本文介绍了允许在一个跃点中完全绑定任何模式的6元组的最小索引集是什么?的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！
问题描述
我正在尝试在wiredtiger上建立一个六元组的商店.元组可以描述如下:
I am trying to build a 6-tuple store on top of wiredtiger. The tuples can be described as follow:
(graph, subject, predicate, object, alive, transaction)
数据库中存储的每个元组都是唯一的.
Every tuple stored in the database is unique.
查询与常规SPARQL查询类似，除了数据库存储6个元组.元组的其他元素中的零可以是可变的.这是一个示例查询，该查询允许检索特定事务 P4X432 引入的所有更改:
Queries are like regular SPARQL queries except that the database store 6 tuples.Zero of more elements of a tuple can be variable. Here is an example query that allows to retrieve all changes introduces by a particular transaction P4X432:
SELECT ?graph ?subject ?predicate ?object ?alive
WHERE
{
?graph ?subject ?predicate ?object ?alive "P4X432"
}
考虑所有可能的模式最终要考虑以下所有组合:
Considering all possible patterns ends up with considering all combinations of:
(graph, subject, predicate, object, alive, transaction)
这是由以下功能给出的:
That is given by the following function:
def combinations(tab):
out = []
for i in range(1, len(tab) + 1):
out.extend(x for x in itertools.combinations(tab, i))
assert len(out) == 2**len(tab) - 1
return out
位置:
print(len(combinations(('graph', 'subject', 'predicate', 'object', 'alive', 'transaction'))))
显示:
63
有6个元组的63个组合.我可以使用缺少的元组项来完成每个索引，例如以下组合:
That is there 63 combinations of the 6-tuples. I can complete each indices with the missing tuple item, e.g. the following combination:
('graph', 'predicate', 'transaction')
将与以下索引相关联:
('graph', 'predicate', 'transaction', 'subject', 'alive', 'object')
但是我知道6元组的所有排列中都有一个较小的子集，具有以下属性:
But I know there is a smaller subset of all permutations of the 6-tuple that has the following property:
换句话说，所有组合的排列都是该集合中一个元素的前缀.
Otherwise said, all combinations have a permutation that is prefix of one element of the set.
我使用蛮力算法发现了具有该属性的大小为25(小于63)的集合:
I found using a brute force algorithm a set of size 25 (inferior to 63) that has that property:
((5 0 1 2 3 4) (4 5 0 1 2 3) (3 4 5 0 1 2) (2 3 4 5 0 1) (1 2 3 4 5 0) (0 1 2 3 4 5) (0 2 1 3 4 5) (0 3 2 1 5 4) (0 4 3 1 5 2) (0 4 2 3 1 5) (2 1 5 3 0 4) (3 2 1 5 0 4) (3 1 4 5 0 2) (3 1 5 4 2 0) (3 0 1 4 2 5) (3 5 2 0 1 4) (4 3 1 0 2 5) (4 2 1 5 3 0) (4 1 0 2 5 3) (4 5 2 1 0 3) (5 4 1 2 3 0) (5 3 0 1 4 2) (5 2 1 3 4 0) (5 1 2 4 0 3) (5 0 2 4 3 1))
这是我用来计算该解决方案的r7rs计划程序:
Here is the r7rs scheme program I use to compute that solution:
(define-library (indices)
(export indices)
(export permutations)
(export combination)
(export combinations)
(export run)
(import (only (chezscheme) trace-define trace-lambda random trace-let))
(import (scheme base))
(import (scheme list))
(import (scheme comparator))
(import (scheme hash-table))
(import (scheme process-context))
(import (scheme write))
(begin
(define (combination k lst)
(cond
((= k 0) '(()))
((null? lst) '())
(else
(let ((head (car lst))
(tail (cdr lst)))
(append (map (lambda (y) (cons head y)) (combination (- k 1) tail))
(combination k tail))))))
(define (factorial n)
(let loop ((n n)
(out 1))
(if (= n 0)
out
(loop (- n 1) (* n out)))))
(define (%binomial-coefficient n k)
;; https://en.wikipedia.org/wiki/Binomial_coefficient#Multiplicative_formula
(let loop ((i 1)
(out 1))
(if (= i (+ k 1))
out
(loop (+ i 1) (* out (/ (- (+ n 1) i) i))))))
(define (memo proc)
(let ((m (make-hash-table (make-equal-comparator))))
(lambda args
(if (hash-table-contains? m args)
(hash-table-ref m args)
(let ((v (apply proc args)))
(hash-table-set! m args v)
v)))))
(define binomial-coefficient
(memo
(lambda (n k)
(cond
((= n k) 1)
((= k 0) 1)
(else (%binomial-coefficient n k))))))
;; k-combination ranking and unranking procedures according to
