158 lines
4.2 KiB
Markdown
158 lines
4.2 KiB
Markdown
---
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type: theoretical
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---
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Can be utilized in algorithms. Here's a rundown of a couple of popular graph algorithms along with their use case and [Complexity](Complexity.md). TSP is [NP](P%20vs.%20NP.md), but it can be approximated/solved using a graph algorithm!
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# Graph Algorithms
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## Shortest Path Algorithms
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### Dijkstra's Algorithm
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Finds the shortest path from a source vertex to all other vertices in a graph with non-negative edge weights.
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1. Initialize distances:
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- Set the distance to the source vertex as 0 and all others as infinity.
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2. Use a priority queue (min-heap) to extract the vertex with the smallest distance.
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3. For each neighboring vertex, calculate the tentative distance via the current vertex:
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- If the tentative distance is smaller than the known distance, update it.
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4. Repeat until all vertices are processed.
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- Using a priority queue: $O((V + E) \log V)$, where $V$ is the number of vertices and $E$ is the number of edges.
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### Floyd-Warshall Algorithm
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Computes shortest paths between all pairs of nodes in a weighted graph.
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1. Initialize distances with edge weights.
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2. Iteratively update distances by considering each node as an intermediate point.
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3. Handle both positive and negative edge weights (no negative cycles allowed).
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- **Complexity**: $O(V^3)$, where $V$ is the number of vertices.
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---
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## Minimum Spanning Tree Algorithms
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### Kruskal's Algorithm
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Finds the minimum spanning tree (MST) of a graph.
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1. Sort all edges by weight.
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2. Add edges to the MST in increasing order of weight.
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3. Skip edges that form a cycle (using Union-Find data structure :) ).
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- **Complexity**: $O(E \log E)$, where $E$ is the number of edges.
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### Prim's Algorithm
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Constructs the minimum spanning tree (MST) of a graph.
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1. Start with an arbitrary node.
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2. Repeatedly add the smallest edge that connects a node in the MST to a node outside it.
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3. Continue until all nodes are included.
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- $O(E + V \log V)$ using a priority queue, where $V$ is the number of vertices and $E$ is the number of edges.
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---
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## Graph Traversal Algorithms
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### Depth-First Search (DFS)
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Explores as far as possible along each branch before backtracking.
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- Detects cycles.
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- Topological sorting.
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- Pathfinding in mazes.
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- **Complexity**: $O(V + E)$.
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---
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### Breadth-First Search (BFS)
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explores all neighbors at the current depth before moving to the next level.
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- Finding shortest paths in unweighted graphs.
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- Level-order traversal of trees.
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- **Complexity**: $O(V + E)$.
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---
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## Topological Sorting
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Arranges the vertices of a directed acyclic graph (DAG) in a linear order such that for every directed edge $(u, v)$, $u$ comes before $v$.
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- Perform a DFS and use a stack to store the vertices in reverse order of completion.
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- **Complexity**: $O(V + E)$.
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---
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## Maximum Flow Algorithms
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### Ford-Fulkerson Algorithm
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**Finds the maximum flow in a flow network.
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1. Initialize the flow to 0.
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2. While there is an augmenting path, increase the flow along the path.
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3. Repeat until no augmenting paths remain.
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- **Complexity**: $O(E \cdot \text{max\_flow})$.
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---
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### Edmonds-Karp Algorithm
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A refinement of the Ford-Fulkerson algorithm that uses BFS to find augmenting paths.
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- **Complexity**: $O(V \cdot E^2)$.
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---
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## Grid-Based Algorithms
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### A* Algorithm
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Finds the shortest path in a weighted grid using a heuristic.
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1. Start from the source node.
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2. Use a priority queue to explore the most promising paths first.
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3. Use a heuristic to guide the search toward the goal.
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- **Complexity**: Depends on the heuristic but generally better than BFS for weighted grids.
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---
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## Graph Coloring
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Assigns colors to vertices such that no two adjacent vertices share the same color.
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- **Algorithms**:
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- Greedy coloring: $O(V^2)$.
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- Backtracking for exact coloring.
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---
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