advanced data structures uvt

Applications of the depth-first search algorithmUndirected Graphs

Lecture 3

Mircea Marin

October 2013

Mircea Marin Applications of the depth-first search algorithm Undirected Graphs

Outline

Applications of the depth-first search algorithmDepth-first spanning forestDAG recognitionTopological sort of DAGsComputation of strong components of digraphs

Undirected graphsDefinition, main notionsMain operations


Depth-first search

Given a graph G = (V ,E) and a node s ∈ VFind all nodes reachable by a path from s, by searching

"deeper" in the graph whenever possible.Edges are explored out of the most recentlydiscovered vertex v that still has unexplorededges leaving it.When all of v ’s edges have been explored,the algorithm "backtracks" to its parent node,to explore other leaving edges.


Depth-first traversal (DFT)Iterative version (see also Lecture 2)

procedure dfs(v : vertex)first-visit-op(v) /* first visit of v */S:= a stack containing only node vwhile S is not empty

u:=S.peek()if u has unvisited neighbors

w:=the first unvisited neighbor of ufirst-visit-op(w) /* first visit of w */mark[w]:=visitedS.push(w)

elsew:=S.pop()last-visit-op(w) /* last visit of w */

NOTES:

I mark is a global array (or map) initialized withmark[w ] := unvisited for all nodes w

I a node is marked as visited when it is first encountered.Mircea Marin Applications of the depth-first search algorithm Undirected Graphs

Compare with breadth-first traversalA simplified version of breadth first-search (see also Lecture 2)

procedure bfs(v : vertex)first-visit-op(v) /* first visit of v */Q:= a queue containing only node vwhile Q is not empty

extract last element u from Qlast-visit-op(u) /* last visit of u */for all neighbours w of u do

if mark[w ] = unvisitedmark[w ] := visitedfirst-visit-op(w) /* first visit of w */Q.push(w)

NOTES:


I a node is marked as visited when it is first encountered.


Depth-first traversal (DFT)Recursive version

DFT is more natural to describe recursively (pseudocode):

procedure dfs(v : vertex)mark[v] := visitedfirst-visit-op(v) /* first visit of v */for each neighbor w of v do

if mark[w]=unvisited thendfs(w) /* recursive call */

last-visit-op(v) /* last visit of v */

NOTES:


I a node is marked as visited when it is first visited.


Applications of the depth-first search algorithmDepth-first spanning forest

Consider the digraph G E

F

G

B

D

A

C

with adjacency listsA 7→ [B,C] B 7→ [C,D] C 7→ [A]D 7→ [C] E 7→ [F ,G]F 7→ [B] G 7→ [F ,D]

Some depth-first traversals of G do not visit all nodes of G:A

BC D

E

F G

I Nodes E ,F ,G are not visited.

By selecting unvisited nodes as long as possible andrestarting depth first traversal from them, we produce adepth-first spanning forest.



Consider the digraph G E

F

G

B

D

A

C

with adjacency listsA 7→ [B,C] B 7→ [C,D] C 7→ [A]D 7→ [C] E 7→ [F ,G]F 7→ [B] G 7→ [F ,D]

Some depth-first traversals of G do not visit all nodes of G:A

BC D

E

F G

I Nodes E ,F ,G are not visited.By selecting unvisited nodes as long as possible andrestarting depth first traversal from them, we produce adepth-first spanning forest.



A1

B2

C 3 D4

E5

F6

G7

dfnumber [A] = 1dfnumber [B] = 2dfnumber [C] = 3dfnumber [D] = 4. . .

Drawing strategy:

The nodes of depth first-search tree are drawn from left to right, in the order oftheir first visit.

Assign a number dfnumber [n] to every node n, starting from 1, in the orderin which they are visited first by the depth-first search.

The trees are drawn from left to right, in the order in which they are generated.

Remarks: There are 4 kinds of arcs

1 The solid arcs connecting nodes in the depth-first search trees are tree arcs.2 The other arcs, which are dashed, are of 3 kinds:

(1) back arcs: connect a node with an ancestor in its tree: C → A, D → A(2) forward arcs: connect a node with a descendant in its tree: A→ C(3) cross arcs: all other arcs: F → B,G→ F ,G→ D



A1

B2

C 3 D4

E5

F6

G7


Drawing strategy:




Remarks: There are 4 kinds of arcs1 The solid arcs connecting nodes in the depth-first search trees are tree arcs.

