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Graphs III: Shortest Paths

Master shortest path algorithms in graphs and learn how to efficiently traverse graph structures. Understand the applications of BFS in pathfinding and distance calculations.

Learning Objectives

Core Concepts

  • Understanding shortest path algorithms
  • Implementing BFS for pathfinding
  • Analyzing unweighted graph distances
  • Path reconstruction techniques
  • Optimizing path finding algorithms

Skills Development

  • Implementing efficient path finding
  • Handling path reconstruction
  • Optimizing graph traversal
  • Solving real-world routing problems
  • Performance analysis and optimization

Prerequisites

Required Knowledge

  • Basic graph traversal algorithms
  • Queue data structure operations
  • Graph representation methods
  • Basic path concepts in graphs

Recommended Preparation

  • Review BFS implementation
  • Practice with adjacency lists
  • Study queue-based algorithms
  • Understand graph properties

Video Lecture

Breadth-First Search for Shortest Paths

Key Concepts Covered:

  • BFS as a shortest path algorithm
  • Distance tracking in unweighted graphs
  • Path reconstruction techniques
  • Applications in routing problems

Code Implementation

BFS Shortest Path Implementation

public class Graph {
    private Map> adjacencyList = new HashMap<>();
    
    public Map shortestPaths(T start) {
        Map distances = new HashMap<>();
        Queue queue = new LinkedList<>();
        Set visited = new HashSet<>();
        
        // Initialize distances
        distances.put(start, 0);
        queue.offer(start);
        visited.add(start);
        
        while (!queue.isEmpty()) {
            T current = queue.poll();
            
            for (T neighbor : adjacencyList.get(current)) {
                if (!visited.contains(neighbor)) {
                    visited.add(neighbor);
                    distances.put(neighbor, distances.get(current) + 1);
                    queue.offer(neighbor);
                }
            }
        }
        
        return distances;
    }
    
    public List findPath(T start, T end) {
        Map parentMap = new HashMap<>();
        Queue queue = new LinkedList<>();
        Set visited = new HashSet<>();
        
        queue.offer(start);
        visited.add(start);
        
        while (!queue.isEmpty() && !queue.peek().equals(end)) {
            T current = queue.poll();
            
            for (T neighbor : adjacencyList.get(current)) {
                if (!visited.contains(neighbor)) {
                    visited.add(neighbor);
                    parentMap.put(neighbor, current);
                    queue.offer(neighbor);
                }
            }
        }
        
        if (!visited.contains(end)) {
            return Collections.emptyList(); // No path exists
        }
        
        // Reconstruct path
        List path = new ArrayList<>();
        T current = end;
        while (current != null) {
            path.add(0, current);
            current = parentMap.get(current);
        }
        
        return path;
    }
}

Path Finding Utilities

public class PathUtils {
    public static  boolean hasPath(Graph graph, T start, T end) {
        return !graph.findPath(start, end).isEmpty();
    }
    
    public static  int shortestDistance(Graph graph, T start, T end) {
        Map distances = graph.shortestPaths(start);
        return distances.getOrDefault(end, -1);
    }
    
    public static  Set reachableNodes(Graph graph, T start) {
        return new HashSet<>(graph.shortestPaths(start).keySet());
    }
}

Guided Practice

For additional hands-on practice and in-depth exercises on graph algorithms, check out our comprehensive guided project:

Practice Exercises

1. Basic Path Finding

Practice fundamental path finding techniques:

2. Intermediate Path Problems

Solve more complex path finding challenges:

3. Advanced Path Applications

Apply path finding to complex scenarios: