By Manoochehr Azmoodeh (auth.)
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For instance, evaluate 2x + 4x 2 + 5x 3 + 13x4 at x =5. Design the necessary data structures and write the programs. Compare their time complexities. 9. 10. 9 by not evaluating xk more than once (for a given k). 11. 10. 12. Write a program to compute e n2 n3 n4 n10 e(n) =n +- +- +- + ... + 2! 3! 4! 10! What is the time complexity of your solution? 13. _ n logn 3 10 log 3n The Complexity of Algorithms 23 (f(n) + 10? 14. Which of the following are valid? 15. Show that if a computer executes algorithm A6 of sizeS in T units of time, then a computer which is 60 times faster would execute the algorithm in T time for size x where x=S+ Hint: n!
Push(s,a)) =last< n Therefore (a) is proved. push(f(s),a)) = (c' > c and (a, c) E elems') This proves (b) above. (a) and (b) above thus ensure that the effects of the procedure push on an a_stack sis exactly the same as the effect of the procedure push on stack f(s). The above proof should be applied to all the other operations of stacks. We have assumed that the a_stack implementation satisfies its own specifications. For this simple implementation it is trivial to verify the pre-conditions and postconditions.
6x 106 s; ~ 2 X10 5 s3 s4 s~ s~ s~ s~ ~ ~ ~ ~ 1897 153 21 10 S5 S6 ~244 ~39 ~ 15 ~8 fore~ Max. problem size in 1 minute after speed up s~ = 60S1 s; ~ 60S 2 (for large S2 ) s~ ~ 8S3 s~ ~ 4S4 S~ ~ S5 +6 S~ ~ S6 +e lo&l60 logeS 6 - 1 Alfred V. Aha/John E. Hopcroft/Jeffrey D. , Reading, Massachusetts. 1. Reprinted with permission of the publisher. Also the maximum problem sizes are given in one hour (36 x 105 ms) of computation: S~, s;, ... , S~. Now, if the speed of the computer is increased 60 times, represents the maximum problem size executable in one minute for the problem Ai.
Abstract Data Types and Algorithms by Manoochehr Azmoodeh (auth.)