By Barry J. Mailloux, Adriaan van van Wijngaarden
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Extra info for Report on the algorithmic language Algol 68
Another instance when recursion is invaluable is when we want to describe a backtracking procedure. But for now we will content ourselves with examining some simple, iterative programs and show how to eliminate the iteration statements and replace them by recursion. This may sound strange, but the objective is not to show that the result is simpler to understand nor more efficient to execute. The main purpose is to make one more familiar with the execution of a recursive procedure. Suppose we start with the sorting algorithm presented in this section.
Let us write out the algorithm in full. 3 SPARSE MATRICES // A is a matrix represented in sparse form// // B is set to be its transpose// 1 (m,n,t) (A(0,l),A(0,2),A(0,3)) 2 (B(0,1),B(0,2),B(0,3)) 3 if t 4 q 5 for col 1 to n do //transpose by columns// 6 for p 1 to t do //for all nonzero terms do// (n,m,t) 0 then return 1 //check for zero matrix// //q is position of next term in B// if A(p,2) = col 7 //correct column// then [(B(q,1),B(q,2),B(q,3)) 8 //insert next term of B// 9 (A(p,2),A(p,1),A(p,3)) 10 11 12 q q + 1] end end 13 end TRANSPOSE The above algorithm makes use of (lines 1, 2, 8 and 9) the vector replacement statement of SPARKS.
T(j) is maintained so that it is always the position in B where the next element in row j is to be inserted. There are four loops in FAST--TRANSPOSE which are executed n, t, n - 1, and t times respectively. Each iteration of the loops takes only a constant amount of time, so the order of magnitude is O(n + t). The computing time of O(n + t) becomes O(nm) when t is of the order of nm. This is the same as when two dimensional arrays were in use. However, the constant factor associated with FAST__TRANSPOSE is bigger than that for the array algorithm.
Report on the algorithmic language Algol 68 by Barry J. Mailloux, Adriaan van van Wijngaarden