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Apr 24, 20246 min read

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Space complexity

The space complexity of each graph representation varies depending on the graph's density, which refers to the ratio of the number of edges to the number of possible edges. Let’s analyze the space complexity for each representation and discuss which type of graph (sparse or dense) each is best suited for.

Adjacency Matrix
  • Space Complexity: (O(V^2)), where (V) is the number of vertices.
  • Best Suited For: Dense graphs. In a dense graph, where the number of edges (E) is close to the maximum possible number of edges ((V(V-1)/2) for an undirected graph), an adjacency matrix is efficient because most of the entries in the matrix will be 1 (indicating the presence of an edge). In such cases, the space used by the matrix is justified by the high connectivity.
Adjacency List
  • Space Complexity: (O(V + E)), where (V) is the number of vertices and (E) is the number of edges.
  • Best Suited For: Sparse graphs. In a sparse graph, where (E) is much less than (V(V-1)/2), an adjacency list is more space-efficient than an adjacency matrix because it only stores information about the existing edges. This makes it particularly suited for graphs where the number of edges per vertex is low, as it avoids the storage of a large number of zeros that would be present in an adjacency matrix.
Edge List
  • Space Complexity: (O(E)), where (E) is the number of edges.
  • Best Suited For: Situations where the graph’s connectivity needs to be enumerated or processed as a list (e.g., in some graph algorithms like Kruskal’s algorithm for finding the minimum spanning tree). It is not particularly suited to dense or sparse graphs per se but is less efficient for operations that require quick access to information about whether a specific edge exists or to enumerate all neighbors of a vertex.
Analysis
  • Dense Graphs In dense graphs, where the number of edges is close to the maximum possible, an adjacency matrix is preferred due to its simplicity and the efficiency of checking the existence of an edge between any two vertices.
  • Sparse Graphs: For sparse graphs, an adjacency list is more space-efficient, as it does not waste space representing edges that do not exist.
  • Edge Lists: This representation can be used for both dense and sparse graphs but is less efficient for operations that require frequent access to adjacency information. It’s more suited for algorithms that need to process edges and might not need to check adjacency often.

Operations in different representations

The efficiency of graph operations like adding or removing edges and finding adjacent vertices significantly depends on the choice of graph representation. Let’s examine how each representation handles these operations:

Adjacency Matrix
  • Adding an Edge: (O(1)). To add an edge between vertices (i) and (j), simply set the matrix entry ([i][j]) and ([j][i]) to 1.
  • Removing an Edge: (O(1)). To remove an edge, set ([i][j]) and ([j][i]) to 0.
  • Finding Adjacent Vertices: (O(V)). You need to iterate through a row (or column) to find all vertices connected to a vertex, where (V) is the total number of vertices.

The adjacency matrix allows for very quick addition and removal of edges, but finding all vertices adjacent to a given vertex is less efficient because it requires scanning through all vertices, regardless of the number of edges.

Adjacency List
  • Adding an Edge: (O(1)) (assuming we’re appending to the list or using a data structure that allows constant time insertion).
  • Removing an Edge: (O(E)) in the worst case, where (E) is the number of edges. This is because, in the worst case, you might need to search through all edges of a vertex to find the specific edge to remove.
  • Finding Adjacent Vertices: (O(1)) for beginning the iteration, but overall complexity depends on the number of adjacent vertices. This operation is efficient because each vertex’s list directly contains all its adjacent vertices.

Adjacency lists provide an efficient way to iterate over adjacent vertices but can be less efficient for removing edges, especially if the edge is not at the beginning of the list.

Edge List
  • Adding an Edge: (O(1)) if you’re appending to the list.
  • Removing an Edge: (O(E)), as you must search through the list to find and remove a specific edge.
  • Finding Adjacent Vertices: (O(E)) in the worst case, since you may need to scan the entire list to find all edges involving a particular vertex.

The edge list is straightforward for adding edges but can be inefficient for finding adjacent vertices or removing edges, as both operations potentially require scanning through the entire list of edges.

Summary
  • Adjacency Matrices are optimal for dense graphs and operations that frequently check for the existence of an edge between two vertices. They are less suitable for operations that require finding all adjacent vertices, especially in sparse graphs.
  • Adjacency Lists are optimal for sparse graphs, as they efficiently support finding all adjacent vertices and adding edges. Removing edges can be less efficient, depending on the underlying data structure.
  • Edge Lists are generally not as efficient for the basic operations discussed, except for adding edges. They can be useful in specific scenarios, like implementing certain algorithms that need to process all edges in the graph.

In practice, the choice between these representations often depends on the specific needs of the application, including the graph’s density and the types of operations that are most critical for performance.

Graph View

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