1. Week 3 – Sorting Machine, Stacks, Recursion, and Partial Maps Jimmy Voss Disclaimer: Not all material is original. Some is taken from the official course slides, and some is taken from Anna’s slides.

2. Sorting Machine Idea Container class for sorting arbitrary Items. Should work with any sorting algorithm. Works in 2 stages: – Insertion phase – place items into a sorting machine. While in insertion phase, inserting is true. – Extraction phase – remove sorted items from Sorting Machine. Note: We make Sorting Machine a Separate component / class.

3. Why A Separate Component? Sorting_Machine will be templated – Allows for basic logic of a particular sort to be implemented regardless of the type being operated on. (Template functions require messier notation for client). – Many different sorting functions are commonly used. – Allows use of Utility functions for Comparison based sorting algorithms. (General_Are_In_Order)

4. Sorting_Machine Insert(x) – requires: inserting Change_To_Extraction_Phase() – requires: inserting Remove_First(x) – requires: not inserting, |self| > 0 Remove_Any(x) – requires: not inserting, |self| > 0 Size() Is_In_Extraction_Phase()

5. Sorting Machine Usage Typical program flow: 1.Insert all objects you want sorted one at a time. 2.Call Change_To_Extraction_Phase() A boolean flag maintains whether this has been called. After this has been called, insertion phase is done. 3.Repeatedly call Remove_First() to get elements in sorted order.

6. Sorting Machine Templates abstract_template < concrete_instance class Item, concrete_instance utility_class Item_Are_In_Order /*! implements abstract_instance General_Are_In_Order !*/ > class Sorting_Machine_Kernel { /* Declarations and Specs */ };

7. Example: Integer Sorting Machine #include “/CT/Sorting_Machine/Kernel_1_C.h” #include “/CI/Integer/Integer_Are_In_Order.h” class Integer_Sorting_Machine : instantiates Sorting_Machine_Kernel_1a_C< Integer, Integer_Are_In_Order_1 > { }; Note: Integer_Are_In_Order_1 is not actually in the Resolve Library

8. Example: Integer Sorting Machine #include “/CT/Sorting_Machine/Kernel_1_C.h” #include “/CI/Integer/Integer_Are_In_Order.h” class Integer_Sorting_Machine : instantiates Sorting_Machine_Kernel_1a_C< Integer, Integer_Are_In_Order_1 > { }; Note: Integer_Are_In_Order_1 is not actually in the Resolve Library Uses a utility class

9. Exercise Suppose you have access to a sorting_machine object instantiated to work with text objects named “sorter”. Write code for the following: global_procedure Lex_Sort(alters Queue_Of_Text& sort_me); /*! ensures sort_me = permutation of #sort_me and for every a, b, c: string of string of character, i, j: string of character where (sort_me = a * * b * * c) (|i| <= |j| and if |i| = |j| then i < j) !*/

10. Exercise global_procedure Lex_Sort(alters Queue_Of_Text& sort_me) { while ( sort_me.Length() > 0 ) { object catalyst Text x; sort_me.Dequeue( x ); sorter.Insert( x ); } sorter.Change_To_Extraction_Phase(); while ( sorter.Size() > 0 ) { object catalyst Text x; sorter.Remove_First( x ); sort_me.Enqueue( x ); }

11. Concept of Recursion

12. Recursion in Programming

13. A recursive function typically refers to a function which calls itself. Base cases – there is some “small” case which the recursive function handles without making a recursive call. Recursive step – The step that reduces the problem to a “smaller” problem, and uses the results of the smaller problem to solve the larger problem.

14. Exercise (Fibonacci)

15. Exercise (Solution) global_function Integer Fibonacci( preserves Integer n ); { if ( n <= 2 ) { // base cases return 1; } else { // recursive case return Fibonacci( n-1 ) + Fibonacci( n-2 ); } Question: Is this very efficient?

16. Stacks Intuitively like a stack of objects. LIFO – Last in first out. Purest form has 2 operations: – Push – add an object to the stack – Pop – Take an object off the top of the stack. Implemented as a template container class.

17. Stacks (Operations) Push ( x ) – consumes x Pop( x ) – produces x Accessor, i.e. [current] – Look at the top element of the stack. Length()

18. Stack (Mathematical Model) Modeled as a string of Item. Push adds an Item to the front of the string. Pop removes an Item from the front of the string. Accessor can be used to access the Item at the front of the string.

19. Stacks are Important Program Layout in Virtual Memory Program stack allocates space for function / procedure calls. Very Easy to implement at a low level (Assembly level programming). Reserved Stack Heap Program Code Memory Addr 0 Large Number

20. Exercise: Reversing a Stack Give a recursive implementation for the following procedure: global_procedure Reverse_Queue( alters Queue_Of_Integer & myQueue ) { // fill in code here } What is the base case? How does one reduce the size of the problem? Give a second implementation without recursion. Hint: use a stack.

21. Exercise (Recursive Solution) global_procedure Reverse_Queue( alters Queue_Of_Integer & myQueue ) { // Base Case if ( myQueue.Length() <= 1 ) { // Do nothing } // Recursive Case else { object catalyst Integer x; myQueue.Dequeue( x ); Reverse_Queue( myQueue ); myQueue.Enqueue( x ); }

22. Exercise (Recursive Solution Trace) myQueue = Reverse_Queue( myQueue) myQueue = Reverse_Queue( myQueue) myQueue = Reverse_Queue( myQueue) myQueue = X = 1 X = 2 X = 3 myQueue =

23. Exercise global_procedure Reverse_Queue( alters Queue_Of_Integer & myQueue ) { object catalyst Stack_Of_Integer S; while ( myQueue.Length() > 0 ) { object catalyst Integer temp; myQueue.Dequeue( temp ); S.Push( temp ); } while ( S.Length() > 0 ) { object catalyst Integer temp; S.Pop( temp ); myQueue.Enqueue( temp ); }

24. Partial Map A map is a data structure which maps an d_item to an r_item. d_items are unique. Both the index and the item mapped to are template types. Example: Webster’s Dictionary maps each word to a set of definitions.

25. Examples of maps Mathematical model: a map is a set of points (d_item, r_item) such that d_item is the index and r_item the item mapped to. Which of these are maps? {(1, 2), (2, 2), (3, 2)} {(1, 2), (1, 3), (3, 3)} {(“hi”, “a greeting”), (“Tom”, “a name”), (“Tom”, “my best friend”)} {(“hi”, {“a greeting”}), (“Tom”, {“a name”, “my best friend”})}

26. Examples of maps Answers: {(1, 2), (2, 2), (3, 2)} – Yes {(1, 2), (1, 3), (3, 3)} – No, 1 is a d_item twice {(“hi”, “a greeting”), (“Tom”, “a name”), (“Tom”, “my best friend”)} – No, “Tom” is a d_item twice. {(“hi”, {“a greeting”}), (“Tom”, {“a name”, “my best friend”})} – Yes

27. requires |self| > 0

28. d_item type r_item type