Simple Graph Algorithms - Graz University of Technology · Birgit Vogtenhuber Simple Graph Algorithms 22 iv Recognizing Bipartite Graphs Observation: A graph G is 2-colorable , G

Birgit Vogtenhuber Simple Graph Algorithms1

Simple Graph Algorithms

Design & Analysis of Algorithms WS 2019/20

Birgit Vogtenhuber


Why Graph Algorithms?

• Graphs are a versatile data structure

• Graph-related algorithms are rather important inalgorithm theory and applications

• Many computational problems can be formulatedefficiently in terms of graphs


Plan

• some basic graph terminology

• possibilities to store graphs

• some simple graph algorithms

Birgit Vogtenhuber Simple Graph Algorithms4 ii

Basic Terminology

• Directed graph G = (V,E): V is the set of vertices andE is the set of edges: E ⊆ V × Vedge (u, v): ordered pair of ordered pair of vertices,

u = startpoint, v = endpoint

Example: V = 1, 2, 3, 4, 5,E = (1, 2), (2, 3), (3, 2), (4, 5), (4, 4)

1 2 4 5

3

• A loop at a vertex v ∈ V is an edge (v, v)• A directed Graph is called simple if it has no loops

Birgit Vogtenhuber Simple Graph Algorithms5 iii

Basic Terminology

• Undirected graph: like a simple directed graph whereE is symmetric, that is, (v, w) ∈ E ⇔ (w, v) ∈ E

• A (directed or undirected) weighted graph is a tripleG = (V,E, g) where g assigns a weight to each edge

25

3

82

11

33 4

16

7 5

3

8

⇔ E is a set of unordered pairs (v, w) with v 6= w ∈ V

Birgit Vogtenhuber Simple Graph Algorithms5 x

Basic Terminology

• A path in G = (V,E) from v to w is a sequenceof vertices v = v0, v1, . . . , vk = w ∈ Vwith edges (vi, vi+1) ∈ E for all 0≤ i< k

• Length of a path: number of its edges (vi, vi+1)

length in weighted graphs: sum of the edge weights

• Simple path: contains every vertex in V at most once

v

w

• Distance dG(v, w) from v to w: length of the shortestpath from v to w in G (or ∞ if no such path exists)

Birgit Vogtenhuber Simple Graph Algorithms5 xvi

Basic Terminology

• A cycle in G=(V,E) is a path v0, v1, . . . , vk with v0 =vk

• Simple cycle: contains every vertex in V at most once,except for v0 = vk

• Trivial cycle: contains only one vertex:or two for undirected graphs:

• Length of a cycle: number of its edges (vi, vi+1)


• Length of a cycle: number of its edges (vi, vi+1)



Basic Terminology

• a subgraph G′ = (E′, V ′) of G = (E, V ) is a graphwith V ′ ⊆ V and E′ ⊆ E

Example :Graph G Subgraph G′


Basic Terminology

• A graph is acyclic if does not contain any subgraphthat is a non-trivial simple cycle

• A graph is connected if it contains a path from v to wfor every pair v, w ∈ V (not necessarily an edge (v, w))

connected : disconnected :

acyclic : not acyclic :

disconnected :


Basic Terminology

• An undirected graph is a forest if it is acyclic

• A an undirected grpah is a tree if it is acyclic andconnected


Basic Terminology

• The in-degree (out-degree) of a vertex v ∈ V is thenumber of edges ending (starting) in v:

in-degree (v) := |w | (w, v) ∈ E|out-degree (v) := |w | (v, w) ∈ E|

• For undirected graphs we have

in-degree(v) = out-degree(v) ∀ v ∈ V .

In this case it called the degree of the vertex v.

Degrees of vertices in a graph G = (V,E):


Basic Terminology

• degree (v) = 0: v is called an isolated vertex of G

• degree (v) = 1: v is called a leaf (end vertex) of G

• degree (v) = r ∀ v ∈ V : The graph G is called aregular graph of degree r

Example: regular graph(s) of degree 3

Degrees of vertices in an undirected graph G = (V,E):


Basic Terminology

• An empty graph has no edges (regular of degree 0)

K6

• A complete graph on n vertices contains all(n2

)possible edges (regular of degree n− 1)

K4

Degrees of vertices in an undirected graph G = (V,E):


How to Store a Graph?

