Lecture 1 - Comparison Sorts

Question 1

What comparison sorts have average+worst time complexity of O(n^2)?

Answer 1

Selection sort
Insertion sort
Bubble sort

Question 2

What comparison sorts have average and worst time complexity of O(nlogn)?`

Answer 2

Merge sort
Heap sort

Question 3

What is the fastest comparison - based sorting algorithm in practice?

Answer 3

Quicksort

Question 4

What is the worst time complexity of quicksort?

Answer 4

O(n^2)

Question 5

What is the average time complexity of quicksort?

Answer 5

O(nlogn)

Question 6

What is the claim on comparison based sorting algorithms?

Answer 6

No comparison based sorting algorithm can have a better complexity than O(nlogn).

Question 7

How to prove comparison based algorithms can’t do better than O(nlogn)?

Answer 7

Via a decision tree

Question 8

What does the number of leafs of a decision tree represent?

Answer 8

Number of outcomes.
This can also be calculated via !n.

Question 9

Why can the complexity not be better than the height of the decision tree?

Answer 9

Since we have to reach a leaf to get a result.

Question 10

What is the maximum number of leaf nodes in a binary tree?

Answer 10

2^(h+1) - 1

Question 11

Is a decision tree a binary tree?

Answer 11

Yes

Question 12

What is the equation to prove the O(nlogn) claim?

Answer 12

!n <= 2^(h+1)-1 < 2^(h+1)

Question 13

How to solve !n <= 2^(h+1)-1 < 2^(h+1).

Answer 13

• take log2 of both sides
  h+1 > log2 (!n)
• since !n > (n/2) ^ (n/2)
  h+1 > log2(n/2)^(n/2)
• simplify
  h+1 > n/2 log2(n/2)
• expand
  h+1 > n/2 log2n - n/2 log22
  h > n/2 log2n - n/2 - 1
  -> this gives O(nlogn) ignoring smaller terms