2 The other arcs, which are dashed, are of 3 kinds:(1) back arcs: connect a node with an ancestor in its tree: C → A, D → A(2) forward arcs: connect a node with a descendant in its tree: A→ C(3) cross arcs: all other arcs: F → B,G→ F ,G→ D



A1

B2

C 3 D4

E5

F6

G7


Drawing strategy:




Remarks: There are 4 kinds of arcs1 The solid arcs connecting nodes in the depth-first search trees are tree arcs.2 The other arcs, which are dashed, are of 3 kinds:

(1) back arcs: connect a node with an ancestor in its tree: C → A, D → A(2) forward arcs: connect a node with a descendant in its tree: A→ C(3) cross arcs: all other arcs: F → B,G→ F ,G→ D



A1

B2

C3

D4

E5

F6

G7


Characterizations of the node relations by vertex numbering

1 A node u is a descendant of v iff dfnumber [v ] ≤ dfnumber [u] ≤ dfnumber [v ]+number of descendants of v . Thus:

forward arcs go from low-numbered nodes to high-numbered nodes.back arcs go from high-numbered nodes to low-numbered nodes.

2 All cross arcs go from high-numbered to low-numbered nodes.



A1

B2

C3

D4

E5

F6

G7


Characterizations of the node relations by vertex numbering1 A node u is a descendant of v iff dfnumber [v ] ≤ dfnumber [u] ≤ dfnumber [v ]+

number of descendants of v . Thus:

forward arcs go from low-numbered nodes to high-numbered nodes.back arcs go from high-numbered nodes to low-numbered nodes.

2 All cross arcs go from high-numbered to low-numbered nodes.


Directed acyclic graphs (DAGs)A directed acyclic graph (DAG for short) is a digraph without cycles.

1 DAGs are useful in representing the syntactic structure of arithmetic expressionswith common subexpressions.

Example (DAG for (a + b) ∗ c + ((a + b) + e) ∗ (e + f )) ∗ (a + b) ∗ c)

a b+

∗c

+

e f

∗+

∗+

2 DAGs can be used to represent partial orders (=irreflexive and transitive).

Example (DAG for the relation ⊂ between subsets of {1, 2, 3})


Applications of depth-first search

1 Test for acyclicity: Determine whether a digraph is acyclic.Solution: Generate a depth-first spanning forest.

The digraph is acyclic iff it contains a back arc.2 Topological sort

Scenario

A large project is often divided into a collection of smaller tasks, some of which have tobe performed in certain specified orders so that we may complete the entire project.For example, a university curriculum may have courses that require other courses asprerequisites. We can model the required orders with a DAG, e.g.:

C1

C2

C3

C4

C5

Requirement: Assign a linear ordering to the nodes of a DAG s.t. if there is an arcfrom i to j then i occurs before j in the linear ordering.The linear ordering is called a topological sort.

Topological sort can be accomplished easily with depth first search.




The digraph is acyclic iff it contains a back arc.

2 Topological sort

Scenario


C1

C2

C3

C4

C5






The digraph is acyclic iff it contains a back arc.2 Topological sort

Scenario


C1

C2

C3

C4

C5




Applications of Depth-First SearchTopological sort using depth-first search

topsort(node v) is the depth-first search procedure adjusted with a statementto print the nodes accessible from a node v in reverse topological order.

Pseudocode

void topsort(node v) {nodeColor[v]=BLACK; // mark v as visitedfor every node w adjacent to v do

if (nodeColor[w] == WHITE) { // if w is not visitedtopsort(w);

}cout << v.getName();

}

I Remark. When topsort finishes searching all nodes adjacent to a node x , itprints x .

⇒ topsort(v) will print in a reverse topological order all nodes of a DAG whichare accessible from v by a path in the DAG.


Applications of Depth-First SearchStrong components

Assumption: G = (V ,E) is a digraph.A strongly connected component of G is a subgraph (V ′,E ′) ofG where V ′ is a maximal subset of V s.t. there is a path fromany node in the set to any other node in the set.

v ≡ w iff there exists a path from v to w and also a pathfrom w to v is an equivalence relation on V

⇒ V can be partitioned into a family {Vi | 1 ≤ i ≤ r} ofequivalence classes w.r.t. ≡, and then build the strongcomponents Gi = (Vi ,Ei) of G.

A digraph with only one strong component is calledstrongly connected.

Example

a b

cdhas 2 strong components:

a b

cd


Applications of Depth-First SearchStrong components

Remarks

Every node of a digraph belongs to a strong component.

Some arcs of a digraph may not belong to a strong component. Such arcs arecross-component arcs.

The strong components of a digraph G can be found by constructing a reducedgraph G′ of G:

I the nodes of G′ are the strong components of GI there is an arc from C in C′ in G′ iff there is an arc in G

from some node in C to some node in C′.

