Recursion Depth Research Articles

Depth of recursion and the ackermann function

The maximum depth of recursion refers to the number of levels of activation of a procedure which exist during the deepest call of the procedure. A re-examination of the maximum depth of recursion of the Ackermann function results in a new formula which takes a full account of the dependence of this property on the parameters. It is shown that the recursive use parameter of the Ackermann function contributes to the depth of recursion, and that this contribution may be reduced by rearranging the order of the parameters. The stack required by the Ackermann function was investigated using ALGOL 68-R.

Algorithm 426: Merge sort algorithm [M1

Sorting by means of a two-way merge has a reputation of requiring a clerically complicated and cumbersome program. This ALGOL 60 procedure demonstrates that, using recursion, an elegant and efficient algorithm can be designed, the correctness of which is easily proved [2]. Sorting n objects gives rise to a maximum recursion depth of [log 2 ( n - 1) + 2]. This procedure is particularly suitable for sorting when it is not desirable to move the n objects physically in store and the sorting criterion is not simple. In that case it is reasonable to take the number of compare operations as a measure for the speed of the algorithm. When n is an integral power of 2, this number will be comprised between ( n × log 2 n )/2 when the objects are sorted to begin with and ( n × log 2 n - n + 1) as an upper limit. When n is not an integral power of 2, the above formulas are approximate.