Question

Part A Analyze the following recurrences and show their time complexity functions using (I) iteration method and (2) Master Theorem. AI. T(n) = 2T 3 A2. T(n) = 3T 2n АЗ. Т(п) — Т(п — 2) + 3 А4. Т(п) — 2Т (п — 1) + 1 A5. T(n)= 4T +n log n A6. T(n) = 3T +n log n n2 A7. T(n) = 27 Part B Do 2.3-4 (р39) and Problem 2-1 (р39) Part C Implement MERGE-SORT() algorithm that reads from a file named "inputHW02.txt" a list of double numbers (max = 3,000,000 numbers), sorts those numbers and indicates time consumption. This programming question will address the advantage of using iteration loops over recursive calls as well as using INSERTION-SORT() as a procedure in MERGE- SORT() Your program must perform the following actions: Opens the given file name and reads all double numbers. For simplicity, we assume this file only contains numbers and nothing else. mplements the function INSERTION-SORT() that only sort an array of maximum 25 numbers. The idea is that INSERTION-SORT() will be used as a sub-procedure to sort any sub-array when its size is small enough Four versions of MERGE-SORT() namely I. 2. 3. MERGE-SORT-A(): Using recursive calls and NO INSERTION-SORT() as a sub-procedure MERGE-SORT-B(): Using ITERATIVE loops (i.e, NO recursion) and NO INSERTION-SORT() a. as a sub- procedure. MERGE-SORT-C(): Using recursive calls and INSERTION-SORT() as a sub-procedure d. MERGE-SORT-D(): Using ITERATIVE loops (i.e, NO recursion) and INSERTION-SORT() c. as a sub

Answer 1

Tn) 2T) +3 + 12 t6 48 $7 -2T/) (4() 2T(1) -t3(k(にリリ) 4 k +3(3+ 2+ k times 2 On'2) By Masturs muthed T(n)- 2 T ) + 3 bE 4 C 2 2 0ln'ne)ae J0(n'3-2) Tin complaxitg sClea Bee

Ton) = 3T () -3 (3T)+) 3(3T () 2n ino =. +2 h 4 + Cn+2り 4 +2 h 1/3-11 3T 2 18ntint2n 27 8IT 5) n 54h t 16h+nt2n Ktems h +31) 2h 21 +9 4 + dn (1+3+ +27 41 T) 2 O (n) + 2 1h -) marter's mathod 3T 2n 4- O(n) casr(3) l0143L

onk mathod TIn)= T(n-2)+3 t b= L mastan con not b appled Th) T(n- 4) +3+3 = T(n-6) t3+3+3 - T(h-8) TR3T33 3+3 TCO) 3 3 teimg 2 O(n) Tn) T Cn) I Tin) (2T(h-2)+1) + -4 T(n-31)+2t 8 (2 TCh-+)t1) +4+ 2+1 = 16 T(h-4) +8+ 4+2+1 +4t 2 +1 2 T(h-+) 4-ten m hT(o)+1+ 2+ 40 ntem = gn To)キ (2h-リ 0lah) b-hmaster's mu od can't bt opplieal 2

ten's mulh od can't b opplieal 5Tn) 4T)+ hln 4 Ta)+1)+h logn 4 T nlg 17 + atenmy (2t1270)nype 2 k mastar mothod 6 2 a4 cos(L) n2nlogn On2)

T(r)= 3T (冬) +hlegn 3(7 T) legn 3T hogh master's mo hod b5 l04 3 h5 casr 3 T(h)= 4T) (2T() thi 2h tn2 =1RT() h V 2 h2 Kt mastul mathod a= 2 <2 10(