⇒ Analysis wants two parameters !

• We distinguish between

dense graphs: m ≈ n2, for example complete graphs

sparse graphs: m n2, for ex. trees (m = n− 1)or hypercubes (m = d

2 · n = O(n log n), where d isthe dimension of the hypercube)

• Size of a graph G = (V,E):

Number of vertices: n = |V | Number of edges: m = |E|, 0 ≤ m ≤ n2

⇒ In total: size Θ(n+m)



Adjacency-matrix: Matrix A[1 . . . n, 1 . . . n] with

A[i, j] =

1 if (i, j) ∈ E0 else

• Memory: Θ(n2)• convenient for dense graphs• test for existence of edge in Θ(1) time

Adjacency-List: Array F [1 . . . n] with pointers,F [i] points to a linear list with all vertices that are incident

to the ith vertex

• Memory: Θ(n+m)• test for existence of edge in Ω(1), O(n) time



1 2

34

A =

0 1 0 01 0 1 10 1 0 10 1 1 0

1 2 3 4

1

2

3

4

F : 1 2 3 4

2 1

3

4

2

4

2

3

↑ symmetric if graph is undirected

← 1 Entry for every node,Θ(n+m) memory for all edges.

The k neighbours of a node canbe obtained in Θ(k) time.

A small example:


Searching in Graphs

• Known from Data Structures and Algorithms (1):

Searching in binary trees

“Walking” through a binary tree:- in-order (symmetrische Reinehfolge)- pre-order (Hauptreihenfolge)- post-order (Nebenreihenfolge)

• Searching in general graphs:

Breadth-first search (BFS):search closeby vertices first

Depth-first search (DFS)search one branch first

Birgit Vogtenhuber Simple Graph Algorithms16 vi

Breadth First Search - BFS

Given: connected graph G = (V,E), startnode s ∈ VIdea: Starting from s, search in all directions uniformly

• visit vertices u with distance dG(s, u) = 1,then with dG(s, u) = 2, and so on.

• traverse a BFS-tree T with root s:branches of T consist of shortest paths to s.

• to build T , assign the following values to each vertex u:

pre(u) predecessor of u in the BFS-Tree Tstate(u) new (unvisited), labelled (visited),

satured (all neighbours visited)

• store visited unsaturated vertices in a queue Q

• initially: Q empty, pre(u) not set, state(u) =new


BFS Algorithm: Pseudo-Code

BFS(G,s) /* G given as adjacency list F */for all u ∈ V

state(u)=newstate(s)=labellednum(s)=1; i=2; pre(s)=nilPUT(Q,s)while Q 6= 0

GET(Q,u)for all v ∈ F [u] /* testing all neighbors of u */

if state(v)==newstate(v)=labelled; num(v)=ipre(v)=u; PUT(Q,v); i = i+ 1

state(u)=satured

Birgit Vogtenhuber Simple Graph Algorithms18 vii

BFS Algorithm: Analysis

Runtime:

• Each vertex is inserted into Q exactly once:Θ(1) time per vertex ⇒ Θ(n) for all vertices

• After removal of a vertex u from Q, the algorithm goesthrough the adjacency-list of u:Θ(degree(u)) time for u ⇒ How much for all vertices?

Every edge contributes to exactly two lists⇒∑

u∈V degree(u) = 2m ⇒ Θ(m) for all vertices

⇒ The whole algorithm: Θ(n+m) time in total

Memory: Θ(n) for Q, vertices ⇒ Θ(n+m) space in total

⇒ Runtime and memory linear in the size of G


All Distances in an Unweighted Graph

Question: How can one compute of all distances betweenpairs of vertices in an unweighted graph G ?

• All the shortest paths from some vertex u to s arecoded in the BFS-tree T via the pre-pointers.

• distances to s can be easily computed during BFS: dG(s, s) = 0 dG(s, u) = dG(s, pre(u)) + 1

⇒ Running BFS for n times (once for each vertex), onecan compute the distances between any u, v ∈ V

⇒ The distance-matrix of G can be computed in Θ(n ·m)time and Θ(n2) space if G is connected.

? What if G is disconnected?

