CPSC 212: Data Structures and Algorithms

1. CPSC 212: Data Structures and Algorithms • Instructor: Harry Plantinga

2. Computer Science 212Data Structures and Algorithms • The Heart of Computer Science • Data structures • Study of important algorithms • Algorithm analysis • Algorithm design techniquesPlus • GUIs and a GUI toolkit • Intelligent systems • Course Information: http://cs.calvin.edu/CS/212/

3. Why study DS & Algorithms? • Some problems are difficult to solve and good solutions are known • Some “solutions” don’t always work • Some simple algorithms don’t scale well • Data structures and algorithms make good tools for addressing new problems • Why not just get a faster computer? • Esthetic beauty!

4. Asymptotic Analysis • What does it mean to say that an algorithm has runtime O(n log n)? • n: Problem size • Big-O: upper bound over all inputs of size n • “Ignore constant factor” (why?) • “as n grows large” g(n) is said to be O(f(n)) if there exist constants c0 and n0 such that g(n) < c0 f(n) for all n > n0 Most algorithms we study will have runtimes of O(1), O(log n), O(n), O(n log n), O(n2), O(n3), O(2n)

5. Connectivity • Building railroads across America • Dutch Bingo • IC connectivity • Telephone network Given an edge from A to B, print it out unless previous edges already connect A and B n edges are added

6. Example 3-4 3-4 4-9 4-9 8-0 8-0 2-3 2-3 5-6 5-6 2-9 [2-3-4-9] 5-9 5-9 7-3 7-3 4-8 4-8 5-6 [5-6] 0-2 [0-8-4-3-2] 6-1 6-1

7. Abstract Data Types • What is an abstract data type? • A collection of data • A set of operations on that data • What abstract data type do we need? • Data: • Operations: Sets of elements Find(x) – what set is x in? Union(A,B) – merge sets A and B

8. Algorithm • How to solve the railroads problem? • How to implement the Union-Find ADT? For road (x,y) if find(x) != find(y) union(find(x),find(y)) print “x – y”

9. Quick Find • For each city, keep a pointer to the “capital” of the set • Find(x): return the set name for x • Runtime: • Union(A,B): change all sets named A (=Find(x)) in the array to B (=Find(y)) • Runtime: • Can we do faster unions? O(1) 10^8 items, 10^9 unions, 100 mips: 10sec O(n) 10^8 items, 10^9 unions, 31 yrs

10. Quick Union • To do a Unite(X,Y) (where X and Y are sets), just point X to Y • Union runtime: • Find runtime: • Total time for adding n roads? Can we do better? O(1) O(n)

11. Weighted Union • Keep track of the number of cities in each set • Insert the smaller set into the larger • Runtime for Unite(A,B): • Runtime for Find(x): What’s the worst-case tree height? O(1) O(?)

12. Weighted UnionWorst case • What sequence of inputs gives the tallest trees? • How tall are the trees? • What is the runtime for find? • What is the runtime for n unions? <= log n (prove by induction) O(log n) O(n log n) Better still: average O(1) per union!

13. Path Compression • When you do a find, make every city on the path to the “capital” point to the capital • Runtime? O(n log* n)

14. What difference does it make? • Small problem: use the simplest algorithm • Large problem: use the best algorithm! N M qf qu wqu pc 1000 6206 14 25 6 5 10000 83857 1216 4577 91 73 100000 1545119 1071 1106 A better algorithm is much more advantageous than a faster computer.

15. Lessons • Start with simple algorithm • Need a good algorithm for big problems • Higher level of abstraction (tree) is helpful • Strive for worst-case performance guarantees • Average-case is better than nothing • Identify fundamental abstraction Show me your code—you’ll have to explain your DS Show me your DS—I don’t need to see your code