1. Course Review COMP171 Spring 2009

2. Hashing / Slide 2 Elementary Data Structures * Linked lists n Types: singular, doubly, circular n Operations: insert, find, delete O(N) * Stack ADT (array or linked-list) n First-in-Last-out n Operations: push, pop * Queue ADT (array or linked-list) n First-in-First-out n Operations: enqueue, dequeue * Priority queue (heap) n Heap-order property n Operations: insert and deleteMin (both O(log N) using array)

3. Hashing / Slide 3 Analysis of Algorithms * Worst case analysis n time and memory space n depends on the input (size, special features) and algorithms n growth rate of the functions * Asymptotic notations O(.): upper bound, “ at most ”, worst case, ≤ n Ω(.): lower bound, “at least”, best case, ≥ Θ(.): tight bound, O(.) && Ω(.), asymptotically “≈ ” growth rate

4. Hashing / Slide 4 Comparison-based Sorting Algorithms Algorithm s MethodWorst timeAverage time Memor y Insertion sort A[1 … p-1] are sorted, then insert A[p] O(N²) O(N + #Inversions) O(1) Mergesort Divide, conquer and merge O(N logN) O(N) Quicksort Pick pivot, partition, recursion O(N²) O(N logN)O(logN) Heapsort Build heap, deleteMin O(N logN) O(1)

5. Hashing / Slide 5 Other Sorting Algorithms * Theorem: Any comparison based sorting algorithm takes Ω(n logn) comparisons to sort a list of n distinct elements. * Non-comparison sorting n Rely on some special features of the input n Counting sort: O(N + Range) n Radix sort: O(N × #digits)

6. Hashing / Slide 6 Overview of the Forrest * Binary Trees n Binary Search Trees  AVL trees * M-ary Trees n B + -Trees * Terms: n root, leaves, child & parent, siblings, internal nodes, n height, paths & length, depth, n subtrees * Main theme: search, insertion, deletion.

7. Hashing / Slide 7 Binary Search Trees * Traversal n Pre-order, in-order, post-order * Average depth O(log N), max depth O(N), * Operations: n Search, findMin, findMax n Insertion n Deletion: 3 cases, recursive * All takes O(height of the tree).

8. Hashing / Slide 8 AVL Trees * Balanced BST n Height of an AVL tree is O(log N). * Search O(log N) * Insertion: 2 types of rebalancing O(log N) n Trace from the new node to the root, locate the unbalanced node, and at most one rotation. Single rotation for “ outside ” insertion Double rotation for “ inside ” insertion * Deletion O(log N) n Trace the deletion path, and more than one node may need rotation.

9. Hashing / Slide 9 B + -Trees * M-ary trees n The root is either a leaf or has 2 to M children (1 to M-1 keys). n Each internal node, except the root, has between  M/2  and M children (  M/2  - 1 and M - 1 keys). n Each leaf has between  L/2  and L keys and corresponding data items. n The data items are stored at leaves * Search * Insertion: always insert to a leaf n Possibly splitting leaf (and internal nodes) n Update the key in a internal node if necessary * Deletion n “Borrow” a key from your siblings n Merge two leaves (or internal nodes) – opposite to splitting

10. Hashing / Slide 10 Graph Essentials * Graph = Vertices + Edges * Representation n Adjacency matrix: O(N 2 ) space, O(1) access time n Adjacency list: O(M+N) space, O(N) access time * Keywords n subgraph, tree, acyclic graph n path, length, cycle, simple n directed, undirected n incident, degree, indegree / outdegree n connected components

11. Hashing / Slide 11 Graph Traversal: BFS * Connectivity and shortest path (from source) * Time: O(M+N) * Use visited table, predecessor list, and distance table * BFS tree

12. Hashing / Slide 12 Graph Traversal: DFS * Connectivity and cycle detection * Time: O(M+N) * Recursive function * DFS tree

13. Hashing / Slide 13 Topological Sort * Topological ordering on DAG and connectivity * Time: O(M+N) * Repeatedly remove zero-degreed vertices and the outgoing arcs * Linear ordering

14. Hashing / Slide 14 Hashing  Hashing is a function that maps a key (K) into an entry in a table (hash table), T[h(K)]. It only supports operations Find, insertion, deletion * The hashing function should be n Computationally fast, and efficient with O(1) complexity n Minimize the collisions, when T[h(K1)] = T[h)K2)] and K1 and K2 are two distinctive keys * Collision Resolution 1: Separate chaining * Collision Resolution 2: Open Addressing n Linear Probing n Quadratic Probing n Double hashing