(i) BigO
(ii) BigW
(Iii) Big \(\mu \)

An algorithm that yields expected output for a valid input is called an algorithmic solution

(i) The way of defining an algorithm is called J algorithmic strategy.
(ii)  For example to calculate factorial for the given value n then it can be done by defining the function to calculate factorial once for the iteration-1 then it can be called recursively until the number of required iteration is reached.

Examples for data structures are arrays, structures, list, tuples, dictionary.

A searching algorithm is the step-by step procedure used to locate specific data among a collection of data. There are two types of searching are.
(i) Linear Search
(ii) Binary Search

The space required by an algorithm is equal to the sum of the following two components: 
(i) A fixed part is defined as the total space required to store certain data and variables for an algorithm. For example, simple variables and constants used in an algorithm.
(ii) A variable part is defined as the total space required by variables, which sizes depends on the problem and its iteration. For example: recursion used to calculate factorial of a given value n.

(i) A space-time or time-memory trade off is a way of solving in less time by using more storage space or by solving a given algorithm in very little space by spending more time.
(ii) To solve a given programming problem, many different algorithms may be used. Some of these algorithms may be extremely time-efficient and others extremely spaceefficient.
(iii) Time/space trade off refers to a situation where you can reduce the use of memory at the cost of slower program execution, or reduce the running time at the cost of increased memory usage.

The algorithm can be written as:
Step 1: start the process
Step 2: get the input x
Step 3: calculate the square by multiplying the input value ie., square \(\leftarrow \)x* x
Step 4: display the result square
Step 5: stop

(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 sub-problems. So that their results can be re-used 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. The solutions of overlapped sub-problems are combined in order to get the better solution.

Asymptotic Notations are languages that uses meaningful statements about time and space complexity. The following three asymptotic notations are mostly used to represent time complexity of algorithms:
(i) Big O: Big O is often used to describe the worst -case of an algorithm.
(ii) Big \(\Omega \):Big Omega is the reverse Big O, if Big O is used to describe the upper bound (worst - case) of a asymptotic function, Big Omega is used to describe the lower bound (best -case).
(iii) Big \(\Theta \):When an algorithm has complexity with lower bound = upper bound, Say that an algorithm has a complexity O(n log n) and \(\Omega \) (n log n), it's actually has the complexity \(\Theta \) (n log n), which means the running time of that algorithm always falls in n log n in the best-case and worst-case.

(i) Insertion sort is a simple sorting algorithm. It works by taking elements from the list one by one and inserting then in their correct position in to a new sorted list.
(ii) This algorithm builds the final sorted array at the end. This algorithm uses n-1 number of passes to get the final sorted list as per the pervious algorithm as we have discussed.
Pseudo for Insertion sort:
Step 1 - If it is the first element, it is already sorted.
Step 2 - Pick next element
Step 3 - Compare with all elements in the sorted sub-list
Step 4 - Shift all the elements in the sorted sublist that is greater than the value to be sorted Step 5 - Insert the value Step 6 - Repeat until list is sorted.

(i) Let us assume a list of n number of values stored in an array. Suppose if we want to search a particular element in this list, the algorithm that search the key element in the list among n elements, by comparing the key element with each element in the list sequentially.
(ii) The best case would be if the first element in the list matches with the key element to be searched in a list of elements. The efficiency in that case would be expressed as 0(1) because only one comparison is enough.
(iii) Similarly, the worst case in this scenario would be if the complete list is searched and the element is found only at the end of the list or is not found in the list. The efficiency of an algorithm in that case would be expressed as O(n) because n comparisons required to complete the search.
(iv) The average case efficiency of an algorithm can be obtained by finding the average number of comparisons as given below: Minimum number of comparisons = 1 Maximum number of comparisons = n If the element not found then maximum number of comparison = n Therefore, average number of comparisons = (n + 1)/2
(v) Hence the average case efficiency will be expressed as 0 (n).

(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.