1. Titles make PowerPoint happy.
This is a title
Titles make PowerPoint happy.

2. Up-Trees (not as much work as it looks)
Perform Disjoint Union-Find operations on Up-Tree with path compression and using weight (# of nodes) in merges.
Starting with empty tree.
Keep track of (sub)tree weight, and pointers into tree (pointers isn’t too hard)
Vaguely have a sense of # of operations (e.g. edge traversals)
For items: a,b,c,d,e,f,g,h,i,j,k
Union (a,b). Union (b,c). Union (d,e). Union (d,f). Union (d,j). Union (d,k). Union (e,g). Union (h,i). Union (c,h). Union (c,f). Find (i). Find (b). Find (c). Find (i). Find (h).

3. Make a nicely-connected Maze
(diagram on next page)
Make a random 3x3 maze with exactly one path between any two points
Simulate up-trees (or don’t if you feel comfortable)
Use edge order (first edge, second edge,…) on next page.
Feel the power of the Dark Side

4. Order of edges:

5. BTW
Show that -notation is an equivalence relation. Formal proofs not necessary.
Contemplate putting functions into up-trees based on -notation (but not for too long…)

6. Hashing functions Hash table size 7, key=integers
Consider this hashing function:
H(X)= X mod 7
What patterns of (distinct) keys would cause only one bucket to be used?
(answer on next page)

7. Hashing functions cont’d
One answer: 0,7,14,21,28,… Integers may have a reason to be multiples of 7 (probably even worse if you’re hashing strings of English words – very biased data)

8. Hashing functions… How about this hashing function:
given a parameters A,B,K:
HA,B,K(X)=(A*X+B mod 7K)/K
What patterns would cause only one bucket to be used?
For a given bad pattern, can you find different values of A,B,K to make this pattern OK (defeat your adversary)?
Given a patterns, if you ran your program many times with random A,B,K, what’s the average performance?
If you knew the data ahead of time, how could you design a really good hashing function?

9. Extendible Hashing
In the extendible hash table on the next page, perform the following inserts:

10. Extendible Hashing: table so far…00 01 10 11
0001
0110
0111
1000
1010
1001
1111
1100
Note: Buckets can store up to 3 keys before overflowing

11. Binomial Queues Merge these binomial queues:
From Introduction to Algorithms, Cormen et. al, pp

12. Tricks & Leaps with Treaps
What happens when you insert this into a BST: 1,2,3,4,5,…
What about a treap?
Create a sequence of random numbers that’d create a perfectly balanced Treap for 1,2,3,4,5…
What should the range of your random priorities be? What are bad properties for the random-number generator?

13. Traverse & not-too Deep: Treaps
Insert key 18 with random priority 2 into this Treap (don’t worry about “proper” way to rotate):