To understand the efficiency of the mergesort algorithm it is useful to separate the merging from the sorting. The sorting takes place indirectly, by repeatedly splitting the data in half until sorted singleton sets are created. The merging then rebuilds the complete, original data set by splicing together the sorted mini-lists. To determine the efficiency of the sorting (breaking down) algorithm, consider how many times the data has to be split. A data set of size 4 has to be split twice, once into two sets of two and then again into four sets of one. A data set of size 8 has to be split 3 times, 16 pieces of data have to be split 4 times, 32 needs 5 splits, and so on. This sort of behavior is reflected by the logarithm:
The breaking down of the data, then, occurs with efficiency (log n). The merging process is linear each time two lists have to be merged, because it is simply done by doing one comparison for each pair of elements at the top of each sublist. For example, to merge the subarrays (2 4) and (0 1 7), the following comparisons have to take place: 0 & 2, 1 & 2, 2 & 7, 4 & 7, and 7 alone. 5 comparisons for 5 elements, efficiency n. Because all log(n) sublists have to be merged, the efficiency of mergesort is O(nlog(n)).
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