How to check if two permutations are symmetric?

Here is the working solution for the question:

def isSymmetric(A, B):
    L = len(A) #assume equivalent to len(B), modifying this would be as simple as checking if len(A) != len(B), return []
    la = L//2 # half-list length
    Al = A[:la]
    Ar = A[la:]
    Bl = B[:la]
    Br = B[la:]
    for i in range(la):
        lai = la - i #just to reduce the number of computation we need to perform
        for j in range(1, lai + 1):
            k = lai - j #same here, reduce computation
            if Al[i] != Br[k] or Ar[k] != Bl[i]: #the key for efficient computation is here: do not proceed unnecessarily
                 continue
            n = i #written only for the sake of clarity. i is n, and we can use i directly
            m = i + j
            if A[n:m] == B[L-m:L-n] and B[n:m] == A[L-m:L-n]: #possibly symmetric
                if A[0:n] == B[0:n] and A[m:L-m] == B[m:L-m] and A[L-n:] == B[L-n:]:
                    return [n, m]
    return []

As you have mentioned, though the idea looks simple, but it is actually quite a tricky one. Once we see the patterns, however, the implementation is straight-forward.

The central idea of the solution is this single line:

if Al[i] != Br[k] or Ar[k] != Bl[i]: #the key for efficient computation is here: do not proceed unnecessarily

All other lines are just either direct code translation from the problem statement or optimization made for more efficient computation.

There are few steps involved in order to find the solution:

Firstly, we need to split the each both list Aand list B into two half-lists (called Al, Ar, Bl, and Br). Each half-list would contain half of the members of the original lists:

Al = A[:la]
Ar = A[la:]
Bl = B[:la]
Br = B[la:]

Secondly, to make the evaluation efficient, the goal here is to find what I would call pivot index to decide whether a position in the list (index) is worth evaluated or not to check if the lists are symmetric. This pivot index is the central idea to find an efficient solution. So I would try to elaborate it quite a bit:

Consider the left half part of the A list, suppose you have a member like this:

Al = [al1, al2, al3, al4, al5, al6]

We can imagine that there is a corresponding index list for the mentioned list like this

Al  = [al1, al2, al3, al4, al5, al6]
iAl = [0,   1,   2,   3,   4,   5  ] #corresponding index list, added for explanation purpose

(Note: the reason why I mention of imagining a corresponding index list is for ease of explanation purposes)

Likewise, we can imagine that the other three lists may have similar index lists. Let's name them iAr, iBl, and iBr respectively and they are all having identical members with iAl.

It is the index of the lists which would really matter for us to look into - in order to solve the problem.

Here is what I mean: suppose we have two parameters:

1. index (let's give a variable name i to it, and I would use symbol ^ for current i)
2. length (let's give a variable name j to it, and I would use symbol == to visually represent its length value)

for each evaluation of the index element in iAl - then each evaluation would mean:

Given an index value i and length value of j in iAl, do something to determine if it is worth to check for symmetric qualifications starting from that index and with that length (Hence the name pivot index come).

Now, let's take example of one evaluation when i = 0 and j = 1. The evaluation can be illustrated as follow:

iAl = [0, 1, 2, 3, 4, 5]
       ^ <-- now evaluate this index (i) = 0
       == <-- now this has length (j) of 1

In order for those index i and length j to be worth evaluated further, then the counterpart iBr must have the same item value with the same length but on different index (let's name it index k)

iBr = [0, 1, 2, 3, 4, 5]
                      ^ <-- must compare the value in this index to what is pointed by iAl
                      == <-- must evaluate with the same length = 1

For example, for the above case, this is a possible "symmetric" permutation just for the two lists Al-Br (we will consider the other two lists Ar-Bl later):

Al = [0, x, x, x, x, x] #x means don't care for now
Br = [x, x, x, x, x, 0]

At this moment, it is good to note that

It won't worth evaluating further if even the above condition is not true

And this is where you get the algorithm to be more efficient; that is, by selectively evaluating only the few possible cases among all possible cases. And how to find the few possible cases?

By trying to find relationship between indexes and lengths of the four lists. That is, for a given index i and length j in a list (say Al), what must be the index k in the counterpart list (in the case is Br). Length for the counterpart list need not be found because it is the same as in the list (that is j).

Having know that, let's now proceed further to see if we can see more patterns in the evaluation process.

