25pcs916.ppt DSA-1 data structure algorithm

That is data structure algorithm

1. The Travelling Salesperson Problem Backtracking 25PCS916

2. Travelling Salesperson Problem (TSP)? The Travelling Salesperson Problem (TSP) is a classic algorithmic challenge that seeks to find the most efficient route among a set of locations. Shortest Possible Route Identify the absolute shortest path that visits each designated city precisely once. Real-World Applications Critical for logistics optimisation, delivery route planning, and efficient manufacturing processes. Complete the Cycle Ensure the journey concludes by returning to the original starting city, forming a closed loop.

3. While the TSP is notoriously complex due to its NP-hard nature, backtracking offers a strategic approach compared to simpler, brute force methods. Why Choose Backtracking for TSP? Beyond Brute Force Avoids the exponential explosion of checking every single possible route (n! permutations). Intelligent Pruning Explores paths depth-first, intelligently abandoning unpromising or overly costly routes early on. Optimal Solution Guarantees identification of the true minimum cost Hamiltonian cycle.

4. Example: Cost Matrix for Three Cities Consider a simple scenario involving three cities (City 0, City 1, City 2) and their inter-city travel costs represented in a matrix: Our objective is to find the minimum cost route starting and ending at City 0. For three cities, the only distinct candidate routes (excluding reversals) are: Route 1: 0 ³ 1 ³ 2 ³ 0 Route 2: 0 ³ 2 ³ 1 ³ 0

5. Backtracking Algorithm: Step-by-Step The backtracking approach systematically explores potential paths, using recursion and pruning to find the optimal solution: Initialise Path Begin at the designated starting city (e.g., City 0) and mark it as visited Explore Recursively Recursively visit all unvisited cities, meticulously accumulating the travel cost along each path segment. Complete Cycle Once all cities have been visited, add the cost required to return to the initial starting city. Track Minimum Cost Continuously update and store the minimum total cost discovered amongst completed cycles so far Intelligent Backtracking Prune paths that either have no valid next moves or whose accumulated cost already exceeds the current minimum known cost Optimal Cycle The algorithm returns the overall minimum cost cycle after systematically exploring all promising paths.

6. Pseudocode Logic: A Conceptual Overview Algorithm Core The pseudocode for backtracking in TSP typically involves a recursive function, findMinRoute ( currentCity , visitedCities , currentCost ). It iterates through unvisited neighbours , making a recursive call for each. A critical optimisation is the pruning step: if currentCost exceeds minOverallCost , the function immediately returns

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