(i) Dynamic programming is an algorithmic design method that can be used when the solution to a problem can be viewed as the result of a sequence of decisions.
(ii) Dynamic programming approach is similar to divide and conquer. The given problem is divided into smaller and yet smaller possible sub-problems.
(iii) Dynamic programming is used whenever problems can be divided into similar subproblems. so that their results can be reused to complete the process.
(iv) Dynamic programming approaches are used to find the solution in optimized way. For every inner subproblem, dynamic algorithm will try to check the results of the previously solved sub-problems.
(v) The solutions of overlapped sub-problems are combined in order to get a better solution.
Steps to doDynamic programming :
(i) The given problem will be divided into smaller overlapping sub-problems.
(ii) An optimum solution for the given problem can be achieved by using result of smaller sub-problem.
(iii) Dynamic algorithms uses Memoization
Fibonacci Series - An example :
(i) Fibonacci series generates the subsequent number by adding two previous numbers. Fibonacci series starts from two numbers -Fib 0 & Fib 1. The initial values of Fib 0 & Fib l can be taken as 0 and 1.
(ii) Fibonacci series satisfies the following conditions:
Fibn = Fib_a-1 + Fib_a-2
(iii) Hence, a Fibonacci series for the n value 8 can look like this
Fib₈ = 0 1 1 2 3 5 8 13
Fibonaeci Iterative Algorithm with Dynamic programning approach : The following example shows a simple Dynamic programning approach for the generation ot Fibonacci series.
Initialize f0 = 0, f1 = 1.
Step- 1: Print the initial values of Fibonacci f0 and f1 
Step- 2: Calculate Fibonacci fib \(\leftarrow \) f0+ f1 
Step- 3: Assign f0 \(\leftarrow \) f1, f1 \(\leftarrow \) fib
Step- 4: Print the next consecutive value of Fibonacci fib
Step- 5: Go to step-2 and repeat until the specified number of terms generated
Example:
if we generate Fibonacci series up to 10 digits, the algorithm will generate the series as shown below:
The Fibonacci series is:
0 1 1 2 3 5 8 13 21 34 55.

Bubble sort algorithm:
(i) Bubble sort algorithm simple sorting algorithm. The algorithm starts at the beginning of the list of values stored in an array. It compares each pair of adjacent elements and swaps them if they are in the unsorted order.
(ii) This comparison and passed to be continued until no swaps are needed, which indicates that the list of values stored in an array is sorted. The algorithm is a comparison sort, is named for the way smaller elements "bubble" to the top of the list.
(iii) Although the algorithm is simple, it is too slow and less efficient when compared to insertion sort and other sorting methods.
(iv) Assume list is an array of n elements. The swap function swaps the values of the given array elements.
Procedure :
(i) Start with the first element i.e., index = 0, compare the current element with the next element of the array.
(ii) If the current element is greater than the next element of the array, swap them.
(iii) If the current element is less than the next or right side of the element, move to the next element. Go to Step 1 and repeat until the end of the index is reached.
(iv) Let's consider an array with values {15, 11, 16, 12, 14, 13} Below, we have a pictorial representation of how bubble sort will sort the given array.

(v) The above pictorial example is for iteration-d. Similarly, remaining iteration can be done. The final iteration will give the sorted array. At the end of all the iterations we will get the sorted values in an array as given below:

Binary Search:
Binary search also called half-interval search algorithm. It finds the position of a search element within a sorted array. The binary search algorithm can be done as a divide- and -conquer search algorithm and executes in logarithmic time.
Pseudo Code:
Start with the middle element:
(i) If the search element is equal to the middle element of the array i.e., the middle value = number of elements in array/2, then return the index of the middle element.
(ii) If not, then compare the middle element with the search value,
(iii) If the search element is greater than the number in the middle index, then select the elements to the right side of the middle index, and go to Step-1.
(iv) If the search element is less than the number in the middle index, then select the elements to the left side of the middle index, and start with Step-1.
(v) When a match is found, display success message with the index of the element matched.
(vi) If no match is found for all comparisons, then display unsuccessful message.
Binary Search Working principles :
(i) List of elements in an array must be sorted first for Binary search. The following example describes the step by step operation of binary search.
(ii) Consider the following array of elements, the array. is being sorted so itenables to do the binary searçh algorithm. Let us assume that the search element is 60 and we need to search the location or index of search element 60 using binary search.

(iii) First, we find index of middle element of. the array byusing this formula:
                         mid = low + (high - low) /2
(iv) Here it is, 0 + (9-0)/2=4 (fractional part ignored). So, 4 is thè mid value of the array.

(v) Now compare the search element with the value stored at mid value location 4. The value stored at location or index 4 is 50, which is not match with search element. As the search value 60 is greater than 50.

(vi)  Now we change our low to mid+1 and find the new mid value again using the formula.
                       low = mid + 1
                       mid = low + (high - low) / 2
(vii) Our new mid is 7 now. We compare the value stored at location 7 with our target value 60.

(viii) The value stored at location or index 7 is not a match with search element, rather it is more than what we are looking for. So, the search element must be in the lower part from the current mid value location.

(ix) The search element still not found. Hence, we calculated the mid again by using the formula.
                               high = mid -1
                               mid = low +(high - low)/2
                               Now the mid value is 5.

(x) Now we compare the value stored at location 5 with our search element. We found that it is a match.

(xi) We can conclude that the search element 60 is found at locationor index 5. For example if we take the search element as 95, For this value this binary search algorithm return unsucessful result.
 

(i) Linear search also called sequential search is a sequential method for finding a particular value in a list.
(ii) This method checks the search element with each element in sequence until the desired element is found or the list is exhausted. In this searching algorithm, list need not be ordered.
procedure :
1. Traverse the array using for loop
2. In every iteration, compare the target search key value with the current value of the list.
(i) If the values do not match, move on to the next array element.
(ii) If the values match, display the current index and value of the array.
3. If no match is found, display the search element not found.
(iii) To search the number 25 in the array given below, a linear search will go step by step in a sequential order starting from the first element in the given array if the search element is found that index is returned otherwise the search is continued till the last index of the array. In this example number 25 is found at index number 3.

Example 1:
Input: values[] = {5, 34, 65, 12, 77, 35)
target = 77
Output: 4
Example 2:
Input: values[] = [101, 392, 1, 54, 32, 22, 90, 93) target = 200
Output: -1 (not found)