Consider now the effect of length (j). For example, if we are to evaluate from index 0, but the length is 2 then the counterpart list would need to have different index k evaluated than when the length is 1

iAl = [0, 1, 2, 3, 4, 5]
       ^ <-- now evaluate this index (i) = 0
       ===== <-- now this has length (j) of 2

iBr = [0, 1, 2, 3, 4, 5]
                   ^ <-- must compare the value in this index to what is pointed by iAl
                   ===== <-- must evaluate with the same length = 2

Or, for the illustration above, what really matters fox i = 0 and y = 2 is something like this:

# when i = 0 and y = 2
Al = [0, y, x, x, x, x] #x means don't care for now
Br = [x, x, x, x, 0, y] #y means to be checked later

Take a look that the above pattern is a bit different from when i = 0 and y = 1 - the index position for 0 value in the example is shifted:

# when i = 0 and y = 1, k = 5
Al = [0, x, x, x, x, x] #x means don't care for now
Br = [x, x, x, x, x, 0]

# when i = 0 and y = 2, k = 4
Al = [0, y, x, x, x, x] #x means don't care for now
Br = [x, x, x, x, 0, y] #y means to be checked later

Thus, length shifts where the index of the counterpart list must be checked. In the first case, when i = 0 and y = 1, then the k = 5. But in the second case, when i = 0 and y = 1, then the k = 4. Thus we found the pivot indexes relationship when we change the length j for a fixed index i (in this case being 0) unto the counterpart list index k.

Now, consider the effects of index i with fixed length j for counterpart list index k. For example, let's fix the length as y = 4, then for index i = 0, we have:

iAl = [0, 1, 2, 3, 4, 5]
       ^ <-- now evaluate this index (i) = 0
       ========== <-- now this has length (j) of 4

iAl = [0, 1, 2, 3, 4, 5]
          ^ <-- now evaluate this index (i) = 1
          ========== <-- now this has length (j) of 4

iAl = [0, 1, 2, 3, 4, 5]
             ^ <-- now evaluate this index (i) = 2
             ========== <-- now this has length (j) of 4

#And no more needed

In the above example, it can be seen that we need to evaluate 3 possibilities for the given i and j, but if the index i is changed to 1 with the same length j = 4:

iAl = [0, 1, 2, 3, 4, 5]
          ^ <-- now evaluate this index (i) = 1
          ========== <-- now this has length (j) of 4

iAl = [0, 1, 2, 3, 4, 5]
             ^ <-- now evaluate this index (i) = 2
             ========== <-- now this has length (j) of 4

Note that we only need to evaluate 2 possibilities. Thus the increase of index i decreases the number of possible cases to be evaluated!

With all the above patterns found, we almost found all the basis we need to make the algorithm works. But to complete that, we need to find the relationship between indexes which appear in Al-Br pair for a given [i, j] => [k, j] with the indexes in Ar-Bl pair for the same [i, j].

Now, we can actually see that they are simply mirroring the relationship we found in Al-Br pair!

(IMHO, this is really beautiful! and thus I think term "symmetric" permutation is not far from truth)

For example, if we have the following Al-Br pair evaluated with i = 0 and y = 2

Al = [0, y, x, x, x, x] #x means don't care for now
Br = [x, x, x, x, 0, y] #y means to be checked later

Then, to make it symmetric, we must have the corresponding Ar-Bl:

Ar = [x, x, x, x, 3, y] #x means don't care for now
Bl = [3, y, x, x, x, x] #y means to be checked later

The indexing of Al-Br pair is mirroring (or, is symmetric to) the indexing of Ar-Bl pair!

Therefore, combining all the pattern we found above, we now could find the pivot indexes for evaluating Al, Ar, Bl, and Br.

We only need to check the values of the lists in the pivot index first. If the values of the lists in the pivot indexes of Al, Ar, Bl, and Br matches in the evaluation then and only then we need to check for symmetric criteria (thus making the computation efficient!)

