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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. 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 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) Hashing – 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.) 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 Graphs – 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