Answer to Assignment 2

Advanced Algorithm Analysis (CP5602)

Due Date: 2nd June 2019 at 11:59pm

Total: 20 marks

Aim: This assignment is designed to evaluate/improve your critical thinking and problem solving skills. It also evaluate/improve your coding skill.

1. For a tree T, let nI denote the number of its internal nodes, and let nE denote the number of its external nodes. Show that if every internal node in T has exactly 3 children, then nE = 2nI + 1.

[2 marks]

2. Insert entries with keys, 2, 7, 3, 12, 5, 20, 14, 6, 11, 8, 15, 17, 1, 19, 23, 14 (in this order), into an empty:

(a) heap. [1 mark]

(b) binary search tree. [1 mark]

(c) AVL tree. [1 mark]

(d) (2, 4) tree. [1 mark]

[4 marks] 3. Although merge sort runs in ????(n lg n) worst-case time and insertion sort runs in ????(n2) worst case time, the constant factors in insertion sort make it faster for small n. Thus, it makes sense to use insertion sort within merge sort when sub problems become sufficiently small. Consider a modification to merge sort in which n/k sub lists of length k are sorted using insertion sort and then merged using the standard merging mechanism, where k is a value to be determined.

i) Show that the n/k sub lists, each of length k, can be sorted by insertion sort in ????(nk) worst-case time. [1 mark]

ii) Show that the sub lists can be merged in ????(n lg(n/k)) worst-case time. [2 marks] iii) Given that the modified algorithm runs in ????(nk + n lg(n/k)) worst-case time, what is the largest asymptotic (????-notation) value of k as a function of n for which the modified algorithm has the same asymptotic running time as standard merge sort. [2 marks] [5 marks]

4. Consider the recurrence T(n) = 3T(?n/2?) + n.

i) Use the master method to give tight asymptotic bound for this recurrence (if the master method cannot be used, explain why). [1 mark]

ii) Use a recursion tree to determine a good asymptotic upper bound on this recurrence.

[2 marks] iii) Use the substitution method to verify your answer. [1 mark]

[4 marks]

5. Show all the steps for performing any of the following algorithms for matching the pattern ‘rithm’ in the text ‘advancedalgorithmanalysis’.

(a) brute-force [1 mark]

(b) Boyer-Moore [2 mark]

(c) Knuth-Morris-Pratt [2 mark]

[5 marks]

Advanced Algorithm Analysis (CP5602)

Due Date: 2nd June 2019 at 11:59pm

Total: 20 marks

Aim: This assignment is designed to evaluate/improve your critical thinking and problem solving skills. It also evaluate/improve your coding skill.

1. For a tree T, let nI denote the number of its internal nodes, and let nE denote the number of its external nodes. Show that if every internal node in T has exactly 3 children, then nE = 2nI + 1.

[2 marks]

2. Insert entries with keys, 2, 7, 3, 12, 5, 20, 14, 6, 11, 8, 15, 17, 1, 19, 23, 14 (in this order), into an empty:

(a) heap. [1 mark]

(b) binary search tree. [1 mark]

(c) AVL tree. [1 mark]

(d) (2, 4) tree. [1 mark]

[4 marks] 3. Although merge sort runs in ????(n lg n) worst-case time and insertion sort runs in ????(n2) worst case time, the constant factors in insertion sort make it faster for small n. Thus, it makes sense to use insertion sort within merge sort when sub problems become sufficiently small. Consider a modification to merge sort in which n/k sub lists of length k are sorted using insertion sort and then merged using the standard merging mechanism, where k is a value to be determined.

i) Show that the n/k sub lists, each of length k, can be sorted by insertion sort in ????(nk) worst-case time. [1 mark]

ii) Show that the sub lists can be merged in ????(n lg(n/k)) worst-case time. [2 marks] iii) Given that the modified algorithm runs in ????(nk + n lg(n/k)) worst-case time, what is the largest asymptotic (????-notation) value of k as a function of n for which the modified algorithm has the same asymptotic running time as standard merge sort. [2 marks] [5 marks]

4. Consider the recurrence T(n) = 3T(?n/2?) + n.

i) Use the master method to give tight asymptotic bound for this recurrence (if the master method cannot be used, explain why). [1 mark]

ii) Use a recursion tree to determine a good asymptotic upper bound on this recurrence.

[2 marks] iii) Use the substitution method to verify your answer. [1 mark]

[4 marks]

5. Show all the steps for performing any of the following algorithms for matching the pattern ‘rithm’ in the text ‘advancedalgorithmanalysis’.

(a) brute-force [1 mark]

(b) Boyer-Moore [2 mark]

(c) Knuth-Morris-Pratt [2 mark]

[5 marks]

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