Putting up all the knowledge above into code, the following is the resulting for-loop Python code to check for symmetricity:

for i in range(len(Al)): #for every index in the list
    lai = la - i #just simplification
    for j in range(1, lai + 1): #get the length from 1 to la - i + 1
        k = lai - j #get the mirror index
        if Al[i] != Br[k] or Ar[k] != Bl[i]: #if the value in the pivot indexes do not match
             continue #skip, no need to evaluate
        #at this point onwards, then the values in the pivot indexes match
        n = i #assign n
        m = i + j #assign m
        #test if the first two conditions for symmetric are passed
        if A[n:m] == B[L-m:L-n] and B[n:m] == A[L-m:L-n]: #possibly symmetric
            #if it passes, test the third condition for symmetric, the rests of the elements must stay in its place
            if A[0:n] == B[0:n] and A[m:L-m] == B[m:L-m] and A[L-n:] == B[L-n:]:                   
                return [n, m] #if all three conditions are passed, symmetric lists are found! return [n, m] immediately!
        #passing this but not outside of the loop means 
        #any of the 3 conditions to find symmetry are failed
        #though values in the pivot indexes match, simply continue
return [] #nothing can be found - asymmetric lists

And there go you with the symmetric test!

(OK, this is quite a challenge and it takes quite a while for me to figure out how.)

I tried to implement 3 different algorithms for this task. All of them have O(N) time complexity and require O(1) additional space. Interesting fact: all other answers (known so far) implement 2 of these algorithms (though they not always keep optimal asymptotic time/space complexity). Here is high-level description for each algorithm:

Algorithm A

1. Compare the lists, group "non-equal" intervals, make sure there are exactly two such intervals (with special case when intervals meet in the middle).
2. Check if "non-equal" intervals are positioned symmetrically, and their contents is also "symmetrical".

Algorithm B

1. Compare first halves of the lists to guess where are "intervals to be exchanged".
2. Check if contents of these intervals is "symmetrical". And make sure the lists are equal outside of these intervals.

Algorithm C

1. Compare first halves of the lists to find first mismatched element.
2. Find this mismatched element of first list in second one. This hints the position of "intervals to be exchanged".
3. Check if contents of these intervals is "symmetrical". And make sure the lists are equal outside of these intervals.

There are two alternative implementations for step 1 of each algorithm: (1) using itertools, and (2) using plain loops (or list comprehensions). itertools are efficient for long lists but relatively slow on short lists.

Here is algorithm C with first step implemented using itertools. It looks simpler than other two algorithms (at the end of this post). And it is pretty fast, even for short lists:

import itertools as it
import operator as op

def test_C(a, b):
    length = len(a)
    half = length // 2
    mismatches = it.imap(op.ne, a, b[:half]) # compare half-lists

    try:
        n = next(it.compress(it.count(), mismatches))
        nr = length - n
        mr = a.index(b[n], half, nr)
        m = length - mr
    except StopIteration: return None
    except ValueError: return None

    if a[n:m] == b[mr:nr] and b[n:m] == a[mr:nr] \
            and a[m:mr] == b[m:mr] and a[nr:] == b[nr:]:
        return (n, m)

This could be done using mostly itertools:

def test_A(a, b):
    equals = it.imap(op.eq, a, b) # compare lists
    e1, e2 = it.tee(equals)
    l = it.chain(e1, [True])
    r = it.chain([True], e2)
    borders = it.imap(op.ne, l, r) # delimit equal/non-equal intervals
    ranges = list(it.islice(it.compress(it.count(), borders), 5))

    if len(ranges) == 4:
        n1, m1 = ranges[0], ranges[1]
        n2, m2 = ranges[2], ranges[3]
    elif len(ranges) == 2:
        n1, m1 = ranges[0], len(a) // 2
        n2, m2 = len(a) // 2, ranges[1]
    else:
        return None

    if n1 == len(a) - m2 and m1 == len(a) - n2 \
            and a[n1:m1] == b[n2:m2] and b[n1:m1] == a[n2:m2]:
        return (n1, m1)

High-level description of this algorithm is already provided in OP comments by @j_random_hacker. Here are some details:

Start with comparing the lists:

A  0 1 2 3 4 5 6 7
B  0 5 6 3 4 1 2 7
=  E N N E E N N E

Then find borders between equal/non-equal intervals:

=  E N N E E N N E
B  _ * _ * _ * _ *

Then determine ranges for non-equal elements:

B  _ * _ * _ * _ *
    [1 : 3] [5 : 7]

Then check if there are exactly 2 ranges (with special case when both ranges meet in the middle), the ranges themselves are symmetrical, and their contents too.

