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Introduction of Programming Algorithms Assignment

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Part A

Problem A.

To identify the spanning trees, a greedy method is utilised in conjunction with prim's algorithm and Kruskal's algorithm. It also seems to be routing the network. It is a mathematical strategy that appears to be extremely simple to produce answers by determining the next step to provide the advantage. The greedy algorithm's drawback is that it can't find the best answer since it doesn't look at all of the data. It can sometimes fail to evaluate the facts and be unaware of future options. The greedy algorithm cannot solve the 0-1 knapsack issue because it is unable to fill the knapsack's capacity; hence, a greedy approach is not optimum here.

To discover the spanning trees, a greedy method is utilised to combine prim's approach and Kruskal's algorithm. It also looks to be routing the network. It is a mathematical strategy that appears to be extremely simple to build answers by determining which step to go next to provide the advantage. The greedy algorithm's drawback is that it can't find the best answer since it doesn't look at all the data. It occasionally fails to evaluate the facts and is unaware of future options. The greedy algorithm cannot solve the 0-1 knapsack issue because it is unable to fill the knapsack's capacity; hence, a greedy method is not optimum in this case.

Different from greedy algorithms, dynamic programming algorithms optimise the answer. The greedy algorithm is only interested in making a profit, not in the past or the future. The dynamic programming approach generates a workable solution that can be controlled. 

Problem B.

Two sets are there one is A that contains animals and the subset of A contains the animals which are not harmful and it doesn't eat. All animals are safe to each other.

A = {b, c, d, ai}

A’ = {e, f, g, h, i, k, j, aj}

If a counts the function a and A’

Then int count = 0;

Count = A’ - 2

Count = count / 2;

Return count;

Problem C.

The sorting algorithm is made up of a set of instructions that define the operation of an array. The Quicksort algorithm is a well-organized algorithm. It's also commonly used as an algorithm. To discover this aim of output to sort the list, all sorting algorithms must run quickly and operate according to the amount of space required. The sorting algorithm has numerous complications that may be used to estimate its time and space complexity.

The time complexity refers to the algorithm's time taken to execute this operation in absolute terms owing to the amount of the input. It can be written in a number of different ways, including Big O notation, Theta notation, and Omega notation.

The space complexity refers to the algorithm's use of all available memory to finish its execution. Both the input and auxiliary memory can be included. Auxiliary memory provides extra room for the algorithm to run in between data inputs. It entails employing an algorithm to calculate the space's complexity.

The order of the data input is too delicate for the Quicksort algorithm. When the component is in increasing order, it performs poorly. It divides the array into two pieces, labelled "1 and n-1." It is not stable since users frequently modify the pivots position component on the report. On the "implementation of the Quick Sort is O(n2)" test, the Quicksort example performs the poorest. When the pivot is a very large component, this happens.

Part B

 Problem A

The dynamic programming algorithm is a method for resolving optimization issues by breaking them down into smaller chunks. It makes use of the best solution for the entire problem and is dependent on its sub-problems. The problem can be solved in two ways. Users that utilise the top-down strategy try to solve the problem by looking for a solution to a sub-problem. When people try to solve the difficulties, they save the solutions that they get regularly. The optimization is conveyed, which is not the case with greedy algorithms. Dynamic algorithms are always the source of a difficulty in the optimization process.

  • A recursive algorithm calls itself simpler and smaller input variables. It obtains the current input for the result.

As an input V is given and it is the positive even number

Output: V natural number.

The algorithm that is stated as

If V = 1, then it will return 0

Else it will return Even (V-1) +2

Another recursive formulation can be done

“int i, V;

i := 1;

V:= 0;

while ( i < k ) {

V := V+ 2;

i := i + 1;

}

return V.”

  • The pseudocode of the program

The i and v value has been taken as an integer

I is equal to 1

V is equal to 0

While i is less than K then

V is equal to V summation with 2

I is equal to i summation with 1

Else

It returns the value of V.

  • The bottom-up approach is totally different from top-down and keeps away from the recursion. Users can solve problems, at first solving all sub-problems.

Time complexity

The time complexity refers to the time it takes an algorithm to execute a task in absolute terms owing to the amount of the input. It can be written in a number of different ways, including Big O notation, Theta notation, and Omega notation.

Space complexity

The space complexity refers to the algorithm's use of all available memory to finish its execution. Both the input and auxiliary memory can be included. Auxiliary memory provides extra room for the algorithm to run in between data inputs. It entails employing an algorithm to calculate the space's complexity.

  • The most important task in the transportation system for linked and autonomous cars is multi-vehicle optimization. Because of the scheduling trajectories, this type of space-time complexity employs dynamic programming and integer programming. adjusting the vehicles to manage the reaction time to the current speed This technique optimises a domain that has been thoroughly explored and includes aeroplane and vehicle surfaces.

Problem B

The recursive formulation of a Depth-first search algorithm has been used here that is connected with the components.

Algorithm DFS (G, m)

if m is already visited

return

Mark m has visited.

for all neighbors x of m

DFS(G, x)

The temporal complexity of this approach is determined on the graph's structure and size. If users begin at the top of the graph, the algorithm will traverse all four edges. Calculate the time it takes for users to complete the complimentary procedure. The mark operation and statement are both executed at the same time, followed by a single loop to the DFS for each duplicate. All m vertices are not visited and tagged before the algorithm executes. This is based on the (|M|) epoch. As a result, the time complexity of this method is (|M| + |E'|).

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