1. Stacks CS 3358 – Data Structures

2. What is a stack? It is an ordered group of homogeneous items of elements. Elements are added to and removed from the top of the stack (the most recently added items are at the top of the stack). The last element to be added is the first to be removed (LIFO: Last In, First Out).

3. Stack Specification Definitions: (provided by the user) – MAX_ITEMS: Max number of items that might be on the stack – ItemType: Data type of the items on the stack Operations – makeEmpty – bool isEmpty – bool isFull – push (ItemType newItem) – pop ()

4. push (ItemType newItem) Function: Adds newItem to the top of the stack. Preconditions: Stack has been initialized and is not full. Postconditions: newItem is at the top of the stack.

5. ItemType pop () Function: Removes topItem from stack and returns it. Preconditions: Stack has been initialized and is not empty. Postconditions: Top element has been removed from stack and item is a copy of the removed element.

7. Stack overflow The condition resulting from trying to push an element onto a full stack. if(!stack.isFull()) stack.push(item); Stack underflow The condition resulting from trying to pop an empty stack. if(!stack.isEmpty()) stack.pop();

8. Example: Convert number bases Use the stack to remember – Let’s convert 35 10 to binary

9. Example: Postfix expressions Postfix notation is another way of writing arithmetic expressions. In postfix notation, the operator is written after the two operands. infix: 2+5 postfix: 2 5 + Expressions are evaluated from left to right. Precedence rules and parentheses are never needed!!

10. Example: postfix expressions (cont.)

11. Postfix expressions: Algorithm using stacks (cont.)

12. Postfix expressions: Algorithm using stacks WHILE more input items exist Get an item IF item is an operand stack.Push(item) ELSE stack.Pop(operand2) stack.Pop(operand1) Compute result stack.Push(result) stack.Pop(result)

13. The N-Queens Problem Suppose you have 8 chess queens......and a chess board

14. The N-Queens Problem Can the queens be placed on the board so that no two queens are attacking each other ?

15. The N-Queens Problem Two queens are not allowed in the same row...

16. The N-Queens Problem Two queens are not allowed in the same row, or in the same column...

17. The N-Queens Problem Two queens are not allowed in the same row, or in the same column, or along the same diagonal.

18. The N-Queens Problem The number of queens, and the size of the board can vary. N Queens N rows N columns

19. The N-Queens Problem We will write a program which tries to find a way to place N queens on an N x N chess board.

20. How the program works The program uses a stack to keep track of where each queen is placed.

21. How the program works Each time the program decides to place a queen on the board, the position of the new queen is stored in a record which is placed in the stack. ROW 1, COL 1

22. How the program works We also have an integer variable to keep track of how many rows have been filled so far. ROW 1, COL 1 1 filled

23. How the program works Each time we try to place a new queen in the next row, we start by placing the queen in the first column... ROW 1, COL 1 1 filled ROW 2, COL 1

24. How the program works...if there is a conflict with another queen, then we shift the new queen to the next column. ROW 1, COL 1 1 filled ROW 2, COL 2

25. How the program works If another conflict occurs, the queen is shifted rightward again. ROW 1, COL 1 1 filled ROW 2, COL 3

26. How the program works When there are no conflicts, we stop and add one to the value of filled. ROW 1, COL 1 2 filled ROW 2, COL 3

27. How the program works Let's look at the third row. The first position we try has a conflict... ROW 1, COL 1 2 filled ROW 2, COL 3 ROW 3, COL 1

28. How the program works...so we shift to column 2. But another conflict arises... ROW 1, COL 1 2 filled ROW 2, COL 3 ROW 3, COL 2

29. How the program works...and we shift to the third column. Yet another conflict arises... ROW 1, COL 1 2 filled ROW 2, COL 3 ROW 3, COL 3

30. How the program works...and we shift to column 4. There's still a conflict in column 4, so we try to shift rightward again... ROW 1, COL 1 2 filled ROW 2, COL 3 ROW 3, COL 4

31. How the program works...but there's nowhere else to go. ROW 1, COL 1 2 filled ROW 2, COL 3 ROW 3, COL 4

32. How the program works When we run out of room in a row: pop the stack, reduce filled by 1 and continue working on the previous row. ROW 1, COL 1 1 filled ROW 2, COL 3

33. How the program works Now we continue working on row 2, shifting the queen to the right. ROW 1, COL 1 1 filled ROW 2, COL 4

34. How the program works This position has no conflicts, so we can increase filled by 1, and move to row 3. ROW 1, COL 1 2 filled ROW 2, COL 4

35. How the program works In row 3, we start again at the first column. ROW 1, COL 1 2 filled ROW 2, COL 4 ROW 3, COL 1

36. Pseudocode for N-Queens  Initialize a stack where we can keep track of our decisions.  Place the first queen, pushing its position onto the stack and setting filled to 0.  repeat these steps – if there are no conflicts with the queens... – else if there is a conflict and there is room to shift the current queen rightward... – else if there is a conflict and there is no room to shift the current queen rightward...

37. Pseudocode for N-Queens  repeat these steps – if there are no conflicts with the queens... Increase filled by 1. If filled is now N, then the algorithm is done. Otherwise, move to the next row and place a queen in the first column.

38. Pseudocode for N-Queens  repeat these steps – if there are no conflicts with the queens... – else if there is a conflict and there is room to shift the current queen rightward... Move the current queen rightward, adjusting the record on top of the stack to indicate the new position.

39. Pseudocode for N-Queens  repeat these steps – if there are no conflicts with the queens... – else if there is a conflict and there is room to shift the current queen rightward... – else if there is a conflict and there is no room to shift the current queen rightward... Backtrack! Keep popping the stack, and reducing filled by 1, until you reach a row where the queen can be shifted rightward. Shift this queen right.

40. Pseudocode for N-Queens  repeat these steps – if there are no conflicts with the queens... – else if there is a conflict and there is room to shift the current queen rightward... – else if there is a conflict and there is no room to shift the current queen rightward... Backtrack! Keep popping the stack, and reducing filled by 1, until you reach a row where the queen can be shifted rightward. Shift this queen right.

41.  Stacks have many applications.  The application which we have shown is called backtracking.  The key to backtracking: Each choice is recorded in a stack.  When you run out of choices for the current decision, you pop the stack, and continue trying different choices for the previous decision. Summary of N-Queens

42. Example using templates // Client code StackType myStack; StackType yourStack; StackType anotherStack; myStack.Push(35); yourStack.Push(584.39);

43. Implementing stacks using templates template class StackType { public: StackType(); void makeEmpty(); bool isEmpty() const; bool isFull() const; void push(const ItemType &); ItemType pop(); private: int top; ItemType items[MAX_ITEMS]; };

44. Implementing stacks using templates Templates allow the compiler to generate multiple versions of a class type or a function by allowing parameterized types. It is similar to passing a parameter to a function (we pass a data type to a class !!)

45. Implementing stacks using dynamic array allocation template class StackType { public: StackType(int); ~StackType(); void makeEmpty(); bool isEmpty() const; bool isFull() const; void push(const ItemType &); ItemType pop(); private: int top; int maxStack; ItemType *items; };

46. Let’s implement a simple stack Using arrays as memory – Stacks grow “up” – Make sure to test copy constructor! Using a linked structure – Always insert at head