Elementary Graph Algorithms

CLRS Chapter 22

Graph

• A graph is a structure that consists of a set of vertices and a set of edges between pairs of vertices

Graph

• A graph is a pair (V, E), where– V is a set of vertices– E is a set of pairs of vertices, called edges

• Example:– V = {A, B, C, D, E}– E = {(A,B), (A,D), (C,E), (D, E)}

Directed graph (digraph)

• Directed edge– ordered pair of vertices (u,v)

• Undirected edge– unordered pair of vertices (u,v)

• A graph with directed edges is called a directed graph or digraph

• A graph with undirected edges is an undirected graph or simply a graph

Weighted Graphs

• The edges in a graph may have values associated with them known as their weights

• A graph with weighted edges is known as a weighted graph

Terminology

• End vertices (or endpoints) of an edge– U and V are the endpoints of

a• Edges incident on a vertex

– a, d, and b are incident on V• Adjacent vertices

– U and V are adjacent• Degree of a vertex

– X has degree 5 • Parallel edges

– h and i are parallel edges• Self-loop

– j is a self-loop

XU

V

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a

c

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h

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j

P1

Terminology (cont.)

• A path is a sequence of vertices in which each successive vertex is adjacent to its predecessor

• In a simple path, the vertices and edges are distinct except that the first and last vertex may be the same

• Examples

– P1=(V,X,Z) is a simple path

– P2=(U,W,X,Y,W,V) is a path that is not simple

XU

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hP2

Terminology (cont.)

• A cycle is a simple path in which only the first and final vertices are the same

• Simple cycle– cycle such that all its

vertices and edges are distinct

• Examples

– C1=(V,X,Y,W,U,V) is a simple cycle

– C2=(U,W,X,Y,W,V,U) is a cycle that is not simple

C1

XU

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hC2

Subgraphs

• A subgraph S of a graph G is a graph such that – The vertices of S are a

subset of the vertices of G– The edges of S are a

subset of the edges of G• A spanning subgraph of G is a

subgraph that contains all the vertices of G

Subgraph

Spanning subgraph

Connectivity

• A graph is connected if there is a path between every pair of vertices

• A connected component of a graph G is a maximal connected subgraph of G

Connected graph

Non connected graph with two connected components

Trees and Forests

• A (free) tree is an undirected graph T such that– T is connected– T has no cycles

• A forest is an undirected graph without cycles

• The connected components of a forest are trees

Tree

Forest

DAG

• A directed acyclic graph (DAG) is a digraph that has no directed cycles

B

A

D

C

E

DAG G

Spanning Trees and Forests

• A spanning tree of a connected graph is a spanning subgraph that is a tree

• A spanning tree is not unique unless the graph is a tree

• A spanning forest of a graph is a spanning subgraph that is a forest

Graph

Spanning tree

Representation of Graphs

Two standard ways:• Adjacency List

– preferred for sparse graphs (|E| is much less than |V|^2)

– Unless otherwise specified we will assume this representation

• Adjacency Matrix– Preferred for dense graphs

Adjacency List

• An array Adj of |V| lists, one per vertex• For each vertex u in V,

– Adj[u] contains all vertices v such that there is an edge (u,v) in E (i.e. all the vertices adjacent to u)

• Space required Θ(|V|+|E|) (Following CLRS, we will use V for |V|

and E for |E|) thus Θ(V+E)

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51 4

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2 3 5

4 1 2

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0 1 0 0 1

1 0 0 1 1

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0 1 1 0 1

1 1 0 1 0

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Adjacency Matrix

Graph Traversals

• For solving most problems on graphs– Need to systematically visit all the vertices and edges of a graph

• Two major traversals– Breadth-First Search (BFS)

– Depth-First Search(DFS)

BFS

• Starts at some source vertex s • Discover every vertex that is reachable from s• Also produces a BFS tree with root s and including all

reachable vertices

• Discovers vertices in increasing order of distance from s– Distance between v and s is the minimum number of edges on a

path from s to v

• i.e. discovers vertices in a series of layers

BFS : vertex colors stored in color[]

