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UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Spring, 2001 Final Review Mon. 5/14-Wed. 5/16 Overview of Next 2 Lectures Review of some key course material Final Exam: Course Grade Logistics, Coverage, Format Handout for basis of 40% of test Course Evaluations Review of Key Course Material What’s It All About? Algorithm: steps for the computer to follow to solve a problem Problem Solving Goals: recognize structure of some common problems understand important characteristics of algorithms to solve common problems select appropriate algorithm & data structures to solve a problem tailor existing algorithms create new algorithms Some Algorithm Application Areas Robotics Bioinformatics Geographic Information Systems Design Analyze Telecommunications Apply Computer Graphics Medical Imaging Astrophysics Tools of the Trade Algorithm Design Patterns such as: binary search divide-and-conquer Data Structures such as: trees, linked lists, hash tables, graphs Theoretical Computer Science principles such as: NP-completeness, Summations Growth of Functions hardness MATH Probability Proofs Sets Recurrences Discrete Math Review Chapters 1-6 Growth of Functions, Summations, Recurrences, Sets, Counting, Probability Topics Discrete Math Review : Chapters 1-6 Solving Summations & Recurrences Sets, Basic Tree & Graph concepts Counting: Permutations/Combinations Probability: Basics, including Expectation of a Random Variable Proof Techniques: Induction Basic Algorithm Analysis Techniques: Chapters 1-6 Asymptotic Growth of Functions Types of Input: Best/Average/Worst Bounds on Algorithm vs. Bounds on Problem Algorithmic Paradigms/Design Patterns: Divide-and-Conquer Analyze pseudocode running time to form summations &/or recurrences What are we measuring? Some Analysis Criteria: Scope “Dimension” Upper? Lower? Both? Type of Input Time Complexity? Space Complexity? Type of Bound The problem itself? A particular algorithm that solves the problem? Best-Case? Average-Case? Worst-Case? Type of Implementation Choice of Data Structure Function Order of Growth 1 lglg(n) lg(n) n n lg(n) n lg2(n) n2 n5 know how to order functions asymptotically (behavior as n becomes large) O( ) upper bound W( ) lower bound Q( ) upper & lower bound know how to use asymptotic complexity notation to describe time or space complexity 2n Types of Algorithmic Input Best-Case Input: of all possible algorithm inputs of size n, it generates the “best” result for Time Complexity: “best” is smallest running time Best-Case Input Produces Best-Case Running Time provides a lower bound on the algorithm’s asymptotic running time (subject to any implementation assumptions) for Space Complexity: “best” is smallest storage Average-Case Input Worst-Case Input these are defined similarly Best-Case Time <= Average-Case Time <= Worst-Case Time Bounding Algorithmic Time (using cases) Using “case” we can discuss lower and/or upper bounds on: best-case running time or average-case running time or worst-case running time 1 lglg(n) T(n) = W(1) lg(n) n n lg(n) n lg2(n) n2 very loose bounds are not very useful! n5 2n T(n) = O(2n) Worst-Case time of T(n) = O(2n) tells us that worst-case inputs cause the algorithm to take at most exponential time (i.e. exponential time is sufficient). But, can the algorithm every really take exponential time? (i.e. is exponential time necessary?) If, for arbitrary n, we find a worst-case input that forces the algorithm to use exponential time, then this tightens the lower bound on the worst-case running time. If we can force the lower and upper bounds on the worst-case time to match, then we can say that, for the worst-case running time, T(n) = Q(2n ) (i.e. we’ve found the minimum upper bound, so the bound is tight.) Bounding Algorithmic Time (tightening bounds) for example... 1 lglg(n) TB(n) = W(1) 1st attempt lg(n) n n lg(n) n lg2(n) TB (n) = O(n) 1st attempt TB(n) = Q(n) 2nd attempt n2 n5 2n n TW (n) = W(n2) TW (n) = O(2 ) 1st attempt 1st attempt TW(n) = Q(n2) 2nd attempt Algorithm Bounds Here we denote best-case time by TB(n); worst-case time by TW(n) Approach Explore the problem to gain intuition: Establish worst-case upper bound on the problem using an algorithm 1 Describe it: What are the assumptions? (model of computation, etc...) Has it already been solved? Have similar problems been solved? (more on this later) What does best-case input look like? What does worst-case input look like? Design a (simple) algorithm and find an upper bound on its worst-case asymptotic running time; this tells us problem can be solved in a certain amount of time. Algorithms taking more than this amount of time may exist, but won’t help us. Establish worst-case lower bound on the problem Tighten each bound to form a worst-case “sandwich” n n2 n3 n4 n5 increasing worst-case asymptotic running time as a function of n 2n Know the Difference! Strong Bound: This worst-case lower bound on the problem holds for every algorithm that solves the problem and abides by our problem’s assumptions. 1 No algorithm for the problem exists that can solve it for worst-case inputs in less than linear time . Weak Bound: This worst-case upper bound on the problem comes from just considering one algorithm. Other, less efficient algorithms that solve this problem might exist, but we don’t care about them! n5 n worst-case bounds on problem An inefficient algorithm for the problem might exist that takes this much time, but would not help us. Both the upper and lower bounds are probably loose (i.e. probably can be tightened later on). 