Struktur Data & Algoritme Objectives Outline Graph Application

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Struktur Data & Algoritme (

Data Structures & Algorithms

)

Denny ([email protected]) Suryana Setiawan ([email protected])

Fakultas Ilmu Komputer Universitas Indonesia Semester Genap - 2004/2005

Version 2.0 - Internal Use Only

Graphs

SDA/TOPIC/V2.0/2

Objectives

Understand the definition, terminology and application of graph

Students can implement graph data structures using Java

Outline

Application

Definition

Terminology

Graph Representation

Graph Application

Networks

Maps

Shortest path

Scheduling (Project Planning)

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Definition

A graph G = (V, E) is composed of:

V: set of vertices/nodes (simpul)

E: set of edges (sisi/busur)connecting the vertices in V

An edge e = (a, b) is a pair of vertices, where a and b

∈V

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Terminology

undirected graph

directed graph

adjacent vertices: connected by an edge

degree (of a vertex): # of adjacent vertices

for directed graph

• in-degree

• out-degree

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Weighted Graph

weighted graph

each edge have weighted/cost

V0 V1

V₂ V₃ V₄

V5 V6

2 3

10

6 2 2

4

5

1

8 4

1

V = {V0, V1, V2, V3, V4, V5, V6}

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|V| = 7; |E| = 12

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

1 , , , 6 , , , 5 , , , 4 , ,

2 , , , 8 , , , 4 , , , 2 , ,

10 , , , 3 , , , 1 , , , 2 , ,

5 6 6 4 5 2 0 2

2 3 5 3 6 3 4 3

4 1 3 1 3 0 1 0

V V V V V V V V

V V V V V V V V E

Weighted Graph

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path: sequence of vertices v₁, v₂,. . .v_ksuch that consecutive vertices viand vi+1are adjacent.

simple path: no repeated vertices

cycle: simple path, except that the last vertex is the same as the first vertex

DAG (Directed Acyclic Graph): directed graph with no cycles.

More Terminology

V₀ V₁

V₂ V₃ V₄

V₅ V₆

2

3 10

6 2 2

4

5

1

8 4

1

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More Terminology

connected graph: any two vertices are connected by some path

subgraph: subset of vertices and edges forming a graph

Representation Representation: Edge List

The edge list structure simply stores the vertices and the edges into unsorted sequences.

Easy to implement.

Finding the edges incident on a given vertex is inefficient since it requires examining the entire edge sequence

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Representation: Adjancency List (traditional)

adjacency list of a vertex v:

sequence of vertices adjacent to v

represent the graph by the adjacency lists of all the vertices

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Representation: Adjancency List (modern)

The adjacency list structure extends the edge list structure by adding incidence containers to each vertex.

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Representation: Adjancency Matrix (traditional)

matrix M with entries for all pairs of vertices

M[i,j] = true means that there is an edge (i,j) in the graph.

M[i,j] = false means that there is no edge (i,j) in the graph.

There is an entry for every possible edge, therefore:

Space = (N²)

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Representation: Adjancency Matrix (modern)

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Representation: Adjancency Matrix (modern)

The adjacency matrix structures augments the edge list structure with a matrix where each row and column corresponds to a vertex.

The space requirement is O(n²+ m)

Topological Sorting

A topological sort of a directed acyclic graph is an ordering on the vertices such that all edges go from left to right.

Only an acyclic graph can have a topological sort, because a directed cycle must eventually return home to the source of the cycle.

However, every DAG has at least one topological sort, and we can use depth-first search to find such an ordering.

