and Sorting selection
Prof. Noah Snavely CS1114 http://cs1114.cs.cornell.edu
Administrivia
Assignment 1 due Friday by 5pm
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Assignment 2 out on Friday Quiz 2 next Thursday 2/12
Coverage through next Tuesday Closed book / closed note
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Administrivia
More robots will be working soon
If all of the working ones are in use, you can test your code with the robot arena
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Logical operators
&& logical "and" || logical "or" if (x == 1 && y == 2) || z == 3 % do something interesting else % do something else end
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What's the difference between...
A(i) = A(i) + 2i; A(i) = A(i) + 2 * i;
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Plotting in Matlab
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Recap from last time
We looked at the "trimmed mean" problem for locating the lightstick
Remove 5% of points on all sides, find centroid
This is a version of a more general problem:
Finding the kth largest element in an array Also called the "selection" problem
We considered an algorithm that repeatedly removes the largest element
How fast is this algorithm?
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Recap from last time
BigO notation allows us to reason about speed without worrying about
Getting lucky on the input Depending on our hardware
BigO of repeatedly removing the biggest element?
Worstcase (k = n/2, i.e., median) is quadratic, O(n2)
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Classes of algorithm speed
Constant time algorithms, O(1)
Do not depend on the input size Example: find the first element
Linear time algorithms, O(n)
Constant amount of work for every input item Example: find the largest element
Quadratic time algorithms, O(n2)
Linear amount of work for every input item Example: repeatedly removing max element
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How to do selection better?
If our input were sorted, we can do better
Given 100 numbers in increasing order, we can easily figure out the 5th biggest or smallest
Very important principle! (encapsulation)
Divide your problem into pieces
One person (or group) can provide sort The other person can use sort
As long as both agree on what sort does, they can work independently Can even "upgrade" to a faster sort
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How to sort?
Sorting is an ancient problem, by the standards of CS
First important "computer" sort used for 1890 census, by Hollerith (the 1880 census took 8 years, 1890 took just one)
There are many sorting algorithms
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How to sort?
Given an array of numbers: [10 2 5 30 4 8 19 102 53 3] How can we produce a sorted array? [2 3 4 5 8 10 19 30 53 102]
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How to sort?
A concrete version of the problem
Suppose I want to sort all actors by height
...
How do I do this?
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Sorting, 1st attempt
Idea: Given n actors
1. Find the shortest actor (D. Devito), put him first 2. Find the shortest actor in the remaining group, put him/her second ... Repeat ... n. Find the shortest actor in the remaining group (one left), put him/her last
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Sorting, 1st attempt
Algorithm 1 1. Find the shortest actor put him first 2. Find the shortest actor in the remaining group, put him/her second ... Repeat ... n. Find the shortest actor in the remaining group put him/her last
What does this remind you of? This is called selection sort After round k, the first k entries are sorted
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Selection sort pseudocode
function [ A ] = selection_sort(A) % Returns a sorted version of array A % by applying selection sort % Uses in place sorting n = length(A); for i = 1:n % Find the smallest element in A(i:n) % Swap that element with something (what?) end
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Filling in the gaps
% Find the smallest element in A(i:n) We pretty much know how to do this
m = 10000; = m_index -1; for j in i:n if A(j) < m m = A(j); m_index = j; end [ 10 13 41 6 51 11 ] end % After round 1, % m = 6, m_index = 4
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Filling in the gaps
% Swap the smallest element with something % Swap element A(m_index) with A(i) A(i) = A(m_index); A(m_index) = A(i); tmp = A(i); A(i) = A(m_index); A(m_index) = tmp;
[ 10 13 41 6 51 11 ] [ 6 13 41 10 51 11 ]
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Putting it all together
function [ A ] = selection_sort(A) % Returns a sorted version of array A len = length(A); for i = 1:len % Find the smallest element in A(i:len) m = 10000; m_index = -1; for j in i:n if A(j) < m m = A(j); m_index = j; end end % Swap element A(m_index) with A(i) tmp = A(i); A(i) = A(m_index); A(m_index) = tmp; end
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Example of selection sort
[ 10 13 41 6 51 11 ] [ 6 13 41 10 51 11 ] [ 6 10 41 13 51 11 ] [ 6 10 11 13 51 41 ] [ 6 10 11 13 51 41 ] [ 6 10 11 13 41 51 ] [ 6 10 11 13 41 51 ]
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Speed of selection sort
Let n be the size of the array How fast is selection sort? O(1) O(n) O(n2) ? How many comparisons (<) does it do? First iteration: n comparisons Second iteration: n 1 comparisons ... nth iteration: 1 comparison
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Speed of selection sort
Total number of comparisons: n + (n 1) + (n 2) + ... + 1
n(n + 1) i = 2 i =1
n
Work grows in proportion to n2 selection sort is O(n2)
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Is this the best we can do?
Let's try a different approach Suppose we tell all the actors and all actors
shorter than 5.5' to move to the left side of the room taller than 5.5' to move to the right side of the room (actors who are exactly 5.5' move to the middle)
[ 6.0 [ 5.4
5.4 5.3
5.5 5.0
6.2 5.5
5.3 6.0
5.0 6.2
5.9 ] 5.9 ]
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Sorting, 2nd attempt
[ 6.0 [ 5.4 5.4 5.3 5.5 5.0 6.2 5.5 5.3 6.0
> 5.5'
5.0 6.2
5.9 ] 5.9 ]
< 5.5'
Not quite done, but it's a start We've put every element on the correct side of 5.5' (the pivot) What next? Do this again on each side: ask shorter group to pivot on (say) 5.3', taller group to pivot on 6.0' Divide and conquer
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How do we select the pivot?
How did we know to select 5.5' as the pivot? Answer: averageish human height In general, we might not know a good value Solution: just pick some va...