# Minimum re-labeling to make an array as close as possible to another one

I have an array with integer labels in [0, n-1], say arr1=[1, 0, 3, 2] with n = 3. And then I have another array again with integer labels in [0, n-1], say arr2=[0, 0, 2, 3]. How can I find the minimum re-labeling of arr2 in order to make it as close as possible to arr1 in terms of number of nonzero elements of their difference? For example, here the new arr2 is arr2* = [0, 0, 3, 2].

Here is the code that I'm already using:

from itertools import permutations
import numpy as np

arr1 = np.array([1, 0, 3, 2])
n_labels = 4
perms = list(permutations(np.arange(n_labels)))
arr2 = np.array([0, 0, 2, 3])

best_perm = None
best_err = 1e8
for perm in perms:
arr2_new = [perm[el] for el in arr2]
err = np.linalg.norm(arr2_new - arr1, ord = 0)
if err < best_err:
best_err = err
best_perm = perm
if best_err == 0:
break

arr2_new = [best_perm[el] for el in arr2]


Is there any way to make it more efficient?

UPDATED: Some notes

1. I need the least number of nonzero elements of arr1-arr2*. That's why I use the l0-norm.

2. I want a re-labeling scheme: every 0 should become 1, for example, and so on and so forth.

3. I don't want the positions of the elements of arr2 to change.

4. I also care about best_perm. Because I have another array, say c = [0.1, 1.1, 2.1, 3.1] in which each element corresponds to a label in arr2. So when I change arr2 to arr2*, I also want to change c to c* = [c[best_perm[j]] for j in range(n_labels)]

5. The arrays arr1 and arr2 may contain repeating elements. But the elements of both are in [0, n-1].

6. The solution may not be unique. We are interested in just one of the possible solutions.

• What will your typical array lengths actually be? Feb 4 at 21:30
• Do you mean integer? Yes. Say 1000-2000.
– ApPs
Feb 4 at 21:31
• Yes, yes, they are integer always.
– ApPs
Feb 4 at 21:34
• I meant, every element in arr2 with value equal to 0 should become 1, for example, and so on and so forth. Regarding best_perm, please see comment 4 in the original question.
– ApPs
Feb 4 at 21:56
• Is arr1 guaranteed to always have unique elements? Feb 6 at 14:47

Yes. First of all, your norm() call implies Frobenius: but any kind of least-squares is totally uncalled for in this problem since you just want integer comparison.

To put it lightly, the description of what this is actually supposed to do is troubled. For instance, rather than "labelling" I would call it "substitution". I hope the extended back-and-forth has prepared you for the kind of information reviewers need to know in future questions.

A better algorithm is definitely possible. First, frame the problem as a bipartite graph, where:

• the left subset nodes are unique values from the original arr2
• the right subset nodes are all possible destination values from 0 through n-1
• the graph starts as being complete and unbalanced; the left cardinality will typically be smaller than the right cardinality
• the edge weights are non-negative integers representing the number of mismatches between the source and destination

The problem becomes: how do you prune this graph such that

• for every left-node, there remains exactly one edge (the left subset becomes 1-regular);
• for every right-node, there remains at most one edge;
• the graph becomes disconnected and acyclic; and
• the edges are chosen such that the total remaining weight is minimised.

This is a known problem - a variant of bipartite maximum matching called Bipartite Min-Cost Perfect Matching, also called the assignment problem. Crucially, the insight from interpreting this as a bipartite matching problem is that it can be solved using mixed-integer linear programming (MIP).

MIP problems are well-understood and have excellent software that already exists to solve them. I recommend glpk and its Python SWIG bindings in swiglpk.

In this strategy,

• c, the cost vector, will be non-negative integers capturing your edge weights
• z, the objective, is the total distance
• xs, the structural variables/columns, will be a flattened 2d matrix of boolean edge choices
• xr, the auxiliary variables/rows, will be the node degree (sum of edge counts per node)
• A, the constraint matrix, will be constructed to sum up the node degrees over both axes

xs, when interpreted as a 2D matrix, must be constrained to have exactly one nonzero per (source) row, and between 0 and 1 nonzeros per (dest) column. In glpk these are expressed with fixed and double bounds, respectively.

It takes a little bit of code to express all of this, but it's quite worth it: when I run a problem of n=500 on my machine it completes in about two seconds.

