Algorithm 如何生成块中的组合
我有一个算法，可以对输入元素的每个组合执行计算，例如，100个元素集中的每个5元素子集。我正在将它移植到GPU，现在我已经准备好了它的初始版本。为了加快速度，我想从本地内存加载数据，本地内存是有限的（例如32KB），可以容纳100个输入元素中的20个。因此，我必须以某种方式划分我的工作，并以块的形式生成组合。现在这是最难的部分，如何做到这一点。很可能我必须首先加载20个元素的数据，然后对这20个元素的5个元素子集执行计算。在此之后，我将不得不用新的替换部分（或全部），并对它们进行计算，然后冲洗并重复。您能告诉我如何在本地内存中选择替换的元素以避免重复工作吗？到目前为止，我得出的结论是，我必须一次更换至少16个，以避免重复工作的问题 编辑：以下是从5个元素中生成2个元素组合的示例。以下是所有可能案例的完整列表：Algorithm 如何生成块中的组合,algorithm,combinations,Algorithm,Combinations,我有一个算法，可以对输入元素的每个组合执行计算，例如，100个元素集中的每个5元素子集。我正在将它移植到GPU，现在我已经准备好了它的初始版本。为了加快速度，我想从本地内存加载数据，本地内存是有限的（例如32KB），可以容纳100个输入元素中的20个。因此，我必须以某种方式划分我的工作，并以块的形式生成组合。现在这是最难的部分，如何做到这一点。很可能我必须首先加载20个元素的数据，然后对这20个元素的5个元素子集执行计算。在此之后，我将不得不用新的替换部分（或全部），并对它们进行计算，然后冲洗并
1, 2
1, 3
1, 4
1, 5
2, 3
2, 4
2, 5
3, 4
3, 5
4, 5
1, 2
1, 3
2, 3
1, 4
1, 5
4, 5
1, 2
1, 3
2, 3
1, 4
1, 5
4, 5
1, 2 // duplicate
1, 4 // ok, new
2, 4 // ok, new
2, 4
2, 5
3, 4
3, 5
通过以这种方式拆分工作，我将能够使用有限的本地内存处理原始输入集的所有组合，该内存只能保存其中的一部分。例如，您有100个元素，您可以在内存中保存20个元素，并且您需要所有5个元素的组合，其中有C（100,5）=75287520 为了能够生成所有的组合，5个元素的每个组合都必须在内存中的某个位置。这可以通过将元素分成20/5=4个元素的组来实现；输入中有25个这样的组，C（25,5）=5个组的53130个组合 对于每个组的组合，我们将首先从五个组中的每个组生成一个元素的组合；这为我们提供了53130 x 45=54405120个独特的组合 我们现在有了组合，其中每个元素来自不同的组，即分区[1,1,1,1,1]。我们仍然需要找到划分[2,1,1]，[2,2,1]，[3,1,1]，[3,2]和[4,1]的组合。最简单的方法是分阶段进行，但最快的方法当然是将它们合并到[1,1,1,1,1]的第一阶段中，因为我们需要的所有组组合都在第一阶段的某个时间加载到内存中 对于分区[2,1,1,1]，我们依次加载每个组，作为包含2个元素的组，然后从其余24个组中加载3个组的每个组合，并从每个组中提取一个元素。这将需要25 x C（24,3）=50600个步骤，每个步骤产生C（4,2）x 43=384个组合，或总共19430400个 像[2,2,1]这样的分区有点不同，因为我们会依次加载每个组，使其成为包含2个元素的第一个组，但只加载后面的组，作为包含2个元素的第二个组，以避免重复。然后，对于其中的每一个，我们将加载其他23个组中的每一个，以获得最终的元素。这需要C（25,2）/2 x 23=6900个步骤，每个步骤产生C（4,2）x C（4,2）x C（4,1）=144个组合，总共99.36万个 分区[3,1,1]需要25 x C（24,2）=25 x 276=6900个步骤，每个步骤产生C（4,3）x 42=64个组合，总共441600个 分区[3,2]需要25 x 24=600个步骤，每个步骤产生C（4,3）x C（4,2）=24个组合，总共14400个 分区[4,1]需要25 x 24=600个步骤，每个步骤产生C（4,4）x C（4,1）=4个组合，总共2400个 因此，我们总共有：
[1,1,1,1,1] -> 54,405,120
[2,1,1,1] -> 19,430,400
[2,2,1] -> 993,600
[3,1,1] -> 441,600
[3,2] -> 14,400
[4,1] -> 2,400
----------
75,287,520 combinations
不同阶段的整合 运行分区[1,1,1,1,1,1]的所有组组合的最简单方法是将组1至21加载为组A，然后将A后的所有组加载为组B，将B后的所有组加载为组C，将B后的所有组加载为组C，将C后的所有组加载为组D，将D后的所有组加载为组E
A B C D E
1 2 3 4 5 <- ABCDE
1 2 3 4 6 <- ABCDE
...
