cs.uni-paderborn.de€¦ · ss 2019 chapter 5 3 basic notation a directed graph g is a pair (v,e),...

Premaster Course Algorithms 1

Chapter 5: Basic Graph Algorithms

Christian ScheidelerSS 2019

Basic Graph Algorithms

Overview:• Basic notation• Graph representations• Breadth-First-Search• Depth-First-Search• Minimum spanning trees

SS 2019 Chapter 5 2

SS 2019 Chapter 5 3

Basic Notation

A directed graph G is a pair (V,E), where V is a finite setand E⊆V×V.

Elements in V are called nodes or vertices, elements in Eare called edges. Correspondingly, V is the node set andE the edge set of G.

Edges are ordered pairs of nodes. Edges of the form (u,u), u∈V, are allowed and called loops.

If (u,v)∈E, we say that the edge leads from u to v. Wealso say that u and v are adjacent.

The degree of a node v is the number of edges (v,w) in E.

SS 2019 Chapter 5 4

Directed Graph

1 2

5

3

4 6

{ }

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }3,6,4,5,5,4,1,4,5,2,4,2,2,2,2,1E

6,5,4,3,2,1V

=

=

SS 2019 Chapter 5 5

Basic Notation

An undirected graph G is a pair (V,E), where V is a finite set and E is a set of subsets of V of size two.

Elements in V are called nodes or vertices, elements in Eare called edges. Correspondingly, V is the node set andE the edge set of G.

Edges have the form {u,v}. Edges of the form {u,u} areusually not allowed.

If {u,v}∈E, then we say that u and v are adjacent.

The degree of u is the number of edges {u,v} in E.

SS 2019 Chapter 5 6

Undirected Graph

1 2

5

3

4 6

V={1,2,3,4,5,6}

E={ {1,2}, {1,5}, {2,5}, {3,6} }

SS 2019 Chapter 5 7

• n: number of nodes, m: number of edges• δ(v,w): distance of w to v in G

- directed graph: number of edges of a shortest directed pathfrom v to w

- undirected draph: number of edges of a shortest path from vto w

• D=maxv,w δ(v,w): diameter of G

ADB C

D=4E

Graph Theory

SS 2019 Chapter 5 8

G undirected: • G is connected: diameter D is finite (i.e., there is a path

from every node to every other node in G)Example: graph that is not connected

ADB C

E

Graph Theory

SS 2019 Chapter 5 9

G directed: • G is weakly connected: diameter D is finite, if all edges

are seen as undirected• G is strongly connected: D is finite

ADB C

E

Graph Theory

SS 2019 Chapter 5 10

Linear list:

• degree 2• large diameter (n-1 for n nodes)→ bad for data structures

Graph Theory


Complete binary tree:

• n=2d+1-1 nodes, degree 3• Diameter is 2d ~ 2 log2 n→ good for data structures

0

d

depth d

Graph Theory


2-dimensional grid:

• n = k2 nodes, maximum degree 4• diameter is 2(k-1) ~ 2 n

1

k

side length k

Graph Theory


1: Sequence of edges

1 2

4 3

(1,2)/ (4,1)(3,4)(2,3)

Graph Representations

Dummy


2: Adjacency list

1 2

4 3 2

3

V

3

4

4 1

1 n

Graph Representations


3: Adjacency matrix

• A[i,j]∈{0,1}• A[i,j]=1 if and only if (i,j)∈E

1 2

4 3

0 1 1 00 0 1 10 0 0 11 0 0 0

A =

Graphrepräsentationen


Central question: How can we traverse thenodes of the graph so that every node isvisited at least once?

Graph Traversal


Central question: How can we traverse thenodes of the graph so that every node isvisited at least once?

Basic strategies:• Breadth-first search• Depth-first search

Graph Traversal


• Start at some node s• Explore graph distance by distance from s

s

Breadth-First Search


• Start from some node s• Explorate graph in depth-first manner

( : current, : still active, : done)

s

Depth-First Search



Breadth-first search algorithm forms the base of manyother graph algorithms.

Given a graph G=(V,E) and a source s∈V, the goal ofbreadth-first search is to find all nodes v∈V that arereachable from s. A node v is reachable from s if there isa (directed) path in G from s to v.

The breadth-first search algorithm can also be used tocompute the distance δ(s,v) of v from s.



If a new node v is discovered while searching for new nodesin Adj[u], then u is called the predecessor of v.(Adj[u]: set of neighbors or adjacency list of u, i.e., the nodesv∈V with {u,v}∈E resp. (u,v)∈E)

In the algorithm: nodes are white, gray, or black.

