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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 less than 2000:
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.
Input
The program constructed according to the grammar given above. Whitespace and newlines can appear anywhere in a program, but not within the keywords BEGIN, END, LOOP and OP or in an integer value. The nesting depth of the LOOP-operators will be at most 10.
Output
Output the run-time of the program in terms of n; this will be a polynomial of degree < 10.
Print the polynomial in the usual way, i.e., collect all terms, and print it in the form
Runtime =
a*n^10+
b*n^9+ . . . +
i*n^2+
j*n+k
,Runtime = 0.
Sample InputBEGIN LOOP n OP 4 LOOP 3 LOOP n OP 1 END OP 2 END OP 1 END OP 17 END |
Sample OutputRuntime = 3*n^2+11*n+17 |
Sample InputBEGIN OP 1997 LOOP n LOOP n OP 1 END END END |
Sample OutputRuntime = n^2+1997 |
Author
from Valladolid, 1997
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