Questions and Answers
What is the principal idea behind backtracking?
The principal idea is to construct solutions one component at a time and evaluate partially constructed candidates.
What is a promising node in a state-space tree?
A promising node is a partially constructed solution that may still lead to a complete solution.
What does NP stand for in computational complexity?
Non-deterministic Polynomial time.
What is the sum of subsets problem?
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What is the main characteristic of NP-complete problems?
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All NP-Complete problems are NP-Hard.
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In the N-Queens problem, no two queens can attack each other by being in the same ______.
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What does the term 'Hamiltonian cycle' refer to?
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Study Notes
Backtracking
- Backtracking involves an exhaustive search technique, generating all potential solutions to identify those meeting specific criteria.
- It constructs solutions incrementally, evaluating each partial solution to see if it can still lead to a valid complete solution.
- Backtracking is implemented using a state-space tree, representing choices made during the solution process.
- Nodes in the state-space tree are characterized as promising (potentially valid solutions) or non-promising (unable to lead to a complete solution).
- Backtracking commonly employs a depth-first search approach to explore potential solutions thoroughly before reverting to previous steps when dead ends are encountered.
- A complete solution can be found and returned, or the algorithm can continue searching for additional solutions if required.
N-Queens Problem
- The N-Queens problem tasks one with placing N queens on an N x N chessboard such that no two queens threaten each other.
- The backtracking algorithm positions queens row by row, trying various column placements while adhering to rules of no shared rows, columns, or diagonals.
- For example, solving the 4-Queens problem demonstrates backtracking by exploring different placements and reverting upon dead ends until a valid configuration is achieved.
- Solutions for n ≥ 4 can be found in linear time, with symmetry in board arrangements also leveraged to explore alternative solutions.
Sum of Subsets Problem
- The objective is to determine a subset of a set A = {a1,..., an} of n positive integers that totals to a given integer d.
- An instance example includes finding subsets of A = {1, 2, 5, 6, 8} summing to d = 9, which has the solutions {1, 2, 6} and {1, 8}.
- Sorting elements of the set aids in efficient solution finding and subset evaluation.
Decision vs Optimization Algorithms
- Optimization problems aim to find the best solution (e.g., minimal weight in traveling salesman).
- Decision problems focus on answering binary questions (e.g., whether a Hamiltonian cycle exists within a weight limit).
- Many problems exist in both forms, enabling transformation from decision to optimization.
NP-Complete and NP-Hard Problems
- P represents decision problems solvable by deterministic algorithms in polynomial time.
- NP refers to decision problems solvable by nondeterministic algorithms in polynomial time, implying P ⊆ NP.
- A problem is NP-complete if it belongs to NP, and every problem in NP is polynomially reducible to it, making it the hardest within NP.
- An illustration of an NP-complete problem is the CNF-satisfiability problem, where boolean variables must be assigned true/false to validate an expression.
- NP-hard problems are those that do not need to be solvable in polynomial time but are at least as hard as NP-complete problems.
- If any NP-complete problem can be solved in polynomial time, all NP problems can be solved in polynomial time, leading to the conjecture whether P = NP.
Additional Notes
- The relationship between P, NP, NP-Complete, and NP-Hard is fundamental in understanding computational complexity.
- The halting problem illustrates an NP problem that isn't NP-complete, emphasizing the complexities in categorizing problems within these classifications.
- Research continues into whether finding a polynomial-time solution for one NP-complete problem implies polynomial-time solutions for all NP problems.
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Description
This quiz covers key concepts from Module-5 of BCS401, focusing on Backtracking techniques in algorithm design. Explore topics such as the N-Queens problem, the 0/1 Knapsack problem, and various Branch and Bound solutions. Test your understanding of these algorithms and their applications.