 3 5 3
238 | 3 2 5 3
239 | 4 1 5 3
240 | 5 0 5 3
241 | 0 6 4 3
242 | 1 5 4 3
243 | 2 4 4 3
244 | 3 3 4 3
245 | 4 2 4 3
246 | 5 1 4 3
247 | 6 0 4 3
248 | 0 7 3 3
249 | 1 6 3 3
250 | 2 5 3 3
251 | 3 4 3 3
252 | 4 3 3 3
253 | 5 2 3 3
254 | 6 1 3 3
255 | 7 0 3 3
256 | 0 8 2 3
257 | 1 7 2 3
258 | 2 6 2 3
259 | 3 5 2 3
260 | 4 4 2 3
261 | 5 3 2 3
262 | 6 2 2 3
263 | 7 1 2 3
264 | 8 0 2 3
265 | 0 9 1 3
266 | 1 8 1 3
267 | 2 7 1 3
268 | 3 6 1 3
269 | 4 5 1 3
270 | 5 4 1 3
271 | 6 3 1 3
272 | 7 2 1 3
273 | 8 1 1 3
274 | 9 0 1 3
275 | 0 10 0 3
276 | 1 9 0 3
277 | 2 8 0 3
278 | 3 7 0 3
279 | 4 6 0 3
280 | 5 5 0 3
281 | 6 4 0 3
282 | 7 3 0 3
283 | 8 2 0 3
284 | 9 1 0 3
285 |10 0 0 3
286 | 0 0 11 2
287 | 0 1 10 2
288 | 1 0 10 2
289 | 0 2 9 2
290 | 1 1 9 2
291 | 2 0 9 2
292 | 0 3 8 2
293 | 1 2 8 2
294 | 2 1 8 2
295 | 3 0 8 2
296 | 0 4 7 2
297 | 1 3 7 2
298 | 2 2 7 2
299 | 3 1 7 2
300 | 4 0 7 2
301 | 0 5 6 2
302 | 1 4 6 2
303 | 2 3 6 2
304 | 3 2 6 2
305 | 4 1 6 2
306 | 5 0 6 2
307 | 0 6 5 2
308 | 1 5 5 2
309 | 2 4 5 2
310 | 3 3 5 2
311 | 4 2 5 2
312 | 5 1 5 2
313 | 6 0 5 2
314 | 0 7 4 2
315 | 1 6 4 2
316 | 2 5 4 2
317 | 3 4 4 2
318 | 4 3 4 2
319 | 5 2 4 2
320 | 6 1 4 2
321 | 7 0 4 2
322 | 0 8 3 2
323 | 1 7 3 2
324 | 2 6 3 2
325 | 3 5 3 2
326 | 4 4 3 2
327 | 5 3 3 2
328 | 6 2 3 2
329 | 7 1 3 2
330 | 8 0 3 2
331 | 0 9 2 2
332 | 1 8 2 2
333 | 2 7 2 2
334 | 3 6 2 2
335 | 4 5 2 2
336 | 5 4 2 2
337 | 6 3 2 2
338 | 7 2 2 2
339 | 8 1 2 2
340 | 9 0 2 2
341 | 0 10 1 2
342 | 1 9 1 2
343 | 2 8 1 2
344 | 3 7 1 2
345 | 4 6 1 2
346 | 5 5 1 2
347 | 6 4 1 2
348 | 7 3 1 2
349 | 8 2 1 2
350 | 9 1 1 2
351 |10 0 1 2
352 | 0 11 0 2
353 | 1 10 0 2
354 | 2 9 0 2
355 | 3 8 0 2
356 | 4 7 0 2
357 | 5 6 0 2
358 | 6 5 0 2
359 | 7 4 0 2
360 | 8 3 0 2
361 | 9 2 0 2
362 |10 1 0 2
363 |11 0 0 2
364 | 0 0 12 1
365 | 0 1 11 1
366 | 1 0 11 1
367 | 0 2 10 1
368 | 1 1 10 1
369 | 2 0 10 1
370 | 0 3 9 1
371 | 1 2 9 1
372 | 2 1 9 1
373 | 3 0 9 1
374 | 0 4 8 1
375 | 1 3 8 1
376 | 2 2 8 1
377 | 3 1 8 1
378 | 4 0 8 1
379 | 0 5 7 1
380 | 1 4 7 1
381 | 2 3 7 1
382 | 3 2 7 1
383 | 4 1 7 1
384 | 5 0 7 1
385 | 0 6 6 1
386 | 1 5 6 1
387 | 2 4 6 1
388 | 3 3 6 1
389 | 4 2 6 1
390 | 5 1 6 1
391 | 6 0 6 1
392 | 0 7 5 1
393 | 1 6 5 1
394 | 2 5 5 1
395 | 3 4 5 1
396 | 4 3 5 1
397 | 5 2 5 1
398 | 6 1 5 1