1 2 3 4 25 <- ABCDE
1 2 3 5 6 <- ABCDE
...
1 2 3 24 25 <- ABCDE
1 2 4 5 6 <- ABCDE
...
1 2 23 24 25 <- ABCDE
1 3 4 5 6 <- ABCDE
...
1 22 23 24 25 <- ABCDE
2 3 4 5 6 <- ABCDE
...
21 22 23 24 25 <- ABCDE
k: [1,1,1 ... 1]
k-1: [2,1 ... 1] [1,2 ... 1] ... [1,1 ... 2]
...
2: [g,k-g] [k-g,g]
p=k-1: ABCDE [2,1,1,1,1] -> [a,a,b,c,d,e] C(g,2)*C(g,1)^4 combinations
[1,2,1,1,1] -> [a,b,b,c,d,e]
[1,1,2,1,1] -> [a,b,c,c,d,e]
[1,1,1,2,1] -> [a,b,c,d,d,e]
[1,1,1,1,2] -> [a,b,c,d,e,e]
ABCDE [2,1,1,1,1] -> [a,a,b,c,d,f]
[1,2,1,1,1] -> [a,b,b,c,d,f]
[1,1,2,1,1] -> [a,b,c,c,d,f]
[1,1,1,2,1] -> [a,b,c,d,d,f]
[1,1,1,1,2] -> [a,b,c,d,f,f]
...
BCDEF [2,1,1,1,1] -> [b,b,c,d,e,f]
[1,2,1,1,1] -> [b,c,c,d,e,f]
[1,1,2,1,1] -> [b,c,d,d,e,f]
[1,1,1,2,1] -> [b,c,d,e,e,f]
[1,1,1,1,2] -> [b,c,d,e,f,f]
A B C D E F
1 2 3 4 5 6
k: [1,1,1 ... 1]
k-1: [2,1 ... 1] [1,2 ... 1] ... [1,1 ... 2]
...
2: [g,k-g] [k-g,g]
p=k-1: ABCDE [2,1,1,1,1] -> [a,a,b,c,d,e] C(g,2)*C(g,1)^4 combinations
[1,2,1,1,1] -> [a,b,b,c,d,e]
[1,1,2,1,1] -> [a,b,c,c,d,e]
[1,1,1,2,1] -> [a,b,c,d,d,e]
[1,1,1,1,2] -> [a,b,c,d,e,e]
ABCDE [2,1,1,1,1] -> [a,a,b,c,d,f]
[1,2,1,1,1] -> [a,b,b,c,d,f]
[1,1,2,1,1] -> [a,b,c,c,d,f]
[1,1,1,2,1] -> [a,b,c,d,d,f]
[1,1,1,1,2] -> [a,b,c,d,f,f]
...
