Python 如何快速计算集合中所有交集的包含顺序
这是以下方面的后续行动： 我有一个有限集合a={A1，…Ak}的有限整数集Ai，我想在Python 如何快速计算集合中所有交集的包含顺序,python,algorithm,Python,Algorithm,这是以下方面的后续行动： 我有一个有限集合a={A1，…Ak}的有限整数集Ai，我想在Python中计算以下内容： A:F={B:B的交集是A}的子集。这就是上面的问题，有一个相当快的解决方案 a。X，Y的所有对（X，Y）都在F中设置，使得X是Y的子集 b。对于X，Y的所有对（X，Y）都在F中设置，使得X是Y的子集，而F中没有集Z，使得Z的子集是Y的子集。换句话说，没有集Z以包含顺序适合于X和Y之间。这样的一对（X，Y）称为覆盖 我为什么要这么做？--我想计算10^7多面体的面格。在所考虑的场景
Python# this is John Coleman's function from above question's answer
def allIntersections(frozenSets):
universalSet = frozenset.union(*frozenSets)
intersections = set([universalSet])
for s in frozenSets:
moreIntersections = set(s & t for t in intersections)
intersections.update(moreIntersections)
return intersections
def all_intersections(lists):
sets = allIntersections([frozenset(s) for s in lists])
return [list(s) for s in sets]
A = [[19, 40, 41, 48], [19, 44, 45, 49], [23, 42, 43, 50], [23, 46, 47, 51], [19, 40, 41, 52], [19, 44, 45, 53], [23, 42, 43, 54], [23, 46, 47, 55], [2, 25, 36, 56], [0, 24, 32, 56], [24, 25, 56, 57], [24, 32, 56, 57], [16, 32, 56, 57], [1, 24, 32, 57], [25, 36, 56, 57], [16, 36, 56, 57], [3, 25, 36, 57], [8, 28, 34, 58], [10, 29, 38, 58], [28, 29, 58, 59], [28, 34, 58, 59], [20, 34, 58, 59], [29, 38, 58, 59], [20, 38, 58, 59], [9, 28, 34, 59], [11, 29, 38, 59], [6, 27, 37, 60], [4, 26, 33, 60], [5, 26, 33, 61], [26, 27, 60, 61], [26, 33, 60, 61], [16, 33, 60, 61], [27, 37, 60, 61], [7, 27, 37, 61], [16, 37, 60, 61], [12, 30, 35, 62], [14, 31, 39, 62], [30, 35, 62, 63], [20, 39, 62, 63], [20, 35, 62, 63], [30, 31, 62, 63], [31, 39, 62, 63], [15, 31, 39, 63], [13, 30, 35, 63], [0, 24, 32, 64], [1, 24, 32, 64], [8, 28, 34, 65], [9, 28, 34, 65], [3, 25, 36, 66], [2, 25, 36, 66], [11, 29, 38, 67], [10, 29, 38, 67], [4, 26, 33, 68], [5, 26, 33, 68], [12, 30, 35, 69], [13, 30, 35, 69], [6, 27, 37, 70], [7, 27, 37, 70], [15, 31, 39, 71], [14, 31, 39, 71], [4, 33, 68, 72], [0, 32, 64, 72], [18, 64, 72, 73], [32, 64, 72, 73], [32, 33, 72, 73], [1, 32, 64, 73], [18, 68, 72, 73], [5, 33, 68, 73], [33, 68, 72, 73], [2, 36, 66, 74], [6, 37, 70, 74], [3, 36, 66, 75], [7, 37, 70, 75], [36, 66, 74, 75], [37, 70, 74, 75], [36, 37, 74, 75], [22, 66, 74, 75], [22, 70, 74, 75], [12, 35, 69, 76], [8, 34, 65, 76], [18, 65, 76, 77], [34, 65, 76, 77], [34, 35, 76, 77], [18, 69, 76, 77], [35, 69, 76, 77], [13, 35, 69, 77], [9, 34, 65, 77], [10, 38, 67, 78], [14, 39, 71, 78], [38, 67, 78, 79], [22, 71, 78, 79], [22, 67, 78, 79], [38, 39, 78, 79], [39, 71, 78, 79], [15, 