Graph algorithms are essential tools for solving complex problems in computer science. They help us navigate through interconnected data structures, finding optimal paths and uncovering hidden patterns. From search techniques to shortest path calculations, these algorithms form the backbone of many real-world applications.
This section dives into various graph algorithms, including depth-first and breadth-first search , topological sorting , and shortest path algorithms. We'll explore their implementations, time complexities, and practical applications, giving you a solid foundation in graph theory and its algorithmic solutions.
Graph Traversal Algorithms
Depth-First and Breadth-First Search
Top images from around the web for Depth-First and Breadth-First Search CS 360: Lecture 16: Breadth-First Search View original
Is this image relevant?
[Java / 알고리즘] 그래프 알고리즘 BFS와 DFS View original
Is this image relevant?
CS 360: Lecture 17: Depth-First Search View original
Is this image relevant?
CS 360: Lecture 16: Breadth-First Search View original
Is this image relevant?
[Java / 알고리즘] 그래프 알고리즘 BFS와 DFS View original
Is this image relevant?
1 of 3
Top images from around the web for Depth-First and Breadth-First Search CS 360: Lecture 16: Breadth-First Search View original
Is this image relevant?
[Java / 알고리즘] 그래프 알고리즘 BFS와 DFS View original
Is this image relevant?
CS 360: Lecture 17: Depth-First Search View original
Is this image relevant?
CS 360: Lecture 16: Breadth-First Search View original
Is this image relevant?
[Java / 알고리즘] 그래프 알고리즘 BFS와 DFS View original
Is this image relevant?
1 of 3
Depth-First Search (DFS) explores graph by going as deep as possible before backtracking
Uses stack data structure to keep track of vertices
Implemented recursively or iteratively
Time complexity : O ( V + E ) O(V + E) O ( V + E ) where V is number of vertices and E is number of edges
Applications include maze solving, detecting cycles, and topological sorting
Breadth-First Search (BFS) explores graph by visiting all neighbors before moving to next level
Uses queue data structure to keep track of vertices
Always implemented iteratively
Time complexity: O ( V + E ) O(V + E) O ( V + E )
Applications include finding shortest path in unweighted graphs and social network analysis
Comparison between DFS and BFS:
DFS uses less memory for deep graphs, BFS uses less for wide graphs
DFS may not find shortest path, BFS always finds shortest path in unweighted graphs
DFS better for decision trees, BFS better for finding nearby nodes
Advanced Graph Algorithms
Topological sorting orders vertices in directed acyclic graph (DAG) such that for every edge (u, v), u comes before v
Uses modified DFS algorithm
Time complexity: O ( V + E ) O(V + E) O ( V + E )
Applications include scheduling tasks, build systems, and resolving dependencies
Strongly Connected Components (SCC) are maximal subgraphs where every vertex is reachable from every other vertex
Kosaraju's algorithm finds SCCs using two DFS passes
Time complexity: O ( V + E ) O(V + E) O ( V + E )
Applications include analyzing social networks and simplifying complex graphs
Shortest Path Algorithms
Single-Source Shortest Path
Dijkstra's algorithm finds shortest path from source to all other vertices in weighted graph with non-negative edges
Uses greedy approach and priority queue
Time complexity: O ( ( V + E ) log V ) O((V + E) \log V) O (( V + E ) log V ) with binary heap implementation
Cannot handle negative edge weights
Applications include GPS navigation and network routing protocols
Bellman-Ford algorithm finds shortest path from source to all other vertices, even with negative edge weights
Uses dynamic programming approach
Time complexity: O ( V E ) O(VE) O ( V E )
Can detect negative cycles
Applications include currency exchange and arbitrage detection in financial markets
All-Pairs Shortest Path
Floyd-Warshall algorithm finds shortest paths between all pairs of vertices in weighted graph
Uses dynamic programming approach
Time complexity: O ( V 3 ) O(V^3) O ( V 3 )
Can handle negative edge weights but not negative cycles
Applications include finding transitive closure of a graph and solving all-pairs shortest path problem in small graphs
Minimum Spanning Tree Algorithms
Fundamentals and Greedy Approaches
Minimum Spanning Tree (MST) connects all vertices in weighted, undirected graph with minimum total edge weight
Properties include:
Contains exactly V-1 edges for V vertices
No cycles in the tree
Unique if all edge weights are distinct
Applications include network design, clustering algorithms, and image segmentation
Prim's algorithm builds MST by iteratively adding cheapest edge that connects tree to a new vertex
Uses priority queue to select cheapest edge
Time complexity: O ( ( V + E ) log V ) O((V + E) \log V) O (( V + E ) log V ) with binary heap implementation
Efficient for dense graphs
Step-by-step process:
Start with arbitrary vertex
Add cheapest edge connecting tree to new vertex
Repeat until all vertices are in tree
Kruskal's algorithm builds MST by iteratively adding cheapest edge that doesn't create a cycle
Uses disjoint-set data structure to detect cycles
Time complexity: O ( E log E ) O(E \log E) O ( E log E ) or O ( E log V ) O(E \log V) O ( E log V ) (since E ≤ V 2 E \leq V^2 E ≤ V 2 )
Efficient for sparse graphs
Step-by-step process:
Sort all edges by weight
Add cheapest edge that doesn't create cycle
Repeat until V-1 edges are added