Other alternative is to use itertools to process only half of each list. This allows slightly simpler (and slightly faster) algorithm because there is no need to handle a special case:

def test_B(a, b):
    equals = it.imap(op.eq, a, b[:len(a) // 2]) # compare half-lists
    e1, e2 = it.tee(equals)
    l = it.chain(e1, [True])
    r = it.chain([True], e2)
    borders = it.imap(op.ne, l, r) # delimit equal/non-equal intervals
    ranges = list(it.islice(it.compress(it.count(), borders), 2))

    if len(ranges) != 2:
        return None

    n, m = ranges[0], ranges[1]
    nr, mr = len(a) - n, len(a) - m

    if a[n:m] == b[mr:nr] and b[n:m] == a[mr:nr] \
            and a[m:mr] == b[m:mr] and a[nr:] == b[nr:]:
        return (n, m)

This does the right thing:

Br = B[L//2:]+B[:L//2]
same_full = [a==b for (a,b) in zip(A, Br)]
same_part = [a+b for (a,b) in zip(same_full[L//2:], same_full[:L//2])]

for n, vn in enumerate(same_part):
    if vn != 2:
        continue
    m = n
    for vm in same_part[n+1:]:
        if vm != 2:
            break
        m+=1

    if m>n:
        print("n=", n, "m=", m+1)

I'm pretty sure you could do the counting a bit bettter, but... meh

I believe the following pseudocode should work:

1. Find the first element i for which A[i] != B[i], set n = i. If no such element, return success. If n >= L/2, return fail.
2. Find the first element i > n for which A[i] == B[i], set m = i. If no such element or m > L/2, set m = L/2.
3. Check so A[0:n] == B[0:n], A[n:m] == B[L-m:L-n], B[n:m] == A[L-m:L-n], A[m:L-m] == B[m:L-m] and A[L-n:L] == B[L-n:L]. If all are true, return success. Else, return fail.

Complexity is O(n) which should be the lowest possible as one always needs to compare all elements in the lists.

I build a map of where the characters are in list B, then use that to determine the implied subranges in list A. Once I have the subranges, I can sanity check some of the info, and compare the slices.

If A[i] == x, then where does x appear in B? Call that position p.

I know i, the start of the left subrange.

I know L (= len(A)), so I know L-i, the end of the right subrange.

If I know p, then I know the implied start of the right subrange, assuming that B[p] and A[i] are the start of a symmetric pair of ranges. Thus, the OP's L - m would be p if the lists were symmetric.

Setting L-m == p gives me m, so I have all four end points.

Sanity tests are:

• n and m are in left half of list(s)
• n <= m (note: OP did not prohibit n == m)
• L-n is in right half of list (computed)
• L-m is in right half (this is a good check for quick fail)

If all those check out, compare A[left] == B[right] and B[left] == A[right]. Return left if true.

def find_symmetry(a:list, b:list) -> slice or None:

    assert len(a) == len(b)
    assert set(a) == set(b)
    assert len(set(a)) == len(a)

    length = len(a)
    assert length % 2 == 0

    half = length // 2
    b_loc = {bi:n for n,bi in enumerate(b)}

    for n,ai in enumerate(a[:half]):
        L_n = length - 1 - n    # L - n
        L_m = b_loc[ai]         # L - m (speculative)

        if L_m < half:         # Sanity: bail if on wrong side
            continue

        m = b_loc[a[L_n]]  # If A[n] starts range, A[m] ends it.

        if m < n or m > half:   # Sanity: bail if backwards or wrong side
            continue

        left = slice(n, m+1)
        right = slice(L_m, L_n+1)

        if a[left] == b[right] and \
            b[left] == a[right]:
            return left

    return None

res = find_symmetry(
        [ 10, 11, 12, 13, 14, 15, 16, 17, ],
        [ 10, 15, 16, 13, 14, 11, 12, 17, ])
assert res == slice(1,3)

res = find_symmetry(
    [ 0, 1, 2, 3, 4, 5, 6, 7, ],
    [ 1, 0, 2, 3, 4, 5, 7, 6, ])
assert res is None

res = find_symmetry("abcdefghijklmn", "nbcdefghijklma")
assert res == slice(0,1)

res = find_symmetry("abcdefghijklmn", "abjklfghicdmen")
assert res == slice(3,4)

res = find_symmetry("abcdefghijklmn", "ancjkfghidelmb")
assert res == slice(3,5)

res = find_symmetry("abcdefghijklmn", "bcdefgaijklmnh")
assert res is None

res = find_symmetry("012345", "013245")
assert res == slice(2,3)

