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

Sec. 14.5 Exercises 48114.14 For the following recurrence, give a closed-form solution. You should notgive an exact solution, but only an asymptotic solution (i.e., using Θ notation).You may assume that n is a power of 2. Prove that your answer iscorrect.T(n) = T(n/2) + √ n for n > 1; T(1) = 1.14.15 Using the technique of expanding the recurrence, find the exact closed-formsolution for the recurrence relationT(n) = 2T(n/2) + n; T(2) = 2.You may assume that n is a power of 2.14.16 Section 5.5 provides an asymptotic analysis for the worst-case cost of functionbuildHeap. Give an exact worst-case analysis for buildHeap.14.17 For each of the following recurrences, find and then prove (using induction)an exact closed-form solution. When convenient, you may assume that n isa power of 2.(a) T(n) = T(n − 1) + n/2 for n > 1; T(1) = 1.(b) T(n) = 2T(n/2) + n for n > 2; T(2) = 2.14.18 Use Theorem 14.1 to prove that binary search requires Θ(log n) time.14.19 Recall that when a hash table gets to be more than about one half full, itsperformance quickly degrades. One solution to this problem is to reinsertall elements of the hash table into a new hash table that is twice as large.Assuming that the (expected) average case cost to insert into a hash table isΘ(1), prove that the average cost to insert is still Θ(1) when this re-insertionpolicy is used.14.20 Given a 2-3 tree with N nodes, prove that inserting M additional nodes requiresO(M + N) node splits.14.21 One approach to implementing an array-based list where the list size is unknownis to let the array grow and shrink. This is known as a dynamic array.When necessary, we can grow or shrink the array by copying the array’s contentsto a new array. If we are careful about the size of the new array, thiscopy operation can be done rarely enough so as not to affect the amortizedcost of the operations.(a) What is the amortized cost of inserting elements into the list if the arrayis initially of size 1 and we double the array size whenever the numberof elements that we wish to store exceeds the size of the array? Assumethat the insert itself cost O(1) time per operation and so we are justconcerned with minimizing the copy time to the new array.