Adjacency Matrix

The rows/columns are the source/target vertex, the matrix is a square matrix with non-negative entries, 0 if there is no edge between the vertices, n if there are n edges from the source to the target vertex. This is efficient for dense multigraphs. For undirected graphs you either extend the matrix as being symmetric or you only use it as a triangular matrix.

Adjacency List

A vector with entries the list of adjacent vertices. If you need to represent a multigraph, then you need a vector of dictionaries where the key is the target vector and the value the multiplicity of the edge. This is efficient for sparse graphs. For undirected graphs you either fix an order of the vertices on an edge or you enter each edge twice (once at each vertex).

Comparison

the list might work better with level 1 cache with adjacency objects (which node, visited, inPath, pathWeight, fromWhere).

Pseudocode

 dfs(vertex w)
   if w has already been marked visited
     return
   mark w as visited
   for each adjacent vertex v
     dfs(v)

Non recursive DFS is more difficult. It requires that each node keep memory of the last child visited, as it descends the current child. One implementation uses a indexed array of iterators, so on visiting a node, the node's number is an index into an array that stores the iterator for the nodes child. Then the first iterator value is pushed onto the job stack. Peek not pop is used to examine the current top of the stack, and pop is only invoked when the iterator for the peeked node is exhausted.

Properties

Classification of Edge

Pseudocode

  bfs ( x ): 
    q insert x;
     while (q not empty )
       y = remove head  q
       visit y
       mark y
       for each z adjacent y 
         q add tail z

Example

Correctness

Analysis

Usage

A breadth first search can be used to explore a database schema, in an attempt to turn it into an xml schema. This is done by naming the root table, and then doing a referential breadth first search . The search is both done on the referring and referred ends, so if another table refers to to the current node being searched, than that table has a one-to-many relationship in the xml schema, otherwise it is many-to-one.

Directed graph cycle detection

In a directed graph, check is *acyclic* by having a second marker on before dfs recursive call and off after, and checking for second marker before original mark check in dfs adjacency traversal. If second marker present, then cycle exists.

Topological Sort of Directed Graph

1. check for cycle as in previous section.
2. dfs the acyclic graph. Reverse postorder by storing on stack instead of queue after dfs calls on adjacent nodes.

The last node on stack must be there from first dfs call, and removing the last node exposes the second node which can only have been reached by last node. Use induction to show topological order.

Strongly Connected Components in Directed Graphs

1. Strong connected components have cycles within and must by acyclic between components ( the kernel directed acyclic graph).
2. Difference between dfs reverse postorder of original graph vs the same on reverse graph is that first node is least dependent in latter. Thus all non strong connected nodes will be removed first by dfs on the original graph in the latter's order, and then dfs will remove only strongly connected nodes by marking , one SC component at each iteration over reverse postorder of reverse graph , visiting unmarked nodes only. Each outgoing edge from a SC component being traversed will go to an already marked node due to reverse postorder on reverse graph.

Articulation Vertex

Bridge

Diameter

Top, Chapters: 1, 2, 3, 4, 5, 6, 7, 8, 9, ACategory:Book:Algorithms#Unweighted%20Graph%20Algorithms%20