This document summarizes key information from a recitation on analysis of algorithms: 1) It orders common functions by asymptotic growth rate and provides solutions for determining tight asymptotic bounds for runtime recurrences using the master method. 2) It examines the "hat-check problem" - the expected number of customers that get their own hat back in a random ordering is 1, shown through indicator random variables. 3) It introduces the problem of searching for a value in an unsorted array by randomly picking indices, and asks for the expected number of picks needed to find 1 value, k values, or to terminate if the value is not present.

1. BLG 335E – Analysis of Algorithms I
Fall 2013, Recitation 2
23.10.2013
R.A. Atakan Aral
aralat@itu.edu.tr – Research Lab 1
R.A. Doğan Altan
daltan@itu.edu.tr – Research Lab 3

2. Warm-up Problem
• Order the following functions by
asymptotic growth rate:
– 𝑛2 + 5𝑛 + 7
– log2 𝑛3
– 9517
– 2log2 𝑛
– 𝑛3
– 𝑛𝑙𝑜𝑔2 𝑛 + 9𝑛
– 4 log2 𝑛
– log2 𝑛 + 3𝑛

3. Warm-up Problem
• Solution:
– 9517
– log2 𝑛3
– 4 log2 𝑛
– 2log2 𝑛
– log2 𝑛 + 3𝑛
– 𝑛𝑙𝑜𝑔2 𝑛 + 9𝑛
– 𝑛2 + 5𝑛 + 7
– 𝑛3

4. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
a. 𝑇 𝑛 = 𝑇 𝑛 − 1 + 𝑛
b. 𝑇 𝑛 = 𝑇
𝑛
2
+ 𝑇
𝑛
4
+ 𝑇
𝑛
8
+ 𝑛
c. 𝑇 𝑛 = 𝑇
9𝑛
10
+ 𝑛
d. 𝑇 𝑛 = 16𝑇
𝑛
4
+ 𝑛2
e. 𝑇 𝑛 = 7𝑇
𝑛
2
+ 𝑛2

5. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
a. 𝑇 𝑛 = 𝑇 𝑛 − 1 + 𝑛
Lower bound (Ω):
𝑇 𝑛 ≥ 𝑐𝑛2 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐 > 0
𝑇 𝑛 ≥ 𝑐 𝑛 − 1 2 + 𝑛
= 𝑐𝑛2 − 2𝑐𝑛 + 𝑐 + 𝑛 ≥ 𝑐𝑛2
𝑡𝑟𝑢𝑒 𝑖𝑓 0 < 𝑐 <
1
2
𝑎𝑛𝑑 𝑛 ≥ 0

6. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
a. 𝑇 𝑛 = 𝑇 𝑛 − 1 + 𝑛
Upper bound (O):
𝑇 𝑛 ≤ 𝑐𝑛2 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐 > 0
𝑇 𝑛 ≤ 𝑐 𝑛 − 1 2 + 𝑛
= 𝑐𝑛2 − 2𝑐𝑛 + 𝑐 + 𝑛 ≤ 𝑐𝑛2
𝑡𝑟𝑢𝑒 𝑖𝑓 𝑐 = 1 𝑎𝑛𝑑 𝑛 ≥ 1

7. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
b. 𝑇 𝑛 = 𝑇
𝑛
2
+ 𝑇
𝑛
4
+ 𝑇
𝑛
8
+ 𝑛

8. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
b. 𝑇 𝑛 = 𝑇
𝑛
2
+ 𝑇
𝑛
4
+ 𝑇
𝑛
8
+ 𝑛
Upper bound (O):
𝑇 𝑛 ≤
𝑐𝑛
2
+
𝑐𝑛
4
+
𝑐𝑛
8
+ 𝑛 =
7𝑐𝑛
8
+ 𝑛 ≤ 𝑐𝑛
𝑡𝑟𝑢𝑒 𝑖𝑓 𝑐 ≥ 8

9. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
b. 𝑇 𝑛 = 𝑇
𝑛
2
+ 𝑇
𝑛
4
+ 𝑇
𝑛
8
+ 𝑛
Lower bound (Ω):
𝑇 𝑛 ≥
𝑐𝑛
2
+
𝑐𝑛
4
+
𝑐𝑛
8
+ 𝑛 =
7𝑐𝑛
8
+ 𝑛 ≥ 𝑐𝑛
𝑡𝑟𝑢𝑒 𝑖𝑓 0 < 𝑐 ≤ 8

10. Master Method

11. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
c. 𝑇 𝑛 = 𝑇
9𝑛
10
+ 𝑛
𝑎 = 1, 𝑏 =
10
9
, 𝑓 𝑛 = 𝑛 = 𝛺 n
log10
9
1+1
𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑦 𝑐𝑎𝑠𝑒 3, 𝑙𝑒𝑡′ 𝑠 𝑐ℎ𝑒𝑐𝑘 𝑐
1
9𝑛
10
≤ 𝑐𝑛 ℎ𝑜𝑙𝑑𝑠 𝑓𝑜𝑟 𝑐 =
9
10
≤ 1
𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑙𝑦 𝑐𝑎𝑠𝑒 3:
𝑇 𝑛 = Θ(n)

12. Problem 1
• Give tight asymptotic bounds for 𝑇(𝑛) in
each of the following recurrences.
d. 𝑇 𝑛 = 16𝑇
𝑛
4
+ 𝑛2 𝑎 = 16, 𝑏 = 4, 𝑓 𝑛 = 𝑛2
𝑛2=Θ 𝑛log4 16 , 𝑐𝑎𝑠𝑒 2:
𝑇 𝑛 = Θ(𝑛2 log2 𝑛)
e. 𝑇 𝑛 = 7𝑇
𝑛
2
+ 𝑛2
𝑎 = 7, 𝑏 = 2, 𝑓 𝑛 = 𝑛2
𝑛2 = 𝑂 𝑛log2 7−ε , 𝑐𝑎𝑠𝑒 1:
𝑇 𝑛 = Θ(𝑛log2 7)

13. Problem 2
• Hat-check problem
• Each of n customers gives a hat to a hat-
check person at a restaurant.
• The hat-check person gives the hats back
to the customers in a random order.
• What is the expected number of customers
that get back their own hat?

14. Problem 2
• Counting Solution:
– Count number of fixed points in all n!
possible permutations
– Divide by n! To find average number
– Answer is 1
• With Indicator Random Variables:
– 𝑋𝑖 = 𝐼 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟 𝑖 𝑔𝑒𝑡𝑠 ℎ𝑖𝑠 𝑜𝑤𝑛 ℎ𝑎𝑡
𝑓𝑜𝑟 𝑖 = 1,2, … , 𝑛
– 𝑋 = 𝑋1 + 𝑋2 + ⋯ + 𝑋 𝑛
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑡ℎ𝑎𝑡 𝑔𝑒𝑡 𝑡ℎ𝑒𝑖𝑟 𝑜𝑤𝑛 ℎ𝑎𝑡

15. Problem 2
• 𝐸 𝑋 = 𝐸 𝑖=1
𝑛
𝑋𝑖 = 𝑖=1
𝑛
𝐸[𝑋𝑖]
• 𝐸 𝑋𝑖 =
1
𝑛
• 𝐸 𝑋 = 𝑖=1
𝑛 1
𝑛
= 1

16. Problem 3
• Searching for a value x in an unsorted
array A consisting of n elements.
• Pick a random index i into A.
• If A[i ] = x, then we terminate;
• Otherwise, we continue the search by
picking a new random index into A.
• We continue picking random indices into
A until we find x or until we have checked
every element of A.

17. Problem 3
• What is the expected number of indices
into A must be picked before x is found?
(There is exactly one x in A)
• What if there are 𝑘 ≥ 1 x values in A?
• If there is no x in A, what is the expected
number of indices into A that must be
picked before search terminates?