Adjacency Lists;Breadth-First Search &

Depth-First Search

Adjacency Lists

For directed graphs:

b

a

c

1 a 1 2 3

2 b 3

3 c 2bc

cb

cbaa

Simple Notation

{ (a,a), (a,b), (a,c), (b,c), (c,b) }010c

100b

111acba

Space: Adjacency Lists vs. Matricies

• Space (n vertices and m edges)– matrix: n2 + n(vertex-name size)

• = matrix size + header size• matrix can be bits, but bits are not directly addressable

– list: n(header-node size) + m(list-node size)• Sparse: few edges 0 in the extreme case

– Matrix fixed size: so no size benefit– List variable size: as little as n(vertex-node size)

• Dense: many edges n2 in the extreme case– Matrix fixed size: so no size loss– List variable size: as much as n(header-node

size) + n2(list-node size)

Operations: Adjacency Lists vs. Matricies

• Operations depend on sparse/dense and what’s being done.

• Examples (n nodes and m edges)– Is there an arc from x to y?

• Matrix: O(1) check value at (x, y)

• List: O(n) index to x, traverse list to y or end

– Get successor nodes of a node.• Matrix: O(n) scan a row

• List: O(n) traverse a linked list

– Get predecessor nodes of a node.• Matrix: O(n) scan a column

• List: O(n+m) traverse all linked lists, which could be as bad as O(n+n2) = O(n2).

Adjacency Lists for Undirected and Weighted Graphs

• Undirected Graphs:

a a b c

b a c

c a b

Make each edge (except loops) go both ways.

b

a

c5

3

7

1

• Weighted Graphs:

a (a,1) (b,5) (c,3)

b (a,5) (c,7)

c (a,3) (b,7)

- add additional field to node- node-weight pairs

Breadth-First Search (BFS)

BFS Algorithm

Search order:

b

c

d

a

a b c d

= O(n+2m)= O(m) if m>>n

Undirected edges: each edge twice

Mark start node and enqueue

While queue is not empty

Dequeue N

For each neighbor X of N

If X is not marked

Mark X and enqueue

Marked N Queuea - a

a -b a bc a b c

b cc -

d c dd -

Depth-First Search (DFS)

DFS Algorithm

= O(n+2m)= O(m) if m>>n

b

c

d

a

a b c d

DFS (start node)

Proc DFS (N)

Mark N

For each neighbor X of N

If X is not marked

DFS (X)

Marked N DFSa

a aa b

b bb c

c cc d

d dcba

Search order:

BFS vs. DFS a b

e c

f

dMarked N Queuea - a

a -b a bc a b ce a b c e

b c ed b c e d

c e df c e d f

e d fd ff -

BFS Order: a b c e d f

BFS vs. DFS

(BFS Order: a b c e d f)

a b

e c

f

dMarked N DFSa

a aa b

b bb c

c cc f

f fcb d

d dba e

e ea

DFS Order: a b c f d e

a

b c d

e f g h i

j k l m n

BFS & DFS with Directed Graphs

BFS DFS

a

b c d

e f g h i

j k l m n

a,b,c,d,e,f,g,h,i,j,k,l,m,n

Same as before, by chance

a,b,e,j,f,k,l,h,c,g,d,i,m,n

Not same as before