如何快速计算集合的所有交集的包含顺序

这是后续如何在python中快速获取集合的所有交集 https://stackoverflow.com/questions/37622153:

我有一个整数有限集合 Ai 的有限集合 A = {A1,...Ak} ，我想计算Python下列：

1. A 子集的所有交集：F = { B 的交集：B 是 A 的子集}。这是上面的问题，有一个相当快的解决方案。

2. A。 X,Y 的所有对 (X,Y) 都在 F 中设置，使得 X 是 Y 的子集。

   b. X,Y 的所有对 (X,Y) 都是 F 中的集合，使得 X 是 Y 的子集，并且 F 中不存在集合 Z，使得 X 是 Y 的 Z 子集的子集。换句话说，没有集合 Z 适合按遏制顺序位于 X 和 Y 之间。这样的一对 (X,Y) 称为cover.

我为什么要这么做？ -- 我想计算 10^7 多面体的面格。在设想的场景中，上面的集合 A 包含 600 套。它确实是著名的600细胞 https://en.wikipedia.org/wiki/600-cell，当前计算大约需要 6 秒，如果可能的话，我希望将其减少 10 倍。

6 秒获得 2.a。简单地通过做来完成

# 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

一个例子是顶点标记为 {1,2,3,4} 的正方形：集合 A 则为 {{1,2},{2,3},{3,4},{4,1}}，交集 F 为 {{},{1},{2},{3},{4},{1,2},{2,3},{3,4},[4,1},{1 ,2,3,4}}，所讨论的对是

({},{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})

一旦你得到了套装F，我认为没有什么比仅仅比较元素更好的了。但我更多地考虑一种使用刚刚相交的内容的知识同时计算 (1) 和 (2) 的算法。

按照下面 David K 的解决方案，给出why，还有两个可以使用的假设：

1. 生成的订单使用唯一的最小元素和唯一的最大元素进行分级。也就是说，覆盖关系的每个最大链 F0

2. 每个 M 阶集合恰好是 2 个 M+1 阶集合的交集。

非常感谢！

这是一个函数，它利用了输入中的列表是抽象多面体的面的假设。不是取面集合的所有交集， 该函数假设输入是 M + 1 级多胞形内的 M 面（M 级多胞形）的完整列表。 然后，它执行一个循环，其中每次迭代都会获取 M 面的完整列表并生成 (M-1) 面的完整列表，同时累积这两个面列表的所有包含对。

该函数的主循环与每对 M 面相交，并构建一个列出每个相交点和包含它的 M 面的结构。 这些交点包括所有 (M-1) 面，但也包括 一些低等的面孔。可以识别较低级别的面孔 通过观察它们中的每一个都是 (M-1) 面的子集， 因此，作为另一个交叉点子集的任何交叉点都将被消除。

运行时间的粗略细分是 40% 用于相交成对的面， 40% 来跟踪哪些 M 面包含每个结果 交叉点，10% 消除等级小于 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 （大约8秒而不是原始功能的6或6.5秒）， 但新函数的最终结果是所有包含的列表 每个等级和下一个等级之间的配对，比之前的快大约 10 倍到 15 倍 产生所有包含对（包括 “跳过”排名的那些）。

请注意，并非每个整数列表列表都是新的有效输入 函数，因为存在不存在的点集的集合 抽象多面体的各个方面。我没有包含检查输入的代码 为了正确性。

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) 形式的对也在列表中， 然后返回删除所有这些对的修改后的列表。 我还通过调用修改了新函数和旧函数sorted()在 他们放入输出中的每个整数列表，以便输出列表 会正确比较。 然后我确认这两个函数产生相同的输出。

顺便说一句，我怀疑这个函数本来就是一个Pythonic。 欢迎提出改进建议的意见。

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