Example:

a b

cdhas the reduced graph

a,b, c

d


Applications of Depth-First SearchComputation of strong components

1 Perform a depth-first search of G and number the nodes inpost-order traversal of the depth-first search tree (that is,assign to the parent node a larger number than to itschildren)

2 Construct the digraph Gr obtained from G by reversing thedirection of all its arcs.

3 Perform depth-first search of Gr , starting the search fromthe node with highest number assigned in step 1. Ifdepth-first search does not reach all nodes, start the nextdepth-first search from the highest-numbered remainingnode.

4 Each tree in the resulting spanning forest is a stronglyconnected component of G.


Computation of strong componentsExample

a b

cdFig. 1

a4 b 3

c1 d 2

Fig. 2

da

c

b3

2

4 1

Fig. 3

I Start performing depth-first search from node a in Fig. 1;number increasingly the vertices during the time of theirlast visit (step 1) and revert the arcs of G (step 2)⇒digraph Gr shown in Fig. 2.

I Perform depth-first search in Gr starting with a because ahas highest assigned number. The first depth-first searchtree will contain nodes a,b, c. Then, continue with root d tobuild the next depth-first search tree . . .

⇒ Depth-first spanning forest shown in Fig. 3.Each tree of the forest forms a strong component of theoriginal graph.


Computation of strong componentsExample

a b

cdFig. 1

a4 b 3

c1 d 2

Fig. 2

da

c

b3

2

4 1

Fig. 3

I Start performing depth-first search from node a in Fig. 1;number increasingly the vertices during the time of theirlast visit (step 1) and revert the arcs of G (step 2)⇒digraph Gr shown in Fig. 2.

I Perform depth-first search in Gr starting with a because ahas highest assigned number. The first depth-first searchtree will contain nodes a,b, c. Then, continue with root d tobuild the next depth-first search tree . . .⇒ Depth-first spanning forest shown in Fig. 3.

Each tree of the forest forms a strong component of theoriginal graph.


Undirected graphsG = (V ,E) where

V = finite set of vertices (or nodes)

E = finite set of edges between nodes.Every edge e ∈ E connects two nodes u, v ∈ V called the nodes incident to e.

An undirected graph can be

simple: there is at most one edge between any two nodes. An edge between

nodes u, v is represented as u−v and drawn asu v .

It is assumed that edges are unordered, thus u−v = v−u.If u−v is an edge than we say nodes u and v are adjacent.

multiple: there can be more than one edge between two nodes.Distinct edges between same nodes are distinguished by labelingthem. An edge with label e between nodes u and v is represented as

ue− v and drawn as

u ve

.

It is assumed that edges are unordered, thus ue− v = v

e− u.

path = sequence of nodes π = v1, . . . , vn such that v1−v2, . . . , vn−1−vn ∈ E .The length of π is n − 1. π is a path between v1 and vn.

simple path = path with distinct nodes, except possibly the first and last.

simple cycle = path of length at least 1 that begins ad ends at the same node.


Examples of undirected graphs

Acquaintanceship graph=simple graph where edges connectpeople that know each other.

Computer network: multigraph where edges are between citieslinked by computer connections.


Auxiliary notions

Assumption: G = (V ,E) undirected graph.Subgraph of G: G′ = (V ′,E ′) where V ′ ⊆ V and E ′ = the set of edges in E

between nodes in V ′.

a b

cd

V = {a,b, c,d}

a b

c

V ′ = {a,b, c}

Two nodes are connected if there is a path between them.

A connected component of G is a subgraph G′ = (V ′,E ′) with V ′ maximal suchthat every u, v ∈ V ′ are connected.

Figure : An unconnected graph with 2 connected components


Undirected graphs: Methods of RepresentationAdjacency matrix, adjacency lists, and edge list

a b

cd

1 Adjacency matrixNote: This is a symmetric matrix.

2 Adjacency lists

3 Edge list: [a−b,b−c, c−d ,d−a,d−b]


Operations on Undirected Graphs

I All operations on directed graphs carry over to undirectedgraphs, with the difference that every undirected edge u−v maybe thought as the pair of directed edges u → v and v → u.

I Terminology specific to unconnected graphs G = (V ,E)

Connected component = maximal connected inducedsubgraph G′ = (V ′,E ′) of G, that is:

B All nodes of V ′ are connectedB There is no path between a node in V ′ and a node in V − V ′.

Free tree = a connected acyclic graph.

Properties of depth-first spanning forests in undirected graphs:1 There is no distinction between forward and backward

edges.2 There are no cross edges, that is, edges v−w such that v

is neither ancestor nor descendant of w .

⇒ all edges are either tree edges or back edges.


References

I Alfred V. Aho et al. Data Structures and Algorithms.Chapters 6 and 7.

I K. H. Rosen. Discrete Mathematics and Its Applications.Chapter 9.


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