is there any condition in which the merge sort can be performed without additional memory, my professor said he had, and he would give a bonus ball.
You want google to host merge sort.
Here is one of the results: http://thomas.baudel.name/Visualisation/VisuTri/inplacestablesort.html
Given that this is a homework question, I can only point out the art of programming. A good programmer should be able to use standard links in our field to investigate such a question.
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Use linked list. This avoids the extra O (n) space needed when merging the two lists. However, you cannot do anything about the space occupied by recursion calls ie O (lg (n)).
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