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Concerning your edits at Counting sort
Hi, I just wanted to draw your attention, that the page to which you refer, http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/Sorting/countingSort.htm, and the wikipedia page on Counting sort, have two different algorithms working in two different directions. The page from kent.edu has c holding the count of all elements with the lowest key, but the algo in WP has c holding 0.
That is why the WP algo adds 1 to it, each time it adds its element to the sorted array, going along the original array from the bottom up. But the algo from kent subtracts 1 when it adds an element, going through the original array from top down.
Both are right, both are different, but one must use (+1), and the other must use (-1). Just changing (+1) to (-1) is wrong - there are other changes that must be made to the algorithm too, and in the end it will be just the same algorithm just working in the opposite direction.