Partition and Composition (combinatorics) implementation in C++

问题

Given a matrix of size M and N, we want to fill in each row with integer value (>=0) so that it sums up to certain value.

Note that the dimension of M and N are pre-computed using certain formula, so that it is guaranteed to match the fill given the desired condition (i.e. sum_val below).

This is implemented in R under Partition library.

library(partitions)

# In this example, we impose condition 
# that each rows must sum up to 2 in total
# And each row has 5 columns
sum_val <- 2
n <- 5
#The above two parameters are predefined.

t(as.matrix(compositions(sum_val, n)))
      [,1] [,2] [,3] [,4] [,5]
 [1,]    2    0    0    0    0
 [2,]    1    1    0    0    0
 [3,]    0    2    0    0    0
 [4,]    1    0    1    0    0
 [5,]    0    1    1    0    0
 [6,]    0    0    2    0    0
 [7,]    1    0    0    1    0
 [8,]    0    1    0    1    0
 [9,]    0    0    1    1    0
[10,]    0    0    0    2    0
[11,]    1    0    0    0    1
[12,]    0    1    0    0    1
[13,]    0    0    1    0    1
[14,]    0    0    0    1    1
[15,]    0    0    0    0    2

Is there any existing implementation in C++?

回答1:

Recursive version

Here is a recursive solution. You have a sequence a where you keep track of the numbers you already have set. Each recursive call will assign valid numbers to one of these elements in a loop, before recursively calling that function for the remainder of the list.

void recurse(std::vector<int>& a, int pos, int remaining) {
  if (remaining == 0) { print(a); return; }
  if (pos == a.size()) { return; }
  for (int i = remaining; i >= 0; --i) {
    a[pos] = i;
    recurse(a, pos + 1, remaining - i);
  }
}

void print_partitions(int sum_val, int n) {
  std::vector<int> a(n);
  recurse(a, 0, sum_val);
}

Proof of concept run visible at http://ideone.com/oJNvmu.

Iterative version

Your comment below indicates a performance problem. While it seems very likely that I/O is eating most of your performance, here is an iterative solution which avoids the function call overhead of the recursive approach.

void print_partitions(int sum_val, int n) {
  int pos = 0, last = n - 1;
  int a[n]; // dynamic stack-allocated arrays are a gcc extension
  for (int i = 1; i != n; ++i)
    a[i] = 0;
  a[0] = sum_val;
  while (true) {
    for (int i = 0; i != last; ++i)
        printf("%3d ", a[i]);
    printf("%3d\n", a[last]);
    if (pos != last) {
      --a[pos];
      ++pos;
      a[pos] = 1;
    }
    else {
      if (a[last] == sum_val)
        return;
      for (--pos; a[pos] == 0; --pos);
      --a[pos];
      int tmp = 1 + a[last];
      ++pos;
      a[last] = 0;
      a[pos] = tmp;
    }
  }
}

The general idea and the order in which things are printed is the same as for the recursive approach. Instead of maintaining a counter remaining, all the tokens (or whatever it is you are partitioning) are immediately dropped in the place where they belong for the next partition to be printed. pos is always the last non-zero field. If that is not the last, then you obtain the next partition by taking one token from pos and moving it to the place after that. If it is the last, then you take all tokens from that last place, find the last non-zero place before that and take one token from there as well, then dump all these tokens onto the place after the one where you took the single token.

Demo run at http://ideone.com/N3lSbQ.

回答2:

You can implement it yourself: such a partition is defined by 6 integers 0 <= x[0] <= x[1] <= x[2] <= x[3] <= 2; the values in the corresponding row are just the differences x[0]-0, x[1]-x[0], x[2]-x[1], etc. If the number of columns (5) is fixed, you have 4 nested loops; it it is not, you can formulate the problem recursively.

来源：https://stackoverflow.com/questions/17461761/partition-and-composition-combinatorics-implementation-in-c