Description: An adjacency set is a data structure used to represent graphs, where each vertex of the graph is associated with a set containing its neighboring vertices. This representation is particularly useful in sparse graphs, where the number of edges is much lower than the square of the number of vertices. In an adjacency set, each vertex is stored alongside a set that includes all the vertices it is directly connected to. This structure allows for easy addition and removal of edges, as well as efficient checking for the existence of a connection between two vertices. Additionally, the use of sets ensures that there are no duplicates in the connections, simplifying data management. The representation using adjacency sets is particularly advantageous in algorithms that require exploring the neighbors of a vertex, such as depth-first search (DFS) or breadth-first search (BFS) algorithms. In terms of complexity, representing a graph using adjacency sets consumes less memory compared to an adjacency matrix, especially in sparse graphs, making it a preferred option in many applications of graph theory and computer science.
Uses: Adjacency sets are used in various applications of graph theory, including the representation of social networks, where users are vertices and the connections between them are edges. They are also useful in graph search and traversal algorithms, such as Dijkstra’s algorithm for finding the shortest path. Additionally, they are employed in modeling transportation systems, optimizing routes, and representing relationships between entities in database management.
Examples: A practical example of an adjacency set is the representation of a graph modeling a road network, where each intersection is a vertex and each road is an edge. In this case, the adjacency set of an intersection would include all the intersections directly connected by roads. Another example is in social networks, where a user may have a set of friends representing their direct connections.
