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.

• 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

Treaps

Skip Lists

– What is a skip list• 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

– Skip list operations• 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)

– Hash table• 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.)

Hashing

– 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

Disjoint Sets

– Graph definitions• 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

Graphs