BCDEF [2,1,1,1,1] -> [b,b,c,d,e,f]
[1,2,1,1,1] -> [b,c,c,d,e,f]
[1,1,2,1,1] -> [b,c,d,d,e,f]
[1,1,1,2,1] -> [b,c,d,e,e,f]
[1,1,1,1,2] -> [b,c,d,e,f,f]
A B C D E F
1 2 3 4 5 7
...
1 2 3 4 5 e/g
A B C D E F
1 2 3 4 6 7
...
1 2 3 4 e/g-1 e/g
A B C D E F
1 2 3 5 6 7
...
1 2 3 e/g-2 e/g-1 e/g
A B C D E F
1 2 3 4 5 7
...
1 2 3 4 5 e/g
p=k: ABCDEF
p=k-1: ABCEF ABDEF ACDEF BCDEF
p=k-2: ABEF ACEF ADEF BCEF BDEF CDEF
...
p=2: EF
A B C D E F
1 2 3 4 6 7
...
1 2 3 4 e/g-1 e/g
p=k: ABCDEF
p=k-1: ABDEF ACDEF BCDEF
p=k-2: ADEF BDEF CDEF
...
p=2: none
A B C D E F
1 2 3 5 6 7
...
1 2 3 e/g-2 e/g-1 e/g
A B C D E F
e/g-5 e/g-4 e/g-3 e/g-2 e/g-1 e/g
p=k: ABCDEF
number of groups in memory: 3 (k)
group size: g = m/k = 3 elements
number of groups: e/g = 7
groups: 1:[1,2,3] 2:[4,5,6] 3:[7,8,9] 4:[10,11,12] 5:[13,14,15] 6:[16,17,18] 7:[19,20,21]
number of element sets loaded into memory: C(e/g,k) = C(7,3) = 35
partitions of k with max part g: [1,1,1] [2,1] [3]
permutations: 3:{[1,1,1]} 2:{[1,2],[2,1]} 1:{[3]}
group sets: 3:{[A,B,C]} 2:{[A,B],[A,C],[B,C]} 1:{[A],[B],[C]}
group sets: 3:{[A,B,C]} 2:{[A,B],[A,C],[B,C]} 1:{[A],[B],[C]} (all)
A B C
1 2 3 -> elements in memory: [1,2,3] [4,5,6] [7,8,9] -> 84 combinations
3: [1,1,1]:[A,B,C] -> [a,b,c] -> [1,4,7] [1,4,8] [1,4,9] [1,5,7] [1,5,8] [1,5,9] [1,6,7] [1,6,8] [1,6,9]
[2,4,7] [2,4,8] [2,4,9] [2,5,7] [2,5,8] [2,5,9] [2,6,7] [2,6,8] [2,6,9]
[3,4,7] [3,4,8] [3,4,9] [3,5,7] [3,5,8] [3,5,9] [3,6,7] [3,6,8] [3,6,9]
2: [1,2]:[A,B] -> [a,b,b] -> [1,4,5] [1,4,6] [1,5,6] [2,4,5] [2,4,6] [2,5,6] [3,4,5] [3,4,6] [3,5,6]
[1,2]:[A,C] -> [a,c,c] -> [1,7,8] [1,7,9] [1,8,9] [2,7,8] [2,7,9] [2,8,9] [3,7,8] [3,7,9] [3,8,9]
[1,2]:[B,C] -> [b,c,c] -> [4,7,8] [4,7,9] [4,8,9] [5,7,8] [5,7,9] [5,8,9] [6,7,8] [6,7,9] [6,8,9]
[2,1]:[A,B] -> [a,a,b] -> [1,2,4] [1,3,4] [2,3,4] [1,2,5] [1,3,5] [2,3,5] [1,2,6] [1,3,6] [2,3,6]
[2,1]:[A,C] -> [a,a,c] -> [1,2,7] [1,3,7] [2,3,7] [1,2,8] [1,3,8] [2,3,8] [1,2,9] [1,3,9] [2,3,9]
[2,1]:[B,C] -> [b,b,c] -> [4,5,7] [4,6,7] [5,6,7] [4,5,8] [4,6,8] [5,6,8] [4,5,9] [4,6,9] [5,6,9]
1: [3]:[A] -> [a,a,a] -> [1,2,3]
[3]:[B] -> [b,b,b] -> [4,5,6]
[3]:[C] -> [c,c,c] -> [7,8,9]
group sets: 3:{[A,B,C]} 2:{[A,C],[B,C]} 1:{[C]} (sets without C removed)
A B C
