Data Structures and Algorithm Design

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1 - University of Applied Sciences - Data Structures and Algorithm Design - CSCI Friedhelm Seutter Institut für Angewandte Informatik Contact Address: Prof. Dr. Friedhelm Seutter Institut für Angewandte Informatik Salzdahlumer Straße 46 / Wolfenbüttel Germany Telephone: (+49) Telefax: (+49) f.seutter@fh-wolfenbuettel.de Internet: Institut für Angewandte Informatik 2 1

2 Contents 1. Analyzing Algorithms and Problems 2. Data Abstraction 3. Recursion and Induction 4. Sorting 6. Dynamic Sets and Searching 7. Graphs and Graph Traversals 8. Optimization and Greedy Algorithms 10. Dynamic Programming 13. NP-Complete Problems Institut für Angewandte Informatik 3 Textbook Sara Baase, Allen Van Gelder: Computer Algorithms Introduction to Design and Analysis. Third Edition. Addison Wesley Longman Inc., Mass., 2000 ISBN Institut für Angewandte Informatik 4 2

3 Another Text T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein: Introduction to Algorithms. Second Edition. The MIT Press Cambridge, Mass. Mc Graw-Hill Book Company, New York, 2001 ISBN Institut für Angewandte Informatik 5 1. Analyzing Algorithms and Problems Correctness Amount of work done Amount of space used Simplicity, clarity Optimality Institut für Angewandte Informatik 6 3

4 Correctness A precise statement is needed about the characteristics of the inputs it is expected to work on, and what result it is to produce for each input. Then the relationship between the preconditions and the postconditions must be proved. But often just an informal argument or explanation is given to justify an algorithm. Institut für Angewandte Informatik 7 Search in an unordered array Problem: Let E be an array containing n entries, called keys, E = (e 1,e 2,,e n ), in no particular order. Find an index of a specified key K, if K is in E. Return -1, if K is not in E. Strategy: Compare K to each entry in turn until a match is found or the array is exhausted. Institut für Angewandte Informatik 8 4

5 Search in an unordered array Institut für Angewandte Informatik 9 Amount of work done A measure for the efficiency of an algorithm is needed, which is independent of the computer, the programming language, the implementation details and the programmer. The measure should aid in comparing two algorithms solving the same problem. The measure is called time complexity. Institut für Angewandte Informatik 10 5

6 Amount of work done Isolate a particular operation fundamental to the algorithm being considered. The total number of operations performed is roughly proportional to the number of basic operations. Count the number of these basic operations. Express this number as a function of the input size. Institut für Angewandte Informatik 11 Examples of basic operations Problem Operation Find a key in an array. Comparison of the key with an array entry. Multiply two matrices. Multiplication of two numbers. Sort an array. Comparison of two array entries. Solve a graph problem. Traversing a node or an edge. Institut für Angewandte Informatik 12 6

7 Examples of the input size Problem Input size Find a key in an array. Number of entries in the array. Multiply two matrices. Dimension of the matrices. Sort an array. Number of entries in the array. Solve a graph problem. Number of nodes or edges. Institut für Angewandte Informatik 13 Best- and worst-case complexity Institut für Angewandte Informatik 14 7

8 Average-case complexity Institut für Angewandte Informatik 15 Search in an unordered array Institut für Angewandte Informatik 16 8

9 Amount of space used A measure for the memory space for executing an algorithm is needed, expressed as a function depending on the input size (space complexity). The space for storing the program itself is not considered, in general. If the amount of extra space, aside from the input data, is constant with respect to the input size, the algorithm is said to work in place. Institut für Angewandte Informatik 17 Amount of space used The space complexity for the search algorithm (alg. 1.1) is linear, dependent on the number of array entries, a storage cell per entry, works in place. Institut für Angewandte Informatik 18 9

10 Simplicity A simple and straightforward algorithm makes verifying its correctness, and it makes implementing, debugging, and modifying a program easier, and the time needed is less. But such an algorithm must not solve the problem the most efficient way. Institut für Angewandte Informatik 19 Optimality An algorithm is optimal if there is no algorithm in the class under study that performs more efficient in the worst case. By algorithms in the class under study all possible algorithms are meant, including those not yet discovered. Optimal doesn t mean the best known, it means the best possible. Institut für Angewandte Informatik 20 10

11 Institut für Angewandte Informatik 21 Decision tree for search algorithms Institut für Angewandte Informatik 22 11

12 Decision tree for search algorithms Institut für Angewandte Informatik 23 Decision tree for search algorithms Institut für Angewandte Informatik 24 12

13 Decision tree for search algorithms Institut für Angewandte Informatik 25 Asymptotic growth rates The complexity of an algorithm is expressed by a function depending on the input size of the algorithm. The value of the complexity function is only roughly proportional to the total number of steps of the algorithm. The complexity function is classified by its asymptotic growth rate or order, and that order is applied to the complexity of the algorithm. Institut für Angewandte Informatik 26 13

14 Asymptotic notation Institut für Angewandte Informatik 27 Asymptotic notation Institut für Angewandte Informatik 28 14

15 Asymptotic notation Institut für Angewandte Informatik 29 Examples of classification Let f(n) = 3n 3 + 2n 2 4. f O(n 3 ) f O(n 2 ) f Ω(n 3 ) f Θ(n 3 ) Institut für Angewandte Informatik 30 15

16 Hierarchy of complexity classes O(n α ) O(n β ) for any 0 α < β O(1) O( log b n) = O( log c n) O(n α ) for any α > 0 (including fractional powers) and any base b,c 2 O(n α ) O(c n ) for any α > 0 and c > 1 Institut für Angewandte Informatik 31 How functions grow Algorithm Time function (µs) 33n 46n lg n 13n 2 3.4n 3 2 n Input size (n) Solution time s.0015 s.0013 s.0034 s.001 s s.03 s.13 s 3.4 s yr 1, s.45 s 13 s.94 hr 10, s 6.1 s 22 min 39 days 100, s 1.3 min 1.5 days 108 yr Institut für Angewandte Informatik 32 16

17 Complexity of SeqSearch B(n) = 1 = Θ(1) W(n) = n = Θ(n) A(n) = n(1 q/2) + q/2 = Θ(n) (0 q 1) (q is the probability that the key is in the array.) Institut für Angewandte Informatik 33 Usefull lemmata Institut für Angewandte Informatik 34 17

18 Some properties Institut für Angewandte Informatik 35 18