(1) How to design algorithms to implement the stack operations.
(2) Write a program to multiply any two matrices. (Using Basic)
and finally,
(3) Consider the algebraic expression E=(2x+y)(5a-b)^3. Draw the tree T which correspond to the expression E and find the pre-order traversal of T.
I'm not exactly sure what (1) is asking. General considerations? An exemplar? Anyway, a stack has only two operations:
push something in;
pop something out.
You will need space in which to store the items on the stack. Start with the items. What data types will you be storing? Stacks are most commonly used for integer addresses, but you need to consider what your implementation will be. The space can be preallocated in one big lump, if you specify a limitation that your stack cannot grow beyond a certain size. That makes your implementation easier than the alternative, which is to allocate new space every time a new item is pushed, and possibly deallocate it when an item is popped, or let it grow, and then deallocate it when some threshold is reached.
Now you can consider the methods themselves. Let's assume you have the easiest case, which involves a fixed-size (or elastic, with the O/S doing the allocation work for you) linear array. You must have set up:
1) an array
2) a pointer to the latest entry in that array
Push implies
a) increment pointer into the array
b) copy item to the array item of current pointer
and Pop implies
a) copy array item at current pointer position into return value
b) decrement pointer
Things to consider: what happens if the pointer is about to exceed its specified limit? what happens if a pop request would decrement the pointer below its zero value?
As for (2), I suggest you start, and ask again about any problems you have. It seems to me that the hardest part of this, like many programming issues, is the input and output - in this case, how to handle the ins and outs and organize the data for matrices of wildly different shapes, and validate that they can be multiplied at all - rather than the matrix multiplication itself, which is relatively easy.
Where to start? Work a simple case, to allow you to think about the method. How do you multiply two 2x2 matrices? Work that theough step by step. What data structure do you need? What algorithm? When you've done that, consider how to adapt it for a different shape of matrix. BASIC - well, most BASICs, I suppose, since I haven't used them all - are a bit limiting in terms of what you can DIM at runtime; in BASIC, I'd be inclined to DIM a whopping big single-dimension array for each of my matrices and manage the dimensions with a couple of pointers.
(3) is really very simple, when compared with the other two.
(1) To design algorithms to implement the stack operations, you can follow these steps:
Step 1: Decide on the data structure to be used to implement the stack. One common choice is to use an array or a linked list.
Step 2: Define the necessary operations of a stack, which typically include:
- Push: Adding an element to the top of the stack.
- Pop: Removing and returning the top element from the stack.
- Peek: Viewing the top element without removing it.
- IsEmpty: Checking if the stack is empty.
- Size: Returning the number of elements in the stack.
Step 3: Define the algorithm for each operation:
- To implement Push, you need to add the element to the top of the stack. This can be achieved by inserting the element at the end of the array or by updating the links in the linked list.
- To implement Pop, you need to remove and return the top element. This can be done by removing the last element from the array or by updating the links in the linked list.
- To implement Peek, you simply return the value of the top element without modifying the stack.
- To implement IsEmpty, you check if the stack is empty by checking if the array or linked list has any elements.
- To implement Size, you return the number of elements in the stack, which can be obtained from the length of the array or by traversing the linked list and counting the elements.
Step 4: Test your implementation by performing a variety of stack operations and verifying that the desired behavior is achieved.
(2) Here's a program in BASIC to multiply any two matrices:
```
10 INPUT "Enter the number of rows in matrix A: ", R1
20 INPUT "Enter the number of columns in matrix A: ", C1
30 INPUT "Enter the number of rows in matrix B: ", R2
40 INPUT "Enter the number of columns in matrix B: ", C2
50 IF C1 <> R2 THEN PRINT "Matrix multiplication is not possible": END
60 DIM A(R1, C1), B(R2, C2), Result(R1, C2)
70 PRINT "Enter elements of matrix A:"
80 FOR I = 1 TO R1
90 FOR J = 1 TO C1
100 INPUT "Enter element at position (" + STR$(I) + "," + STR$(J) + "): ", A(I, J)
110 NEXT J
120 NEXT I
130 PRINT "Enter elements of matrix B:"
140 FOR I = 1 TO R2
150 FOR J = 1 TO C2
160 INPUT "Enter element at position (" + STR$(I) + "," + STR$(J) + "): ", B(I, J)
170 NEXT J
180 NEXT I
190 PRINT "Product of the two matrices:"
200 FOR I = 1 TO R1
210 FOR J = 1 TO C2
220 Result(I, J) = 0
230 FOR K = 1 TO C1
240 Result(I, J) = Result(I, J) + A(I, K) * B(K, J)
250 NEXT K
260 PRINT Result(I, J);
270 NEXT J
280 PRINT
290 NEXT I
```