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Final Exam Answer Key MCIS 503 Dr. Jerry A. Smith 10 problems, each worth 10 points. Problem 1) Give asymptotic upper and lower bounds to T (n) = 16T (n / 4) + n 2 . Make your bounds as tight as possible and justify your answer. This is a divide-and-conquer recurrence with a = 16, b = 4, f(n) = n2, and n logb a = n log 4 16 = n 2 . Since n 2 = !(n log 4 16 ) , case 2 of the master theorem applies, and T (n) = !(n 2 lg n) . Problem 2) Give asymptotic upper and lower bounds to T (n) = T ( n ) + 1 . Make your bounds as tight as possible and justify your answer. Let m = lg n and S(m) = T(2m), thus T(2m) = T(2m/2)+1 and S(m) = S(m/2)+1. Using the Master Theorum, n log 2 1 = n 0 = 1 and f(n) = 1. Since 1 = Θ(1), case 2 applies and S(m) = Θ(lg m). Therefore, T(n) = Θ(lg lg n). Problem 3) Using figure 6.3 as a model, illustrate the operation of Build-Max-Heap on the array <4, 2, 15, 55, 34, 32, 9, 22, 3> 1 4 2 3 2 i 4 5 55 34 8 22 15 9 3 6 32 7 9 1 4 2 3 2 15 4 5 55 34 8 i 6 7 32 9 9 22 3 1 4 2 i 3 2 32 4 5 55 34 8 6 7 15 9 9 22 3 1 i 4 2 3 55 4 5 22 34 8 2 32 9 3 6 15 7 9 1 55 2 3 34 32 4 5 22 6 4 8 7 15 9 9 2 3 Problem 4) Using CLRS figure 8.4 as a model, illustrate the operation of Bucket-Sort on the array <0.10, 0.15, 0.34, 0.67, 0.54, 0.59, 0.45, 0.11>. Length[A] = 8 A B 1 .10 0 .10 2 .15 1 .15 / 3 .34 2 .34 / 4 .67 3 .45 / 5 .54 4 .54 6 .59 5 .67 7 .45 6 / 8 .11 7 / .11 / .59 / Problem 5) Draw a picture of a sequence <15, 2, 22, 10, 19, 3, 1> stored as a doubly linked list using the multiple-array representation. Do the same for the single-array representation. 1 next 1 2 3 4 5 6 7 8 2 3 4 5 6 7 / / 22 10 19 3 1 2 5 6 15 2 key prev / 1 3 4 1 1 2 3 4 5 6 7 15 4 / 2 7 1 2 key 8 9 10 11 12 10 4 10 13 7 13 14 15 19 16 10 16 17 18 19 20 21 3 19 13 1 / 16 prev next Problem 6) Consider inserting the keys <22, 14, 10, 88, 44, 47, 50, 9, 31, 18 > into a hash table of length m = 9 using open addressing with auxiliary hash function h’(k) = k mod m. Illustrate the result of inserting these keys using linear and quadratic probing with c1= 1 and c2=5. Linear Probing: m 9 h(k) 0 1 2 3 4 5 6 7 8 9 10 Hash(k) Key 22 4 4 14 5 5 10 1 1 88 7 7 keys 44 8 8 47 2 2 50 5 5 6 9 0 0 31 4 4 5 6 7 8 0 1 2 3 0 9 1 10 2 47 3 31 4 22 5 14 6 50 7 88 8 44 18 0 0 1 2 3 4 5 6 7 8 Error Error 18 overflow Quadratic Probing: m c1 c2 Quadratic 9 1 5 h(k) 0 1 2 3 4 5 6 7 8 9 10 Hash(k) Key 22 4 4 14 5 5 10 1 1 88 7 7 keys 44 8 8 47 2 2 50 5 5 2 0 9 0 0 6 31 4 4 1 8 7 7 8 1 4 8 4 error 0 50 1 10 2 47 3 18 4 22 5 14 6 9 7 88 8 44 18 0 0 6 4 3 Problem 7) Demonstrate the insertion of keys <5, 10, 27, 38, 21, 16, 9, 4> into a hash table with collision resolution by chaining. Let the table have 11 slots, and let the hash function be h(k) = k mod 11. A 0 / 1 / 2 / 3 / 4 4 5 5 6 / 7 / 8 / 9 10 9 10 / 27 / 21 38 16 / Problem 8) For the set of keys < 2, 4, 1, 23, 17, 3, 10, 9>, show the binary search tree of height 3, 4, and 5. Height 3: 4 2 10 1 3 9 17 23 Height 4: 2 1 10 9 3 23 4 Height 5: 17 1 1 3 4 9 17 10 23