让 S(n) 表示所有有效的排序顺序，其中 n 对应于包含 [0,n] 的对。
S(n) = []
for each order in S(n-1)
for each combination of n-th pair
if pair can be inserted in order, add the order after insertion to S(n)
else don't include the order in S(n)
一对配对可以通过两种方式（正常配对和反向配对）的任意一种方式插入到订单中。
Maximum orderings = O(2^n)
我对摊销排序不是很确定，但请听我解释。
对于一个序列和一对数，我们有四种方式在插入后得到排序的序列（两种顺序，一种正常，一种反转，零种）
摊销排序数量 = (1/4)*2 + (1/4)*1 + (1/4)*1 + (1/4)*0 = 1
Amortized orderings = O(1)
同样，时间复杂度将是O(n^2)，但不确定。
以下程序使用插入排序的变体来查找排序顺序。
debug = False
(LEFT, RIGHT, ERROR) = range(3)
def position(first, second):
""" Returns the position of first pair when compared to second """
x,y = first
a,b = second
if x <= a and b <= y:
return LEFT
if x >= a and b >= y:
return RIGHT
else:
return ERROR
def insert(pair, order):
""" A pair can be inserted in normal order or reversed order
For each order of insertion we will get one solution or none"""
solutions = []
paircombinations = [pair]
if pair[0] != pair[1]:
paircombinations.append(pair[::-1])
for _pair in paircombinations:
insertat = 0
if debug: print "Inserting", _pair,
for i,p in enumerate(order):
pos = position(_pair, p)
if pos == LEFT:
break
elif pos == RIGHT:
insertat += 1
else:
if debug: print "into", order,"is not possible"
insertat = None
break
if insertat != None:
if debug: print "at",insertat,"in", order
solutions.append(order[0:insertat] + [_pair] + order[insertat:])
return solutions
def swapsort(pairs):
"""
Finds all the solutions of pairs such that ending vector
of pairs are be sorted non decreasingly by the first number in
each pair and non increasingly by the second in each pair.
"""
solutions = [ pairs[0:1] ]
for pair in pairs[1:]:
newsolutions = []
for solution in solutions:
sols = insert(pair, solution)
if sols:
newsolutions.extend(sols)
if newsolutions:
solutions = newsolutions
else:
return None
return solutions
if __name__ == "__main__":
groups = [ [(1,5), (7,1), (3,8), (5,6)],
[(1,5), (2,3), (3,3), (3,4), (2,4)],
[(3,5), (6,6), (7,4)],
[(1,4), (2,5)] ]
for pairs in groups:
print "Solutions for",pairs,":"
solutions = swapsort(pairs)
if solutions:
for sol in solutions:
print sol
else:
print "not possible"
输出：
Solutions for [(1, 5), (7, 1), (3, 8), (5, 6)] :
[(1, 7), (1, 5), (6, 5), (8, 3)]
Solutions for [(1, 5), (2, 3), (3, 3), (3, 4), (2, 4)] :
[(1, 5), (2, 4), (2, 3), (3, 3), (4, 3)]
[(1, 5), (2, 3), (3, 3), (4, 3), (4, 2)]
[(1, 5), (2, 4), (3, 4), (3, 3), (3, 2)]
[(1, 5), (3, 4), (3, 3), (3, 2), (4, 2)]
Solutions for [(3, 5), (6, 6), (7, 4)] :
not possible
Solutions for [(1, 4), (2, 5)] :
[(1, 4), (5, 2)]
7,1变成了1,7），这是有意为之吗？ - Björn Pollex(2,9) (5,1)怎么样？ - NPE