1. Problem Complexity Review

2. Problem Complexity

3. Online Survey The Spring term course/instructor opinion survey will be available during the period Monday, April 17th through Friday, April 28th from 6am to 11:59pm each day: http://www.coursesurvey.gatech.edu LB

4. Final Exam Schedule CS1311 Sections L/M/N Tuesday/Thursday 10:00 A.M. Exam Scheduled for 8:00 Friday May 5, 2000 Physics L1 LB

5. Final Exam Schedule CS1311 Sections E/F Tuesday/Thursday 2:00 P.M. Exam Scheduled for 2:50 Wednesday May 3, 2000 Physics L1 LB

6. Relations between Problems, Algorithms, and Programs Problem Algorithm Program....

7. Cost and Complexity Algorithm complexity can be expressed in Order notation, e.g. “at what rate does work grow with N?”: –O(1)Constant –O(logN)Sub-linear –O(N)Linear –O(NlogN)Nearly linear –O(N 2 )Quadratic –O(X N )Exponential But, for a given problem, how do we know if a better algorithm is possible?

8. The Problem of Sorting For example, in discussing the problem of sorting: Two algorithms to solve: –Bubblesort – O(N 2 ) –Mergesort – O(N Log N) Can we do better than O(N Log N)?

9. Algorithm vs. Problem Complexity Algorithmic complexity is defined by analysis of an algorithm Problem complexity is defined by –An upper bound – defined by an algorithm –A lower bound – defined by a proof

10. The Upper Bound Defined by an algorithm Defines that we know we can do at least this good Perhaps we can do better Lowered by a better algorithm –“For problem X, the best algorithm was O(N 3 ), but my new algorithm is O(N 2 ).”

11. The Lower Bound Defined by a proof Defines that we know we can do no better than this It may be worse Raised by a better proof –“For problem X, the strongest proof showed that it required O(N), but my new, stronger proof shows that it requires at least O(N 2 ).”

12. Upper and Lower Bounds The Upper bound is the best algorithmic solution that has been found for a problem. –“What’s the best that we know we can do?” The Lower bound is the best solution that is theoretically possible. –“What cost can we prove is necessary?”

13. Changing the Bounds Upper bound Lowered by better algorithm Lower bound Raised by better proof

14. Open Problems The upper and lower bounds differ. Upper bound Lowered by better algorithm Lower bound Raised by better proof Unknown

15. Closed Problems The upper and lower bounds are identical. Upper bound Lower bound

16. Closed Problems Better algorithms are still possible Better algorithms will not provide an improvement detectable by “Big O” Better algorithms can improve the constant costs hidden in “Big O” characterizations

17. Tractable vs. Intractable Problems are tractable if the upper and lower bounds have only polynomial factors. –O (log N) –O (N) –O (N K ) where K is a constant Problems are intractable if the upper and lower bounds have an exponential factor. –O (N!) –O (N N ) –O (2 N )

18. Problems that “Cross the Line” The upper bound implies intractable The lower bound implies a tractable Could go either way… Next time!!!

19. Terminology Polynomial algorithms are reasonable Polynomial problems are tractable Exponential algorithms are unreasonable Exponential problems are intractable

20. Terminology Tractable Intractable Reasonable Unreasonable ProblemsAlgorithms Polynomial Exponential

21. Questions?

22. Review The Algorithmic Model –Algorithm defined –Properties of good algorithms –How to describe algorithms –Relating problems to algorithms to programs (hierarchy needed)

23. Review View of programming languages/algorithms –Data vs. instructions –Built-in vs. user-defined –Complex vs. atomic Data –Type vs. variable (Declaration and Initialization of variables) –The 4 atomic built-in types (Num, Characters, Booleans, Pointers) –The complex built in type String

24. Review Operations –Assignment –Arithmetic +, -, x, /, div, mod Precedence rules Using parenthesis to modify precedence –Input and output Print Read

25. Review Conditionals –Purpose & defined –Relational operators (, =, <>, >=, <=) –Boolean operators (AND, OR, NOT) –Boolean expressions (Simple & Complex) –Control flow of the if-then-else statement –Elseif as shorthand Writing an algorithm (how to begin)

26. Review Program maintenance Software Engineering facts about program cost Documentation Benefits of Constants

27. Review Procedural Abstraction Why modularity (Benefits) Need for interface Scope of variables Contract (Pre, post, and purpose statements for every module) Information flow – in, out, in/out Parameters intro (In, out, in/out) Types of modules

28. Review Procedure Rules when to use a procedure Declaration Function Rules when to use a function Declaration Returning values via the “returns”

