Consider the set, S, of all binary vectors of length n where each contains exactly m ones; so there are n-m zeros in each vector.
My goal is to construct a number, k, of vectors from S such that these vectors are as different as possible from each other.
As a simple example, take n=4, m=2 and k=2, then a possible solution is: [1,1,0,0] and [0,0,1,1].
It seems that this is an open problem in the coding theory literature (?).
Is there any way (i.e. algorithm) to find a suboptimal yet good solution ?
Is Hamming distance the right performance measure to use in this case ?
Some thoughts:
In this paper, the authors propose a couple of algorithms to find the subset of vectors such that the pairwise Hamming distance is >= a certain value, d.
I have implemented the Random approach as follows: take a set SS, which is initialized by any vector from S. Then, I consider the remaining vectors
in S. For each of these vectors, I check if this vector has at least a distance d with respect to each vector in SS. If so, then it is added to SS.
By taking the maximal possible d, if the size of SS is >= k, then I consider SS as an optimal solution, and I choose any subset of k vectors from SS.
Using this approach, I think that the resulting SS will depend on the identity of the initial vector in SS; i.e. there are multiple solutions(?).
But how to proceed if the size of SS is < k ?
From the proposed algorithms in the paper, I have only understood the Random one. I am interested in the Binary lexicographic search (section 2.3) but I don't know how to implement it (?).
Maybe you find this paper useful (I wrote it). It contains algorithms that efficiently create permutations of bitstrings.
For example, the inc() algorithm:
long inc(long h_in , long m0 , long m1) {
long h_out = h_in | (~m1); //pre -mask
h_out ++;
// increment
h_out = (h_out & m1) | m0; //post -mask
return h_out;
}
It takes an input h_in and return the next higher value that is at least 1 larger than h_in and 'matches' the boundaries m0 and m1. 'Matching' means: the result has a 1 whereever m0 has a 1, and the result has a 0 whereever m1 has a 0. Not that h_in MUST BE a valid value with regards to mo and m1! Also, note that m0 has to be bitwise smaller than m1, which means that m0 cannot have a 1 in a position where m1 has a 0.
This could be used to generate permutations with a minimum edit distance to a given input string:
Let's assume you have 0110, you first NEGATE it to 1001 (edit distance = k).
Set 'm0=1001' and 'm1=1001'. Using this would result only on '1001' itself.
Now to get all values with edit distance k-1, you can do the following, simply flip one of the bits of m0 or m1, then inc() will return an ordered series of all bitstring that have a difference of k or k-1.
I know, not very interesting yet, but you can modify up to k bits, and inc() will always return all permutations with the maximum allowed edit difference with regard to m0 and m1.
Now, to get all permutations, you would have to re-run the algorithm with all possibly combinations of m0 and m1.
Example: To get all possible permutations of 0110 with edit distance 2, you would have to run inc() with the following permutations of m0=0110 and m1=0110 (to get permutations, a bit position has to be expanded, meaning that m0 is set to 0 and m1 is set to 1:
- Bit 0 and 1 expanded:
m0=0010andm1=1110 - Bit 0 and 2 expanded:
m0=0100andm1=1110 - Bit 0 and 3 expanded:
m0=0110andm1=1111 - Bit 1 and 2 expanded:
m0=0000andm1=0110 - Bit 1 and 3 expanded:
m0=0010andm1=0111 - Bit 2 and 3 expanded:
m0=0100andm1=0111
As starting value for h_0 I suggest to use simply m0. Iteration can be aborted once inc() returns m1.
Summary
The above algorithm generates in O(x) all x binary vectors that differ in at least y bits (configurable) from a given vector v.
Using your definition of n=number of bits in a vector v, setting y=n generates exactly 1 vector which is the exact opposite of the input vector v. For y=n-1, it will generate n+1 vectors: n vectors which differ in all but one bits and 1 vector that differs in all bits. And so on different values of y.
**EDIT: Added summary and replaced erroneous 'XOR' with 'NEGATE' in the text above.
I don't know if maximizing the sum of the Hamming distances is the best criterion to obtain a set of "maximally different" binary vectors, but I strongly suspect it is. Furthermore I strongly suspect that the algorithm that I'm going to present yields exactly a set of k vectors that maximizes the sum of Hamming distances for vectors of n bits of with m ones and n - m zeroes. Unfortunately I don't have the time to prove it (and, of course, I might be wrong – in which case you would be left with a “suboptimal yet good” solution, as per your request.)