Birgit Vogtenhuber Simple Graph Algorithms20 iv

Bipartite Graphs

A graph G(V,E) is called bipartite, if there exists apartition of V into V1, V2 such that all (u, v) ∈ E have oneendpoint in V1 and the other one in V2

V1 V2

In other words:G contains no edges within V1 or V2


k-Colorability of a Graph

A graph G(V,E) is called k-colorable if its nodes can becolored with ≤ k colors, so that no nodes of the samecolor share an edge.

k = 3 colors are enough

The chromatic number χ(G) (spoken ’Chi of G’) of agraph G is the minimum k such that G is k-colorable.

Example: How many colors are needed?

Examples:

χ(cycle) =

2, n even3, nodd

χ(K4) = 4


Recognizing Bipartite Graphs

Observation: A graph G is 2-colorable ⇔ G is bipartite

Algorithm:• Choose arbitrary vertex s and color it blue• Traverse G in BFS-Order, starting from s• For each vertex u that is removed from the queue Q: u is colored red (blue) check for all colored neighbors that they are blue

(red). If no: return false color all its uncolored neighbors in blue (red)

• After processing all vertices: return true.

Question: How to determine whether a graph is bipartite?

Questions: Correctness? Runtime & Memory?

Birgit Vogtenhuber Simple Graph Algorithms23 v

Depth First Search - DFS

• Idea:DFS explores the graph, starting at the last visitedvertex having unvisited neighbors.

• Special case: G is a tree ⇒ DFS-Order = pre-order

a

b

d e

c

f g h

i j

• Maintain Stack STthat contains allvisited but not yetsatured nodes.

• Rest similar to BFS

Question: What is the pre-order for this tree?

a b d e c f i j g h


DFS Algorithm: Pseudo-Code

DFS(G) /* G given as adjacency list F */for all u ∈ V

state(u)=newpre(u)=nil

for all u ∈ V /* loop not necessary for connected graphs */if state(u)==new

DEPTH(u)

DEPTH(u)state(u)=visited; write(u)for all v ∈ F [u] /* test all neighbors of u */

if state(v)==newpre(v)=uDEPTH(v) ← recursion can be replaced by stack

state(u)=saturated


DFS Algorithm: Example

1 2 3

5 6 7

root

Example: F : 1 2 3 4

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6

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8

5 6 7 8

2

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3 4

4

8

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root1

6

arrows point to predecessors


DFS Algorithm: Example

Further Observations:

• The pre-pointers form a set of trees (DFS-forest);

• every call of DEPTH in the main programm results in anew root and tree.

• For connected graphs there is only one root and tree.

1 2 3

5 6 7

root

Example: F : 1 2 3 4

1

6

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7

8

5 6 7 8

2

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3 4

4

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root1

6


DFS Algorithm: Analysis

Runtime Analysis:

• DEPTH ist called exactly once per node (only for newnodes, that are immediately marked as ”visited”).

• A call of DEPTH(v) takes O(degree(v)) time

⇒ Θ(n+m) time in total

Space: Θ(n+m) space in total

Correctness:• vertex that is set to visited: put on stack when removed from stack, all neighbors are considered

⇒ every vertex set to visited exactly once


Next

Planar and Plane Graphs:

Definition, Recognition, Properties


Isomorphic Graphs

Two graphs G = (V,E) and G′(V ′, E′) are calledisomorphic if there exists a bijection φ : V → V ′, so that(v, w) ∈ E ⇔ (φ(v), φ(w)) ∈ E′

2 3

1 4

Example:

4

1

2

3

⇔

5 55


Planar Graphs

Definition 1:A graph G = (V,E) is called planar if there exists aninjective mapping of V to n = |V | distinct points in theEuclidian plane and a mapping of E to pairwise disjointopen curves in the Euclidian plane, so that:

1. e = (xi, xj)⇒ curve (e) connects point (xi) withpoint (xj)

2. e = (xi, xj)⇒ curve (e) does not contain any otherpoint (xk), k 6= i, j

Definition 2:A graph is called planar if it can be drawn in the planewithout intersections between its edges (except in endpoints).


Plane & Embeddable Drawings

A drawing of a graph is called plane if it is free ofintersections.

A non-plane drawing of a graph is called embeddable if itis isomorphic to a planar graph.

A plane graph is a plane drawing of a planar graph.

