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| 1 | +# Topological Sorting in Directed Graphs |
| 2 | + |
| 3 | +Topological sorting is a fundamental algorithm used in graph theory to order the vertices of a directed acyclic graph (DAG) in such a way that for every directed edge `(u, v)`, vertex `u` comes before `v` in the ordering. This ordering helps to represent dependencies between tasks or elements, making it a valuable tool in various applications. |
| 4 | + |
| 5 | +## How Topological Sorting Works |
| 6 | + |
| 7 | +The algorithm for topological sorting typically uses depth-first search (DFS) to explore the graph. During the DFS traversal, vertices are visited in a specific order, and the order of finishing times is used to determine the topological order. The key idea is to start from vertices with no incoming edges and gradually process vertices with dependencies. |
| 8 | + |
| 9 | +## Algorithm Overview |
| 10 | + |
| 11 | +1. **DFS Traversal:** |
| 12 | + - Start DFS traversal from any unvisited vertex. |
| 13 | + - Mark the current vertex as visited. |
| 14 | + - Recursively visit all unvisited neighbors. |
| 15 | + |
| 16 | +2. **Finish Times:** |
| 17 | + - Assign finishing times to vertices during DFS. |
| 18 | + |
| 19 | +3. **Ordering:** |
| 20 | + - Order vertices in decreasing order of finishing times. |
| 21 | + |
| 22 | +## Time and Space Complexity |
| 23 | + |
| 24 | +- **Time Complexity:** O(V + E) |
| 25 | + - Each vertex and each edge are processed once during DFS. |
| 26 | + |
| 27 | +- **Space Complexity:** O(V) |
| 28 | + - Storage for color, distance, predecessor, etc., associated with each vertex. |
| 29 | + |
| 30 | +## Applications of Topological Sorting |
| 31 | + |
| 32 | +1. **Task Scheduling:** In project management and task scheduling, topological sorting is used to schedule tasks based on dependencies. Each task represents a vertex, and the dependencies between tasks are represented as directed edges. |
| 33 | + |
| 34 | +2. **Dependency Resolution:** In software development, topological sorting is used to resolve dependencies between modules or components. It ensures that modules are compiled or executed in the correct order. |
| 35 | + |
| 36 | +3. **Course Prerequisites:** In academic course planning, topological sorting can be applied to determine the order in which courses should be taken based on prerequisites. |
| 37 | + |
| 38 | +4. **Build Systems:** Topological sorting is commonly used in build systems to determine the order in which software components or modules should be built. |
| 39 | + |
| 40 | +5. **Network Routing:** In networking, topological sorting can be used to determine the order in which data should be transmitted through a network based on dependencies between routers or switches. |
| 41 | + |
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