## Suggested

import random
from collections import defaultdict
from timeit import timeit
from typing import Optional, Iterator

import swiglpk as lp

class OptError(Exception):
pass

def make_map() -> tuple[tuple[int, list[int]], ...]:
"""
From arr1 and arr2, produce an ordered map in tuple-of-pairs format
where the key is the unique source arr2 value, and the values are all
(potentially non-unique) arr1 values to which a match will be attempted.
"""
targets = defaultdict(list)
for source, dest in zip(arr2, arr1):
targets[source].append(dest)
return tuple(targets.items())

def fill_sources() -> None:
"""
For each of our unique arr2 source values,
- Fill some the GLPK problem's "auxiliary rows" of count n_source. These
rows correspond to constraints on the arr2 source values.
- Name those rows and apply fixed bounds: each source must get exactly one
match.

For each combination of source and candidate destination,
- Reinterpret the source/destination matrix as a flat, horizontal vector
- Fill all of the GLPK problem's "structural columns" of count
n_source*n_dest.
- Name those columns and set the structural variable type as binary-value.
- Set the corresponding objective coefficient to the mismatch cost of
choosing this pair.
- Populate the constraint matrix to enforce that each source gets one dest,
and every dest gets up to one source
"""
row_indices = lp.intArray(3)
row_values = lp.doubleArray(3)
row_values[1] = row_values[2] = 1

# For each source in the map
for y, (source, dests) in enumerate(target_map):
lp.glp_set_row_name(problem, y+1, f'{source}->')

# First n_source auxiliary rows are fixed-bounded at 1
lp.glp_set_row_bnds(
P=problem, i=y+1, type=lp.GLP_FX, lb=1, ub=1,
)

# For each possible destination (not just those in the map)
for candidate_dest in range(n_dest):
x = y*n_dest + candidate_dest + 1

# This structural variable is a Binary Variable
lp.glp_set_col_kind(problem, x, lp.GLP_BV)
lp.glp_set_col_name(problem, x, f"{source}->{candidate_dest}")

# Objective coefficient is the sum of mismatched elements
cost = sum(
candidate_dest != actual_dest
for actual_dest in dests
)
lp.glp_set_obj_coef(problem, x, cost)

# Set the constraint matrix elements (sparse)
row_indices[1] = 1 + y                          # sum the sources
row_indices[2] = 1 + n_source + candidate_dest  # sum the destinations
lp.glp_set_mat_col(
P=problem, j=x, len=2, ind=row_indices, val=row_values,
)

def fill_dests() -> None:
"""
For each of our candidate destination values,
- Fill the rest of the GLPK problem's "auxiliary rows" of count n_dest.
These rows correspond to constraints on the arr1 dest values.
- Name those rows and apply double bounds: each dest must get up to one
match.
"""

# For each possible destination
for candidate_dest in range(n_dest):
y = 1 + n_source + candidate_dest
lp.glp_set_row_name(problem, y, f'->{candidate_dest}')

# Last n_dest auxiliary rows are double-bounded between 0 and 1
lp.glp_set_row_bnds(
P=problem, i=y, type=lp.GLP_DB, lb=0, ub=1,
)

def create() -> None:
"""
Set some problem parameters, including objective characteristics; set the
structural and auxiliary size; call into subroutines to fill all problem
constraints.
"""
lp.glp_set_prob_name(problem, 'proximal_substitution')
lp.glp_set_obj_name(problem, 'distance')
lp.glp_set_obj_dir(problem, lp.GLP_MIN)

lp.glp_add_cols(problem, n_source * n_dest)
lp.glp_add_rows(problem, n_source + n_dest)
fill_sources()
fill_dests()

def check_code(code: int) -> None:
"""
If an optimisation function returned a failure code, attempt to convert
that into a code name and raise an OptError
"""
if code != 0:
codes = {
getattr(lp, k): k
for k in dir(lp)
if k.startswith('GLP_E')
}
raise OptError(f'glpk returned {codes[code]}')

def check_status(expected: int, actual: int) -> None:
"""
If an optimisation status is not as expected, attempt to convert the status
into a status name and raise an OptError
"""
if expected != actual:
statuses = {
getattr(lp, f'GLP_{stat}'): stat
for stat in ('OPT', 'FEAS', 'INFEAS', 'NOFEAS', 'UNBND', 'UNDEF')
}
raise OptError(
f'Optimisation failed: status {statuses.get(actual)}'
)

def solve(log_level: int = lp.GLP_MSG_ON) -> None:
"""
Perform a two-pass linear programming solver, with a simplex basis and then
a mixed-integer programming (MIP) discretisation.
"""

