python - 如何快速计算一组集合的所有交集的包含顺序

Question

这是 How to get all intersections of sets in python fast 的后续行动:

我有一个有限集合 A = {A1,...Ak} 的有限集合 Ai 的整数，我想在 Python 中计算以下内容:

1. A 的子集的所有交集: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. 一个。 X,Y 的所有对 (X,Y) 在 F 中设置，使得 X 是 Y 的子集。

   X,Y 的所有对 (X,Y) 在 F 中设置，使得 X 是 Y 的子集，并且 F 中不存在集合 Z，使得 Y 的 Z 子集的 X 子集。换句话说，没有集合 Z 适合按收容顺序在 X 和 Y 之间。这样的一对 (X,Y) 称为覆盖。

Why do I want to do that? -- I want to compute face lattices of 10^7 polytopes. In the scenario in mind, the collection A above contains 600 sets. It is indeed the famous 600-cell, the computation currently takes about 6 secs, and I would like that to go down by a factor of 10, if possible.

获得 2.a 的 6 秒。只需通过做就可以完成

# 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

一个例子是顶点标有 {1,2,3,4} 的正方形:集合 A 则为 {{1,2},{2,3},{3,4},{4,1 }}，交点 F 是 {{},{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})

一旦您获得了集合 F，我认为没有什么比仅仅比较元素更好的了。但我更多的是考虑一种算法，它使用关于刚刚相交的东西的知识同时计算 (1) 和 (2)。

根据下面 David K 的解决方案，给出原因，还有两个可以使用的假设:

1. 生成的订单使用唯一的最小值和唯一的最大值元素进行评分。也就是说，覆盖关系的每个最大链 F0 < F1 < ... < Fm 具有相同的长度，F0 是空集，Fm 是输入集 A 的并集。我们称集合 Fi 为秩 i，考虑到分级性，这是明确定义的。

2. 每个秩 M 集合恰好是 2 个秩 M+1 集合的交集。

非常感谢!

Answer 1

这是一个函数，它利用了输入中的列表是抽象多面体的各个方面的假设。不是取所有方面的集合的交集，此函数假定输入是 M + 1 阶多胞形内的 M 面(M 阶多胞形)的完整列表。然后它执行一个循环，其中每次迭代都采用完整的 M 面列表并生成完整的 (M-1) 面列表，同时为这两个面列表累积所有包含对。

函数的主循环与每对 M 面相交，并构建一个结构，列出每个交点和包含它的 M 面。这些交叉点包括所有的 (M-1)-faces，但也包括一些低级别的面孔。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)-face 的子集，因此任何作为另一个交叉路口子集的交叉路口都将被消除。

运行时间的粗略分割是 40% 与成对的面相交，40% 用于跟踪哪些 M 面包含每个结果交叉点，10% 以消除等级小于 M - 1 的面，和 10% 将包含对写入输出列表。我的电脑好像比你的慢(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，比速度快 10 到 15 倍生成所有包含对的原始函数(包括那些“跳过”排名的)。

请注意，并非每个整数列表列表都是新的有效输入函数，因为存在点集的集合不是抽象多胞体的面。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)代码来查找(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，然后返回一个修改后的列表，其中删除了所有这些对。我还通过调用 sorted() 修改了新旧函数他们放入输出中的每个整数列表，以便输出列表会比较正确。然后我确认这两个函数产生了相同的输出。

顺便说一下，我怀疑这个函数是不是 pythonic 的。欢迎提出改进意见。

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