如何快速計算集合集合的所有交叉點的包含順序
[英]How to compute fast the containment order of all intersections of a collection of sets
問題描述
這是如何快速獲取python中所有集合交集的后續內容：
我有一個有限集合A = {A1，... Ak}的整數有限集Ai，我想在Python計算如下：
A的子集的所有交叉點：F = {B的交叉點：B是A的子集}。 這是上述問題，解決方案速度非常快。
一個。 X，Y的所有對（X，Y）在F中設置，使得X是Y的子集。
灣 X，Y的所有對（X，Y）在F中設置，使得X是Y的子集，並且在F中沒有集合Z，使得Y的Z子集的X子集。換句話說，所以沒有集合Z適合在收容順序中的X和Y之間。 這樣的一對（X，Y）被稱為封面 。
我為什么要那樣做？ - 我想計算10 ^ 7多面體的面格子。 在該場景中，上面的集合A包含600組。 它確實是着名的600單元 ，計算目前需要大約6秒，如果可能的話，我希望它下降10倍。
得到2.a的6秒 簡單地完成
# 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
一個例子是頂點標有{1,2,3,4}的正方形：集合A則為{{1,2}，{2,3}，{3,4}，{4,1}}，交叉點F為{{}，{1}，{2}，{3}，{4}，{1,2}，{2,3}，{3,4}，[4,1}，{1 ，2,3,4}}，以及有問題的對是
({},{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})
一旦給出了集合F ，我認為除了比較元素之外沒有什么比這更好了。 但我更想到一種算法，它使用關於剛剛相交的東西的知識同時計算（1）和（2）。
根據David K的解決方案，鑒於原因 ，還有兩個可以使用的假設：
生成的順序將使用唯一的最小元素和唯一的最大元素進行分級。 也就是說，封面關系的每個最大鏈F0 <F1 <... <Fm具有相同的長度，F0是空集，Fm是輸入集A的並集。我們將集合Fi稱為等級i，考慮到分級，這是明確定義的。
每個等級M組是恰好2級M + 1組的交集。
非常感謝！
3 個解決方案
解決方案1
2 2016-06-05 18:44:16
這是一個利用輸入中的列表是抽象多面體的方面的假設的函數。 此函數假定輸入是M +面（M級的多面體）的完整列表，而不是采用面的集合的所有交集，而是在M + 1的多面體內執行。然后執行循環，其中每次迭代完成M-faces列表並生成（M-1）面的完整列表，同時累積這兩個面列表的所有包含對。
函數的主循環與每對M面相交，並構建一個列出每個交點和包含它的M面的結構。 這些交叉點包括所有（M-1）面，但也包括一些較低等級的面。 可以通過觀察它們中的每一個是（M-1）面的子集來識別較低等級的面部，因此消除了作為另一個交叉點的子集的任何交叉點。
運行時間的粗略分解為40％與面對相交，40％用於跟蹤哪些M面包含每個得到的交叉點，10％用於消除等級小於M-1的面，10％用於將包含對寫入輸出列表。 我的電腦似乎比你的慢（大約8秒而不是原始功能的6或6.5秒），但新功能的最終結果是每個等級和下一個等級之間的所有包含對的列表，大約10x比產生所有包含對的原始函數（包括“跳過”排名的那些）快15倍。
請注意，並非每個整數列表列表都是新函數的有效輸入，因為有一組點集合不是抽象多面體的方面。 我沒有包含檢查輸入正確性的代碼。
為了檢查輸出的正確性，我在原始函數中添加了一些（相當慢的）代碼來查找（原始）輸出列表中的所有對（s，t），使得形式（s，u）和（u， t）也在列表中，然后返回修改后的列表，刪除所有這些對。 我還通過在輸出的每個整數列表上調用sorted()來修改new和old函數，以便輸出列表可以正確比較。 然后我確認兩個函數產生相同的輸出。
順便說一句，我懷疑這個函數是否像它本來就是pythonic。 建議改進該問題的評論是值得歡迎的。
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
解決方案2
0 2016-06-04 14:56:44
你能告訴我以下變化的運行時間嗎？
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)]
解決方案3
0 2016-06-06 19:00:24
這是一種替代算法，它僅使用上面的分級假設。 這個想法就是這樣
除了
A並集之外的每個交叉點都有一個可以快速計算的覆蓋這會產生一個生成樹，該生成樹從頂部元素開始，由
A的並集給出然后，一個人連續計算等級，並僅比較相差1的等級的元素
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 # ie, 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)))
對於600個單元的測試數據，David K的算法要快得多：
sage: %time X = generatePairs(A)
CPU times: user 363 ms, sys: 28.2 ms, total: 392 ms
Wall time: 373 ms
sage: %time Y = generate_covers(A)
CPU times: user 1.09 s, sys: 25.3 ms, total: 1.12 s
Wall time: 1.08 s
sage: set(X) == set(Y)
True
但是這些集合越“密集”，其他算法就越好：
10個頂點上的單形：
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
和14個頂點上的單形：
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
我主要在這里發布算法用於文檔，但我當然非常感謝有關改進的建議。
查找四個集合相交的所有線段成員（阿拉維恩圖）
[英]Finding all segments members of four sets intersections (ala Venn diagram)
計算 2 個集合的笛卡爾積，是否可以強制集合中元素的順序？
[英]compute Cartesian product of 2 sets, Is it possible to force the order of elements in a set?
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