Topological sorting proves very useful in scheduling jobs in their proper sequence

Prerequisite of subjects in your Syllabus Book! Î precedence graph

V₀ V₁

V2 V3 V4

V₅ V₆

0

1 1

2

2 3

3

Topological Sorting

Start from vertex that have in-degree 0

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V0 V1

V₂ V₃ V₄

V5 V6

0

0 1

2

2 2

2

Topological Sorting

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V0 V1

V₂ V₃ V₄

V5 V6

0

0 0

2

2 1

2

Topological Sorting

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V₀ V₁

V2 V3 V4

V₅ V₆

0

0 0

1

2 0

2

Topological Sorting

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V₀ V₁

V2 V3 V4

V₅ V₆

0

0 0

0

1 0

1

Topological Sorting

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V0 V1

V₂ V₃ V₄

V5 V6

0

0 0

0

0 0

1

Topological Sorting

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V0 V1

V₂ V₃ V₄

V5 V6

0

0 0

0

0 0

0

Topological Sorting

V₀ V₁

V2 V3 V4

V₅ V₆

0

0 0

0

0 0

0 V⁵

Topological Sorting

V₀ V₁

V2 V3 V4

V₅ V₆

4

2

2 1

1

5 6

10

8 4

3

Topological Sorting

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V0 V1

V₂ V₃ V₄

V5 V6

4

2

2 2

1

5 6

10

8 4

3

Topological Sorting

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Shortest Path: unweighted graph

V₀ V₁

V2 V3 V4

V₅ V₆

V2

V₆

Starting vertex: V₂

Use BFS (Breath First Search) instead of DFS (Depth First Search) to find shortest path in unweighted graph (or each edge have the same weight)

0

1

2

3

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Shortest Path: positive weighted graph

In many applications, e.g., transportation networks, the edges of a graph have different weights.

Dijkstra’s algorithm finds shortest paths from a start vertex s to all the other vertices in a graph with

non-negative edge weights

Dijkstra’s algorithm uses a greedy method

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Dijkstra’s Algorithm

the algorithm computes for each vertex v the distance of v from the start vertex s, that is, the weight of a shortest path between s and v.

the algorithm keeps track of the set of vertices for which the shortest path has been computed (the white cloud W), the distance has been computed (the grey cloud G), and the distance has not been

computed (the black cloud B)

the algorithm uses a label D[v] to store an approximation of the distance between s and v

when a vertex v is added to the cloud, its label D[v] is equal to the actual distance between s and v

initially, the cloud W contains s, and we set

D[s] = 0

D[v] = ∞for v ≠s

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Dijkstra’s Algorithm: stages

V0 V1

V₂ V₃ V₄

V5 V6

5

2 1 2

5 8

1 4

2

3 10

1

∞ ∞

∞

∞ ∞

∞ 0

D[V₂] = 0, others ∞

Add V2into the white cloud (note: in this slide, the white cloud is the green cloud)

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Expanding the White Cloud

meaning of D[z]: length of shortest path from s to z that uses only intermediate vertices in the white cloud

after a new vertex u is added to the cloud, we need to check whether u is a better routing vertex to reach z

let u be a vertex not in the white cloud that has smallest label D[u]

we add u to the white cloud W

we update the labels of the adjacent vertices of u as follows

for each vertex z adjacent to u do if z is not in the cloud W then

if D[u] + weight(u,z) < D[z] then D[z] = D[u] + weight(u,z)

Dijkstra’s Algorithm: stages

V₀ V₁

V₂ V₃ V₄

V₅ V₆

5

2 1 2

5 8

1 4

2

3 10

1

5 ∞

2

5 ∞

∞ 0

after V₂added into the white cloud, update D[V_x] which V_xis the adjacent of V₂

Dijkstra’s Algorithm: stages

V₀ V₁

V₂ V₃ V₄

V₅ V₆

5

2 1 2

5 8

1 4

2

3 10

1

5 ∞

2

5 ∞

∞ 0

add the vertex that have minimum total weight (V₃) into the white cloud

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V0 V1

V₂ V₃ V₄

V5 V6

5

2 1 2

5 8

1 4

2

3 10

1

5 ∞

2

5 6

0 4

after V₃added into the white cloud, update D[V_x] which Vxis the adjacent of V3

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Dijkstra’s Algorithm: stages

