CS 171: Introduction to Computer Science II

Simple Sorting

Ymir Vigfusson

Roadmap

Algorithm Analysis Simple sorting algorithms

Bubble sort Selection sort Insertion sort

Advanced sorting algorithms

Warm-up questions Give the big Oh cost of the following code fragments:

a. int sum = 0; for (int i = 1; i < N; i *= 2)

for (int j = 0; j < N; j++) sum++;

b. int sum = 0; for (int i = 1; i < N; i *= 2)

for(int j = 0; j < i; j++) sum++;

Consult previous tattoo! “Remembered these, you have, hmm?”

∑𝑖=1

𝑛

𝑖=¿1+2+3+⋯+𝑛=𝑛(𝑛+1)

2¿

∑𝑖=0

𝑛

𝑥 𝑖=¿ 𝑥0+𝑥1+⋯+𝑥𝑛= 𝑥𝑛+1−1𝑥−1

¿

∑𝑖=0

log (𝑛)

2𝑖=¿1+2+4+⋯+2 log (𝑛)≈2𝑁 ¿

Triangular sum

Geometric sum

More tattoos (other arm?)

Harmonic sum

Stirling’s approximation

∑𝑖=1

𝑛1𝑖=¿1+ 1

2+ 1

3+⋯+ 1

𝑛≈ log (𝑛)¿

log (𝑛 ! )=log (1 )+ log (2 )+⋯+ log (𝑛)≈𝑛 log (𝑛)

Roadmap

Algorithm Analysis Simple sorting algorithms

Bubble sort Selection sort Insertion sort

Advanced sorting algorithms

Sorting problem

Sorting Problem

Rules of the Game Goal

rearrange items such that their keys are in ascending (increasing) order

Algorithms Based on comparisons of and exchanges between

items Sorting cost model

count compares and exchanges Whether extra memory needed

in place (no extra memory) need extra memory to hold another copy of the

array to be sorted.

Java Comparable Interface

We can sort any data items as long as items implements Comparable interface v.compareTo(w) returns an integer that is negative,

zero, or positive when v<w, v=w, v>w

Example class that implements Comparable interface: Date.java

Two useful sorting abstractions

Sorting Problem

Bubble Sort and Selection Sort

Intuition: Find the biggest (smallest) number Find the second biggest (smallest) number … keep going

Bubble sort Repeatedly swap two adjacent numbers in each

pass ▪ Biggest number pushed to the end!

Selection sort Select the biggest (smallest) number in each pass

• After one pass, we find the biggest number.

• It’s like the biggest ‘bubble’ floats to the topof the surface, hence the name ‘bubble sort’.

• Examples: 3 5 2 7 1 6 4

Bubble Sort

• In the second pass, we repeat the sameprocess, but now we only have N-1 numbersto work on.

• The third pass is the same, with only N-2numbers.

• …• Repeat until all players are in order.

Bubble Sort

Analysis of Bubble Sort

• Number of comparisons?

• Number of swaps?

= O(N )

= O(N )

Analysis of Bubble Sort

• Number of comparisons?

• Number of swaps?best case:

worst cast:

average:

2N(N -1)

2

= O(N2)

2

N(N -1)

2

N(N -1)

4

O(1)

Selection Sort

1. Keep track of the index of the smallestnumber in each round.

2. Swap the smallest number towards thebeginning of the array.

3. Repeat the above two steps.

Selection Sort

Selection Sort

Selection Sort Implementation

Selection Sort

• Number of comparisons?

• Number of swaps?

O(N2)

Selection Sort

• Number of comparisons?

• Number of swaps?

O(N)

Selection Sort

Online demo http://www.sorting-algorithms.com/selection-sort

Gypsy dance demo http://www.youtube.com/watch?v=Ns4TPTC8wh

w

Insertion Sort

• How do you sort a hand of poker cards?

Insertion Sort

• Idea– Assume the left portion of the array is partially

sorted (however, unlike selection sort, theelements are not necessarily in their finalpositions)

– For each remaining element on the right portion,insert it to the left portion (similar to insertion inan ordered array).

– Repeat until done.

Insertion Sort

Insertion Sort

Insertion Sort Implementation

Insertion Sort

Online demo http://www.sorting-algorithms.com/insertion-sort

Romanian dance demo http://www.youtube.com/watch?v=ROalU379l3U

Insertion Sort in Class!

Each class row to be sorted by height Tallest people to my left

Need two people for i and j in each row Count number of comparisons and swaps

Insertion Sort

• Number of comparisons?

• Number of exchanges?

Insertion sort

Best case N-1 comparisons 0 exchanges

Worst case ~N2/2 comparisons ~N2/2 exchanges

Foreshadowing

http://www.sorting-algorithms.com/random-initial-order

How would sort faster?

Summary

Bubble sort uses repeated comparisons and swaps to find the

biggest (or smallest) element in each pass Selection sort

selects the smallest element in each pass and reduces the cost of exchanges

Insertion sort inserts each element in the left assorted portion

and reduces the cost of comparisons All have average comparison cost of O(N2)