29. Review Module invocation Parameters Formal vs. actual Parameter matching How parameters work (In, Out, & In/out) Tracing Activation stack Frames Rules of parameter matching (In, Out, & In/out) Examples

30. Review Recursion (Intro) Purpose – repetition Characteristics – calls itself, terminating condition, move closer 2 forms – final action or not Examples Tracing recursive modules Recursion (Advanced) Mutual recursion Design by Contract

31. Review Data Abstraction –Records Declaring records Creating variables of record type Accessing fields of a record variable (the ‘.’ operator) Avoid anonymous types Combining records (records inside records)

32. Review Static vs. dynamic memory/data Pointers Simple example of ptr toa num Following the pointer via the ‘^’ operator Dynamic data structures Linked Lists Defined/properties Proper Record declaration Accessing information in a linked list via pointers

33. Review Linked Lists (continued) –Adding nodes –Simple – no recursion (To front) –Insertion recursive method –To end –In middle (when sorted) –Deleting nodes Simple - no recursion (From front) Deletion recursive method

34. Review Doubly-linked lists –Defined –Example methods (simple insertion) Stack –Defined/properties –Push –Pop –Tracing changes (example of algorithm using push & pop)

35. Review Queue Defined/properties Enqueue Dequeue Tracing changes (example of algorithm using enqueue and dequeue) Trees Defined/properties Binary trees (Record declaration) Binary search trees

36. Review Trees (Continued) Recursive insertion in BST Deleting from BST (conceptually) Graphs Defined/properties Record definition

37. Review Static data structures Arrays Defined Need of constants Accessing elements via the index Multi-dimensional arrays (Declaring and accessing)

38. Review Iteration Defined, use for repetition How looping works Loop Endloop Exitif Combined types Examples – array of lists, list of array, tree of lists, etc.

39. Review Methods of Accessing Data Structures Traversal Lists Recursive on lists Iterative on lists Trees In-order Pre-order Post-order Miscellaneous Traversals (Rt before Left)

40. Review Breadth-first vs. depth- first Arrays Iterative on arrays Recursive on arrays Search Linear Recursive on lists Traversal-search on binary tree (not BST) Linear, iterative search on array

41. Review Binary, recursive search on BST Binary, iterative search on sorted array Sort Bubble sort Merge sort

42. Review Conversion Defined Helper module advantages List to Tree Array to List Tree to List

43. Review Optimization Defined & example problems Greedy algorithms Dynamic programming Minimum spanning trees Defined Prim’s algorithm Kruskal’s algorithm

44. Review Object-Oriented Defined Behavioral abstraction (Defined & Advantages) Encapsulation Abstract data types Queue & stack examples Need for formal programmatic construct

45. Review Classes Defined Advantages Structure - public & protected Role for each Benefits Examples Scope of attributes in protected

46. Review Initialize (not built-in, but useful) Using classes Declaring objects Objects vs. classes Accessing methods via the ‘.’ Operator Example Classes Airplane example Queue example Pile example

47. Review Generic classes Defined & benefits Search and replace type then use a “magic” keyword Syntax Defining objects in algorithm Use cases Clone vs. copy Inheritance Defined & benefits Extension Redefinition

48. Review Resolving method reference ambiguity Class hierarchies Deferred class & methods Polymorphism Defined & benefits Simple assignment example Complex collection example Pure OO

49. Review Algorithm Cost and Complexity Measuring performance (space & time) Measuring work (counting instructions) Best, worst, average Measuring work growth O() notation Complex O() rules (drop constants, keep dominant term, etc.)

50. Review Analysis –Linear vs. binary search –Traversals –Data structures Compare on traversals, search, & insert –Bubblesort vs. mergesort Exponential growth –Hanoi, Monkey tiling, wise peasant Reasonable vs. unreasonable Using O() analysis in data structure design of solution

51. Review Systems –Concurrency vs. sequential processing (what we’ve done so far) Defined & advantages Multiprogramming Multiprocessing Multitasking Distributed systems

52. Review Issues –Protection, mutual exclusion –Starvation, fairness –Deadlock –Time –Synchronization –Overhead costs (context switch)

53. Review Parallelism –Defined & advantages –Pipeline processing –Product complexity Dependencies Precedence –Dependence Graphs –Precedence Graphs

54. Review Problem Complexity –Defined –Problems vs. algorithms –Upper bound –Lower bound –Open vs. closed –Tractable vs. intractable

55. Review NP-Complete –Defined –Examples –Certificates –Oracles –Deterministic vs. nondeterministic Decidable vs. undecidable Highly vs. partially

56. Questions? Review slides available at http://prism.gatech.edu/~wl48 Double U Ell