;; https://en.wikipedia.org/wiki/Combinatorial_number_system
(define (ranking lst)
(let loop ((lst (sort < lst)) ;; increasing sequence
(k 1)
(out 0))
(if (null? lst)
out
(loop (cdr lst) (+ k 1) (+ out (binomial-coefficient (car lst) k))))))
(define (%unranking k N)
(let loop ((n (- k 1)))
(if (< N (binomial-coefficient (+ n 1) k))
n
(loop (+ n 1)))))
(define (unranking k N)
(let loop ((k k)
(N N)
(out '()))
(if (= k 0)
out
(let ((m (%unranking k N)))
(loop (- k 1) (- N (binomial-coefficient m k)) (cons m out))))))
(define fresh-random
(let ((memo (make-hash-table (make-eqv-comparator))))
(lambda (n)
(when (= (hash-table-size memo) n)
(error 'oops "no more fresh number" n
))
(let loop ()
(let ((r (random n)))
(if (hash-table-contains? memo r)
(loop)
(begin (hash-table-set! memo r #t) r)))))))
(define (random-k-combination k n)
(unranking k (fresh-random (binomial-coefficient n k))))
(define (combinations lst)
(if (null? lst) '(())
(let* ((head (car lst))
(tail (cdr lst))
(s (combinations tail))
(v (map (lambda (x) (cons head x)) s)))
(append s v))))
;; (define (combinations lst)
;; (append-map (lambda (k) (combination k lst)) (iota (length lst))))
(define (permutations s)
;; http://rosettacode.org/wiki/Permutations#Scheme
(cond
((null? s) '(()))
((null? (cdr s)) (list s))
(else ;; extract each item in list in turn and permutations the rest
(let splice ((l '()) (m (car s)) (r (cdr s)))
(append
(map (lambda (x) (cons m x)) (permutations (append l r)))
(if (null? r) '()
(splice (cons m l) (car r) (cdr r))))))))
(define (shift lst index)
(append (drop lst index) (take lst index)))
(define (rotations lst)
(reverse! (map (lambda (index) (shift lst index)) (iota (length lst)))))
(define (prefix? lst other)
"Return #t if LST is prefix of OTHER"
(let prefix ((lst lst)
(other other))
(if (null? lst)
#t
(if (= (car lst) (car other))
(prefix (cdr lst) (cdr other))
#f))))
(define (indices lst)
(let ((candidates (permutations lst)))
(let loop ((out (rotations lst)) ;; all rotations are solutions
(combinations (combinations lst)))
(if (null? combinations)
(reverse! out)
(let ((permutations (permutations (car combinations))))
(if (any (lambda (s) (any (lambda (p) (prefix? p s)) permutations)) out)
;; there is an existing "solution" for the
;; permutations of COMBINATION move to the next
;; combination
(loop out (cdr combinations))
(loop (cons (find (lambda (c) (if (member c out)
#f
(any (lambda (p) (prefix? p c)) permutations)))
candidates)
out)
(cdr combinations))))))))
(define (permutation-prefix? c o)
(any (lambda (p) (prefix? p o)) (permutations c)))
(define (ok? combinations candidate)
(every (lambda (c) (any (lambda (p) (permutation-prefix? c p)) candidate)) combinations))
(define (run)
(let* ((n (string->number (cadr (command-line))))
(N (iota n))
(solution (indices N))
(min (length solution))
(rotations (rotations N))
(R (length rotations))
;; other stuff
(cx (combinations N))
(px (filter (lambda (x) (not (member x rotations))) (permutations N)))
;; other other stuff
(pn (length px))
(PN (iota pn)))
(display "(length solution) => ") (display (length solution))
(display "\n")
(display "(length rotations) => ") (display R)
(display "\n")
(let try ((x (- (length solution) 1)))
(let ((count (binomial-coefficient pn (- x R))))
(let loop ((index 0)
(cxx (map (lambda (x) (list-ref px x)) (random-k-combination (- x R) pn))))
(when (= (modulo index (expt 10 5)) 0)
(display "n=") (display n) (display " x=") (display x)
(display " ")
(display index) (display "/") (display count) (display "\n"))
(let ((candidate (append rotations cxx)))
(let ((continue? (not (ok? cx candidate))))
(if continue?