White nodes are nodes that have not been discovered yet.

Gray nodes are nodes that have already been discovered but whose adjacency list has not completely been explored yet.

Black nodes are nodes that have already been discoveredand fully processed.


Pseudo-code for Breadth-First Search

BFS uses queue for gray nodes. [ ] =:ucolor stores color of v. Initially

WHITE. [ ] =:ud field for distance to s. Initially ∞.

[ ] =π :u field for predecessor. Initially NIL.

BFS(G,S)1 for each node u∈V\{s} do2 color[u]←WHITE3 d[u] ← ∞4 p[u] ← NIL5 color[s] ← GRAY6 d[s] ← 07 π[s] ← NIL8 Q ← { }9 Enqueue(Q,s)10 while Q ≠ { } do11 u ← Dequeue(Q)12 for each v∈Adj[u] do13 if color[v]=WHITE then14 color[v] ← GRAY15 d[v] ← d[u]+116 π[v] ← u17 Enqueue(Q,v)18 color[u] ← BLACK

· BFS uses queue for gray nodes.

·

[

]

=

:

u

color

stores color of v. Initially WHITE.

·

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]

=

:

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d

field for distance to s. Initially

¥

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]

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p

:

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field for predecessor. Initially NIL.

_1180339330.unknown

_1180339381.unknown

_1180339402.unknown

_1180339283.unknown


Pseudo-code for Breadth-First Search

initialize BFS

BFS(G,S)1 for each node u∈V\{s} do2 color[u]←WHITE3 d[u] ← ∞4 p[u] ← NIL5 color[s] ← GRAY6 d[s] ← 07 π[s] ← NIL8 Q ← { }9 Enqueue(Q,s)10 while Q ≠ { } do11 u ← Dequeue(Q)12 for each v∈Adj[u] do13 if color[v]=WHITE then14 color[v] ← GRAY15 d[v] ← d[u]+116 π[v] ← u17 Enqueue(Q,v)18 color[u] ← BLACK

BFS uses queue for gray nodes. [ ] =:ucolor stores color of v. Initially

WHITE. [ ] =:ud field for distance to s. Initially ∞.

[ ] =π :u field for predecessor. Initially NIL.

· BFS uses queue for gray nodes.

·

[

]

=

:

u

color

stores color of v. Initially WHITE.

·

[

]

=

:

u

d

field for distance to s. Initially

¥

.

·

[

]

=

p

:

u

field for predecessor. Initially NIL.

_1180339330.unknown

_1180339381.unknown

_1180339402.unknown

_1180339283.unknown

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BFS(G,s)1. „initialize BFS“2. while Q≠∅ do3. u ← Dequeue(Q)4. for v∈Adj[u] do5. if color[v]=WHITE then6. color[v] ← GRAY7. d[v] ← d[u]+1; π[v] ← u8. Enqueue(Q,v)9. color[u] ← BLACK

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Breadth-First Search Example

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Breadth-First Search Runtime

Theorem 5.1: Given a graph G=(V,E) and a sources, algorithm BFS has a runtime of

( )EV +O


Depth-First Search

BFS(G,S)1 for each node u∈V\{s} do2 color[u]←WHITE3 p[u] ← NIL4 color[s] ← GRAY5 π[s] ← NIL6 Q ← { }7 Enqueue(Q,s)8 while Q ≠ { } do9 u ← Dequeue(Q)10 for each v∈Adj[u] do11 if color[v]=WHITE then12 color[v] ← GRAY13 π[v] ← u14 Enqueue(Q,v)15 color[u] ← BLACK

DFS(G,S)1 for each node u∈V\{s} do2 color[u]←WHITE3 p[u] ← NIL4 color[s] ← GRAY5 π[s] ← NIL6 Q ← { }7 Push(Q,s)8 while Q ≠ { } do9 u ← Pop(Q)10 for each v∈Adj[u] do11 if color[v]=WHITE then12 color[v] ← GRAY13 π[v] ← u14 Push(Q,v)15 color[u] ← BLACK


Instead of an iterative approach we will look at a recursiveapproach of realizing depth-first search.

