technische universiteit eindhoven 2IL50 Data Structures Assignment 2 Spring 2015 Due at 23:59 on 22–02–2015 Requirements: Assignments have to be handed in by each student separately. Please write your name and student number on the top of the first sheet that you hand in. Remember that assignments have to be typeset in English and sent by email as .pdf to your instructor on the due date. Always justify your answers! Note: Whenever you are asked to describe an algorithm, you should present three things: the algorithm, a proof of its correctness, and a derivation of its running time. Exercise 1: (a) [1 points] Prove that the solution of T (n) = T (dn/3e) + T (b2n/3c) + n with T (1) = 1 is T (n) = O(n log n). √ √ (b) [1 points] Prove that the solution of T (n) = nT (b nc) + n with T (1) = T (2) = T (3) = 0 is T (n) = O(n log log n). (c) [2 points] Can the master method be applied to T (n) = 4T (n/2) + n2 log n? Why or why not? Give a tight asymptotic upper bound for this recurrence. Note: You might want to study the material in Chapter 4.3 of the textbook. Using the substitution method is a good option for this question. Exercise 2: [3 points] Give asymptotic upper and lower bounds for T (n) in each of the following recurrences. Assume that T (n) is constant for n ≤ 2. Make your bounds as tight as possible, and justify your answers. (a) T (n) = 7T (n/2) + n2 (b) T (n) = 7T (n/3) + n2 √ (c) T (n) = 2T (n/4) + n Exercise 3: (a) [2 points] Illustrate the execution of HeapSort on the following input: (3, 11, 22, 37, 16, 24, 13) . Show every step that changes the array. (One call of Max-Heapify counts as one step.) (b) [1 points] Where in a max-heap might the smallest element reside, assuming that all elements are distinct? (c) [1 points] Is the sequence h30, 14, 25, 10, 1, 19, 22, 2, 4, 3i a max-heap? (d) [2 points] Prove by induction that a heap with n vertices has exactly dn/2e leaves. (e) [3 points] Show that there are at most dn/2h+1 e nodes of height h in any n-element heap. (Note: if you assume that n is a power of two minus one – that is, n = 2k − 1 for a positive integer k – you will receive only partial points.) /department of mathematics and computer science 1 2IL50 Data Structures technische universiteit eindhoven Exercise 4: [4 points] Consider the following task scheduling problem: You are given n tasks Ti , 1 ≤ i ≤ n. Each task has an (integer) arrival time ai , an (integer) priority pi , and an (integer) length li . All arrival times and priorities are distinct. The processor schedules only one task at a time. Whenever it schedules a new task, it schedules the task with the highest priority among the tasks which have already arrived. Tasks cannot be interrupted once they are running. Describe an O(n log n) algorithm which computes at which time the processor has processed all tasks. Don’t forget to analyze the running time and prove the correctness of your algorithm. 2 /department of mathematics and computer science