week 13 - wednesday. what did we talk about last time? np-completeness
TRANSCRIPT
CS221Week 13 - Wednesday
Last time
What did we talk about last time? NP-completeness
Questions?
Project 4
Assignment 6
SortingStudent Lecture
What do we want from sorting?
Characteristics of a sort Running time
Best case Worst case Average case
Stable Will elements with the same value get reordered?
Adaptive Will a mostly sorted list take less time to sort?
In-place Can we perform the sort without additional memory?
Simplicity of implementation Relates to the constant hidden by Big Oh
Online Can sort as numbers arrive
Insertion Sort
Insertion sort
Pros: Best case running time of O(n) Stable Adaptive In-place Simple implementation (one of the fastest
sorts for 10 elements or fewer!) Online
Cons: Worst case running time of O(n2)
Insertion sort algorithm
We do n rounds For round i, assume that the elements 0
through i – 1 are sorted Take element i and move it up the list of
already sorted elements until you find the spot where it fits
Insertion sort example
7
45
0
54
37108
51
7
45
0
54
37108
51
0
7
45
54
37108
51
0
7
45
54
37108
51
0
7
37
45
54108
51
0
7
37
45
54108
51
0
7
37
45
51
54108
Insertion Sort Implementation
Merge Sort
Merge sort Pros:
Best, worst, and average case running time ofO(n log n)
Stable Ideal for linked lists
Cons: Not adaptive Not in-place▪ O(n) additional space needed for an array▪ O(log n) additional space needed for linked lists
Merge sort algorithm
Take a list of numbers, and divide it in half, then, recursively: Merge sort each half After each half has been sorted, merge
them together in order
Merge sort example7
45
0
45
0
0
45
0
7
45
54
37108
37
108
37108
37
54108
7
45
0
54
37108
0
7
37
45
54108
7
45
0
54
37108
Merge Sort Implementation
Quicksort
Quicksort
Pros: Best and average case running time of
O(n log n) Very simple implementation In-place Ideal for arrays
Cons: Worst case running time of O(n2) Not stable
Quicksort algorithm
1. Pick a pivot2. Partition the array into a left half
smaller than the pivot and a right half bigger than the pivot
3. Recursively, quicksort the left half4. Recursively quicksort the right half
Partition algorithm Input: array, index, left, right Set pivot to be array[index] Swap array[index] with array[right] Set index to left For i from left up to right – 1
If array[i] ≤ pivot▪ Swap array[i] with array[index]▪ index++
Swap array[index] with array[right] Return index //so that we know where pivot
is
Quicksort Example
7
45
0
54
37108
0
7
45
54
37108
0
7
45
54
37108
0
7
37
45
54108
0
7
37
45
54108
0
7
37
45
54108
0
7
37
45
54108
Quicksort issues Everything comes down to picking the right pivot
If you could get the median every time, it would be great A common choice is the first element in the range as
the pivot Gives O(n2) performance if the list is sorted (or reverse
sorted) Why?
Another implementation is to pick a random location Another well-studied approach is to pick three random
locations and take the median of those three An algorithm exists that can find the median in linear
time, but its constant is HUGE
Heaps
Heaps
A maximum heap is a complete binary tree where The left and right children of the root
have key values less than the root The left and right subtrees are also
maximum heaps
We can define minimum heaps similarly
Heap example
10
9 3
0 1
How do you know where to add?
Easy! We look at the location of the new
node in binary Ignore the first 1, then each 0 is for
going left, each 1 is for going right
New node
Location: 6 In binary: 110 Right then left
10
9 3
0 1
Add 15
Oh no! 10
9 3
0 1 15
After an add, bubble up
10
9 3
0 1 15
10
9 15
0 1 3
15
9 10
0 1 3
Only the root can be deleted
10
9 3
0 1
Replace it with the “last” node
9 3
0 1
9 3
0
1
Then, bubble down
9 3
0
1
1 3
0
9
Operations
Heaps only have: Add Remove Largest Get Largest
Which cost: Add: O(log n) Remove Largest: O(log n) Get Largest: O(1)
Heaps are a perfect data structure for a priority queue
Priority queue implementation
Priority queue
A priority queue is an ADT that allows us to insert key values and efficiently retrieve the highest priority one
It has these operations: Insert(key) Put the key into the priority queue Max() Get the highest value key Remove Max() Remove the highest
value key
Implementation
It turns out that a heap is a great way to implement a priority queue
Although it's useful to think of a heap as a complete binary tree, almost no one implements them that way
Instead, we can view the heap as an array
Because it's a complete binary tree, there will be no empty spots in the array
Array implementation of priority queue
public class PriorityQueue {
private int[] keys = new int[10];
private int size = 0;…
}
Array view
Illustrated:
The left child of element i is at 2i + 1 The right child of element i is at 2i + 2
10
9 3
0 1
10 9 3 0 1
0 1 2 3 4
Insert
public void insert(int key)
Always put the key at the end of the array (resizing if needed)
The value will often need to be bubbled up, using the following helper method
private void bubbleUp(int index)
Max
public int max()
Find the maximum value in the priority queue
Hint: this method is really easy
Remove Max
public int removeMax()
Store the value at the top of the heap (array index 0)
Replace it with the last legal value in the array This value will generally need to be bubbled
down, using the following helper method Bubbling down is harder than bubbling up, because
you might have two legal children!
private void bubbleDown(int index)
Upcoming
Next time…
Heapsort TimSort Counting sort Radix sort
Reminders
Work on Project 4Work on Assignment 6
Due on Friday Read sections 2.4 and 5.1