Recursive approach:• Initially: all nodes are white• Discovered nodes become gray• Fully processed nodes become blackTwo time stamps: d[v] and f[v] (lie between 1 and 2|V|)• d[v]: time at which v is discovered• f[v]: time at which v has been processed

Depth-First Search

85

DFS(G)1. for each vertex u∈V do color[u] ← WHITE ; π[u] ← nil2. time ← 03. for each vertex u∈V do4. if color[u]=WHITE then DFS-Visit(u)

DFS-Visit(u)1. color[u] ← GRAY2. time ← time +1; d[u] ← time4. for each v∈Adj[u] do5. if color[v] = WHITE then π[v] ← u ; DFS-Visit(v)6. color[u] ← BLACK7. time ← time+1; f[u] ← time


Depth-First Search

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Depth-First Search

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Theorem 5.2: Given a graph G=(V,E), algorithm DFS has a runtime of ( )EV +O .

Depth-First Search Runtime

Theorem 5.2: Given a graph G=(V,E), algorithm DFS has a runtime of

(

)

E

V

+

O

.

_1180524334.unknown

Classification of edges:• Tree edges are edges of the DFS-forest in G• Back edges are edges (u,v) that connect node u with

one of its ancestors in the DFS-tree• Forward edges are non-tree edges (u,v) that connect

node u with one of its descendents in the DFS-tree• Cross edges are the other edges

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Depth-First Search




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Depth-First Search




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Depth-First Search


Classification of edges (v,w):

Edge type d[v]< d[w] f[v]>f[w]

tree & forward yes yes

back no no

cross no yes

Depth-First Search


Application:• Recognition of acyclic directed graphs

(DAG)

Property: no directed cycles

Depth-First Search


Theorem 5.3: The following statements areequivalent:1. G is a DAG2. The DFS-tree does not contain a back edge3. ∀(v,w)∈E: f[v]>f[w]

Topological sort: Assign numbers s(v) to the nodes vso that for all edges (v,w)∈E, s(v)


Minimum Spanning Trees

Definition 5.4:

1. A weighted undirected graph (G,w) is an undirected graph G=(V,E) with a weightfunction w:E→ℝ

2. For any subgraph H=(U,F) (i.e., U⊆V, F⊆E) ofG, the weight w(H) of H is defined as

w(H) = Σ w(e)e∈F


Definition 5.5:

1. A subgraph H of an undirected graph G iscalled a spanning tree of G=(V,E) if H is a treeover all nodes of G.

2. A spanning tree S of a weighted undirectedgraph G is called a minimum spanning tree(MST) of G if the weight of S is minimal among all spanning trees of G.



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Goal: Given a weighted undirected graph (G,w) withG=(V,E), find a minimum spanning tree of (G,w).

Approach: We successively extend the edge setA⊆E to a minimum spanning tree.

1. Initially, A={ }.2. In each step, we extend A to A∪{ {u,v} }, where

{u,v} is an A-safe edge, until |A|=|V|-1.

Definition 5.6: {u,v} is called A-safe if under theassumption that A can be extended to an MST, also A∪{ {u,v} } can be extended to an MST.



Generic MST-Algorithm

Generic-MST(G,w)1. A ← { }2. while A is not a spanning tree do3. find an A-safe edge {u,v}4. A ← A ∪ { {u,v} }5. return A

Correctness: results from the definition of A-safe edges


Cuts in GraphsDefinition 5.7: 1. A cut (C,V\C) in a graph G=(V,E) is a partition of the node

set V of the graph. 2. An edge of G is crossing a cut (V,V\C) if one node of it is in

C and the other node in V\C.3. A cut (C,V\C) respects a subset A⊆E if no edge in A crosses

the cut.4. An edge crossing the cut (C,V\C) is called light if it has a

minimal weight among all edges crossing (C,V\C).

CV\C

A


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Cuts in Graphs

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Characterization of Safe Edges

Theorem 5.8: Let (G,w) with G=(V,E) be a weightedundirected graph, and let A⊆E be contained in a minimum spanning tree of (G,w). Moreover, let(C,V\C) be a cut respecting A and {u,v} be a light edge crossing (C,V\C). Then {u,v} is an A-safe edge.

Corollary 5.9: Let (G,w) with G=(V,E) be a weightedundirected graph, and let A⊆E be contained in a minimum spanning tree of (G,w). If {u,v} is an edge of mimimal weight that connects a connectedcomponent C of GA=(V,A) with the rest of thegraph GA, then {u,v} is an A-safe edge.


Prim‘s Algorithm

• At any time of the algorithm, GA=(V,A) consists of a treeTA and a set of isolated nodes IA.

• A edge of minimal weight that connects IA with TA isadded to A.

TANode v with minimum w(u,v) over all edges crossing the cut

u


• At any time of the algorithm, GA=(V,A) consists of a treeTA and a set of isolated nodes IA.