39, 71, 79], [11, 38, 67, 79], [0, 40, 48, 80], [19, 40, 48, 80], [19, 48, 49, 80], [8, 44, 49, 80], [19, 44, 49, 80], [2, 40, 52, 81], [10, 44, 53, 81], [19, 52, 53, 81], [19, 40, 52, 81], [19, 44, 53, 81], [19, 40, 80, 81], [19, 44, 80, 81], [23, 42, 50, 82], [23, 50, 51, 82], [1, 42, 50, 82], [23, 46, 51, 82], [9, 46, 51, 82], [23, 54, 55, 83], [3, 42, 54, 83], [23, 42, 54, 83], [23, 42, 82, 83], [11, 46, 55, 83], [23, 46, 55, 83], [23, 46, 82, 83], [19, 45, 49, 84], [12, 45, 49, 84], [4, 41, 48, 84], [19, 41, 48, 84], [19, 48, 49, 84], [19, 45, 84, 85], [19, 41, 84, 85], [6, 41, 52, 85], [19, 41, 52, 85], [14, 45, 53, 85], [19, 45, 53, 85], [19, 52, 53, 85], [23, 43, 50, 86], [5, 43, 50, 86], [23, 50, 51, 86], [23, 47, 51, 86], [13, 47, 51, 86], [7, 43, 54, 87], [23, 43, 54, 87], [23, 43, 86, 87], [23, 54, 55, 87], [23, 47, 86, 87], [15, 47, 55, 87], [23, 47, 55, 87], [8, 28, 65, 88], [0, 24, 64, 88], [9, 28, 65, 89], [28, 65, 88, 89], [17, 28, 88, 89], [17, 24, 88, 89], [1, 24, 64, 89], [24, 64, 88, 89], [64, 65, 88, 89], [4, 26, 68, 90], [12, 30, 69, 90], [5, 26, 68, 91], [13, 30, 69, 91], [26, 68, 90, 91], [21, 26, 90, 91], [68, 69, 90, 91], [30, 69, 90, 91], [21, 30, 90, 91], [10, 29, 67, 92], [2, 25, 66, 92], [29, 67, 92, 93], [66, 67, 92, 93], [11, 29, 67, 93], [17, 29, 92, 93], [25, 66, 92, 93], [17, 25, 92, 93], [3, 25, 66, 93], [14, 31, 71, 94], [6, 27, 70, 94], [21, 31, 94, 95], [21, 27, 94, 95], [15, 31, 71, 95], [31, 71, 94, 95], [70, 71, 94, 95], [27, 70, 94, 95], [7, 27, 70, 95], [2, 25, 56, 96], [0, 80, 88, 96], [0, 40, 56, 96], [2, 40, 81, 96], [2, 40, 56, 96], [0, 40, 80, 96], [40, 80, 81, 96], [2, 81, 92, 96], [17, 25, 92, 96], [2, 25, 92, 96], [0, 24, 88, 96], [0, 24, 56, 96], [24, 25, 56, 96], [17, 24, 88, 96], [17, 24, 25, 96], [28, 29, 58, 97], [80, 88, 96, 97], [80, 81, 96, 97], [44, 80, 81, 97], [8, 28, 88, 97], [8, 28, 58, 97], [8, 44, 58, 97], [8, 80, 88, 97], [8, 44, 80, 97], [81, 92, 96, 97], [17, 29, 92, 97], [17, 92, 96, 97], [17, 28, 29, 97], [17, 28, 88, 97], [17, 88, 96, 97], [10, 29, 92, 97], [10, 29, 58, 97], [10, 44, 58, 97], [10, 44, 81, 97], [10, 81, 92, 97], [6, 41, 85, 98], [6, 41, 60, 98], [4, 41, 60, 98], [6, 85, 94, 98], [4, 41, 84, 98], [4, 84, 90, 98], [41, 84, 85, 98], [6, 27, 94, 98], [6, 27, 60, 98], [26, 27, 60, 98], [4, 26, 90, 98], [4, 26, 60, 98], [21, 27, 94, 98], [21, 26, 90, 98], [21, 26, 27, 98], [14, 45, 85, 99], [21, 30, 31, 99], [14, 31, 62, 99], [30, 31, 62, 99], [14, 45, 