2 3

1 4

4

1

2

3

⇔2 3

1 4

Example of an embeddable / plane drawing:

4

1

2

3

⇔


Recognizing Planar Graphs

Examples of non-embeddable drawings:

K3,3 K5

K3,3 K5

Non-embeddable drawing ⇔ drawing of a non-planar graph




K3,3 K5

K3,3 K5

Embeddable drawing ⇔ drawing of a planar graph

Birgit Vogtenhuber Simple Graph Algorithms31 ix



K3,3 K5

Kuratowski’s Theorem: A graph G is planar if and only ifit does not contain a subgraph G′ that is isomorphic to asubdivision of K3,3 or K5, respectively. [ without proof ]

Subdivision of a graph G: some vertices are added“on edges” of G (some edges of G replaced by paths)

additional vertices: degree 2

embeddability is invariant undersubdivision of edges

Birgit Vogtenhuber Simple Graph Algorithms32 viii

Plane Graphs: Euler’s Formula

Observation:A plane graph (= plane drawing of a planar graph) dividesthe plane into regions called the faces of the drawing

Theorem: Euler’s Formula (on plane graphs)Given a connected, plane graph with exactly f faces,n = |V |,m = |E|. Then it holds that:

n+ f = m+ 2In other words: vertices - edges + faces = 2

4 faces

12 edges

10 vertices

10+4=12+2 X


Planar Graphs: Example

1011

12

13

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8

76

3

2

4

15

Vertices: n = 14Edges: m = 13Faces: f = 1

14 + 1 = 13 + 2 X


Planar Graphs: Example

1

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3

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69

1011

12

14

1715

5

18

13

Vertices: n = 18Edges: m = 21Faces: f = 5

16

18 + 5 = 21 + 2 X

Note: All plane drawings of the same graph have thesame number of faces


Planar Graphs

Theorem: For a planar connected graph G = (V,E) with|V | = n ≥ 3 and |E| = m it holds that m ≤ 3 · n− 6

Proof: Consider a plane drawing of G with f faces. Letei, i = 1, . . . , f be the number of edges which bound face i.

•∑f

i=1 ei ≥∑f

i=1 3 = 3 · f (since ei ≥ 3)

•∑f

i=1 ei = 2 ·m(since each edge is counted twice – once from each side)

⇒ Combine the two results: 3 · f ≤ 2 ·m⇒ Combine with Euler’s Formula (multiplied by 3):

3 ·m+ 6 = 3 · n+ 3 · f ≤ 3 · n+ 2 ·m⇔ m ≤ 3 · n− 6 X


Non-Planarity of K5

Claim: K5 is not planar.

Proof:• K5 has n = 5 vertices and m =

(52

)= 10 edges

• Assume K5 is planar

⇒ We can apply the previous theorem

m ≤ 3 · n− 6, with n = 5,m = 10

10 ≤ 3 · 5− 6 = 9 contradiction!

⇒ K5 cannot be planar. X

Question:Can we show in the same way that K3,3 is not planar?


Non-Planarity of K3,3

Theorem: For a bipartite planar connected graph G = (V,E)with |V | = n ≥ 3 and |E| = m it holds that m ≤ 2 · n− 4

Proof: Consider a plane drawing of G with f faces. Letei, i = 1, . . . , f be the number of edges which bound face i.

• G bipartite ⇒ every face is 2-colorable ⇒ ei ≥ 4

•∑f

i=1 ei ≥∑f

i=1 4 = 4 · f ,∑f

i=1 ei = 2 ·m⇒ Combine the two results: 4 · f ≤ 2 ·m ⇔ 2 · f ≤ m⇒ Combine with Euler’s Formula (multiplied by 2):

2 ·m+ 4 = 2 · n+ 2 · f ≤ 2 · n+m⇔ m ≤ 2 · n− 4 X


Non-Planarity of K3,3

Claim: K3,3 is not planar.

Proof:• K3,3 has n = 6 vertices and m = 3 · 3 = 9 edges

• Assume K3,3 is planar

⇒ We can apply the previous theorem

m ≤ 2 · n− 4, with n = 6,m = 9

9 ≤ 2 · 6− 4 = 8 contradiction!

⇒ K3,3 cannot be planar. X

Question:Can we use Euler’s formula to show how many edges andtriangles a triangulation of n points in the plane has ?

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Simple Graph Algorithms - Graz University of Technology · Birgit Vogtenhuber Simple Graph Algorithms 22 iv Recognizing Bipartite Graphs Observation: A graph G is 2-colorable , G