As we previously know, the merge sort algorithm needs two circuits, i.e. one for merging and second for sorting the sequences. Thus, the sorting circuit has been derived from the above-discussed merging circuit. The useful steps followed by the circuit are given below:
i) The given input series of length n is separated into two sub-sequences of length n/2 each.
ii) The two sub series are recursively sorted.
iii) The two sorted sub series are merged (n/2,n/2) using a merging circuit in order to finally get the sorted series of length n.
Now, let us take an instance for sorting the n numbers say 4,2,10,12 8,7,6,9. The circuit of sorting + merging given series is illustrated in Figure.
Analysis of Merge Sort
i) The width of the sorting + merging circuit is equivalent to the maximum number of devices needs in a stage is O (n/2). As in the above figure the maximum number of devices for a given stage is 4 which is n/2or8/2.
ii) The circuit has two sorting circuits for sorting series of length n/2 and after that one merging circuit for merging of the two sorted sub series (see stage 4th in the above figure). Let the functions Tm and Ts denote the time complexity of merging and sorting in terms of its depth.
The Ts can be calculated as follows: Ts (n) =Ts (n/2) + Tm (n/2) Ts (n) =Ts (n/2) + log (n/2),
Thus, Ts (n) is equal to O(log2 n).
Sorting + Merging Circuit