假设：
- 所有矩阵都是
symmetric（即正方形；等于它的转置；matrix[x,y]=matrix[y,x]）
-
matrix[x,x]=1 为所有 x
- 所有矩阵条目都是
0 或1
- 对单元素集群不感兴趣
- 对同一簇的排列不感兴趣（即
A,B 与B,A 相同）
- 因为我们不必担心排列，所以我们可以专注于按照它们在矩阵中出现的顺序处理元素（例如，我们处理
A,B,C 并忽略 A,C,B、B,A,C 的等价物, B,C,A, C,A,B 和 C,B,A);这使我们能够专注于处理矩阵的上/右半部分（在恒等/对角线上方），并按从左到右的顺序；这将大大减少我们需要评估的排列数量
- 如问题所示，组成一个簇的元素可以在矩阵中向上/向左移动，以便用
1's 填充矩阵的顶部/左侧（这在处理过程中发挥作用，对于每个新的元素，我们只需要测试添加到矩阵顶部/左侧部分的新列/行的等效项）
关于最后一个假设......假设我们有集群A,D，我们现在要测试A,D,F；我们只需要测试新的列/行条目 (?)：
Current Cluster New Cluster ?
A D A D F
A 1 1 A 1 1 ? # if matrix is symmetric then only need to test
D 1 1 D 1 1 ? # the new column *OR* the new row, not both;
F ? ? 1 # bottom/right == 1 == matrix[F,F] per earlier assumption
一个使用递归函数和两个GNU awk's 功能的想法：a) 数组数组（又称多维数组）和 b) PROCINFO["sorted_in"]将集群自定义排序到标准输出：
awk '
######
# load matrix into memory
FNR==1 { n=(NF-1) # number of elements to be processed
for (i=2;i<=NF;i++)
label[i-1]=$i # save labels
next
}
{ for (i=2;i<=NF;i++)
m[FNR-1][i-1]=$i # populate matrix array m[row#][column#]
}
######
# define our recursive function
function find_cluster(cluster, i, clstrcount, stackseq, j, k, corrcount) {
# cluster : current working cluster (eg, "A,B,C")
# i : index of latest element (eg, for "A,B,C" => latest element is "C" so i = 3
# clstrcount : number of elements in current cluster
# stackseq : sequence number of stack[] array
# : stack[] contains list of indexes for current cluster (for "A,B,C" stack = "1,2,3")
# j,k,corrcount : declaring additional variables as "local" to this invocation of the function
clstrcount++ # number of elements to be processed at this call/level
for (j=i+1;j<=n;j++) { # process all elements/indexes greater than i
corrcount=1 # reset correlation count; always start with 1 since m[j][j]=1
# check the new column/row added to the top/left of the matrix to see if it extends the current cluster (ie, all entries are "1")
for (k in stack) { # loop through element/indexes in stack
if (m[stack[k]][j]) # check column entries
corrcount++
if (m[j][stack[k]]) # check row entries; not necessary if matrix is symmetric but we will add it here to show the m[][] references
corrcount++
}
if (corrcount == (stackseq*2 +1) ) { # if we have all "1"s we have a new cluster of size clstrcount
stack[++stackseq]=j # "push" current element/index on stack; increment stack seq/index
cluster=cluster "," label[j] # add current element/label to cluster
max= (clstrcount>max) ? clstrcount : max # new max(cluster count) ?
clusters[clstrcount][++clsterseq]=cluster # add new cluster to our master list: clusters[cluster_count][seq]
find_cluster(cluster, j, clstrcount, stackseq) # recursive call to check for next element(s)
delete stack[stackseq--] # back from recursive call so "pop" curent element (j) from stack
gsub(/[,][^,]+$/,"",cluster) # remove current element/label from cluster to make way for next element/label to be tested
}
}
}
######
# start looking for clusters of size 2+
END { max=2 # not interested in clusters of "1"
for (i=1;i<n;i++) { # loop through list of elements
clstrcount=1 # init cluster count = 1
clstrseq=0 # init clusters[...][seq] sequence seed
cluster=label[i] # reset cluster to current element/label
stackseq=1 # reset stack[seq] sequence seed
stack[stackseq]=i # "push" current element on stack
find_cluster(cluster, i, clstrcount, stackseq) # start recursive calls looking for next element in cluster
}
######
# for now just display clusters with size > 2; adjust this next line to add/remove cluster sizes from stdout
if (max>2) # print list of clusters with length > 2
for (i=max;i>2;i--) { # print from largest to smallest and ...
PROCINFO["sorted_in"]="@val_str_asc" # in alphabetical order
printf "####### clusters of size %s:\n", i
for (j in clusters[i]) # loop through all entries for clusters of size "i"
print clusters[i][j]
}
}
' matrix.dat
注意：当前版本（诚然）有点冗长......我正在研究细节时记下第一次解决方案的结果；通过进一步分析，可以减少代码；话虽如此，在这个 11 元素矩阵中找到所有 2+ 大小的簇所花费的时间并不算太糟糕：
real 0m0.084s
user 0m0.031s
sys 0m0.046s
这会生成：
####### clusters of size 6:
A,B,C,D,F,K
A,B,C,E,H,I
####### clusters of size 5:
A,B,C,D,F
A,B,C,D,J
A,B,C,D,K
A,B,C,E,H
A,B,C,E,I
A,B,C,F,I
A,B,C,F,K
A,B,C,H,I
A,B,C,H,J
A,B,C,H,K
A,B,D,F,K
A,B,E,H,I
A,C,D,F,K
A,C,E,H,I
B,C,D,F,K
B,C,E,H,I
####### clusters of size 4:
A,B,C,D
A,B,C,E
A,B,C,F
A,B,C,G
A,B,C,H
A,B,C,I
A,B,C,J
A,B,C,K
A,B,D,F
A,B,D,J
A,B,D,K
A,B,E,H
A,B,E,I
A,B,F,I
A,B,F,K
A,B,H,I
A,B,H,J
A,B,H,K
A,C,D,F
A,C,D,J
A,C,D,K
A,C,E,H
A,C,E,I
A,C,F,I
A,C,F,K
A,C,H,I
A,C,H,J
A,C,H,K
A,D,F,K
A,E,H,I
B,C,D,F
B,C,D,J
B,C,D,K
B,C,E,H
B,C,E,I
B,C,F,I
B,C,F,K
B,C,H,I
B,C,H,J
B,C,H,K
B,D,F,K
B,E,H,I
C,D,F,K
C,E,H,I
####### clusters of size 3:
A,B,C
A,B,D
A,B,E
A,B,F
A,B,G
A,B,H
A,B,I
A,B,J
A,B,K
A,C,D
A,C,E
A,C,F
A,C,G
A,C,H
A,C,I
A,C,J
A,C,K
A,D,F
A,D,J
A,D,K
A,E,H
A,E,I
A,F,I
A,F,K
A,H,I
A,H,J
A,H,K
B,C,D
B,C,E
B,C,F
B,C,G
B,C,H
B,C,I
B,C,J
B,C,K
B,D,F
B,D,J
B,D,K
B,E,H
B,E,I
B,F,I
B,F,K
B,H,I
B,H,J
B,H,K
C,D,F
C,D,J
C,D,K
C,E,H
C,E,I
C,F,I
C,F,K
C,H,I
C,H,J
C,H,K
D,F,K
E,H,I