62, 99], [21, 90, 98, 99], [21, 30, 90, 99], [84, 90, 98, 99], [45, 84, 85, 99], [84, 85, 98, 99], [12, 30, 62, 99], [12, 45, 62, 99], [12, 45, 84, 99], [12, 30, 90, 99], [12, 84, 90, 99], [85, 94, 98, 99], [21, 94, 98, 99], [14, 85, 94, 99], [14, 31, 94, 99], [21, 31, 94, 99], [3, 83, 93, 100], [1, 42, 82, 100], [3, 42, 57, 100], [1, 42, 57, 100], [42, 82, 83, 100], [3, 42, 83, 100], [1, 82, 89, 100], [1, 24, 89, 100], [17, 24, 89, 100], [1, 24, 57, 100], [17, 25, 93, 100], [3, 25, 57, 100], [3, 25, 93, 100], [17, 24, 25, 100], [24, 25, 57, 100], [17, 93, 100, 101], [82, 83, 100, 101], [11, 83, 93, 101], [83, 93, 100, 101], [11, 29, 59, 101], [11, 29, 93, 101], [17, 29, 93, 101], [9, 82, 89, 101], [82, 89, 100, 101], [17, 89, 100, 101], [11, 46, 83, 101], [11, 46, 59, 101], [9, 46, 59, 101], [9, 46, 82, 101], [46, 82, 83, 101], [9, 28, 59, 101], [17, 28, 29, 101], [28, 29, 59, 101], [17, 28, 89, 101], [9, 28, 89, 101], [5, 43, 86, 102], [5, 86, 91, 102], [7, 43, 61, 102], [5, 43, 61, 102], [21, 27, 95, 102], [7, 27, 95, 102], [7, 27, 61, 102], [5, 26, 61, 102], [26, 27, 61, 102], [21, 26, 27, 102], [21, 26, 91, 102], [5, 26, 91, 102], [43, 86, 87, 102], [7, 43, 87, 102], [7, 87, 95, 102], [86, 91, 102, 103], [86, 87, 102, 103], [15, 31, 63, 103], [30, 31, 63, 103], [15, 31, 95, 103], [87, 95, 102, 103], [15, 87, 95, 103], [15, 47, 63, 103], [15, 47, 87, 103], [47, 86, 87, 103], [13, 30, 63, 103], [13, 30, 91, 103], [13, 86, 91, 103], [13, 47, 63, 103], [13, 47, 86, 103], [21, 91, 102, 103], [21, 30, 91, 103], [21, 30, 31, 103], [21, 95, 102, 103], [21, 31, 95, 103], [0, 48, 72, 104], [4, 33, 72, 104], [4, 33, 60, 104], [4, 41, 60, 104], [4, 48, 72, 104], [4, 41, 48, 104], [32, 33, 72, 104], [0, 32, 72, 104], [0, 32, 56, 104], [0, 40, 56, 104], [40, 41, 48, 104], [0, 40, 48, 104], [16, 32, 56, 104], [16, 32, 33, 104], [16, 33, 60, 104], [40, 41, 104, 105], [40, 41, 52, 105], [41, 60, 104, 105], [16, 60, 104, 105], [40, 56, 104, 105], [16, 56, 104, 105], [2, 40, 56, 105], [2, 40, 52, 105], [2, 36, 56, 105], [16, 36, 56, 105], [16, 37, 60, 105], [16, 36, 37, 105], [2, 52, 74, 105], [36, 37, 74, 105], [2, 36, 74, 105], [6, 52, 74, 105], [6, 41, 52, 105], [6, 41, 60, 105], [6, 37, 60, 105], [6, 37, 74, 105], [12, 35, 76, 106], [12, 45, 62, 106], [12, 35, 62, 106], [8, 44, 49, 106], [8, 49, 76, 106], [12, 49, 76, 106], [44, 45, 49, 106], [12, 45, 49, 106], [20, 35, 62, 106], [8, 44, 58, 106], [20, 34, 58, 106], [8, 34, 58, 106], [20, 34, 35, 