1. Linear search
A linear search sequentially moves through your list looking for a matching value.
1. The element is found, i.e. a[i] = x.
2. The entire array has been scanned, and no match was found.
i=0;
while (i<n && a[i]≠x) i++;
O(n)

2. There are three conditions: А[mid]==key A[mid]<key A[mid]>key
Binary Search
The key idea is to inspect a middle element and to compare it with the search argument key.
There are three conditions:
А[mid]==key
A[mid]<key
A[mid]>key

3. Example:
А[10], key=16.
left= 0
right= 9
mid = 4
16>A[mid]
left= 5
mid = 7
16<A[mid]
right= 6
mid = 5
16=A[mid]

4. SORTING
Sorting is the process of rearranging a set of objects in a specific order.
Internal sorting (sorting of arrays)
External sorting (sorting of sequential files)
Simple Sorting methods:
insertion Sort
selection Sort
exchange Sort

5. Insertion sort
The items are divided into a sorted sequence a[0] ... a[i-1] and a source sequence a[i] ... a[n-1].
FOR i := 1 TO n-1 DO
x := a[i];
insert x at the appropriate place in a[0] ... a[i]
END

6. Insertion sort
А = 18, 20, 5, 13, 15

7. Insertion sort for (i=1; i<n; i++) { j=i; temp=a[i];
while(j>0 && temp<a[j-1])
a[j]=a[j-1];
j--; }
a[j]=temp;
//i determines sequence a[0]...a[i]
//moving
// insert

8. Insertion sort
O(n2)

9. Binary insertion sort O(n2) void binsort(int a[], int n) {int item;
for (int i=1; i<n; i++) {
item=a[i];
int l=0;
int r=i-1;
while (l<=r)
int m=(l+r)/2;
if (item<a[m]) r=m-1;
else l=m+1; }
for (int j=i-1; j>=l; j--)
a[j+1]=a[j];
a[l]=item; }}
O(n2)

10. Insertion Sort by Diminishing Increment
1, 4, 13, 40, 121, ...
hk-1 = 3hk+1, ht-1 = 1,
t = log3(n) - 1.
1, 3, 7, 15, 31, ...
hk-1 = 2hk+1, ht-1 = 1,
t = log2(n) - 1.