Warning: In the following I'm assuming that, as a further condition, the result set may not contain the same vector twice.
The algorithm I propose is the following:
Starting from a result set with just one vector, repeatedly add one of those remaining vectors that have the maximum sum of Hamming distances from all the vectors that are already in the result set. Stop when the result set contains k vectors or all available vectors have been added.
Please note that the sum of Hamming distances of the result set does not depend on the choice of the first or any subsequent vector.
I found a “brute force” approach to be viable, given the constraints you mentioned in a comment:
n<25, 1<m<10, 10<k<100 (or 10<k<50)
The “brute force” consists in precalculating all vectors in “lexicographical” order in an array, and also keeping up-to-date an array of the same size that contains, for each vector with the same index, the total Hamming distance of that vector to all the vectors that are in the result set. At each iteration the total Hamming distances are updated, and the first (in “lexicographical” order) of all vectors that have the maximum total Hamming distance from the current result set is chosen. The chosen vector is added to the result set, and the arrays are shifted in order to fill in its place, effectively decreasing their size.
Here is my solution in Java. It's meant to be easily translatable to any procedural language, if needed. The part that calculates the combinations of m items out of n can be replaced by a library call, if one is available. The following Java methods have a corresponding C/C++ macro that uses fast specialized processor instructions on modern CPUs:
Long.numberOfTrailingZeros→__builtin_ctzl, Long.bitCount→__builtin_popcountl.
package waltertross.bits;
public class BitsMain {
private static final String USAGE =
"USAGE: java -jar <thisJar> n m k (1<n<64, 0<m<n, 0<k)";
public static void main (String[] args) {
if (args.length != 3) {
throw new IllegalArgumentException(USAGE);
}
int n = parseIntArg(args[0]); // number of bits
int m = parseIntArg(args[1]); // number of ones
int k = parseIntArg(args[2]); // max size of result set
if (n < 2 || n > 63 || m < 1 || m >= n || k < 1) {
throw new IllegalArgumentException(USAGE);
}
// calculate the total number of available bit vectors
int c = combinations(n, m);
// truncate k to the above number
if (k > c) {
k = c;
}
long[] result = new long[k]; // the result set (actually an array)
long[] vectors = new long[c - 1]; // all remaining candidate vectors
long[] hammingD = new long[c - 1]; // their total Hamming distance to the result set
long firstVector = (1L << m) - 1; // m ones in the least significant bits
long lastVector = firstVector << (n - m); // m ones in the most significant bits
result[0] = firstVector; // initialize the result set
// generate the remaining candidate vectors in "lexicographical" order
int size = 0;
for (long v = firstVector; v != lastVector; ) {
// See http://graphics.stanford.edu/~seander/bithacks.html#NextBitPermutation
long t = v | (v - 1); // t gets v's least significant 0 bits set to 1
// Next set to 1 the most significant bit to change,
// set to 0 the least significant ones, and add the necessary 1 bits.
v = (t + 1) | (((~t & -~t) - 1) >>> (Long.numberOfTrailingZeros(v) + 1));
vectors[size++] = v;
}
assert(size == c - 1);
// chosenVector is always the last vector added to the result set
long chosenVector = firstVector;
// do until the result set is filled with k vectors
for (int r = 1; r < k; r++) {
// find the index of the new chosen vector starting from the first
int chosen = 0;
// add the distance to the old chosenVector to the total distance of the first
hammingD[0] += Long.bitCount(vectors[0] ^ chosenVector);
// initialize the maximum total Hamming distance to that of the first
long maxHammingD = hammingD[0];
// for all the remaining vectors
for (int i = 1; i < size; i++) {
// add the distance to the old chosenVector to their total distance
hammingD[i] += Long.bitCount(vectors[i] ^ chosenVector);
// whenever the calculated distance is greater than the max,
// update the max and the index of the new chosen vector
if (maxHammingD < hammingD[i]) {
maxHammingD = hammingD[i];
chosen = i;
}
}
// set the new chosenVector to the one with the maximum total distance
chosenVector = vectors[chosen];
// add the chosenVector to the result set
result[r] = chosenVector;
// fill in the hole left by the chosenVector by moving all vectors
// that follow it down by 1 (keeping vectors and total distances in sync)
System.arraycopy(vectors, chosen + 1, vectors, chosen, size - chosen - 1);
System.arraycopy(hammingD, chosen + 1, hammingD, chosen, size - chosen - 1);
size--;
}
// dump the result set
for (int r = 0; r < k; r++) {
dumpBits(result[r], n);
}
}
private static int parseIntArg(String arg) {
try {
return Integer.parseInt(arg);
} catch (NumberFormatException ex) {
throw new IllegalArgumentException(USAGE);
}
}
private static int combinations(int n, int m) {
// calculate n over m = n! / (m! (n - m)!)