V0 V1

V₂ V₃ V₄

V5 V6

5

2 1 2

5 8

1 4

2

3 10

1

5 ∞

2

5 6

0 4

add the vertex that have minimum total weight (V₄) into the white cloud

SDA/TOPIC/V2.0/39

V₀ V₁

V₂ V₃ V₄

V₅ V₆

5

2 1 2

5 8

1 4

2

3 10

1

5 ∞

2

5 5

0 4

after V₄added into the white cloud, update D[V_x] which V_xis the adjacent of V₄

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V₀ V₁

V₂ V₃ V₄

V₅ V₆

5

2 1 2

5 8

1 4

2

3 10

1

5 ∞

2

5 5

0 4

add the vertex that have minimum total weight (V₀) into the white cloud

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SDA/TOPIC/V2.0/41

Dijkstra’s Algorithm: stages

V0 V1

V₂ V₃ V₄

V5 V6

5

2 1 2

5 8

1 4

2

3 10

1

5 7

2

5 5

0 4

after V₀added into the white cloud, update D[V_x] which Vxis the adjacent of V0

SDA/TOPIC/V2.0/42

V0 V1

V₂ V₃ V₄

V5 V6

5

2 1 2

5 8

1 4

2

3 10

1

5 7

2

5 5

0 4

add the vertex that have minimum total weight (V₅) into the white cloud

after V5added into the white cloud, update D[Vx] which V_xis the adjacent of V₅

Dijkstra’s Algorithm: stages

V₀ V₁

V₂ V₃ V₄

V₅ V₆

5

2 1 2

5 8

1 4

2

3 10

1

5 7

2

5 5

0 4

add the vertex that have minimum total weight (V₆) into the white cloud

after V₆added into the white cloud, update D[V_x] which V_xis the adjacent of V₆

V₀ V₁

V₂ V₃ V₄

V₅ V₆

5

2 1 2

5 8

1 4

2

3 10

1

5 7

2

5 5

0 4

add the vertex that have minimum total weight (V₁) into the white cloud

after V₁added into the white cloud, update D[V_x] which V_xis the adjacent of V₁

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SDA/TOPIC/V2.0/45

V0 V1

V₂ V₃ V₄

V5 V6

2 3

-10

6 2 2

4

5

1

8 4

1

Minimum Spanning Tree (MST)

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V₁ 2 V₂

V₃ V₄

V6 V7

3

10

6 7 2

4

5

1

8 4

1

V₅

Minimum Spanning Tree: a graph

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V1 2 V2

V₃ V₄

V₆

6 2

1 4 1

V₅

V₇

Minimum Spanning Tree

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Prim’s Algorithm

starts from one vertex

grows the rest of the tree one edge at a time.

repeatedly selects the smallest weight edge that will enlarge the tree

greedy algorithms:

we make the decision of what to do next by selecting the best local option from all available choices without regard to the global structure.

