1. CMSC 341
Graphs 3

2. Adjacency List Keep list of adjacent vertices for each vertex.
Array of lists of indices: Each element of array[i] is a list of the indices of the vertices adjacent to vertex i.
List of lists: The i-th element of L is associated with vertex vi and is a List Li of the elements of L adjacent to vi.
Lists in Array (NIL sentinels): Each entry a[i,j] is either the index of the j-th vertex adjacent to vertex I or a NIL sentinel indicating end-of-list.
Lists in Array (with valence array): Instead of using NIL sentinels to mark theend of the list in the array, a separate array Valence is kept indicating the number of entries in each row of the array.

3. Adjacency Lists (cont.)
Storage requirement:
Performance: Array of List of Lists in Lists in
Lists Lists Array (NIL) Array (val)
Storage: O(|V| + |E|)
getDegree(u) -- O(D(u)) -- requires counting list length in each case except
“Lists in Array (with valence)” which is O(1)
getInDegree(u) -- O(|V|+|E|) -- requires searching each list for the presence of
vertex u
getOutDegree(u) -- O(D(u)) -- requires counting list length in each case except
getAdjacent(u) -- O(D(u)) -- requires construction of a list of each vertex of
each vertex on the adjacency list of vertex u
getAdjacentFrom(u) -- O(|V|+|E|) for “Array of lists” and “List of Lists”
O(|V|^2) for “Lists in Array”
requires construction of a list of vertices by searching the
adjacency lists of each vertex
isConnected(u) - O(D(u)) -- requires searching the adjacency list of vertex u
for the presence of vertex v
Storage -- O(|V|+|E|) for “Array of Lists” and “List of Lists” -- entry for each
vertex and list of entries for each adjacent vertex
O(|V|^2) for “Lists in Array” -- entry for each vertex

4. Directed Acyclic Graphs
A directed acyclic graph is a directed graph with no cycles.
A partial order R on a set S is a binary relation such that
for all aS, aRa is false (irreflexive property)
for all a,b,c S, if aRb and bRc then aRc is true (transitive property)
To represent a partial order with a DAG:
represent each member of S as a vertex
for each pair of vertices (a,b), insert an edge from a to b if and only if aRb.
Example: DAG for CS courses

5. More Definitions
Vertex i is a predecessor of vertex j if and only if there is a path from i to j.
Vertex i is an immediate predecessor if vertex j if and only if (i,j) is an edge in the graph.
Vertex j is a successor of vertex i if and only if there is a path from i to j.
Vertex j is an immediate predecessor if vertex i if and only if (i,j) is an edge in the graph.

6. Topological Ordering
A topological ordering of the vertices of a DAG G=(V,E) is a linear ordering such that, for vertices i,j V, if i is a predecessor of j, then i precedes j in the linear order.
In general, there is more than one topo ordering for a graph.
Ex: legitimate orders to take CS courses

7. Topological Sort void TopSort(Graph G) { unsigned int counter = 1 ;
Queue q = new Queue();
Vertex indegree[|V|];
for each Vertex v {
indegree[v] = getInDegree(v);
if (indegree[v] == 0) q.enqueue(v); }
while (!q.isEmpty()){
v = q.dequeue();
Put v on the topological ordering;
counter++;
for each Vertex v adjacent from v {
indegree[w] -=1;
if (indegree[w]==0) q.enqueue(w); }
if (counter <= G.numVertices())
declare an error -- G has a cycle
Q: In what circumstances can counter be equal to NumVerts?
A: One circumstance would be a graph with a single vertex and a directed edge from the vertex to itself.

8. 1 2 3 4 5 6 7 8 9 10 Indegree array at start: 1 2 3 4 5 6 7 8 9 10
Indegree array at start:
Step Queue v counter
0 1
1 4,7,10 1
2 7,10 4 2
4 1,3 10 4
5 3,5 1 5
6 5,6,2 3 6
7 6,2,9 5 7
8 2,9 6 8