# C Programming/Beginning exercises
## Variables
### Naming

* Can a variable name start with a number?
* Can a variable name start with a typographical symbol(e.g. #, *, _)?
* Give an example of a C variable name that would not work. Why doesn't it work?

### Data Types

* List at least three data types in C
** On your computer, how much memory does each require?
** Which ones can be used in place of another? Why?
*** Are there any limitations on these uses?
*** If so, what are they?
*** Is it necessary to do anything special to use the alternative?
* Can the name we use for a data type (e.g. 'int', 'float') be used as a variable?

### Assignment

* How would you assign the value 3.14 to a variable called pi?
* Is it possible to assign an int to a double?
** Is the reverse possible?

### Referencing

* A common mistake for new students is reversing the assignment statement. Suppose you want to assign the value stored in the variable "pi" to another variable, say "pi2":
** What is the correct statement?
** What is the reverse? Is this a valid C statement (even if it gives incorrect results)?
** What if you wanted to assign a constant value (like 3.1415) to "pi2":
    a. What would the correct statement look like?
    b. Would the reverse be a valid or invalid C statement?

## Simple I/O
### String manipulation

1. Write a program that prompts the user for a string (pick a maximum length), and prints its reverse.
2. Write a program that prompts the user for a sentence (again, pick a maximum length), and prints each word on its own line.

### Loops

1. Write a function that outputs a right isosceles triangle of height and width n, so n = 3 would look like
2. Write a function that outputs a sideways triangle of height 2n-1 and width n, so the output for n = 4 would be:
3. Write a function that outputs a right-side-up triangle of height n and width 2n-1; the output for n = 6 would be:

## Program Flow

1. Build a program where control passes from main to four different functions with 4 calls.
2. Now make a while loop in main with the function calls inside it. Ask for input at the beginning of the loop. End the while loop if the user hits Q
3. Next add conditionals to call the functions when the user enters numbers, so 1 goes to function1, 2 goes to function 2, etc.
4. Have function 1 call function a, which calls function b, which calls function c
5. Draw out a diagram of program flow, with arrows to indicate where control goes

## Functions

1. Write a function to check if an integer is negative; the declaration should look like bool is_positive(int i);
2. Write a function to raise a floating point number to an integer power, so for example to when you use it

## Math

1. Write a function to calculate if a number is prime. Return 1 if it is prime and 0 if it is not a prime.
2. Write a function to determine the number of prime numbers below n.
3. Write a function to find the square root by using Newton's method.
4. Write functions to evaluate the trigonometric functions.
5. Try to write a random number generator.
6. Write a function to determine the prime number(s) between 2 and 100.

## Recursion
#### Merge sort

1. Write a C program to generate a random integer array with a given length n , and sort it recursively using the Merge sort algorithm.
* The merge sort algorithm is a recursive algorithm .
- sorting a one element array is easy.
- sorting two one-element arrays, requires the merge operation. The merge operation looks at two sorted arrays as lists, and compares the head of the list , and which ever head is smaller, this element is put on the sorted list and the head of that list is ticked off, so the next element becomes the head of that list. This is done until one of the lists is exhausted, and the other list is then copied onto the end of the sorted list.
- the recursion occurs, because merging two one-element arrays produces one two-element sorted array, which can be merged with another two-element sorted array produced the same way. This produces a sorted 4 element array, and the same applies for another 4 element sorted array.
- so the basic merge sort, is to check the size of list to be sorted, and if it is greater than one, divide the array into two, and call merge sort again on the two halves. After wards, merge the two halves in a temporary space of equal size, and then copy back the final sorted array onto the original array.

#### Binary heaps

2. Binary heaps :
* A binary max-heap or min-heap, is an ordered structure where some nodes are guaranteed greater than other nodes, e.g. the parent vs two children. A binary heap can be stored in an array , where ,
- given a position i (the parent) , i*2 is the left child, and i*2+1 is the right child.
- ( C arrays begin at position 0, but 0 * 2 = 0, and 0 *2 + 1= 1, which is incorrect , so start the heap at position 1, or add 1 for parent-to-child calculations, and subtract 1 for child-to-parent calculations ).
* try to model this using with a pencil and paper, using 10 random unsorted numbers, and inserting each of them into a "heapsort" array of 10 elements.
* To insert into a heap, end-add and swap-parent if higher, until parent higher.
* To delete the top of a heap, move end-to-top, and defer-higher-child or sift-down , until no child is higher.
* try it on a pen and paper the numbers 10, 4, 6 ,3 ,5 , 11.
* the answer was 11, 5, 10, 3, 4 , 6.
* EXERCISE: Now try removing each top element of 11, 5, 10, 3, 4, 6 , using end-to-top and sift-down ( or swap-higher-child) to get the numbers
in descending order.
* a priority queue allows elements to be inserted with a priority , and extracted according to priority. ( This can happen usefully, if the element has a paired structure, one part is the key, and the other part the data. Otherwise, it is just a mechanism for sorting ).
* EXERCISE: Using the above technique of insert-back/challenge-parent, and delete-front/last-to-front/defer-higher-child, implement either heap sort or a priority queue.

