Merge-sort algorithm

To promptly estimate the carbon supply in forests of the U.S.A., a computer algorithm, called Cole s parallel merge sort algorithm, was developed and used to specify all available data on the carbon budget inventory (Cole, 1988). 

This is the so-called Merge-Sort algorithm. It is non-trivial in the sense that a new predicate, here conveniently called mergelh, needs to be invented from scratch. ... 

Let us consider the route our development process would have taken if we had decided to decompose sort instances into two smaller instances of roughly equal size. Let us further suppose that the left half of the sequence is one of the instances created and the right half is the other. For our example we get the instances (10,18) and (8,12,9). These are sorted independently to get the sequence (10,18) and (8.9,12). Next, the two sorted sequence are combined to get the sequence (8,9,10,12,18). This combination process is called merging. The resulting sort algorithm is called merge sort. 

The omega notation is used to provide a lower bound, while the theta notation is used when the obtained bound is both a lower and an upper bound. The little oh notation is a very precise notation that does not find much use in the asymptotic analysis of algorithms. With these additional notations available, the solution to the recurrence for insertion and merge sort are, respectively, 0(n ) and 0(n logn). The definitions of O, 2, 0, and o are easily extended to include functions of more than one variable. For example, f(n,m) = 0(g(n, m)) if there exist positive constants c, uq and mo such that /(n, m) < cg(n, m) for all n> no and all m > mo. As in the case of the big oh notation, there are several functions g(n) for which /(n) = Q(g(n)). The g(n) is only a lower bound on f(n). The 0 notation is more precise that both the big oh and omega notations. The following theorem obtains a very useful result about the order of f(n) when f(n) is a polynomial in n. 

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