Recurrence Relations
As you arrive: Get out a piece of paper
and and pen. We’re gonna do some
math in class today and you’d want to
follow along. Put your name at the top.
After class today…
• You will be able to explain the derivation of 2
example recurrence relations
• You will be able to use recurrence relations to
compute the big-O of recursive functions
Warm Up
public void whatIsMyBigO(String[] strings)
{
int current = 0;
while(strings[current].equals("ninja")) {
current++;
//this loop will eventually end because
//I know there is at least one non-ninja
//string in the list
}
}
1.
2.
3.
4.
5.
O(n)
O(n log n)
O(1)
O(n2)
O(log n)
Warm Up: Harder
//finds an element in a sorted array
public int mikeSearch(int[] sorted, int first, int upto, int key) {
while (first < upto) {
int mid = (first + upto) / 2; // Compute mid point.
if (key < sorted[mid]) {
upto = mid;
// repeat search in bottom half.
} else if (key > sorted[mid]) {
first = mid + 1; // Repeat search in top half.
} else {
return mid;
// Found it. return position
}
}
return -(first + 1);
// Failed to find key
}
1.
2.
3.
4.
5.
O(n)
O(n log n)
O(1)
O(n2)
O(log n)
The problem
private boolean containsNodeBST(int value,
MikesIntTreeNode current) {
if(current == null) return false;
if(current.value == value) return true;
if(value < current.value) {
return containsNode(value, current.left);
} else {
return containsNode(value, current.right);
}
}
What is the Big O assuming the Binary Search Tree is height balanced?
What is the Big O not assuming the Binary Search Tree is height balanced?
A.
B.
C.
D.
E.
Priority Queue
• You add to it like a normal queue
• But when you remove, it gets removed in
order of size (smallest first)
PriorityQueue<String> ex = new PriorityQueue<String>();
ex.add("hello");
ex.add("cs100");
ex.add("class");
while(!ex.isEmpty()) {
System.out.println(ex.remove());
}
// Prints:
// class
// cs100
// hello
Priority Queue
What does this print out?
PriorityQueue<Integer> ex = new PriorityQueue<Integer>();
ex.add(2);
ex.add(13);
ex.add(9);
ex.add(75);
ex.add(4);
while(!ex.isEmpty()) {
System.out.println(ex.remove());
}
Priority Queue Problem
• Snarf the code
• Imagine we’ve got a class BestPrice, that tracks the
best price of an item we’re looking to buy. Everytime
we find a new price we call add to add the price we’re
looking for. Then when someone calls buyCheapest(n)
we buy n items, using the cheapest offer we have then
the second cheapest, etc. We return the total cost to
buy n items.
• Take a look at the sample code in main if this in
unclear.
• Hint: make 1 instance variable – a priority queue