Multisets¶

This chapter describes functions for creating and manipulating multisets. A multiset

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is represented by an array of

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integers in the range 0 to

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, where each value

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may occur more than once. The multiset

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corresponds to indices of

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elements chosen from an

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element vector with replacement. In mathematical terms,

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is the cardinality of the multiset while

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is the maximum multiplicity of any value. Multisets are useful, for example, when iterating over the indices of a

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-th order symmetric tensor in

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-space.

The functions described in this chapter are defined in the header file gsl_multiset.h.

The Multiset struct¶

gsl_multiset¶

A multiset is defined by a structure containing three components, the values of

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and

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, and a pointer to the multiset array. The elements of the multiset array are all of type size_t, and are stored in increasing order. The gsl_multiset structure looks like this:

typedef struct
{
  size_t n;
  size_t k;
  size_t *data;
} gsl_multiset;

Multiset allocation¶

gsl_multiset * gsl_multiset_alloc(size_t n, size_t k)¶: This function allocates memory for a new multiset with parameters n, k. The multiset is not initialized and its elements are undefined. Use the function gsl_multiset_calloc() if you want to create a multiset which is initialized to the lexicographically first multiset element. A null pointer is returned if insufficient memory is available to create the multiset.

gsl_multiset * gsl_multiset_calloc(size_t n, size_t k)¶: This function allocates memory for a new multiset with parameters n, k and initializes it to the lexicographically first multiset element. A null pointer is returned if insufficient memory is available to create the multiset.

void gsl_multiset_init_first(gsl_multiset * c)¶

This function initializes the multiset c to the lexicographically first multiset element, i.e.

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repeated

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times.

void gsl_multiset_init_last(gsl_multiset * c)¶

This function initializes the multiset c to the lexicographically last multiset element, i.e.

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repeated

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times.

void gsl_multiset_free(gsl_multiset * c)¶: This function frees all the memory used by the multiset c.

int gsl_multiset_memcpy(gsl_multiset * dest, const gsl_multiset * src)¶: This function copies the elements of the multiset src into the multiset dest. The two multisets must have the same size.

Accessing multiset elements¶

The following function can be used to access the elements of a multiset.

size_t gsl_multiset_get(const gsl_multiset * c, const size_t i)¶

This function returns the value of the i-th element of the multiset c. If i lies outside the allowed range of 0 to

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then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.

Multiset properties¶

size_t gsl_multiset_n(const gsl_multiset * c)¶

This function returns the range (

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) of the multiset c.

size_t gsl_multiset_k(const gsl_multiset * c)¶

This function returns the number of elements (

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) in the multiset c.

size_t * gsl_multiset_data(const gsl_multiset * c)¶: This function returns a pointer to the array of elements in the multiset c.

int gsl_multiset_valid(gsl_multiset * c)¶

This function checks that the multiset c is valid. The k elements should lie in the range 0 to

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, with each value occurring in nondecreasing order.

Multiset functions¶

int gsl_multiset_next(gsl_multiset * c)¶: This function advances the multiset c to the next multiset element in lexicographic order and returns GSL_SUCCESS. If no further multisets elements are available it returns GSL_FAILURE and leaves c unmodified. Starting with the first multiset and repeatedly applying this function will iterate through all possible multisets of a given order.

int gsl_multiset_prev(gsl_multiset * c)¶: This function steps backwards from the multiset c to the previous multiset element in lexicographic order, returning GSL_SUCCESS. If no previous multiset is available it returns GSL_FAILURE and leaves c unmodified.

Reading and writing multisets¶

The library provides functions for reading and writing multisets to a file as binary data or formatted text.

int gsl_multiset_fwrite(FILE * stream, const gsl_multiset * c)¶: This function writes the elements of the multiset c to the stream stream in binary format. The function returns GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

int gsl_multiset_fread(FILE * stream, gsl_multiset * c)¶

This function reads elements from the open stream stream into the multiset c in binary format. The multiset c must be preallocated with correct values of

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and

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since the function uses the size of c to determine how many bytes to read. The function returns GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

int gsl_multiset_fprintf(FILE * stream, const gsl_multiset * c, const char * format)¶: This function writes the elements of the multiset c line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size_t. In ISO C99 the type modifier z represents size_t, so "%zu\n" is a suitable format [1]. The function returns GSL_EFAILED if there was a problem writing to the file.

int gsl_multiset_fscanf(FILE * stream, gsl_multiset * c)¶

This function reads formatted data from the stream stream into the multiset c. The multiset c must be preallocated with correct values of

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and

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since the function uses the size of c to determine how many numbers to read. The function returns GSL_EFAILED if there was a problem reading from the file.

Examples¶

The example program below prints all multisets elements containing the values

System Message: WARNING/2 ({0,1,2,3})

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

ordered by size. Multiset elements of the same size are ordered lexicographically.

#include <stdio.h>
#include <gsl/gsl_multiset.h>

int
main (void)
{
  gsl_multiset * c;
  size_t i;

  printf ("All multisets of {0,1,2,3} by size:\n") ;
  for (i = 0; i <= 4; i++)
    {
      c = gsl_multiset_calloc (4, i);
      do
        {
          printf ("{");
          gsl_multiset_fprintf (stdout, c, " %u");
          printf (" }\n");
        }
      while (gsl_multiset_next (c) == GSL_SUCCESS);
      gsl_multiset_free (c);
    }

  return 0;
}

Here is the output from the program,

All multisets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 0 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 1 }
{ 1 2 }
{ 1 3 }
{ 2 2 }
{ 2 3 }
{ 3 3 }
{ 0 0 0 }
{ 0 0 1 }
{ 0 0 2 }
{ 0 0 3 }
{ 0 1 1 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 2 }
{ 0 2 3 }
{ 0 3 3 }
{ 1 1 1 }
{ 1 1 2 }
{ 1 1 3 }
{ 1 2 2 }
{ 1 2 3 }
{ 1 3 3 }
{ 2 2 2 }
{ 2 2 3 }
{ 2 3 3 }
{ 3 3 3 }
{ 0 0 0 0 }
{ 0 0 0 1 }
{ 0 0 0 2 }
{ 0 0 0 3 }
{ 0 0 1 1 }
{ 0 0 1 2 }
{ 0 0 1 3 }
{ 0 0 2 2 }
{ 0 0 2 3 }
{ 0 0 3 3 }
{ 0 1 1 1 }
{ 0 1 1 2 }
{ 0 1 1 3 }
{ 0 1 2 2 }
{ 0 1 2 3 }
{ 0 1 3 3 }
{ 0 2 2 2 }
{ 0 2 2 3 }
{ 0 2 3 3 }
{ 0 3 3 3 }
{ 1 1 1 1 }
{ 1 1 1 2 }
{ 1 1 1 3 }
{ 1 1 2 2 }
{ 1 1 2 3 }
{ 1 1 3 3 }
{ 1 2 2 2 }
{ 1 2 2 3 }
{ 1 2 3 3 }
{ 1 3 3 3 }
{ 2 2 2 2 }
{ 2 2 2 3 }
{ 2 2 3 3 }
{ 2 3 3 3 }
{ 3 3 3 3 }

All 70 multisets are generated and sorted lexicographically.

Footnotes

[1]	In versions of the GNU C library prior to the ISO C99 standard, the type modifier `Z` was used instead.