// without using arbitrary precision numbers
if (n <= 0 || m <= 0 || m > n) {
throw new IllegalArgumentException();
}
// possibly avoid unnecessary calculations by swapping m and n - m
if (m * 2 < n) {
m = n - m;
}
if (n == m) {
return 1;
}
// primeFactors[p] contains the power of the prime number p
// in the prime factorization of the result
int[] primeFactors = new int[n + 1];
// collect prime factors of each term of n! / m! with a power of 1
for (int term = n; term > m; term--) {
collectPrimeFactors(term, primeFactors, 1);
}
// collect prime factors of each term of (n - m)! with a power of -1
for (int term = n - m; term > 1; term--) {
collectPrimeFactors(term, primeFactors, -1);
}
// multiply the collected prime factors, checking for overflow
int combinations = 1;
for (int f = 2; f <= n; f += (f == 2) ? 1 : 2) {
// multiply as many times as requested by the stored power
for (int i = primeFactors[f]; i > 0; i--) {
int before = combinations;
combinations *= f;
// check for overflow
if (combinations / f != before) {
String msg = "combinations("+n+", "+m+") > "+Integer.MAX_VALUE;
throw new IllegalArgumentException(msg);
}
}
}
return combinations;
}
private static void collectPrimeFactors(int n, int[] primeFactors, int power) {
// for each candidate prime that fits in the remaining n
// (note that non-primes will have been preceded by their component primes)
for (int i = 2; i <= n; i += (i == 2) ? 1 : 2) {
while (n % i == 0) {
primeFactors[i] += power;
n /= i;
}
}
}
private static void dumpBits(Long bits, int nBits) {
String binary = Long.toBinaryString(bits);
System.out.println(String.format("%"+nBits+"s", binary).replace(' ', '0'));
}
}
The algorithm's data for n=5, m=2, k=4:
result
00011 00101 00110 01001 01010 01100 10001 10010 10100 11000 vectors
0→2 0→2 0→2 0→2 0→4 0→2 0→2 0→4 0→4 hammingD
^ chosen
00011 00101 00110 01001 01010 10001 10010 10100 11000
01100 2→4 2→4 2→4 2→4 2→6 2→6 4→6 4→6
^
00011 00101 00110 01001 01010 10010 10100 11000
01100 4→6 4→8 4→6 4→8 6→8 6→8 6→8
10001 ^
00011 00101 01001 01010 10010 10100 11000
01100 6 6 8 8 8 8
10001
00110
Sample output (n=24, m=9, k=20):
[wtross ~/Dropbox/bits]$ time java -jar bits-1.0-SNAPSHOT.jar 24 9 20
000000000000000111111111
000000111111111000000000
111111000000000000000111
000000000000111111111000
000111111111000000000000
111000000000000000111111
000000000111111111000000
111111111000000000000000
000000000000001011111111
000000111111110100000000
111111000000000000001011
000000000000111111110100
001011111111000000000000
110100000000000000111111
000000001011111111000000
111111110100000000000000
000000000000001101111111
000000111111110010000000
111111000000000000001101
000000000000111111110010
real 0m0.269s
user 0m0.244s
sys 0m0.046s
The toughest case within your constraints (n=24, m=9, k=99) takes ~550 ms on my Mac.
The algorithm could be made even faster by some optimization, e.g., by shifting shorter array chunks. Remarkably, in Java I found shifting "up" to be considerably slower than shifting "down".