(loop (+ index 1)
(map (lambda (x) (list-ref px x)) (random-k-combination (- x R) pn)))
(begin (display "new solution n=") (display n)
(display " length=") (display x)
(display " ") (display candidate)
(display "\n")
(try (- x 1)))))))))))
))
有了排列列表，我可以查询任何模式.
With that list of permutations I can query any pattern.
我想知道是否有一个较小的集合，是否有确定的算法来计算这种集合.
I am wondering if there is a smaller set and whether there is definitive algorithm to compute that kind of set.
推荐答案
基于此答案 https://math.stackexchange.com/a/3146793/23663
以下程序产生的解决方案是根据math™的最小解决方案:
The following program yields a solution that is a minimal solution according to math ™:
import itertools
import math
f = math.factorial
bc = lambda n, k: f(n) // f(k) // f(n-k) if k<n else 0
def pk(*args):
print(*args)
return args[-1]
def stringify(iterable):
return ''.join(str(x) for x in iterable)
def combinations(tab):
out = []
for i in range(1, len(tab) + 1):
out.extend(stringify(x) for x in itertools.combinations(tab, i))
assert len(out) == 2**len(tab) - 1
return out
def ok(solutions, tab):
cx = combinations(tab)
px = [stringify(x) for x in itertools.permutations(tab)]
for combination in cx:
pcx = [''.join(x) for x in itertools.permutations(combination)]
# check for existing solution
for solution in solutions:
if any(solution.startswith(p) for p in pcx):
# yeah, there is an existing solution
break
else:
print('failed with combination={}'.format(combination))
break
else:
return True
return False
def run(n):
tab = list(range(n))
cx = list(itertools.combinations(tab, n//2))
for c in cx:
L = [(i, i in c) for i in tab]
A = []
B = []
while True:
for i in range(len(L) - 1):
if (not L[i][1]) and L[i + 1][1]:
A.append(L[i + 1][0])
B.append(L[i][0])
L.remove((L[i + 1][0], True))
L.remove((L[i][0], False))
break
else:
break
l = [i for (i, _) in L]
yield A + l + B
for i in range(7):
tab = stringify(range(i))
solutions = [stringify(x) for x in run(i)]
assert ok(solutions, tab)
print("n={}, size={}, solutions={}".format(i, len(solutions), solutions))
上面的程序输出是:
n=0, size=1, solutions=['']
n=1, size=1, solutions=['0']
n=2, size=2, solutions=['01', '10']
n=3, size=3, solutions=['012', '120', '201']
n=4, size=6, solutions=['0123', '2031', '3012', '1230', '1302', '2310']
n=5, size=10, solutions=['01234', '20341', '30142', '40123', '12340', '13402', '14203', '23410', '24013', '34021']
n=6, size=20, solutions=['012345', '301452', '401253', '501234', '203451', '240513', '250314', '340521', '350124', '450132', '123450', '142503', '152304', '134502', '135024', '145032', '234510', '235104', '245130', '345210']
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