- Commonly used data structures for graphs:
- Adjacency list
- Adjacency matrix
- Mapping
- Nodes in the graph are mapped to integers 0, 1, 2, ..., (n−1)
Example:
- Storing nodes:
- Nodes are
represented:
- the index of an one-dimensional array
- Nodes are
represented:
- Storing edges:
- Edges are stored as a linked list
- Example:
- Graph:
- Representation:
- Graph:
- Java classes:
/* ======================================= List element that represents an edge ======================================= */ class Edge { int NodeID; // The neighbor node Node next; // Next Edge in the list } /* ======================================= The nodes ======================================= */ Edge[] node; // Node[i] contains a list of neighbor for i
- Storing nodes:
- Nodes are
represented:
- the index of a two-dimensional array (= matrix)
- Nodes are
represented:
- Storing edges:
- Edges are represented as a value in the matrix
- Example:
- Graph:
- Representation:
0 1 2 3 4 5 6 7 8 +- -+ | 0 1 0 1 0 0 0 0 1 | // 0 | 1 0 0 0 0 0 0 1 0 | // 1 | 0 0 0 1 0 1 0 1 0 | // 2 | 1 0 1 0 1 0 0 0 0 | // 3 M = | 0 0 0 1 0 0 0 0 1 | // 4 | 0 0 1 0 0 0 1 0 0 | // 5 | 0 0 0 0 0 1 0 0 0 | // 6 | 0 1 1 0 0 0 0 0 0 | // 7 | 1 0 0 0 1 0 0 0 0 | // 8 +- -+
- Graph:
- Java variables:
/* ======================================= The edges ======================================= */ int[][] M; // M[i][j] > 0 <==> edge (i,j)
- The selection of a
data structure has
consequences to the
running times of
algorithms
- Many algorithms solve problems
by traversing all edges
in the graph
- Assume that a graph has:
- n nodes
- m edges
- Running time to
visit all edges:
- Adjacency list:
- O(m + n)
Because:
We must:
- Visit each array element: n steps
- Visit all "edges" (the blue squares): m steps
- O(m + n)
- Adjacency matrix:
- O(n2)
Because:
0 1 2 3 4 5 6 7 8 +- -+ | 0 1 0 1 0 0 0 0 1 | // 0 | 1 0 0 0 0 0 0 1 0 | // 1 | 0 0 0 1 0 1 0 1 0 | // 2 | 1 0 1 0 1 0 0 0 0 | // 3 M = | 0 0 0 1 0 0 0 0 1 | // 4 | 0 0 1 0 0 0 1 0 0 | // 5 | 0 0 0 0 0 1 0 0 0 | // 6 | 0 1 1 0 0 0 0 0 0 | // 7 | 1 0 0 0 1 0 0 0 0 | // 8 +- -+We must visit all elements in an n × n matrix
- O(n2)
- Adjacency list:
- Note:
- When a graph has
a large number of edges,
we will have
m ~= n × n
- Then no matter which representation you choose, the running time will be O(n × n)
- When a graph has
a large number of edges,
we will have
m ~= n × n
- Ease of programming- Choice of data structure:
- Adjacency matrix
usually result in
simpler algorithms
because you deal with
one data structure (matrix)
- The Adjacency list is a composite structure with an array and a list (or 2 lists)
- Adjacency matrix
usually result in
simpler algorithms
because you deal with
one data structure (matrix)