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## CGAL::sorted_matrix_search

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### Definition

The function sorted_matrix_search selects the smallest entry in a set of sorted matrices that fulfills a certain feasibility criterion.

More exactly, a matrix (over a totally ordered set S) is sorted, iff

Now let be a set of n sorted matrices over S and f be a monotone predicate on S, i.e.

If we assume there is any feasible element in one of the matrices in , there certainly is a smallest such element. This is the one we are searching for.

The feasibility test as well as some other parameters can (and have to) be customized through a traits class.

#include <CGAL/sorted_matrix_search.h>

template < class RandomAccessIterator, class Traits >
Traits::Value |
sorted_matrix_search (
| RandomAccessIterator f, RandomAccessIterator l, Traits t)

returns the element x in one of the sorted matrices from the range [ f, l ), for which t.is_feasible( x) is true and t.compare( x, y) is true for all other y values from any matrix for which t.is_feasible( y) is true.

Precondition:

1. All matrices in [f, l) are sorted according to Traits::compare_non_strictly.
2. There is at least one entry x in a matrix M [f, l) for which Traits::is_feasible(x) is true.

Requirement:

1. Traits is a model for SortedMatrixSearchTraits.
2. Value type of RandomAccessIterator is Traits::Matrix.

### See Also

SortedMatrixSearchTraits

### Implementation

The implementation uses an algorithm by Frederickson and Johnson[FJ83, FJ84] and runs in (n · k + f · log(n · k)), where n is the number of input matrices, k denotes the maximal dimension of any input matrix and f the time needed for one feasibility test.

### Example

In the following program we build a random vector a = (ai)i = 1,...,5 (elements drawn uniformly from { 0,...,99 }) and construct a Cartesian matrix M containing as elements all sums ai + aj, i,j {1,...,5}. If a is sorted, M is sorted as well. So we can apply sorted_matrix_search to compute the upper bound for the maximal entry of a in M.

    #include <CGAL/Random.h>
    #include <CGAL/Cartesian_matrix.h>
    #include <CGAL/sorted_matrix_search.h>
    #include <CGAL/functional.h>
    #include <vector>
    #include <algorithm>
    #include <iterator>

    typedef int                                     Value;
    typedef std::vector<Value>                      Vector;
    typedef Vector::iterator                        Value_iterator;
    typedef std::vector<Vector>                     Vector_cont;
    typedef CGAL::Cartesian_matrix<std::plus<int>,
                                   Value_iterator,
                                   Value_iterator>  Matrix;

    int main()
    {
      // set of vectors the matrices are build from:
      Vector_cont vectors;

      // generate a random vector and sort it:
      Vector a;
      const int n = 5;
      for (int i = 0; i < n; ++i)
        a.push_back(CGAL::default_random(100));
      std::sort(a.begin(), a.end());
      std::cout << "a = ( ";
      std::copy(a.begin(), a.end(), std::ostream_iterator<int>(std::cout," "));
      std::cout << ")\n";

      // build a Cartesian matrix from a:
      Matrix M(a.begin(), a.end(), a.begin(), a.end());

      // search for an upper bound for max(a):
      Value bound = a[n-1];
      Value upper_bound =
      CGAL::sorted_matrix_search(
        &M, &M + 1,
        CGAL::sorted_matrix_search_traits_adaptor(
          CGAL::bind_2(std::greater_equal<Value>(), bound), M));
      std::cout << "Upper bound for " << bound << " is "
                << upper_bound << "." << std::endl;

      return 0;
    }

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The CGAL Project . Tue, December 21, 2004 .