1 2 4 -> elements in memory: [1,2,3] [4,5,6] [10,11,12] -> 64 combinations
3: [1,1,1]:[A,B,C] -> [a,b,c] -> [1,4,10] [1,4,11] [1,4,12] [1,5,10] [1,5,11] [1,5,12] [1,6,10] [1,6,11] [1,6,12]
[2,4,10] [2,4,11] [2,4,12] [2,5,10] [2,5,11] [2,5,12] [2,6,10] [2,6,11] [2,6,12]
[3,4,10] [3,4,11] [3,4,12] [3,5,10] [3,5,11] [3,5,12] [3,6,10] [3,6,11] [3,6,12]
2: [1,2]:[A,C] -> [a,c,c] -> [1,10,11] [1,10,12] [1,11,12] [2,10,11] [2,10,12] [2,11,12] [3,10,11] [3,10,12] [3,11,12]
[1,2]:[B,C] -> [b,c,c] -> [4,10,11] [4,10,12] [4,11,12] [5,10,11] [5,10,12] [5,11,12] [6,10,11] [6,10,12] [6,11,12]
[2,1]:[A,C] -> [a,a,c] -> [1,2,10] [1,3,10] [2,3,10] [1,2,11] [1,3,11] [2,3,11] [1,2,12] [1,3,12] [2,3,12]
[2,1]:[B,C] -> [b,b,c] -> [4,5,10] [4,6,10] [5,6,10] [4,5,11] [4,6,11] [5,6,11] [4,5,12] [4,6,12] [5,6,12]
1: [3]:[C] -> [c,c,c] -> [10,11,12]
A B C
1 2 5 -> elements in memory: [1,2,3] [4,5,6] [13,14,15] -> 64 combinations
1 2 6 -> elements in memory: [1,2,3] [4,5,6] [16,17,18] -> 64 combinations
1 2 7 -> elements in memory: [1,2,3] [4,5,6] [19,20,21] -> 64 combinations
group sets: 3:{[A,B,C]} 2:{[B,C]} (sets without B removed)
A B C
1 3 4 -> elements in memory: [1,2,3] [7,8,9] [10,11,12] -> 45 combinations
3: [1,1,1]:[A,B,C] -> [a,b,c] -> [1,7,10] [1,7,11] [1,7,12] [1,8,10] [1,8,11] [1,8,12] [1,9,10] [1,9,11] [1,9,12]
[2,7,10] [2,7,11] [2,7,12] [2,8,10] [2,8,11] [2,8,12] [2,9,10] [2,9,11] [2,9,12]
[3,7,10] [3,7,11] [3,7,12] [3,8,10] [3,8,11] [3,8,12] [3,9,10] [3,9,11] [3,9,12]
2: [1,2]:[B,C] -> [b,c,c] -> [7,10,11] [7,10,12] [7,11,12] [8,10,11] [8,10,12] [8,11,12] [9,10,11] [9,10,12] [9,11,12]
[2,1]:[B,C] -> [b,b,c] -> [7,8,10] [7,9,10] [8,9,10] [7,8,11] [7,9,11] [8,9,11] [7,8,12] [7,9,12] [8,9,12]
A B C
1 3 5 -> elements in memory: [1,2,3] [7,8,9] [13,14,15] -> 45 combinations
1 3 6 -> elements in memory: [1,2,3] [7,8,9] [16,17,18] -> 45 combinations
1 3 7 -> elements in memory: [1,2,3] [7,8,9] [19,20,21] -> 45 combinations
1 4 5 -> elements in memory: [1,2,3] [7,8,9] [13,14,15] -> 45 combinations
1 4 6 -> elements in memory: [1,2,3] [7,8,9] [16,17,18] -> 45 combinations
1 4 7 -> elements in memory: [1,2,3] [7,8,9] [19,20,21] -> 45 combinations
1 5 6 -> elements in memory: [1,2,3] [7,8,9] [16,17,18] -> 45 combinations
