Question

(12 pts) Solve each of the following Do not only state your solution- Show how you obtained it. That is, if you use substitution, you must present the complete inductive proof that your solution is correct. If you obtained the solution from the tree. Note that you are to prove matching upper and lower bounds recurrences using substitution or a recursion tree. use a recursion tree, show the recursion tree and discuss how you (а) Т(п) — 4T (п/2) + п? (b) T(п) — Т(п/4) + T(п/2) - n

Answer 1

a) Tn) 4T(2) Solving sing subsituhon methocd, we ges n2- n2+n2 43 -T n2 4T 2 + 2 п? + n? K times 2 и -t K times k= log21 t wi stop when J, 2K 2 +nRogan 4 D'ntog.n)

PRoOf using in duction then TLL2 12L0g, Case Bse cose)- when exactty true Incluctive Hypothe sis when which ts uen n 2 Tu)T (2J yug22) D(A22) 0(1*40д) 2H 24 t2 2 Inducve step- when n 2' then Tn)T (2-742 220н-) 4T2) 2942 F 4 2( 2 2 OL(U22) 11 20, fo пoдк, Tu) is aleotue. case Tn s is aleo true, all uе Theree Tw-DLrthogn)

b) T TnA) t Splviy sing neuurson tree Lost 3n 4 2 4.2 2-4 4 2K deay ko lead bigger яесиноn tree tree has height k Cupper bourd). k-l0gn onsidur that we in leat Total LDst TW = nt 3nt DecreoingGp Alays Deeveasing 6tP O) OLn)