#### Dijkstra's algorithm

Dijkstra's algorithm is a searching algorithm using a priority queue. It begins with inserting the start node with a priority value of 0. All other nodes are inserted with priority values of large N. Each node has an adjacency list of other nodes, a current distance to start node, and previous pointer to previous node used to calculate current node. Alternative to an adjacency list, is an adjacency matrix, which needs n x n boolean adjacencies.
The algorithm basically iterates over the priority queue, removing the front node, examining the adjacent nodes, and updating with a distance equal to the sum of the front nodes distance for each adjacent node , and the distance given by the adjacency information for an adjacent node.
After each node's update, the extra operation "update priority" is used on that node :
while the node's distance is less than it's parents node ( for this priority queue, parents have lesser distances than the children), the node is swapped with the parent.
After this, while the node is greater distance than one or more of its children, it is swapped with the least distant child, so the least distant child becomes parent of its greater distant sibling, and parent to the greater distant current node.
With updating the priority, the adjacent node to the current node has a back pointer changed to the current node.
The algorithm ends when the target node becomes the current node removed, and the path to the start node can be recorded in an array by following back pointers, and then doing something like a quick sort partition to reverse the order of the array , to give the shortest path to target node from the start node.

#### Quick sort

3. Write a C program to recursively sort using the Quick sort partition exchange algorithm.
* you can use the "driver", or the random number test data from Q1. on mergesort. This is "re-use", which is encouraged in general.
- an advantage of reuse is less writing time, debugging time, testing time.
* the concept of partition exchange is that a partition element is (randomly) selected, and every thing that needs sorted is put into 3 equivalence
classes : those elements less than the partition value, the partition element, and everything above (and equal to ) the partition element.
* this can be done without allocating more space than one temporary element space for swapping two elements. e.g a temporary integer for integer data.
* However, where the partition element should be using the original array space is not known.
* This is usually implemented with putting the partition on the end of the array to be sorted, and then putting two pointers , one at the start of the array,
and one at the element next to the partition element , and repeatedly scanning the left pointer right, and the right pointer left.
* the left scan stops when an element equal to or greater than the partition is found, and the right scan stops for a smaller element than the partition value,
and these are swapped, which uses the temporary extra space.
* the left scan will always stop if it reaches the partition element , which is the last element; this means the entire array is less than partition value.
* the right scan could reach the first element, if the entire array is greater than the partition , and this needs to be tested for, else the scan doesn't stop.
* the outer loop exits when then left and right pointers cross. Testing for pointer crossing and outer loop exit
should occur before swapping, otherwise the right pointer may be swapping a less-than-partition element previously scanned by the left pointer.
* finally, the partition element needs to be put between the left and right partitions, once the pointers cross.
* At pointer crossing, the left pointer may be stopped at the partition element's last position in the array, and the right pointer not progressed past the
element just before the last element. This happens when all the elements are less than the partition.
- if the right pointer is chosen to swap with the partition, then an incorrect state results where the last element of the left array becomes less than the partition element value.
- if the left pointer is chosen to swap with the partition, then the left array will be less than the partition, and partition will have swapped with an element with value greater than the partition or the partition itself.
* The corner case of quicksorting a 2 element out-of-order array has to be examined.
- The left pointer stops on the first out of order element. The right pointer begins on the first out-of-order element, but the outer loop exits because this is the leftmost element. The partition element is then swapped with the left pointer's first element, and the two elements are now in order.
- In the case of a 2 element in order array, the leftmost pointer skips the first element which is less than the partition, and stops on the partition. The right pointer begins on the first element and exits because it is the first position. The pointers have crossed so the outer loop exits. The partition swaps with itself, so the in-ordering is preserved.
* After doing a swap, the left pointer should be incremented and right pointer decremented, so the same positions aren't scanned again, because an endless loop can result ( possibly when the left pointer exits when the element is equal to or greater than the partition, and the right element is equal to the partition value). One implementation, Sedgewick, starts the pointers with the left pointer minus one and right pointer
the plus one the intended initial scan positions, and use the pre-increment and pre-decrement operators e.g. ( ++i, --i) .