Efficiently looking for permutations that contain permutations through array operations?

Question

I have a set of integers, for example S = {1, ..., 10} and two matrices N and M, whose rows represent some (but not necessarily all possible) permutations of elements from S orders, for example, 3 and 5, respectively for example N = [1 2 3; 2 5 3; ...], M = [1 2 3 4 5; 2 4 7 8 1; ...].

A permutation Q of a permutation P is simply an indexed subset of P for which the order of indices of the elements of Q coincides with the order of their indices in P. Example: [2,4,7] is a permutation [2,3,4,6,7,1] , but [1,2,3] is not a permutation of the latter.

I need an efficient way (e.g. how to vectorize as much as possible and as little as possible for loops) to find

(1) all permutations from M that have subsets of N

and

(2) how many times each permutation of N must be found in M.

So far, what I have is a vectorized code that checks to see if the given single sub-permutation is contained in M ​​(and how many times), but then I have to use the parr loop through N, which slows down for very large Ns. Note that if N is not too large, you can also attack the problem by simply building valid 5-tuples from the given 3-tuples and compare the result with M, but this can quickly become much slower than simply forcing if N is large enough.

: , N M , .. N, M.

, , - , MATLAB.

EDIT: k- M: []

function [A,f] = my_function(x,M)
%// returns all rows in M that contain x and the absolute frequency of x in M
%// suboptimal for checking combinations rather than permutations byy at least ~ 50%
k = size(x,2);
m = size(M,1);
R = zeros(m,k);
I = R;
Z = I;
    for j = 1:k   
        [R(:,j),I(:,j)] = max((M == x(j)),[],2); 
        Z(:,j) = R(:,j).*I(:,j);
    end
z = zeros(m,k-1);
    for j = 1:(k-1)
        z(:,j) = (Z(:,j) > 0 & Z(:,j) < Z(:,j+1)); 
    end
[v,~] = find(sum(z,2) == k-1);    
A = M(v,:);
f = length(v);
end

, N (par) for-loop, .

Answer 1

№ 1

[val,ind] = max(bsxfun(@eq,permute(M,[4 2 1 3]),permute(N,[2 3 4 1])),[],2)
matches = squeeze(all(diff(ind,1)>0,1).*all(val,1))
out1 = any(matches,2) %// Solution - 1
out2 = sum(matches,1) %// Solution - 2

---

# 2

, permuting N longish N -

[val,ind] = max(bsxfun(@eq,N,permute(M,[3 4 1 2])),[],4)
matches = squeeze(all(diff(ind,[],2)>0,2).*all(val,2))
out1 = any(matches,1) %// Solution - 1
out2 = sum(matches,2) %// Solution - 2

---

№ 3

-scroogey -

out1 = false(size(M,1),1);  %// Storage for Solution - 1
out2 = zeros(size(N,1),1);  %// Storage for Solution - 2
for k=1:size(N,1)
    [val3,ind3] = max(bsxfun(@eq,N(k,:),permute(M,[1 3 2])),[],3);
    matches = all(diff(ind3,[],2)>0,2).*all(val3,2);
    out1 = or(out1,matches);
    out2(k) = sum(matches);
end

---

№ 4

-scroogey GPU -

gM = gpuArray(M);
gN = gpuArray(N);

gout1 = false(size(gM,1),1,'gpuArray');  %// GPU Storage for Solution - 1
gout2 = zeros(size(gN,1),1,'gpuArray');  %// GPU Storage for Solution - 2
for k=1:size(gN,1)
    [val3,ind3] = max(bsxfun(@eq,gN(k,:),permute(gM,[1 3 2])),[],3);
    matches = all(diff(ind3,[],2)>0,2).*all(val3,2);
    gout1 = or(gout1,matches);
    gout2(k) = sum(matches);
end
out1 = gather(gout1);  %// Solution - 1
out2 = gather(gout2);  %// Solution - 2

GPU . M : 320000X5 N : 2100X3 ( , ), . GTX 750 Ti 13.867873 seconds!! , GPU , .