• A edge of minimal weight that connects IA with TA isadded to A.

• The nodes in IA are organized in a Min-Heap. The keyd[v] of a node v∈IA is given by the minimal weight of an edge that connects v with TA.

Prim‘s Algorithm


Prim(G,w,s)1. for each vertex v∈V do d[v]←∞; π[v]←nil2. d[s]←0; Q←BuildHeap(V)3. while Q≠∅ do4. u←deleteMin(Q)5. for each vertex v∈Adj[u] do6. if v∈Q and d[v]>w(u,v) then7. DecreaseKey(Q,v,w(u,v)); π[v]←u

Prim‘s Algorithm

DecreaseKey(Q,v,d) operation:Reduces d[v] of v in Q to d and performs heapifyUpfrom the current position of v to repair heap property.


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Prim‘s Algorithm


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Correctness of Prim‘s algorithm for general instances(G,w): results from correctness of Generic-MST andCorollary 5.9.

Prim‘s Algorithm

Premaster Course Algorithms 1��Chapter 5: Basic Graph AlgorithmsBasic Graph AlgorithmsBasic NotationDirected GraphBasic NotationUndirected GraphFoliennummer 7Foliennummer 8Foliennummer 9Foliennummer 10Foliennummer 11Foliennummer 12Foliennummer 13Foliennummer 14Foliennummer 15Foliennummer 16Foliennummer 17Foliennummer 18Foliennummer 19Breadth-First SearchBreadth-First SearchPseudo-code for Breadth-First SearchPseudo-code for Breadth-First SearchFoliennummer 24Foliennummer 25Foliennummer 26Foliennummer 27Foliennummer 28Foliennummer 29Foliennummer 30Foliennummer 31Foliennummer 32Foliennummer 33Foliennummer 34Foliennummer 35Foliennummer 36Foliennummer 37Foliennummer 38Foliennummer 39Foliennummer 40Foliennummer 41Foliennummer 42Foliennummer 43Foliennummer 44Foliennummer 45Foliennummer 46Foliennummer 47Foliennummer 48Foliennummer 49Foliennummer 50Foliennummer 51Foliennummer 52Foliennummer 53Foliennummer 54Foliennummer 55Foliennummer 56Foliennummer 57Foliennummer 58Foliennummer 59Foliennummer 60Foliennummer 61Foliennummer 62Foliennummer 63Foliennummer 64Foliennummer 65Foliennummer 66Foliennummer 67Foliennummer 68Foliennummer 69Foliennummer 70Foliennummer 71Foliennummer 72Foliennummer 73Foliennummer 74Foliennummer 75Foliennummer 76Foliennummer 77Foliennummer 78Foliennummer 79Foliennummer 80Foliennummer 81Breadth-First Search RuntimeDepth-First SearchFoliennummer 84Foliennummer 85Foliennummer 86Foliennummer 87Foliennummer 88Foliennummer 89Foliennummer 90Foliennummer 91Foliennummer 92Foliennummer 93Foliennummer 94Foliennummer 95Foliennummer 96Foliennummer 97Foliennummer 98Foliennummer 99Foliennummer 100Foliennummer 101Foliennummer 102Foliennummer 103Foliennummer 104Foliennummer 105Foliennummer 106Foliennummer 107Foliennummer 108Foliennummer 109Foliennummer 110Foliennummer 111Foliennummer 112Foliennummer 113Foliennummer 114Foliennummer 115Foliennummer 116Foliennummer 117Foliennummer 118Foliennummer 119Foliennummer 120Foliennummer 121Foliennummer 122Foliennummer 123Foliennummer 124Foliennummer 125Foliennummer 126Foliennummer 127Foliennummer 128Foliennummer 129Foliennummer 130Foliennummer 131Foliennummer 132Foliennummer 133Foliennummer 134Foliennummer 135Foliennummer 136Foliennummer 137Foliennummer 138Foliennummer 139Foliennummer 140Foliennummer 141Foliennummer 142Minimum Spanning TreesMinimum Spanning TreesMinimum Spanning TreesMinimum Spanning TreesMinimum Spanning TreesGeneric MST-AlgorithmCuts in GraphsCuts in GraphsCharacterization of Safe EdgesPrim‘s AlgorithmPrim‘s AlgorithmPrim‘s AlgorithmPrim‘s AlgorithmPrim‘s AlgorithmPrim‘s AlgorithmPrim‘s AlgorithmPrim‘s Algorithm

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