V known dV pV

V₁ 0 0 0

V2 0 ∞ 0

V3 0 ∞ 0

V4 0 ∞ 0

V5 0 ∞ 0

V 0 ∞ 0

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

V known dV pV

V1 1 0 0

V2 0 2 V1

V3 0 4 V1

V4 0 1 V1

V5 0 ∞ 0

V6 0 ∞ 0

V 0 ∞ 0

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

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SDA/TOPIC/V2.0/53

V known dV pV

V1 1 0 0

V2 0 2 V1

V3 0 2 V4

V4 1 1 V1

V5 0 7 V4

V6 0 8 V4

V7 0 4 V4

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

SDA/TOPIC/V2.0/54

V known dV pV

V1 1 0 0

V2 1 2 V1

V3 0 2 V4

V4 1 1 V1

V5 0 7 V4

V6 0 8 V4

V7 0 4 V4

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

SDA/TOPIC/V2.0/55

V known dV pV

V1 1 0 0

V2 1 2 V1

V3 1 2 V4

V4 1 1 V1

V5 0 7 V4

V6 0 5 V3

V7 0 4 V4

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

SDA/TOPIC/V2.0/56

V known dV pV

V1 1 0 0

V2 1 2 V1

V3 1 2 V4

V4 1 1 V1

V5 0 6 V7

V6 0 1 V7

V7 1 4 V4

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

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SDA/TOPIC/V2.0/57

V known dV pV

V1 1 0 0

V2 1 2 V1

V3 1 2 V4

V4 1 1 V1

V5 0 6 V7

V6 1 1 V7

V7 1 4 V4

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

SDA/TOPIC/V2.0/58

V known dV pV

V1 1 0 0

V2 1 2 V1

V3 1 2 V4

V4 1 1 V1

V5 1 6 V7

V6 1 1 V7

V7 1 4 V4

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂

Prim’s Algorithm

V known dV pV

V1 1 0 0

V2 1 2 V1

V3 1 2 V4

V4 1 1 V1

V5 1 6 V7

V6 1 1 V7

V 1 4 V

V₁ 2

V₃ V₄

V₆ V₇ 6

2 1

4 1

V₅ V₂

Prim’s Algorithm Kruskal’s Algorithm

add the edges one at a time, by increasing weight

accept an edge if it does not create a cycle

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V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

V₅ V₂ Edge Weight Action

(V1, V4) 1 - (V6, V7) 1 - (V1, V2) 2 - (V3, V4) 2 - (V2, V4) 3 - (V1, V3) 4 - (V4, V7) 4 - (V3, V6) 5 - (V5, V7) 6 -

Kruskal’s Algorithm

SDA/TOPIC/V2.0/62

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

(V1, V4) 1 A (V6, V7) 1 - (V1, V2) 2 - (V3, V4) 2 - (V2, V4) 3 - (V1, V3) 4 - (V4, V7) 4 - (V3, V6) 5 - (V5, V7) 6 -

SDA/TOPIC/V2.0/63

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

(V1, V4) 1 A (V6, V7) 1 A (V1, V2) 2 - (V3, V4) 2 - (V2, V4) 3 - (V1, V3) 4 - (V4, V7) 4 - (V3, V6) 5 - (V5, V7) 6 -

Kruskal’s Algorithm

SDA/TOPIC/V2.0/64

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

(V1, V4) 1 A (V6, V7) 1 A (V1, V2) 2 A (V3, V4) 2 - (V2, V4) 3 - (V1, V3) 4 - (V4, V7) 4 - (V3, V6) 5 - (V5, V7) 6 -

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SDA/TOPIC/V2.0/65

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

(V1, V4) 1 A (V6, V7) 1 A (V1, V2) 2 A (V3, V4) 2 A (V2, V4) 3 - (V1, V3) 4 - (V4, V7) 4 - (V3, V6) 5 - (V5, V7) 6 -

SDA/TOPIC/V2.0/66

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

(V1, V4) 1 A (V6, V7) 1 A (V1, V2) 2 A (V3, V4) 2 A (V2, V4) 3 R (V1, V3) 4 R (V4, V7) 4 A (V3, V6) 5 - (V5, V7) 6 -

V₁ 2

V₃ V₄

V₆ V₇

3 10

6 7 2

4

5

1

8 4

1

(V1, V4) 1 A

(V6, V7) 1 A

(V1, V2) 2 A

(V3, V4) 2 A

(V2, V4) 3 R

(V1, V3) 4 R

(V4, V7) 4 A

(V3, V6) 5 R

(V5, V7) 6 A

V₁ 2

V₃ V₄

V₆ V₇ 6

2 1

4 1

(V1, V4) 1 A

(V6, V7) 1 A

(V1, V2) 2 A

(V3, V4) 2 A

(V2, V4) 3 R

(V1, V3) 4 R

(V4, V7) 4 A

(V3, V6) 5 R

(V5, V7) 6 A

Struktur Data & Algoritme Objectives Outline Graph Application