# First pass to calculate the basis segment: do not use the default MIP
# pre-solver because we want to use the faster primal method instead of dual
parm = lp.glp_smcp()
lp.glp_init_smcp(parm)
parm.msg_lev = log_level
check_code(lp.glp_simplex(problem, parm))
check_status(lp.GLP_FEAS, lp.glp_get_prim_stat(problem))
check_status(lp.GLP_OPT, lp.glp_get_status(problem))

# Second (very fast) MIP pass
parm = lp.glp_iocp()
lp.glp_init_iocp(parm)
parm.msg_lev = log_level
check_code(lp.glp_intopt(problem, parm))
check_status(lp.GLP_OPT, lp.glp_mip_status(problem))

def construct_substitution() -> Iterator[tuple[int, int]]:
for x in range(n_source * n_dest):
matched = lp.glp_mip_col_val(problem, x+1)
if matched:
i_source, dest = divmod(x, n_dest)
source, _ = target_map[i_source]
yield source, dest

def output(substitution: dict[int, int], log_file: Optional[str] = None) -> None:
if log_file:
lp.glp_print_mip(problem, log_file)

print('arr2, arr1, arr2new:')
print(' '.join(f'{d:3}' for d in arr2))
print(' '.join(f'{d:3}' for d in arr1))

print(' '.join(
f'{substitution[d]:3}'
for d in arr2
))

def main():
create()
t = timeit(solve, number=1)
print(f'\nSolution in {t:.2}s')
substitution = dict(construct_substitution())
output(substitution)

if __name__ == '__main__':
N = 500
random.seed(0)
arr1 = [random.randrange(N) for _ in range(N)]
arr2 = [random.randrange(N) for _ in range(N)]
target_map = make_map()
n_dest = len(arr1)
n_source = len(target_map)
problem = lp.glp_create_prob()
main()


## Output

Shown with GLP_MSG_ALL:

GLPK Integer Optimizer 5.0
319 rows, 23800 columns, 47600 non-zeros
23800 integer variables, all of which are binary
Preprocessing...
319 rows, 23800 columns, 47600 non-zeros
23800 integer variables, all of which are binary
Scaling...
A: min|aij| =  1.000e+00  max|aij| =  1.000e+00  ratio =  1.000e+00
Problem data seem to be well scaled
Constructing initial basis...
Size of triangular part is 319
Solving LP relaxation...
GLPK Simplex Optimizer 5.0
319 rows, 23800 columns, 47600 non-zeros
0: obj =   2.000000000e+02 inf =   1.180e+02 (1)
Perturbing LP to avoid stalling [299]...
352: obj =   1.990000000e+02 inf =   0.000e+00 (0)
Removing LP perturbation [796]...
*   796: obj =   1.020000000e+02 inf =   0.000e+00 (0) 2
OPTIMAL LP SOLUTION FOUND
Integer optimization begins...
Long-step dual simplex will be used
+   796: mip =     not found yet >=              -inf        (1; 0)
+   796: >>>>>   1.020000000e+02 >=   1.020000000e+02   0.0% (1; 0)
+   796: mip =   1.020000000e+02 >=     tree is empty   0.0% (0; 1)
INTEGER OPTIMAL SOLUTION FOUND
Writing MIP solution to 'problem-log.txt'...
arr2, arr1, arr2new:
89 111  46  15 128 119  10 152  25 179 100  51  66  91 187 120 145  43 178 172  52 196  14 173  40  41  87 135  64  30 152 113 170  44   3 120 174 104 145 130  79 166  91  99 168  64  39 143 176   3 117 189  20  85 189  11 139  71  34  61 195 123  90 156  73 172  91 151 162 158  33 183  79  99 191 106 166  20   0 152  49 178  85  40  61  57 163 114  96 181 172 145 106   8 102 179 145 107 197 169 181  11  42 114  16  66 179  40 114 135 124 143 154 193   0   9 126  83  79 119  12 106  48 140 162  21 185  33   3 102 173 106  80   0  54   3 183 193   0 172 135 156  25  48  30 155 166  50  77  71 176  46  25 121 101 160  20   5  70 115  29  65  34 167 133 166 165  88  29  39  71   4  10  10  52 174  66 142  80  93 145  10 191 179 155 167 126 182 164 117 163 111  95 137  45  53  96 150  74   2
98 194 107  10  66 130 124 103  77 122  91 149  55 129  35  72  35 193  24 158  64 136 180 154  37  79  25 186  18 175  84 120 143  25  90 111  80 156 163  52 141 122 113 133  66  15 140   3  23 184 102 181 171 160   0 156 126  85  62 186  83 180  16  48 145  56  61  36 139 114  23  20  81 130 125  27  77 141  74 180  31 140  85 138  52 154 140 150  73 113  23 152  98  81 147  61  74  47  48  47   8 156 168  66 121  17  22 173 193  33  38   9  20 179 138 174 100 180 134  70 133  60  55 173 150 107 148  70 115 126 169 164 179  91  21  83 156  29 124 150 161  85  48  62   4 187  69  29 180  56  95  43  85 109  15  25  37 178  56  11 146 162 136 154 174  18   6  31 162  48 155 147  30 100  23  94  29   9 155   5  49  47 183  31 122  53 186  15 173   5 139 108 158  25  66  17  56  18 165  77
98 108  43  10   4  70  30  84  48  61  91 149  12 129  35 111  49 193  24 150  64 136  22 169 138  79  26  33  18 175  84 120 143  32  90 111  94 105  49  52 134  69 129 130  40  18 140   3  95  90 102 181  37 160 181 156 126  56  62 186  83  54  16  85 145 150 129  36 139 114  23  20 134 130 125  60  69  37  74  84  46  24 160 138 186 154  68  66  73 113 150  49  60  81   8  61  49  72  75  47 113 156 168  66 121  12  61 138  66  33  38   3  80 179  74  82 100 180 134  70 133  60  55  87 139 107 148  23  90   8 169  60 155  74  21  90  20 179  74 150  33  85  48  55 175 122  69  29  27  56  95  43  48 109  96  25  37 178  99  11 146 162  62  53 174  69   6  31 146 140  56 147  30  30  64  94  12   9 155   5  49  30 125  61 122  53 100  15 173 102  68 108 158 112 199  17  73 116 165  77


