Graphs--Data Structure

Definition

An (undirected) Graph is a collection V of vertices, and a collection E ofedgeseach of which connects a pair of vertices.

Vertices: A,B,C,D
Edges: (A,B), (A,C), (A,D), (C,D) 

 

Edge List

List of all edges:

Edges: (A,B), (A,C), (A,D), (C,D) 

Adjacency Matrix

Matrix. Entries 1 if there is an edge, 0 ifthere is not.

      A  B  C  D

A   0   1   1   1

B   1   0   0   0

C   1   0   0   1

D   1   0   1   0

Adjacency List

For each vertex, a list of adjacent vertices.

A adjacent to B,C,D

B adjacent to A

C adjacent to A, D
D adjacent to A, C 

Op.

Is Edge?     List Edge    List Nbrs.         

Adj. Matrix

Edge List

Adj. List

Θ(1)              Θ(|V |2)        Θ(|V |)  

Θ(|E |)           Θ(|E |)          Θ(|E |)

Θ(deg)          Θ(|E |)          Θ(deg)

2. Dense Graphs

In dense graphs, |E | ≈ |V |2.

A large fraction of pairs of vertices are connected by edges. 

Sparse Graphs

In sparse graphs, |E | ≈ |V |.

Each vertex has only a few edges. 

Pseudocode

Component(s)

DiscoveredNodes ← {s}
while there is an edge e leavingDiscoveredNodes that has not beenexplored:

    add vertex at other end of e to

    DiscoveredNodes

return DiscoveredNodes

Depth First Ordering 

Explore(v)

visited(v) ← true

for (v,w)∈E:

  if not visited(w):

    Explore(w )

DFS(G)

for all v ∈ V :   mark v unvisited

for v ∈ V :

  if not visited(v):

    Explore(v )

Previsit and Postvisit Functions

Explore(v)

visited(v) ← true

previsit(v)
for (v,w)∈E:

  if not visited(w):

    explore(w )

postvisit(v)

Initialize clock to 1.

previsit(v)

pre(v) ← clock

clock ← clock + 1

postvisit(v)

post(v) ← clock

clock ← clock + 1

3. Directed DFS

4. Toplogical sort

Find sink.
Put at end of order.

Remove from graph.

Repeat. 

LinearOrder(G)

while G non-empty:
   Follow a path until cannot extend

   Find sink v
   Put v at end of order
   Remove v from G

O(|V|) paths.
Each takes O(|V|) time.

Runtime O(|V |2). 

Better algorithm:

TopologicalSort(G)

DFS(G )
sort vertices by reverse post-order

5. connectivity:

Explore(v )

visited(v) ← true

CCnum(v) ← cc
for (v,w)∈E:

  if not visited(w):

    Explore(w )

DFS(G )

for all v ∈ V mark v unvisited

cc ← 1
for v ∈ V :

  if not visited(v):

    Explore(v )
    cc ← cc + 1

Correctness

Each new explore finds new connected component.

Eventually nd every vertex.

Runtime still O (|V | + |E |). 

6.Strongly Connected Components

De nition

Two vertices v,w in a directed graph are connectedif you can reach v from w and canreach w from v.

Summary

Can partition vertices into strongly connected components.

Metagraph describes how strongly connected components connect to each other.

Metagraph always a DAG. 

EasySCC(G)

for each vertex v:
   run explore(v) to determine vertices reachable from v

for each vertex v:

   find the u reachable from v that can also reach v

these are the SCCs

Runtime O (|V |2+ |V ||E |). Want faster. 

SCCs(G)

run DFS(GR)
let v have largest post number

run Explore(v)
vertices found are first SCC

Remove from G and repeat

New algorithm:

SCCs(G)

Run DFS(GR)
for v ∈ V in reverse postorder:

  if not visited(v):

    Explore(v )
    mark visited vertices as new SCC

Essentially DFS on GRand then on G.

Runtime O(|V|+|E|).