106], [8, 34, 76, 106], [34, 35, 76, 106], [20, 62, 106, 107], [20, 38, 39, 107], [20, 39, 62, 107], [10, 38, 78, 107], [38, 39, 78, 107], [10, 53, 78, 107], [20, 58, 106, 107], [20, 38, 58, 107], [10, 38, 58, 107], [44, 58, 106, 107], [10, 44, 58, 107], [10, 44, 53, 107], [14, 39, 62, 107], [14, 39, 78, 107], [14, 53, 78, 107], [14, 45, 53, 107], [44, 45, 106, 107], [44, 45, 53, 107], [14, 45, 62, 107], [45, 62, 106, 107], [16, 32, 57, 108], [1, 32, 57, 108], [16, 32, 33, 108], [16, 33, 61, 108], [5, 33, 61, 108], [1, 32, 73, 108], [32, 33, 73, 108], [1, 50, 73, 108], [5, 33, 73, 108], [5, 50, 73, 108], [1, 42, 50, 108], [1, 42, 57, 108], [5, 43, 61, 108], [5, 43, 50, 108], [42, 43, 50, 108], [7, 37, 61, 109], [3, 36, 57, 109], [3, 42, 57, 109], [7, 43, 61, 109], [42, 43, 108, 109], [43, 61, 108, 109], [42, 57, 108, 109], [16, 36, 57, 109], [16, 36, 37, 109], [16, 57, 108, 109], [16, 61, 108, 109], [16, 37, 61, 109], [36, 37, 75, 109], [7, 37, 75, 109], [3, 36, 75, 109], [3, 42, 54, 109], [42, 43, 54, 109], [7, 43, 54, 109], [3, 54, 75, 109], [7, 54, 75, 109], [34, 35, 77, 110], [13, 35, 63, 110], [13, 35, 77, 110], [13, 47, 63, 110], [9, 34, 77, 110], [9, 51, 77, 110], [13, 51, 77, 110], [9, 46, 51, 110], [13, 47, 51, 110], [46, 47, 51, 110], [20, 35, 63, 110], [20, 34, 35, 110], [9, 34, 59, 110], [20, 34, 59, 110], [9, 46, 59, 110], [11, 38, 59, 111], [11, 38, 79, 111], [15, 47, 63, 111], [11, 55, 79, 111], [15, 47, 55, 111], [15, 55, 79, 111], [11, 46, 59, 111], [46, 47, 55, 111], [11, 46, 55, 111], [38, 39, 79, 111], [15, 39, 79, 111], [15, 39, 63, 111], [20, 38, 39, 111], [20, 39, 63, 111], [20, 38, 59, 111], [47, 63, 110, 111], [20, 59, 110, 111], [20, 63, 110, 111], [46, 59, 110, 111], [46, 47, 110, 111], [8, 65, 88, 112], [18, 65, 76, 112], [8, 65, 76, 112], [8, 49, 76, 112], [0, 64, 88, 112], [64, 65, 88, 112], [18, 64, 65, 112], [18, 64, 72, 112], [0, 64, 72, 112], [0, 48, 72, 112], [8, 49, 80, 112], [8, 80, 88, 112], [48, 49, 80, 112], [0, 48, 80, 112], [0, 80, 88, 112], [4, 68, 90, 113], [4, 84, 90, 113], [12, 84, 90, 113], [18, 68, 69, 113], [68, 69, 90, 113], [12, 69, 90, 113], [4, 68, 72, 113], [18, 68, 72, 113], [18, 69, 76, 113], [12, 69, 76, 113], [4, 48, 84, 113], [4, 48, 72, 113], [12, 49, 76, 113], [48, 49, 84, 113], [12, 49, 84, 113], [18, 76, 112, 113], [49, 76, 112, 113], [18, 72, 112, 113], [48, 49, 112, 113], [48, 72, 112, 113], [2, 66, 92, 114], [66, 67, 92, 114], [52, 53, 81, 114], [2, 52, 81, 114], [2, 81, 92, 