Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
COMP171 Spring 2009 Course Review Hashing / Slide 2 Elementary Data Structures Linked lists Types: singular, doubly, circular Operations: insert, find, delete O(N) Stack ADT (array or linked-list) First-in-Last-out Operations: push, pop Queue ADT (array or linked-list) First-in-First-out Operations: enqueue, dequeue Priority queue (heap) Heap-order property Operations: insert and deleteMin (both O(log N) using array) Hashing / Slide 3 Analysis of Algorithms Worst case analysis time and memory space depends on the input (size, special features) and algorithms growth rate of the functions Asymptotic notations O(.): upper bound, “at most”, worst case, ≤ Ω(.): lower bound, “at least”, best case, ≥ Θ(.): tight bound, O(.) && Ω(.), asymptotically “≈” growth rate Hashing / Slide 4 Comparison-based Sorting Algorithms Algorithm s Method Worst time Average 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 logN) O(N) Quicksort Pick pivot, partition, recursion O(N²) O(N logN) O(logN) Heapsort Build heap, deleteMin O(N logN) O(N logN) O(1) 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 Rely on some special features of the input Counting sort: O(N + Range) Radix sort: O(N × #digits) Hashing / Slide 6 Overview of the Forrest Binary Trees Binary Search Trees AVL trees M-ary Trees Terms: B+-Trees root, leaves, child & parent, siblings, internal nodes, height, paths & length, depth, subtrees Main theme: search, insertion, deletion. Hashing / Slide 7 Binary Search Trees Traversal Pre-order, in-order, post-order Average depth O(log N), max depth O(N), Operations: Search, findMin, findMax Insertion Deletion: 3 cases, recursive All takes O(height of the tree). Hashing / Slide 8 AVL Trees Balanced BST Height of an AVL tree is O(log N). Search O(log N) Insertion: 2 types of rebalancing O(log 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) Trace the deletion path, and more than one node may need rotation. Hashing / Slide 9 B+-Trees M-ary trees The root is either a leaf or has 2 to M children (1 to M-1 keys). Each internal node, except the root, has between M/2 and M children ( M/2 - 1 and M - 1 keys). Each leaf has between L/2 and L keys and corresponding data items. The data items are stored at leaves Search Insertion: always insert to a leaf Possibly splitting leaf (and internal nodes) Update the key in a internal node if necessary Deletion “Borrow” a key from your siblings Merge two leaves (or internal nodes) – opposite to splitting Hashing / Slide 10 Graph Essentials Graph = Vertices + Edges Representation Adjacency matrix: O(N2) space, O(1) access time Adjacency list: O(M+N) space, O(N) access time Keywords subgraph, tree, acyclic graph path, length, cycle, simple directed, undirected incident, degree, indegree / outdegree connected components 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 Hashing / Slide 12 Graph Traversal: DFS Connectivity and cycle detection Time: O(M+N) Recursive function DFS tree 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 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 Computationally fast, and efficient with O(1) complexity 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 Linear Probing Quadratic Probing Double hashing