I'm confused some question.

In Q2, why RHS just need to from log2 n to log2 n/2.

In Q4 (2), how to prove that?

In Q5, how to solve this problem

Worksheet 3 CSE 331: Algorithms and Data Structures 1. Prove log2(n!)=O(nlogn) . loga (n!): log, hcn-1)(n-2)... (1) = log hrlogen-1) + logen-2) + ... + lug. 2 logn logant... loy." con logan) 2. Prove log2(n!)=12(nlogn) log,6!): loganin-cn-26-(1) = login + log." "... luga' > logif + loga! +12 + log2 (2+2) .... tlayan or > loyn+log (n-lit-tyt log 2 3d- logo 3. A binary tree with depth d has at most 2d leaves. (or: A binary tree with L leaves has depth at least logall). doo 2:1 d=1 ,2-2=2 dtl = 2.20 = 24 . Prove (1) (21 1) = N2 (2)X1 = i (Hint: by induction) Diil, 1- il 1:1 1:21+3:n +8+27tooth (1+2t... tho ji... +3+5t... (2n-1) in" ktl in turmala lett: addktlith in formula, . 148+22+...+ k + k + U=1712&-+kkktif) 2ck+1)-1=2kt (1+2+-+kt (kt) 1+3+5+ ... +(2k-1) +k+): HT K+(2k+1) 143 +5+ ... + 12k+1) = (k+U . for niktl 5. Assuming N = 1000000 and M = 100000, how many merge operations will be neces- sary for a polyphase external merge sort that performs a 6 way merge? Page 1 of 1 Worksheet 3 CSE 331: Algorithms and Data Structures 1. Prove log2(n!)=O(nlogn) . loga (n!): log, hcn-1)(n-2)... (1) = log hrlogen-1) + logen-2) + ... + lug. 2 logn logant... loy." con logan) 2. Prove log2(n!)=12(nlogn) log,6!): loganin-cn-26-(1) = login + log." "... luga' > logif + loga! +12 + log2 (2+2) .... tlayan or > loyn+log (n-lit-tyt log 2 3d- logo 3. A binary tree with depth d has at most 2d leaves. (or: A binary tree with L leaves has depth at least logall). doo 2:1 d=1 ,2-2=2 dtl = 2.20 = 24 . Prove (1) (21 1) = N2 (2)X1 = i (Hint: by induction) Diil, 1- il 1:1 1:21+3:n +8+27tooth (1+2t... tho ji... +3+5t... (2n-1) in" ktl in turmala lett: addktlith in formula, . 148+22+...+ k + k + U=1712&-+kkktif) 2ck+1)-1=2kt (1+2+-+kt (kt) 1+3+5+ ... +(2k-1) +k+): HT K+(2k+1) 143 +5+ ... + 12k+1) = (k+U . for niktl 5. Assuming N = 1000000 and M = 100000, how many merge operations will be neces- sary for a polyphase external merge sort that performs a 6 way merge? Page 1 of 1