1. Review for Final Exam Non-cumulative, covers material since exam 2
Data structures covered:
Treaps
Skip lists
Hash tables
Disjoint sets
Graphs
For each of these data structures
Basic idea of data structure and operations
Be able to work out small example problems
Prove related theorems
Advantages and limitations
Asymptotic time performance
Comparison
Review questions are available on the web.

2. Treaps Definition Two values associated with each node
Key: making it a BST
Priority: making is binary min heap
Priorities are randomly generated
Making treap a BST constructed from a randomly ordered sequence of keys (why?)
Main advantages
High probability to be balanced (h = O(logn))
Compare with splay tree and RB tree
Operations
Find: according to key values as if it is a BST
Insert: as a leaf first as in BST, then rotate it up to satisfy heap order
Delete: rotate the node to be deleted down according to heap order until it becomes a leaf, then delete it.
Support set union, partition

3. Skip Lists What is a skip list Why need skip lists
Nodes with different size (different # of skip pointers)
Node size distribution according to the associated probability p
Nodes with different size do not have to follow a rigid pattern
The expected # of nodes with exactly k pointers (pk-1(1- p))
How to determine the size of the head node (log1/p N)
Why need skip lists
Expected time performance O(lg N) for find/insert/remove
Probabilistically determining node size facilitate insert/remove operations
Advantages over sorted arrays, sorted list, BST, balanced BST

4. Skip list operations Performance find
insert (how to determine the size of the new node)
Set pointers in insert and remove operations (backLook node)
Performance
Expected time performance O(lg N) for find/insert/remove (very small prob. of poor performance when N is large)
Expected # of pointers per node: 1/(1 - p)

5. Hashing Hash table Hashing functions Collision management
Trading space for time
Table size (primes)
Hashing functions
Properties making a good hashing function
Examples of division and multiplication hashing functions
Operations (insert/remove/find/)
Collision management
Separate chaining
Open addressing (different probing techniques, clustering problem)
Worst case time performance:
O(1) for find/insert/delete if  is small and hashing function is good
Limitations
Hard to answer order based queries (successor, min/max, etc.)

6. Disjoint Sets Equivalence relation and equivalence class
definitions and examples
Disjoint sets and up-tree representation
representative of each set
direction of pointers
Union-find operations
basic union and find operation
path compression (for find) and union by weight heuristics
time performance when the two heuristics are used:
O(m lg* n) for m operations (what does lg* n mean)
O(1) amortized time for each operation

7. Graphs Graph definitions Adjacency and representation
G = (V, E), directed and undirected graphs, DAG
path, path length (with/without weights), cycle, simple path
connectivity, connected component, connected graph, complete graph, strongly and weakly connectedness.
Adjacency and representation
adjacency matrix and adjacency lists, when to use which
time performance with each
Graph traversal: DF and BF
Single source shortest path
Breadth first (with unweighted edges)
Dijkstra’s algorithm (with weighted edges)
Topological order (for DAG)
What is a topological order (definitions of predecessor, successor, strict partial order)
Algorithm for topological sort