2n Master Theorem n T (n) aT ( ) f (n) b Master Theorem : LetT (n) aT ( n ) f (n) with a > 1 and b > 1 . b Then : Use ratio test to Case 1: If f(n) = O ( n (log b a) - e ) for some e > odistinguish between cases: then T ( n ) = Q ( n log b a ) f(n)/ n log b a Case 2: If f (n) = Q (n log b a ) then T ( n ) = Q (n log b a * log n ) Look for “polynomially larger” dominance. Case 3: If f ( n ) = W (n (log b a) + e ) for some e > o and if a f( n/b) < c f ( n ) for some c < 1 , n > N0 then T ( n ) = Q ( f ( n ) ) Sorting Chapters 7-10 Heapsort, Quicksort, LinearTime-Sorting, Medians Topics Sorting: Chapters 7-10 Sorting Algorithms: [Insertion & MergeSort from Chapters 1-6)], Heapsort, Quicksort, LinearTime-Sorting, Medians Comparison-Based Sorting and its lower bound Breaking the lower bound using special assumptions Tradeoffs: Selecting an appropriate sort for a given situation Time vs. Space Requirements Comparison-Based vs. Non-Comparison-Based Comparison-Based Sorting Time: BestCase Algorithm: InsertionSort AverageCase WorstCase W(n lg n) O(n2) MergeSort W(n lg n) O(n lg n) QuickSort Q(n lg n) HeapSort W(n lg n) O(n lg n) Q(n2) O(n lg n) In algebraic decision tree model, comparison-based sorting of n items requires W(n lg n) time. To breaking the lower bound and obtain linear time, forego direct value comparisons and/or make stronger assumptions about input. Data Structures Chapters 11-14 Stacks, Queues, LinkedLists, Trees, HashTables, Binary Search Trees, Balanced Trees Topics Data Structures: Chapters 11-14 Abstract Data Types: their properties/invariants Stacks, Queues, LinkedLists, (Heaps from Chapter 7), Trees, HashTables, Binary Search Trees, Balanced (Red/Black) Trees Implementation/Representation choices -> data structure Dynamic Set Operations: Query [does not change the data structure] Manipulate: [can change data structure] Search, Minimum, Maximum, Predecessor, Successor Insert, Delete Running Time & Space Requirements for Dynamic Set Operations for each Data Structure Tradeoffs: Selecting an appropriate data structure for a situation Time vs. Space Requirements Representation choices Which operations are crucial? Advanced Techniques Chapters 16-17 Dynamic Programming, Greedy Algorithms Topics Advanced Techniques: Chapters 16-17 Algorithmic Paradigms/Design Patterns: Using Dynamic Programming &/or Greedy Algorithms to solve Optimization Problems Divide-and-Conquer Dynamic Programming Greedy Algorithms Brute-Force/Naive Optimal Substructure Greedy Choice Property: Locally optimal -> Globally optimal Tradeoffs: Selecting an appropriate paradigm to solve a problem Tackling a problem using a sequence of paradigms: Brute-Force (high running time) then improve... Problem Characteristics Divide-and-Conquer Dynamic Programming Greedy Algorithms Modular Independent pieces Modular Modular Optimization Optimization Optimal substructure: Optimal substructure: optimal solution contains optimal solution optimal solutions to contains optimal subproblems solutions to subproblems Overlapping subproblems Greedy choice property: locally optimal choices lead to global optimum Graph Algorithms Chapters 23-25 DFS/BFSTraversals, Topological Sort, MinimumSpanningTrees, Shortest Paths Topics Graph Algorithms: Chapters 23-25 Undirected, Directed Graphs Connected Components of an Undirected Graph Representations: Adjacency Matrix, Adjacency List Traversals: DFS and BFS Differences in approach: DFS: LIFO/stack vs. BFS:FIFO/queue Forest of spanning trees Vertex coloring, Edge classification: tree, back, forward, cross Shortest paths (BFS) Topological Sort Weighted Graphs MinimumSpanningTrees: 2 different approaches Shortest Paths: Single source: Dijkstra’s algorithm Tradeoffs: Representation Choice: Adjacency Matrix vs. Adjacency List Traversal Choice: DFS or BFS Traversals: DFS, BFS DFS backtracks visit most recently discovered vertex LIFO structure stack data structure BFS vertices close to v are visited before those further away FIFO structure queue data structure FINAL EXAM Logistics, Coverage, Format Handout for basis of 40% of test Course Grading Homework 40% Exam 1 15% (closed book) Midterm 20% (open book) Final Exam 25% (open book) Results are scaled if necessary. Check grade status with us before final! Final Exam: Logistics Friday, 12/18 Olsen 311: 8:00-11:00 a.m. Open book, open notes Closed computers, neighbors Cumulative Worth 25% of grade Note change from registrar’s room number Text/Chapter/Topic Coverage Discrete Math Review & Basic Algorithm Analysis Techniques : Chapters 1-6 Sorting: Chapters 7-10 Stacks, Queues, LinkedLists, Trees, HashTables, Binary Search Trees, Balanced (Red/Black) Trees Advanced Techniques: Chapters 16-17 Heapsort, Quicksort, LinearTime-Sorting, Medians Data Structures: Chapters 11-14 Summations, Recurrences, Sets, Trees, Graph, Counting, Probability, Growth of Functions, Divide-and-Conquer Dynamic Programming, Greedy Algorithms Graph Algorithms: Chapters 23-25 Traversal, MinimumSpanningTrees, Shortest Paths Format 60% Mixture of questions of the following types: 1) Multiple Choice 2) True/False 3) Short Answer 4) Analyze Pseudo-Code and/or Data Structure 5) Solve a Problem by Designing an Algorithm 40% Select an appropriate paradigm/ design pattern Select appropriate data structures Write pseudo-code Justify correctness Analyze asymptotic complexity