Here's an O(N) solution which passes the test code:

def sym_check(a, b):
    cnt = len(a)

    ml = [a[i] == b[i] for i in range(cnt)]

    sl = [i for i in range(cnt) if (i == 0 or ml[i-1]) and not ml[i]]
    el = [i+1 for i in range(cnt) if not ml[i] and (i == cnt-1 or ml[i+1])]

    assert(len(sl) == len(el))

    range_cnt = len(sl)

    if range_cnt == 1:
        start1 = sl[0]
        end2 = el[0]

        if (end2 - start1) % 2 != 0:
            return None

        end1 = (start1 + end2) // 2
        start2 = end1

    elif range_cnt == 2:

        start1, start2 = sl
        end1, end2 = el

    else:
        return None

    if end1 - start1 != end2 - start2:
        return None

    if start1 != cnt - end2:
        return None

    if a[start1:end1] != b[start2:end2]:
        return None

    if b[start1:end1] != a[start2:end2]:
        return None

    return start1, end1

I only tested it with Python 2, but I believe it will also work with Python 3.

It identifies the ranges where the two lists differ. It looks for two such ranges (if there is only one such range, it tries to divide it in half). It then checks that both ranges are the same length and in the proper positions relative to each other. If so, then it checks that the elements in the ranges match.

Yet another version:

def compare(a, b):
    i_zip = list(enumerate(zip(a, b)))
    llen = len(a)
    hp = llen // 2

    def find_index(i_zip):
        for i, (x, y) in i_zip:
            if x != y:
                return i
        return i_zip[0][0]

    # n and m are determined by the unmoved items:
    n = find_index(i_zip[:hp])
    p = find_index(i_zip[hp:])
    m = llen - p
    q = llen - n
    # Symmetric?
    if a[:n] + a[p:q] + a[m:p] + a[n:m] + a[q:] != b:
        return None
    return n, m

This solution is based on:

1. All validly permuted list pairs A, B adhering to the symmetry requirement will have the structure:

   A = P1 + P2 + P3 + P4 + P5 B = P1 + P4 + P3 + P2 + P5 ^n ^m ^hp ^p ^q <- indexes

   ,len(P1) == len(P5) and len(P2) == len(P4)

2. Therefore the 3 last lines of the above function will determine the correct solution provided the indexes n, m are correctly determined. (p & q are just mirror indexes of m & n)
3. Finding n is a matter of determining when items of A and B start to diverge. Next the same method is applied to finding p starting from midpoint hp. m is just mirror index of p. All involved indexes are found and the solution emerges.

Make a list (ds) of indices where the first halves of the two lists differ. A possible n is the first such index, the last such index is m - 1. Check if valid symmetry. len(ds) == m - n makes sure there aren't any gaps.

import itertools as it
import operator as op

def test(a, b):
    sz = len(a)
    ds = list(it.compress(it.count(), map(op.ne, a[:sz//2], b[:sz//2])))
    n,m = ds[0], ds[-1]+1
    if a[n:m] == b[sz-m:sz-n] and b[n:m] == a[sz-m:sz-n] and len(ds) == m - n:
        return n,m
    else:
        return None

Here's a simple solution that passes my tests, and yours:

1. Compare the inputs, looking for a subsequence that does not match.
2. Transform A by transposing the mismatched subsequence according to the rules. Does the result match B?

The algorithm is O(N); there are no embedded loops, explicit or implicit.

In step 1, I need to detect the case where the swapped substrings are adjacent. This can only happen in the middle of the string, but I found it easier to just look out for the first element of the moved piece (firstval). Step 2 is simpler (and hence less error-prone) than explicitly checking all the constraints.

def compare(A, B):
    same = True
    for i, (a, b) in enumerate(zip(A,B)):
        if same and a != b:  # Found the start of a presumed transposition
            same = False
            n = i
            firstval = a  # First element of the transposed piece
        elif (not same) and (a == b or b == firstval):  # end of the transposition
            m = i
            break

    # Construct the transposed string, compare it to B
    origin = A[n:m] 
    if n == 0:  # swap begins at the edge
        dest = A[-m:]
        B_expect = dest + A[m:-m] + origin
    else:
        dest = A[-m:-n]
        B_expect = A[:n] + dest + A[m:-m] + origin + A[-n:]

    return bool(B_expect == B)

Sample use:

>>> compare("01234567", "45670123")
True

Bonus: I believe the name for this relationship would be "symmetric block transposition". A block transposition swaps two subsequences, taking ABCDE to ADCBE. (See definition 4 here; I actually found this by googling "ADCBE"). I've added "symmetric" to the name to describe the length conditions.