Design and Analysis of Algorithms
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Design and Analysis of Algorithms
2078 Boards
Section A
Answer any two questions.
1
What are the elementary properties of algorithm? Explain. Why do you need algorithm? Discuss about analysis of the RAM model for analysis of algorithm with suitable example.
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2
Explain about the divide and conquer paradigm for algorithm design with suitable example. Write the Quick sort algorithm using randomized approach and explain its time complexity.
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3
Explain in brief about the Dynamic Programming Approach for algorithm design. How it differs with recursion? Explain the algorithm for solving the 0/1 Knapsack problem using the dynamic programming approach and explain its complexity.
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Section B
Answer any eight questions.
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Explain the recursion tree method for solving the recurrence relation. Solve following recurrence relation using this method.
T(n)=2T(n/2) +1 for n> 1, T(n) =1 for n =1
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5
Write an algorithm to find the maximum element of an array and analyze its time complexity.
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6
Write the algorithm for bubble sort and explain its time complexity.
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7
What do you mean by optimization problem? Explain the greedy strategy for algorithm design to solve optimization problems.
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8
Explain the algorithm and its complexity for solving job sequencing with deadline problem using greedy strategy.
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9
What do you mean by memorization strategy? Compare memorization with dynamic programing.
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Memorization is an optimization technique used primarily in recursive algorithms. It involves storing the results of expensive function calls and reusing them when the same inputs occur again. This avoids redundant computations, significantly improving performance.
Comparison with Dynamic Programming:
| Feature | Memorization | Dynamic Programming |
|---|---|---|
| Approach | Top-down | Bottom-up |
| Implementation | Recursive with storage | Iterative with table |
| Order of Solution | Solves subproblems as needed | Solves all possible subproblems |
| Storage | Stores results of function calls | Stores results in a table |
| Suitability | Problems where not all subproblems need to be solved | Problems where all subproblems need to be solved |
| Overhead | Function call overhead | Table creation and maintenance overhead |
| Example | Calculating Fibonacci numbers recursively with caching | Calculating Fibonacci numbers iteratively using an array |
Both memorization and dynamic programming aim to optimize solutions to overlapping subproblems. Memorization is often easier to implement for existing recursive solutions, while dynamic programming can be more efficient due to the absence of recursive call overhead. Dynamic programming guarantees that all necessary subproblems are solved, whereas memorization only solves those encountered during the recursive process.
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Explain the concept of backtracking. How it differ with recursion?
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11
Explain in brief about the complexity classes P, NP and NP Complete.
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P (Polynomial Time):
- Contains decision problems solvable by a deterministic Turing machine in polynomial time (O(nk), where n is the input size and k is a constant).
- Considered "tractable" or efficiently solvable.
- Example: Sorting a list of numbers.
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NP (Nondeterministic Polynomial Time):
- Contains decision problems where a solution can be verified in polynomial time.
- A solution may not be easily found, but if presented, its correctness can be quickly checked.
- Example: The Traveling Salesperson Problem (TSP) – verifying a proposed tour is fast, but finding the optimal tour is hard.
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NP-Complete:
- A subset of NP.
- Problems are the "hardest" in NP.
- If a polynomial-time algorithm is found for any NP-Complete problem, then P = NP.
- Every problem in NP can be reduced to an NP-Complete problem in polynomial time.
- Example: Boolean Satisfiability (SAT), TSP, Vertex Cover.
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Write short notes on:
a. NP Hard Problems and NP Completeness
b. Problem Reduction
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a. NP-Hard Problems and NP Completeness
- NP (Nondeterministic Polynomial time): Problems solvable in polynomial time by a nondeterministic Turing machine. Equivalently, solutions can be verified in polynomial time.
- NP-Complete: Problems in NP where every other problem in NP can be reduced to them in polynomial time. If a polynomial-time algorithm is found for any NP-Complete problem, then P = NP.
- NP-Hard: Problems that are at least as hard as the hardest problems in NP. They don’t necessarily have to be in NP themselves. An NP-Hard problem can be solved in polynomial time if and only if P = NP.
b. Problem Reduction
- Definition: Transforming one problem (A) into another problem (B) such that a solution to B can be used to solve A.
- Polynomial-Time Reduction: The transformation from A to B must be computable in polynomial time. Denoted as A ≤p B.
- Purpose: Used to prove NP-Completeness. If a problem B is NP-Complete and A ≤p B, then A is also NP-Complete.
- Example: The Traveling Salesperson Problem (TSP) can be reduced to the Hamiltonian Cycle problem.