Discuss the Characteristics and features of Problems which are solved using following Design Techniques i. Divide and Conquer ii. Greedy iii. Dynamic Programming Also Compare these Design Techniques.

 Divide and Conquer is a design technique that involves dividing a problem into smaller subproblems, solving those subproblems independently, and then combining the solutions to the subproblems to solve the original problem. This technique is effective for problems that can be broken down into smaller subproblems that are similar to the original problem and for which it is possible to combine the solutions to the subproblems in a simple way.

Some characteristics and features of problems that are typically solved using the divide and conquer technique include:

1. The problem can be divided into smaller subproblems that are similar to the original problem.

2. The subproblems can be solved independently of each other.

3. It is possible to combine the solutions to the subproblems in a simple way to obtain a solution to the original problem.

4. The divide and conquer technique is often used to solve problems that have an inherent recursive structure, such as sorting algorithms or recursive tree traversals.

Some examples of problems that can be solved using the divide and conquer technique include:

• Sorting algorithms, such as merge sort and quick sort
• Finding the maximum subarray sum in an array
• Finding the closest pair of points in a set of points
• Matrix multiplication
• The Tower of Hanoi problem

Greedy is a design technique that involves making the locally optimal choice at each step with the hope of arriving at a global optimum. This technique is effective for problems that have the property of "optimal substructure," meaning that the globally optimal solution can be arrived at by making locally optimal (greedy) choices.

Some characteristics and features of problems that are typically solved using the greedy technique include:

1. The problem has the property of "optimal substructure," meaning that the globally optimal solution can be arrived at by making locally optimal (greedy) choices.

2. It is possible to make a locally optimal choice at each step of the problem.

3. The greedy technique is often used to solve optimization problems, such as finding the shortest path or the minimum cost.

Some examples of problems that can be solved using the greedy technique include:

• The Knapsack problem
• Huffman coding for data compression
• The Minimum Spanning Tree problem
• Dijkstra's algorithm for finding the shortest path in a graph

Dynamic Programming is a design technique that involves solving a complex problem by breaking it down into smaller subproblems and storing the solutions to these subproblems in a table, so they can be reused (dp stands for "dynamic programming"). This technique is effective for problems that can be broken down into smaller subproblems and for which the solutions to the subproblems can be reused.

Some characteristics and features of problems that are typically solved using dynamic programming include:

1. The problem can be divided into smaller subproblems.

2. The solutions to the subproblems can be reused, rather than recomputed.

3. The subproblems are typically solved in a specific order, often following a recursive structure.

4. The dynamic programming technique is often used to solve optimization problems, such as finding the shortest path or the minimum cost.

Some examples of problems that can be solved using dynamic programming include:

• The Fibonacci sequence
• 
  
• The Knapsack problem
• The Longest Common Subsequence problem
• The Shortest Path problem
• The Matrix Chain Multiplication problem

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