399 | 7 0 5 1
400 | 0 8 4 1
401 | 1 7 4 1
402 | 2 6 4 1
403 | 3 5 4 1
404 | 4 4 4 1
405 | 5 3 4 1
406 | 6 2 4 1
407 | 7 1 4 1
408 | 8 0 4 1
409 | 0 9 3 1
410 | 1 8 3 1
411 | 2 7 3 1
412 | 3 6 3 1
413 | 4 5 3 1
414 | 5 4 3 1
415 | 6 3 3 1
416 | 7 2 3 1
417 | 8 1 3 1
418 | 9 0 3 1
419 | 0 10 2 1
420 | 1 9 2 1
421 | 2 8 2 1
422 | 3 7 2 1
423 | 4 6 2 1
424 | 5 5 2 1
425 | 6 4 2 1
426 | 7 3 2 1
427 | 8 2 2 1
428 | 9 1 2 1
429 |10 0 2 1
430 | 0 11 1 1
431 | 1 10 1 1
432 | 2 9 1 1
433 | 3 8 1 1
434 | 4 7 1 1
435 | 5 6 1 1
436 | 6 5 1 1
437 | 7 4 1 1
438 | 8 3 1 1
439 | 9 2 1 1
440 |10 1 1 1
441 |11 0 1 1
442 | 0 12 0 1
443 | 1 11 0 1
444 | 2 10 0 1
445 | 3 9 0 1
446 | 4 8 0 1
447 | 5 7 0 1
448 | 6 6 0 1
449 | 7 5 0 1
450 | 8 4 0 1
451 | 9 3 0 1
452 |10 2 0 1
453 |11 1 0 1
454 |12 0 0 1
455 | 0 0 13 0
456 | 0 1 12 0
457 | 1 0 12 0
458 | 0 2 11 0
459 | 1 1 11 0
460 | 2 0 11 0
461 | 0 3 10 0
462 | 1 2 10 0
463 | 2 1 10 0
464 | 3 0 10 0
465 | 0 4 9 0
466 | 1 3 9 0
467 | 2 2 9 0
468 | 3 1 9 0
469 | 4 0 9 0
470 | 0 5 8 0
471 | 1 4 8 0
472 | 2 3 8 0
473 | 3 2 8 0
474 | 4 1 8 0
475 | 5 0 8 0
476 | 0 6 7 0
477 | 1 5 7 0
478 | 2 4 7 0
479 | 3 3 7 0
480 | 4 2 7 0
481 | 5 1 7 0
482 | 6 0 7 0
483 | 0 7 6 0
484 | 1 6 6 0
485 | 2 5 6 0
486 | 3 4 6 0
487 | 4 3 6 0
488 | 5 2 6 0
489 | 6 1 6 0
490 | 7 0 6 0
491 | 0 8 5 0
492 | 1 7 5 0
493 | 2 6 5 0
494 | 3 5 5 0
495 | 4 4 5 0
496 | 5 3 5 0
497 | 6 2 5 0
498 | 7 1 5 0
499 | 8 0 5 0
500 | 0 9 4 0
501 | 1 8 4 0
502 | 2 7 4 0
503 | 3 6 4 0
504 | 4 5 4 0
505 | 5 4 4 0
506 | 6 3 4 0
507 | 7 2 4 0
508 | 8 1 4 0
509 | 9 0 4 0
510 | 0 10 3 0
511 | 1 9 3 0
512 | 2 8 3 0
513 | 3 7 3 0
514 | 4 6 3 0
515 | 5 5 3 0
516 | 6 4 3 0
517 | 7 3 3 0
518 | 8 2 3 0
519 | 9 1 3 0
520 |10 0 3 0
521 | 0 11 2 0
522 | 1 10 2 0
523 | 2 9 2 0
524 | 3 8 2 0
525 | 4 7 2 0
526 | 5 6 2 0
527 | 6 5 2 0
528 | 7 4 2 0
529 | 8 3 2 0
530 | 9 2 2 0
531 |10 1 2 0
532 |11 0 2 0
533 | 0 12 1 0
534 | 1 11 1 0
535 | 2 10 1 0
536 | 3 9 1 0
537 | 4 8 1 0
538 | 5 7 1 0
539 | 6 6 1 0
540 | 7 5 1 0
541 | 8 4 1 0
542 | 9 3 1 0
543 |10 2 1 0
544 |11 1 1 0
545 |12 0 1 0
546 | 0 13 0 0
547 | 1 12 0 0
548 | 2 11 0 0
549 | 3 10 0 0
550 | 4 9 0 0
551 | 5 8 0 0
552 | 6 7 0 0
553 | 7 6 0 0
554 | 8 5 0 0
555 | 9 4 0 0
556 |10 3 0 0
557 |11 2 0 0
558 |12 1 0 0
559 |13 0 0 0
]]>