• Initially all undiscovered: white

• When first discovered: gray– They represent the frontier of vertices between discovered

and undiscovered

– Frontier vertices stored in a queue

– Visits vertices across the entire breadth of this frontier

• When processed: black

Additional info stored (some applications of BFS need this info)

• pred[u]: points to the vertex which first discovered u

• d[u]: the distance from s to u

BFS(G=(V,E),s) for each (u in V)

color[u] = whited[u] = infinitypred[u] = NIL

color[s] = grayd[s] = 0Q.enqueue(s)while (Q is not empty) do

u = Q.dequeue()for each (v in Adj[u]) do

if (color(v] == white) thencolor[v] = gray //discovered but not yet

processedd[v] = d[u] + 1pred[v] = uQ.enqueue(v) //added to the frontier

color[u] = black //processed

Analysis

• Each vertex is enqued once and dequeued once : Θ(V)

• Each adjacency list is traversed once:

• Total: Θ(V+E)

Vu

Eu )()deg(

BFS Tree

• predecessor pointers after BFS is completed define an inverted tree– Reverse edges: BFS tree rooted at s

– The edges of the BFS tree are called : tree edges

– Remaining graph edges are called: cross edges

BFS and shortest paths

• Theorem: Let G=(V,E) be a directed or undirected graph, and suppose BFS is run on G starting from vertex s. During its execution BFS discovers every vertex v in V that is reachable from s. Let δ(s,v) denote the number of edges on the shortest path form s to v. Upon termination of BFS, d[v] = δ(s,v) for all v in V.

DFS

• Start at a source vertex s• Search as far into the graph as possible and backtrack

when there is no new vertices to discover– recursive

color[u] and pred[u] as before

• color[u]– Undiscovered: white

– Discovered but not finished processing: gray

– Finished: black

• pred[u]– Pointer to the vertex that first discovered u

Time stamps,

• We store two time stamps:– d[u]: the time vertex u is first discovered (discovery time)

– f[u]: the time we finish processing vertex u (finish time)

DFS(G)

for each (u in V) do

color[u] = white

pred[u] = NIL

time = 0

for each (u in V) do

if (color[u] == white)

DFS-VISIT(u)

DFS-VISIT(u)

//vertex u is just discovered

color[u] = gray

time = time +1

d[u] = time

for each (v in Adj[u]) do

//explore neighbor v

if (color[v] == white) then

//v is discovered by u

pred[v] = u

DFS-VISIT(v)

color[u] = black // u is finished

f[u] = time = time+ 1

DFS-VISIT invoked by each vertex exactly once.

Θ(V+E)

DFS Forest

• Tree edges: inverse of pred[] pointers – Recursion tree, where the edge

(u,v) arises when processing a vertex u, we call DFS-VISIT(v)

Remaining graph edges are classified as:• Back edges: (u, v) where v is an

ancestor of u in the DFS forest (self loops are back edges)

• Forward edges: (u, v) where v is a proper descendent of u in the DFS forest

• Cross edges: (u,v) where u and v are not ancestors or descendents of one another. D

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s

If DFS run on an undirected graph

• No difference between back and forward edges: all called back edges

• No cross edges: can you see why?

Parenthesis Theorem

• In any DFS of a graph G= (V,E), for any two vertices u and v, exactly one of the following three conditions holds:

– The intervals [d[u],f[u]] and [d[v],f[v]] are entirely disjoint and neither u nor v is a descendent of the other in the DFS forest

– The interval [d[u],f[u]] is contained within interval [d[v],f[v]] and u is a descendent of v in the DFS forest

– The interval [d[v],f[v]] is contained within interval [d[u],f[u]] and v is a descendent of u in the DFS forest

How can we use DFS to determine whether a graph contains any cycles?