Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 4.7 Projects 1414.16 Re-implement function fibr from Exercise 2.11, using a stack to replacethe recursive call as described in Section 4.2.4.4.17 Write a recursive algorithm to compute the value of the recurrence relationT(n) = T(⌈n/2⌉) + T(⌊n/2⌋) + n; T(1) = 1.Then, rewrite your algorithm to simulate the recursive calls with a stack.4.18 Let Q be a non-empty queue, and let S be an empty stack. Using only thestack and queue ADT functions and a single element variable X, write analgorithm to reverse the order of the elements in Q.4.19 A common problem for compilers and text editors is to determine if theparentheses (or other brackets) in a string are balanced and properly nested.For example, the string “((())())()” contains properly nested pairs of parentheses,but the string “)()(” does not, and the string “())” does not containproperly matching parentheses.(a) Give an algorithm that returns true if a string contains properly nestedand balanced parentheses, and false otherwise. Use a stack to keeptrack of the number of left parentheses seen so far. Hint: At no timewhile scanning a legal string from left to right will you have encounteredmore right parentheses than left parentheses.(b) Give an algorithm that returns the position in the string of the first offendingparenthesis if the string is not properly nested and balanced.That is, if an excess right parenthesis is found, return its position; ifthere are too many left parentheses, return the position of the first excessleft parenthesis. Return −1 if the string is properly balanced andnested. Use a stack to keep track of the number and positions of leftparentheses seen so far.4.20 Imagine that you are designing an application where you need to performthe operations Insert, Delete Maximum, and Delete Minimum. Forthis application, the cost of inserting is not important, because it can be doneoff-line prior to startup of the time-critical section, but the performance ofthe two deletion operations are critical. Repeated deletions of either kindmust work as fast as possible. Suggest a data structure that can support thisapplication, and justify your suggestion. What is the time complexity foreach of the three key operations?4.21 Write a function that reverses the order of an array of n items.4.7 Projects4.1 A deque (pronounced “deck”) is like a queue, except that items may be addedand removed from both the front and the rear. Write either an array-based orlinked implementation for the deque.