Mata kuliah
Tahun
:K0144/ Matematika Diskrit
:2008
Tree Graph
Pertemuan 21 :
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Learning Outcomes
• Mahasiswa dapat menunjukkan tree yg meliputi konsep dasar tree, komponen
tree, jenis-jenisnya dan contoh tentang penyelesain masalah dgn menggunakan
tree atau ordered rooted tree.
• Mahasiswa dapat mengkombinasikan materi Ordered Rooted Tree dgn disiplin
ilmu yg lain serta menerapkannya untuk berbagai kasus..
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Outline Materi:
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Pengertian Tree
Spanning Tree
Cabang, akar & daun
Titik berat, Berat pohon & Pusat berat
Ordered Rooted Tree
Pengertian Ordered rooted tree
Leksikografik order
Prefix form/ Pre order
Posfix form/Post order
Infix form/ In order
Contoh pemakaian
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Pengertian Pohon
• Suatu graph yang banyak vertexnya sama dengan n (n>1) disebut pohon, jika :
• Graph tersebut tidak mempunyai lingkar (cycle free) dan banyaknya rusuk (n-1).
• Graph tersebut terhubung .
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pohon
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bukan pohon
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Pengertian Pohon (2)
• Diagram pohon, diagram pohon merupakan digraph, yang : mempunyai source
tak ada vertex yang indegreenya lebih dari satu.
• hutan (forest) = himpunan pohon
• diagram pohon dapat digunakan sebagai alat untuk memecahkan masalah
dengan menggambarkan semua alternatif pemecahan.
• Hutan : banyaknya titik = n
banyaknya pohon = k
banyaknya rusuk = n-k
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Applications of Graphs: Topological Sorting
• Topological order
– A list of vertices in a directed graph without cycles such that vertex x precedes vertex y if there
is a directed edge from x to y in the graph
– There may be several topological orders in a given graph
• Topological sorting
– Arranging the vertices into a topological order
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Topological Sorting
Figure 13.14
A directed graph without
cycles
Figure 13.15
The graph in Figure 13-14
arranged according to the
topological orders a) a, g, d,
b, e, c, f and b) a, b, g, d, e,
f, c
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Topological Sorting
• Simple algorithms for finding a topological order
– topSort1 (Example: Figure 13-16)
• Find a vertex that has no successor
• Remove from the graph that vertex and all edges that lead to it, and add the vertex to the
beginning of a list of vertices
• Add each subsequent vertex that has no successor to the beginning of the list
• When the graph is empty, the list of vertices will be in topological order
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Topological Sorting
• Simple algorithms for finding a topological order (Continued)
– topSort2 (Example: Figure 13-17)
• A modification of the iterative DFS algorithm
• Strategy
– Push all vertices that have no predecessor onto a stack
– Each time you pop a vertex from the stack, add it to the beginning of a list of vertices
– When the traversal ends, the list of vertices will be in topological order
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Spanning Trees
• A tree
– An undirected connected graph without cycles
• A spanning tree of a connected undirected graph G
– A subgraph of G that contains all of G’s vertices and enough of its edges to form a tree
– Example: Figure 13-18
• To obtain a spanning tree from a connected undirected graph with cycles
– Remove edges until there are no cycles
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Spanning Trees
• You can determine whether a connected graph contains a cycle by counting its
vertices and edges
– A connected undirected graph that has n vertices must have at least n – 1 edges
– A connected undirected graph that has n vertices and exactly n – 1 edges cannot
contain a cycle
– A connected undirected graph that has n vertices and more than n – 1 edges
must contain at least one cycle
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Spanning Trees
Figure 13.19
Connected graphs that each have four vertices and three edges
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The DFS Spanning Tree
• To create a depth-first search (DFS) spanning tree
– Traverse the graph using a depth-first search and mark the edges that you
follow
– After the traversal is complete, the graph’s vertices and marked edges form the
spanning tree
– Example: Figure 13-20
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The BFS Spanning Tree
• To create a breath-first search (BFS) spanning tree
– Traverse the graph using a breadth-first search and mark the edges that you
follow
– When the traversal is complete, the graph’s vertices and marked edges form
the spanning tree
– Example: Figure 13-21
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Minimum Spanning Tree
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Minimum spanning tree
– A spanning tree for which the sum of its edge weights is minimal
Prim’s algorithm
– Finds a minimal spanning tree that begins at any vertex
– Strategy (Example: Figures 13-22 and 13-23)
• Find the least-cost edge (v, u) from a visited vertex v to some unvisited vertex u
• Mark u as visited
• Add the vertex u and the edge (v, u) to the minimum spanning tree
• Repeat the above steps until there are no more unvisited vertices
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Tree Traversals
• Definition: A traversal is the process for “visiting” all of the vertices in a tree
– Often defined recursively
– Each kind corresponds to an iterator type
– Iterators are implemented non-recursively
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Preorder Traversal
• Visit vertex, then visit child vertices (recursive definition)
• Depth-first search
– Begin at root
– Visit vertex on arrival
• Implementation may be recursive, stack-based, or nested loop
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Preorder Traversal
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Preorder Traversal
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Postorder Traversal
• Visit child vertices, then visit vertex (recursive definition)
• Depth-first search
– Begin at root
– Visit vertex on departure
• Implementation may be recursive, stack-based, or nested loop
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Postorder Traversal
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Postorder Traversal
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Postorder Traversal
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Postorder Traversal
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Levelorder Traversal
• Visit all vertices in level, starting with level 0 and increasing
• Breadth-first search
– Begin at root
– Visit vertex on departure
• Only practical implementation is queue-based
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Levelorder Traversal
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Levelorder Traversal
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Tree Traversals
• Preoder: depth-first search (possibly stack-based), visit on arrival
• Postorder: depth-first search (possibly stack-based), visit on departure
• Levelorder: breadth-first search (queue-based), visit on departure
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Binary Trees
• Definition: A binary tree is a rooted tree in which no vertex has more than two
children
• Definition: A binary tree is complete iff every layer but the bottom is fully
populated with vertices.
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Binary Tree Traversals
• Inorder traversal
– Definition: left child, visit, right child (recursive)
– Algorithm: depth-first search (visit between children)
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Inorder Traversal
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Inorder Traversal
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Inorder Traversal
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Inorder Traversal
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