Analyzing the run-time complexity of algorithms is an important tool for designing efficient programs that solve a problem. An algorithm that runs in linear time is usually much faster than an algorithm that takes quadratic time for the same task, and thus should be preferred.
Generally, one determines the run-time of an algorithm in relation to the `size' n of the input, which could be the number of objects to be sorted, the number of points in a given polygon, and so on. Since determining a formula dependent on n for the run-time of an algorithm is no easy task, it would be great if this could be automated. Unfortunately, this is not possible in general, but in this problem we will consider programs of a very simple nature, for which it is possible. Our programs are built according to the following rules (given in BNF), where < number > can be any non-negative integer:
< Program > ::= "BEGIN" < Statementlist > "END"
< Statementlist > ::= < Statement > | < Statement > < Statementlist >
< Statement > ::= < LOOP-Statement > | < OP-Statement >
< LOOP-Statement > ::= < LOOP-Header > < Statementlist > "END"
< LOOP-Header > ::= "LOOP" < number > | "LOOP n"
< OP-Statement > ::= "OP" < number >
The run-time of such a program can be computed as follows: the execution of an OP-statement costs as many time-units as its parameter specifies. The statement list enclosed by a LOOP-statement is executed as many times as the parameter of the statement indicates, i.e., the given constant number of times, if a number is given, and n times, if n is given. The run-time of a statement list is the sum of the times of its constituent parts. The total run-time therefore generally depends on n.