## Abridged Log

Problem:    proximal_substitution
Rows:       319
Columns:    23800 (23800 integer, 23800 binary)
Non-zeros:  47600
Status:     INTEGER OPTIMAL
Objective:  distance = 102 (MINimum)

No.   Row name        Activity     Lower bound   Upper bound
------ ------------    ------------- ------------- -------------
1 89->                        1             1             =
2 111->                       1             1             =
3 46->                        1             1             =
4 15->                        1             1             =
5 128->                       1             1             =
...
23796 2->195       *              0             0             1
23797 2->196       *              0             0             1
23798 2->197       *              0             0             1
23799 2->198       *              0             0             1
23800 2->199       *              0             0             1

Integer feasibility conditions:

KKT.PE: max.abs.err = 0.00e+00 on row 0
max.rel.err = 0.00e+00 on row 0
High quality

KKT.PB: max.abs.err = 0.00e+00 on row 0
max.rel.err = 0.00e+00 on row 0
High quality

End of output

• Maybe I wasn't very clear, I'm sorry about it: 1) I need the least number of nonzero elements of arr1-arr2*. That's why I use the l0-norm. 2) I want a re-labeling scheme: every 0 should become 1, for example, and so on and so forth. 3) I don't want the positions of the elements of arr2 to change. Does your solution address the above already? Could you provide a code snippet for the included example? Thanks!
– ApPs
Feb 4 at 21:26
• I think you are missing something. My original code is correct. n_labels should be equal to the unique elements in arr1 (or arr2, equivalently). This is why, [best_perm[el] for el in arr2] did not make sense to you, because you were using a different definition of n_labels, and consequently of perms. As an example where your code fails: for arr1 = [1, 0, 3, 2, 1] and arr2 = [0, 0, 2, 3, 1], we should get arr2* = [0, 0, 3, 2, 1], i.e., every 0 in arr2 should become 1. Your code produces arr2* = [1, 0, 3, 2, 0]. In other words, the labeling of arr2 should stay the same.
– ApPs
Feb 5 at 0:33
• If every 0 in arr2 became 1, that would produce 11231 which makes even less sense. You're going to have to spend some time in your question describing what you mean by labelling. Feb 5 at 4:36
• n_labels should be equal to the unique elements in arr1 - if so, your code is wrong because it hard-codes for 4 and performs no unique operation. Feb 5 at 4:53
• Thanks, works perfectly.
– ApPs
Feb 10 at 16:35