114], [22, 66, 67, 114], [22, 67, 78, 114], [2, 66, 74, 114], [2, 52, 74, 114], [22, 66, 74, 114], [10, 53, 81, 114], [10, 53, 78, 114], [10, 67, 78, 114], [10, 81, 92, 114], [10, 67, 92, 114], [6, 85, 94, 115], [6, 52, 85, 115], [52, 53, 85, 115], [14, 85, 94, 115], [14, 53, 85, 115], [52, 53, 114, 115], [6, 52, 74, 115], [52, 74, 114, 115], [14, 71, 94, 115], [70, 71, 94, 115], [6, 70, 94, 115], [6, 70, 74, 115], [22, 74, 114, 115], [22, 70, 74, 115], [22, 70, 71, 115], [14, 71, 78, 115], [53, 78, 114, 115], [14, 53, 78, 115], [22, 78, 114, 115], [22, 71, 78, 115], [18, 64, 65, 116], [18, 65, 77, 116], [50, 51, 82, 116], [1, 50, 82, 116], [9, 51, 82, 116], [9, 51, 77, 116], [9, 65, 77, 116], [18, 64, 73, 116], [1, 64, 73, 116], [1, 50, 73, 116], [64, 65, 89, 116], [1, 64, 89, 116], [9, 65, 89, 116], [1, 82, 89, 116], [9, 82, 89, 116], [18, 73, 116, 117], [18, 77, 116, 117], [51, 77, 116, 117], [18, 69, 77, 117], [13, 51, 86, 117], [13, 51, 77, 117], [13, 69, 77, 117], [18, 68, 69, 117], [18, 68, 73, 117], [13, 69, 91, 117], [13, 86, 91, 117], [68, 69, 91, 117], [5, 86, 91, 117], [5, 68, 73, 117], [5, 68, 91, 117], [50, 51, 116, 117], [5, 50, 86, 117], [50, 51, 86, 117], [5, 50, 73, 117], [50, 73, 116, 117], [11, 55, 83, 118], [3, 66, 75, 118], [3, 54, 75, 118], [3, 54, 83, 118], [54, 55, 83, 118], [11, 55, 79, 118], [11, 67, 79, 118], [66, 67, 93, 118], [3, 83, 93, 118], [3, 66, 93, 118], [11, 67, 93, 118], [11, 83, 93, 118], [22, 66, 67, 118], [22, 67, 79, 118], [22, 66, 75, 118], [54, 55, 87, 119], [54, 55, 118, 119], [55, 79, 118, 119], [54, 75, 118, 119], [22, 75, 118, 119], [22, 79, 118, 119], [22, 71, 79, 119], [22, 70, 71, 119], [70, 71, 95, 119], [22, 70, 75, 119], [7, 54, 75, 119], [7, 54, 87, 119], [7, 70, 75, 119], [7, 87, 95, 119], [7, 70, 95, 119], [15, 71, 79, 119], [15, 55, 79, 119], [15, 55, 87, 119], [15, 71, 95, 119], [15, 87, 95, 119]]
from itertools import combinations
F = all_intersections(A) # all intersections: function from other question
# takes 415 ms
F = sorted(F,lambda x,y: cmp(len(x),len(y)))
pairs = [ (x,y) for x,y in combinations(F,2) if set(y).issuperset(x) ]
# takes ~6 sec
({},{1}),({},{2}),({},{3}),({},{4}),
({1},{1,2}),({1},{4,1}),
({2},{1,2}),({2},{2,3}),
({3},{2,3}),({3},{3,4}),
({4},{3,4}),({4},{4,1}),
({1,2},{1,2,3,4}),({2,3},{1,2,3,4}),({3,4},{1,2,3,4}),({4,1},{1,2,3,4})
F非常感谢！您能告诉我以下更改的运行时间吗
def all_intersections(lists):
sets = allIntersections([frozenset(s) for s in lists])
return list(sets)
A = ...