1 5 7 -> elements in memory: [1,2,3] [7,8,9] [19,20,21] -> 45 combinations
1 6 7 -> elements in memory: [1,2,3] [7,8,9] [19,20,21] -> 45 combinations
group sets: 3:{[A,B,C]} (sets without A removed)
A B C
2 3 4 -> elements in memory: [4,5,6] [7,8,9] [10,11,12] -> 27 combinations
3: [1,1,1]:[A,B,C] -> [a,b,c] -> [4,7,10] [4,7,11] [4,7,12] [4,8,10] [4,8,11] [4,8,12] [4,9,10] [4,9,11] [4,9,12]
[5,7,10] [5,7,11] [5,7,12] [5,8,10] [5,8,11] [5,8,12] [5,9,10] [5,9,11] [5,9,12]
[6,7,10] [6,7,11] [6,7,12] [6,8,10] [6,8,11] [6,8,12] [6,9,10] [6,9,11] [6,9,12]
A B C
2 3 5 -> elements in memory: [4,5,6] [7,8,9] [13,14,15] -> 27 combinations
2 3 6 -> elements in memory: [4,5,6] [7,8,9] [16,17,18] -> 27 combinations
2 3 7 -> elements in memory: [4,5,6] [7,8,9] [19,20,21] -> 27 combinations
2 4 5 -> elements in memory: [4,5,6] [10,11,12] [13,14,15] -> 27 combinations
2 4 6 -> elements in memory: [4,5,6] [10,11,12] [16,17,18] -> 27 combinations
2 4 7 -> elements in memory: [4,5,6] [10,11,12] [19,20,21] -> 27 combinations
2 5 6 -> elements in memory: [4,5,6] [13,14,15] [16,17,18] -> 27 combinations
2 5 7 -> elements in memory: [4,5,6] [13,14,15] [19,20,21] -> 27 combinations
2 6 7 -> elements in memory: [4,5,6] [16,17,18] [19,20,21] -> 27 combinations
3 4 5 -> elements in memory: [7,8,9] [10,11,12] [13,14,15] -> 27 combinations
3 4 6 -> elements in memory: [7,8,9] [10,11,12] [16,17,18] -> 27 combinations
3 4 7 -> elements in memory: [7,8,9] [10,11,12] [19,20,21] -> 27 combinations
3 5 6 -> elements in memory: [7,8,9] [13,14,15] [16,17,18] -> 27 combinations
3 5 7 -> elements in memory: [7,8,9] [13,14,15] [19,20,21] -> 27 combinations
3 6 7 -> elements in memory: [7,8,9] [16,17,18] [19,20,21] -> 27 combinations
4 5 6 -> elements in memory: [10,11,12] [13,14,15] [16,17,18] -> 27 combinations
4 5 7 -> elements in memory: [10,11,12] [13,14,15] [19,20,21] -> 27 combinations
4 6 7 -> elements in memory: [10,11,12] [16,17,18] [19,20,21] -> 27 combinations
5 6 7 -> elements in memory: [13,14,15] [16,17,18] [19,20,21] -> 27 combinations
Phase 1: 84 = 84 combinations
Phase 2: 4 x 64 = 256 combinations
Phase 3: 10 x 45 = 450 combinations
Phase 4: 20 x 27 = 540 combinations
----
1330 combinations = C(21,3)