Run of a sequence

In computer science, a run of a sequence is a non-decreasing range of the sequence that cannot be extended. The number of runs of a sequence is the number of increasing subsequences of the sequence. This is a measure of presortedness, and in particular measures how many subsequences must be merged to sort a sequence.

Definition

Let $X=\langle x_{1},\dots ,x_{n}\rangle$ be a sequence of elements from a totally ordered set. A run of $X$ is a maximal increasing sequence $\langle x_{i},x_{i+1},\dots ,x_{j-1},x_{j}\rangle$. That is, $x_{i-1}>x_{i}$ and $x_{j}>x_{j+1}$ assuming that $x_{i-1}$ and $x_{j+1}$ exists. For example, if $n$ is a natural number, the sequence $\langle n+1,n+2,\dots ,2n,1,2,\dots ,n\rangle$ has the two runs $\langle n+1,\dots ,2n\rangle$ and $\langle 1,\dots ,n\rangle$.

Let ${\mathtt {runs}}(X)$ be defined as the number of positions $i$ such that $1\leq i<n$ and $x_{i+1}<x_{i}$. It is equivalently defined as the number of runs of $X$ minus one. This definition ensure that ${\mathtt {runs}}(\langle 1,2,\dots ,n\rangle )=0$, that is, the ${\mathtt {runs}}(X)=0$ if, and only if, the sequence $X$ is sorted. As another example, ${\mathtt {runs}}(\langle n,n-1,\dots ,1\rangle )=n-1$ and ${\mathtt {runs}}(\langle 2,1,4,3,\dots ,2n,2n-1\rangle )=n$.

Sorting sequences with a low number of runs

The function ${\mathtt {runs}}$ is a measure of presortedness. The natural merge sort is ${\mathtt {runs}}$-optimal. That is, if it is known that a sequence has a low number of runs, it can be efficiently sorted using the natural merge sort.

Long runs

A long run is defined similarly to a run, except that the sequence can be either non-decreasing or non-increasing. The number of long runs is not a measure of presortedness. A sequence with a small number of long runs can be sorted efficiently by first reversing the decreasing runs and then using a natural merge sort.

References

• Powers, David M. W.; McMahon, Graham B. (1983). "A compendium of interesting prolog programs". DCS Technical Report 8313 (Report). Department of Computer Science, University of New South Wales.
• Mannila, H (1985). "Measures of Presortedness and Optimal Sorting Algorithms". IEEE Trans. Comput. (C-34): 318–325. doi:10.1109/TC.1985.5009382.