F = all_intersections(A)
F.sort(key=len)
pairs = [(x,y) for x,y in combinations(F,2) if y.issuperset(x)]
这是一个函数，它利用了一个假设，即输入中的列表是抽象多面体的面， 此函数假定输入是秩为M+1的多面体中的M面（秩为M的多面体）的完整列表。 然后，它执行一个循环，在该循环中，每个迭代获取一个完整的M面列表，并生成一个完整的（M-1）面列表，同时累积这两个面列表的所有包含对 函数的主循环与每对M面相交，并构建一个结构，列出每个相交点和包含它的M面。 这些交点包括所有（M-1）面，但也包括 一些级别较低的面。可以识别级别较低的面 通过观察它们中的每一个都是（M-1）-面的子集， 因此，作为另一个交点子集的任何交点都被消除 运行时间的大致细分为40%，以相交于成对的面， 40%用于跟踪哪个M面包含每个结果 交叉点，10%以消除秩小于M-1的面， 10%将包含对写入输出列表。 我的电脑似乎比你的慢 （约8秒，而不是原始功能的6秒或6.5秒）， 但新函数的最终结果是列出了所有的包容 在每一列和下一列之间配对，大约比第一列快10-15倍 生成所有包含对的原始函数（包括 那些“跳过”等级的） 请注意，并不是每个整数列表都是新函数的有效输入 函数，因为存在不存在的点集集合 抽象多面体的面。我没有包含检查输入的代码 为了正确 为了检查输出的正确性，我在原始函数中添加了一些（相当慢的）代码，以查找（原始）输出列表中的所有对（s，t） 这样（s，u）和（u，t）的成对形式也在列表中， 然后返回一个修改后的列表，删除所有这些对。 我还通过调用上的
sorted（）from collections import defaultdict
import sys
def generatePairs(A):
# It is assumed that A consists exactly of all the facets of an abstract
# polytope of rank N; that is, the abstract polytope is a graded poset
# in which the minimal element is the empty set and has rank -1, the
# maximal element is the polytope's body, which has rank N, and A
# contains all facets of the polytope, which have rank N - 1.
# Then within the graded poset,
# each element of rank 0 is a point and has cardinality 1;
# each element of rank 1 is an edge and has cardiality 2;
# each element of rank M (where M > 1) is a rank-M polytope and has
# cardinality at least M + 1, but may have greater cardinality.
# We start with the facets (rank N-1).
rank_to_intersect = [frozenset(s) for s in A]
# Construct the body (rank N).
polytope_body = list(frozenset.union(*rank_to_intersect))
body_size = len(polytope_body)
# covering_pairs will be all the pairs of polytopes (s,t) such that
# rank(s) + 1 == rank(t) and s is a subset of t. Initially we populate
# it with just the pairs whose ranks are respectively N-1 and N.
covering_pairs = [(s, polytope_body) for s in A]
while (len(rank_to_intersect) > 0) and (len(rank_to_intersect[0]) > 2):
# For some integer M such that M > 1, rank_to_intersect contains all
# the polytopes of rank M. At the end of each iteration of the loop,
# rank_to_intersect will contain all the polytopes of rank M - 1.
# Also, all the pairs (x,y) where rank(x) = M - 1 and rank(y) = M
# will have been added to covering_pairs.
container_map = defaultdict(list)
while rank_to_intersect:
s = rank_to_intersect.pop()
for t in rank_to_intersect:
x = s & t
if len(x) > 1:
container_map[x].extend([s, t])
# Note that the list container_map[x]
# may contain duplicates
# The keys of container_map, consisting of all pairwise
# intersections of polytopes of rank M, include all polytopes
# of rank M - 1 but also some polytopes of lower ranks.
# Any polytope of a lower rank, however, is a subset of
# a polytope of rank M - 1 that is also in the list.
min_size = min([len(s) for s in container_map.keys()])
max_size = max([len(s) for s in container_map.keys()])
size_range = range(min_size, max_size + 1)
candidates = dict([(i, []) for i in size_range])
for s in container_map.keys():
candidates[len(s)].append(s)
# Repopulate rank_to_intersect with the polytopes of rank M - 1.
for set_size in size_range:
larger_sizes = range(set_size + 1, max_size + 1)
for s in candidates[set_size]:
if not any(any(t >= s for t in candidates[i])
for i in larger_sizes):
# We now know that s has rank M - 1, not a lower rank.
rank_to_intersect.append(s)
# Add all the (rank-(M - 1), rank-M) pairs to covering_pairs.
for s in rank_to_intersect:
# container_map[s] may contain duplicates; avoid them.
containers = frozenset(container_map[s])
covering_pairs.extend([(list(s), list(t)) for t in containers])
