Question

I would like to know if there’s an efficient algorithm to find the greatest m elements in an N x N matrix, with a method header like this:

double[] greatestValues(double[][] matrix, int numberOfElements);

Any ideas?

Answer 1

If you treat the N x N matrix as an array of N x N items you can apply one of the following techniques:

Direct application of the quick sort based selection algorithm The quick
sort based selection algorithm can be
used to find k smallest or k largest
elements. To find k smallest elements
find the kth smallest element using
the median of medians quick sort based
algorithm. After the partition that
finds the kth smallest element, all
the elements smaller than the kth
smaller element will be present left
to the kth element and all element
larger will be present right to the
kth smallest element. Thus all
elements from 1st to kth element
inclusive constitute the k smallest
elements. The time complexity is
linear in n, the total number of
elements.

Data structure based solutions Another simple method is to add each
element of the list into an ordered
set data structure, such as a heap or
self-balancing binary search tree,
with at most k elements. Whenever the
data structure has more than k
elements, we remove the largest
element, which can be done in O(log k)
time. Each insertion operation also
takes O(log k) time, resulting in
O(nlog k) time overall.

It is possible to transform the list
into a heap in Θ(n) time, and then
traverse the heap using a modified
Breadth-first search algorithm that
places the elements in a Priority
Queue (instead of the ordinary queue
that is normally used in a BFS), and
terminate the scan after traversing
exactly k elements. As the queue size
remains O(k) throughout the traversal,
it would require O(klog k) time to
complete, leading to a time bound of
O(n + klog k) on this algorithm.

From here.