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Design and Analysis of Algorithms Review Haidong Xue Summer 2012, at GSU Measurement of running time Analysis Asymptotic notations Probabilistic analysis Divide & Conquer Design Algorithms Dynamic Programming Greedy Classic problems and their algorithms Dynamic set model Data structures Hash tables Binary search trees NPCompleteness 1. Algorithm analysis • What is an algorithm? • What are we interested about an algorithm? • How to measure the running time of an algorithm? • What are asymptotic notations? • Can you tell the asymptotic relation between two functions? 2. Algorithm design • What is a divide-and-conquer algorithm? • How to design a divide-and-conquer algorithm? • How to code a divide-and-conquer algorithm? – Recursion • How to calculate the running time of a divideand-conquer algorithm? – Recurrence equation 2. Algorithm design • Problems efficiently solved by divide-andconquer – Maximum sub-array – Matrix multiplication – Sorting • Quick sort, merge sort 2. Algorithm design • When to apply dynamic programming strategy? • What is a dynamic programming algorithm? • What are the two manners to dynamic programming algorithms? • Problems efficiently solved by dynamic programming – – – – Matrix chain multiplication Longest common sequence Bellman-Ford algorithm Floyd-Warshall algorithm 2. Algorithm design • What is a greedy algorithm? • When is a greedy algorithm correct? • Problems efficiently solved by greedy algorithms – – – – – – Coin changing with certain coin denominations Activity selection Fractional knapsack Dijkstra’s algorithm (single-source shortest path) Kruskal’s algorithm (MST) Prim’s algorithm (MST) 3. Sorting algorithm • What are comparison based sorting algorithms? • What is the lower bound of comparison based sorting algorithms? • Classic comparison based sorting algorithms – – – – Insertion sort Merge sort Quick sort Heap sort • Non-comparison based sorting algorithms – Counting sort 4. Graph related algorithms • • • • What is a directed graph? What is a undirected graph? What are the two representations of graphs? BFS, DFS, BFS-Traversal, DFS-Traversal – Topological sort, unweighted shortest path • MST – Kruskal; Prim • Single-source shortest path – Bellman-Ford; Dijkstra • All-pair shortest path – Floyd-Warshall • Vertex-cover – NP-complete • Travelling salesman problem – NP-complete 5. Data structures • What is the dynamic set model? • What are the operations on a dynamic set? • Hash tables – What are hash tables? – What is the worst case running time of each operation? – Hash functions • Multiplication hashing; division hashing; universal hashing • Open address • Binary search tree – Binary search tree properties – Algorithms of each dynamic set operation – Red-black tree 6. NP-Complete • What are P and NP problems? • What are NP-complete problems? – NP-Hard • How to prove a problem is NP-complete? • What are approximation algorithms for NPcomplete problems? • What is the approximation ratio? 6. NP-Complete • Classic approximation algorithms for some NPC problems – Vertex cover – Set cover – TSP Database Algorithms Software Computer Data Structure Programming Language Compiler Principles of Compilers Machine Language Operating System Computer Architecture Computer Composition Hardware Microchip Interfaces VLSI Design Applications Software Engineering Network Data mining Wireless Network Web programming Signal processing AI Automata Robotics Security Graphics