# At the end of the loop, rank_to_intersect contains the rank-1
# polytopes, that is, the edges.
# Each edge contains each of its two endpoints.
points_with_duplicates = []
for e in rank_to_intersect:
covering_pairs.extend([([p], list(e)) for p in e])
points_with_duplicates.extend(e)
# List the containment pairs of the empty set without duplicating points.
points = frozenset(points_with_duplicates)
covering_pairs.extend([([], [p]) for p in points])
return covering_pairs
这里有一个替代算法，它只使用上面的分级假设
a的并集给出from collections import defaultdict
# this is John Coleman's function from
# http://stackoverflow.com/questions/37622153
# compute all intersections including the empty intersection
# corresponding to the union of all sets in `A`.
def all_intersections(A):
# using frozensets for intersections
A = map(frozenset,A)
A.sort(key=len)
# the union of A as the ground set
universalSet = frozenset.union(*A)
# computing the intersections successively
intersections = set([universalSet])
for s in A:
moreIntersections = set(s & t for t in intersections)
intersections.update(moreIntersections)
return intersections
def ranked_pieces(intersections):
# this is to shortcut the length tests below
lens = { s:len(s) for s in intersections }
V = sorted(intersections, lambda x,y:cmp(lens[x],lens[y]))
m = len(V)
lower_covs = defaultdict(set)
# we first compute the spanning tree...
for i,x in enumerate(V):
for j in range(i+1,m):
y = V[j]
if lens[x] < lens[y] and y.issuperset(x):
lower_covs[y].add(x)
# since V is sorted according to the size,
# we have surely found a cover and stop
break
# ... and then the level sets
level = set([V[-1]])
level_sets = [level]
while level:
level = set.union(*[lower_covs[v] for v in level])
level_sets.append(level)
# the level sets are ordered backwards now
# i.e., level_sets[0] is the biggest set
# and level_sets[-1] is the empty set
return level_sets
def ranked_pieces_to_covers(ranked_pieces):
# this is because we want tuples and not sets
# and we want to compute them only once
back_ref = { f:tuple(sorted(f)) for i in range(len(ranked_pieces)) for f in ranked_pieces[i] }
covs = []
for i in range(len(ranked_pieces)-1):
high = ranked_pieces[i]
low = ranked_pieces[i+1]
for x in low:
for y in high:
if y.issuperset(x):
covs.append((back_ref[x], back_ref[y]))
return covs
def generate_covers(A):
return ranked_pieces_to_covers(ranked_pieces(all_intersections(A)))
sage: m = 10; A = [ [i for i in range(m) if i != j] for j in range(m) ]
sage: %time X = generatePairs(A)
CPU times: user 151 ms, sys: 4.33 ms, total: 156 ms
Wall time: 144 ms
sage: %time Y = generate_covers(A)
CPU times: user 118 ms, sys: 17.3 ms, total: 136 ms
Wall time: 106 ms
sage: set(X) == set(Y)
True
sage: m = 14; A = [ [i for i in range(m) if i != j] for j in range(m) ]
sage: %time X = generatePairs(A)
CPU times: user 56.3 s, sys: 136 ms, total: 56.5 s
Wall time: 56.6 s
sage: %time Y = generate_covers(A)
CPU times: user 31 s, sys: 65.4 ms, total: 31 s
Wall time: 31 s
sage: m = 10; A = [ [i for i in range(m) if i != j] for j in range(m) ]
sage: %time X = generatePairs(A)
CPU times: user 151 ms, sys: 4.33 ms, total: 156 ms
Wall time: 144 ms
sage: %time Y = generate_covers(A)
CPU times: user 118 ms, sys: 17.3 ms, total: 136 ms
Wall time: 106 ms
sage: set(X) == set(Y)
True
sage: m = 14; A = [ [i for i in range(m) if i != j] for j in range(m) ]
sage: %time X = generatePairs(A)
CPU times: user 56.3 s, sys: 136 ms, total: 56.5 s
Wall time: 56.6 s
sage: %time Y = generate_covers(A)
CPU times: user 31 s, sys: 65.4 ms, total: 31 s
Wall time: 31 s
我主要在这里发布算法以供文档使用，但我当然非常感谢您提出改进建议。所有交叉点（A）需要多长时间接受输入？我假设
所有交叉点