---

№ 5

--scroogey GPU -

gM = gpuArray(M);
gN = gpuArray(N);

gout1 = false(size(gM,1),1,'gpuArray');  %// GPU Storage for Solution - 1
gout2 = zeros(size(gN,1),1,'gpuArray');  %// GPU Storage for Solution - 2
for k=1:size(gN,1)
    [val2,ind2] = max(bsxfun(@eq,gM,permute(gN(k,:),[1 3 2])),[],2);
    matches = all(diff(ind2,[],3)>0,3).*all(val2,3);
    gout1 = or(gout1,matches);
    gout2(k) = sum(matches);
end
out1 = gather(gout1);  %// Solution - 1
out2 = gather(gout2);  %// Solution - 2

Answer 2

?

n = size(N,1);
m = size(M,1);
p = size(N,2);
pattern = (1:p).'; %'// will be used for checking if it a subpermutation or not
result = false(m,n); %// preallocate result, and initiallize to 0
for k = 1:size(N,1) %// loop over columns of (transposed) N
    [~, loc] = ismember(M, N(k,:)); %// entries of M equal to a value of N(:,k)
    ind = find(sum(loc>0,2)==p); %// we need p matches per row of M
    t = reshape(nonzeros(loc(ind,:).'),p,[]); %'// for those rows, take matches
    ind2 = all(bsxfun(@eq,t,pattern)); %// ... see if they are in the right order
    result(ind(ind2),k) = true; %// ... and if so, take note in result matrix
end

result 1 r, s, s- N - r- M.

result1 = any(result,2);
result2 = sum(result,1);

:

M =
     8     9     4     1    10
     6     5     2     7     8
     4     1     9     2    10
     3     4     5     1     2

N =
     4     1     2
     4     9    10
     3     5     9

result =
     0     0     0
     0     0     0
     1     1     0
     1     0     0

result1 =
     0
     0
     1
     1

result2 =
     2     1     0

Answer 3

N,M, , , parfor vs. for . :

    %//Test 1: size(N) = 2263x3, size(M) = 5000x6

    %//My approach (parfor): 0.650626 sec
    %//Divakar Approach 1: 1.870144 sec
    %//Divakar Approach 2: 1.164088 sec
    %//Divakar Approach 3: 0.380915 sec (with parfor)
    %//Divakar Approach 4: 2.643659 sec (gpu)
    %//Luis Mendo Approach: 1.681007 sec


    %//Test 2: size(N) = 2263x3, size(M) = 25000x6

    %//My approach (parfor): 6.137823 sec
    %//Divakar Approach 1: 8.342699 sec
    %//Divakar Approach 2: 5.784426 sec
    %//Divakar Approach 3: 2.251888 sec (with parfor)
    %//Divakar Approach 4: 6.272578 sec (gpu)
    %//Luis Mendo Approach: 11.514548 sec

    %//Test 3: size(N) = 2100x3, size(M) = 20000x5

    %//My approach (parfor): 3.417432 sec
    %//Divakar Approach 1: 5.732680 sec
    %//Divakar Approach 2: 4.107909 sec
    %//Divakar Approach 3: 1.393052 sec (with parfor)
    %//Divakar Approach 4: 3.145183 sec (gpu)
    %//Luis Mendo Approach: 5.668326 sec

    %//Test 4: size(N) = 2100x3, size(M) = 324632x5

    %//Divakar Approach 3: 54.231878 sec (with parfor)
    %//Divakar Approach 4: 15.111936 sec (gpu) 

    %//Test 5: size(N) = 2263x3, size(M) = 1000000x6

    %//Divakar Approach 3: 210.853515 sec (with parfor)
    %//Divakar Approach 4: 49.529794 sec (gpu) 
    %//Divakar Approach 5: 49.874444 sec (gpu)

    %//Test 6: size(N) = 2263x3, size(M) = 5000000x6

    %//Divakar Approach 3: 1137.606244 sec (with parfor)
    %//Divakar Approach 4: stopped it after 15 min and heavy interrupts/DCPs activity
    %//Divakar Approach 5: 267.169307 sec

, gpu, Divakar 3 . gpu .