UPDATED ANSWER
Looking at the example output of Walter Tross's code, I think that generating a random solution can be simplified to this:
Take any vector to start with, e.g. for n=8, m=3, k=5:
A: 01001100
After every step, sum the vectors to get the number of times each position has been used:
SUM: 01001100
Then, for the next vector, place the ones at positions that have been used least (in this case zero times), e.g.:
B: 00110001
to get:
A: 01001100
B: 00110001
SUM: 01111101
Then, there are 2 least-used positions left, so for the 3 ones in the next vector, use those 2 positions, and then put the third one anywhere:
C: 10010010
to get:
A: 01001100
B: 00110001
C: 10010010
SUM: 11121111 (or reset to 00010000 at this point)
Then for the next vector, you have 7 least-used positions (the ones in the sum), so choose any 3, e.g.:
D: 10100010
to get:
A: 01001100
B: 00110001
C: 10010010
D: 10100010
SUM: 21221121
And for the final vector, choose any of the 4 least-used positions, e.g.:
E: 01000101
To generate all solutions, simply generate every possible vector in each step:
A: 11100000, 11010000, 11001000, ... 00000111
Then, e.g. when A and SUM are 11100000:
B: 00011100, 00011010, 00011001, ... 00000111
Then, e.g. when B is 00011100 and SUM is 11111100:
C: 10000011, 01000011, 00100011, 00010011, 00001011, 00000111
Then, e.g. when C is 10000011 and SUM is 21111111:
D: 01110000, 01101000, 01100100, ... 00000111
And finally, e.g. when D is 01110000 and SUM is 22221111:
E: 00001110, 00001101, 00001011, 00000111
This would result in C(8,3) × C(5,3) × C(8,1) × C(7,3) × C(4,3) = 56 × 10 × 8 × 35 × 4 = 627,200 solutions for n=8, m=3, k=5.
Actually, you need to add a method to avoid repeating the same vector, and avoid painting yourself into a corner; so I don't think this will be simpler than Walter's answer.
INITIAL ANSWER - HAS MAJOR ISSUES
(I will assume than m is not greater than n/2, i.e. the number of ones is not greater than the number of zeros. Otherwise, use a symmetrical approach.)
When k×m is not greater than n, there obviously are optimal solutions, e.g.:
n=10, m=3, k=3:
A: 1110000000
B: 0001110000
C: 0000001110
where the Hamming distances are all 2×m:
|AB|=6, |AC|=6, |BC|=6, total=18
When k×m is greater than n, solutions where the difference in Hamming distances between consecutive vectors are minimized offer the greatest total distance:
n=8, m=3, k=4:
A: 11100000
B: 00111000
C: 00001110
D: 10000011
|AB|=4, |AC|=6, |AD|=4, |BC|=4, |BD|=6, |CD|=4, total=28
n=8, m=3, k=4:
A: 11100000
B: 00011100
C: 00001110
D: 00000111
|AB|=6, |AC|=6, |AD|=6, |BC|=2, |BD|=4, |CD|=2, total=26
So, practically, you take m×k and see how much greater it is than n, let's call it x = m×k−n, and this x is the number of overlaps, i.e. how often a vector will have a one in the same position as the previous vector. You then spread out the overlap over the different vectors as evenly as possible to maximize the total distance.
In the example above, x = 3×4−8 = 4 and we have 4 vectors, so we can spread out the overlap evenly and every vector has 1 one in the same position as the previous vector.
To generate all unique solutions, you could:
- Calculate x = m×k−n and generate all partitions of x into k parts, with the lowest possible maximum value:
n=8, m=3, k=5 -> x=7
22111, 21211, 21121, 21112, 12211, 12121, 12112, 11221, 11212, 11122
(discard partitions with value 3)
- Generate all vectors to be used as vector A, e.g.:
A: 11100000, 11010000, 11001000, 11000100, ... 00000111
- For each of these, generate all vectors B, which are lexicographically smaller than vector A, and have the correct number of overlapping ones with vector A (in the example that is 1 and 2), e.g.:
A: 10100100
overlap=1:
B: 10011000, 10010010, 10010001, 10001010, 10001001, 10000011, 01110000, ... 00000111
overlap=2:
B: 10100010, 10100001, 10010100, 10001100, 10000110, 10000101, 01100100, ... 00100101
- For each of these, generate all vectors C, and so on, until you have sets of k vectors. When generating the last vector, you have to take into account the overlapping with the previous as well as the next (i.e. first) vector.
I assume it's best to treat the partitions of x into k as a binary tree:
1 2
11 12 21 22
111 112 121 122 211 212 221
1112 1121 1122 1211 1212 1221 2111 2112 2121 2211
11122 11212 11221 12112 12121 12211 21112 21121 21211 22111
and traverse this tree while creating solutions, so that each vector only needs to be generated once.
I think this method only works for some values of n, m and k; I'm not sure it can be made to work for the general case.
来源：https://stackoverflow.com/questions/50292530/finding-a-number-of-maximally-different-binary-vectors-from-a-set