資料型態 (Data Type)- 程式語言的變數所能表

示的資料種類

* 基本型態 (Primitive Type):

integer, real, boolean, character(byte)

* 結構化型態 (Structured Type):

string, array, record/structure

int i, j;

char c;

float r;

int i1[10], j1[10];

float r1[10];

i1[0]=20;

i2[1]=90;

…

串列 (List)

* 有序串列可以是空的 () 或寫成 (a1, a2, ..., an)

串列的表示法 (representation of lists)

* 順序對應 (Sequential mapping)-array

* 鏈結串列 (Linked list)-pointer

堆疊與佇列 (Stacks & Queues)

* 堆疊是一個有序串列，所有的 insertion 和 deletion 動作均在頂端 (top) 進行* 具有後進先出 (LIFO-last in first out) 特性

* 佇列是一個有序串列，所有的 insertion 和 deletion 是發生在串列的不同端，加入的一端 稱為尾端 (rear) ，刪除的一端稱為前端 (front)

* 具有先進先出 (FIFO-first in first out) 特性

Example of a stack and a queue

EDCBA

top

stack

A B C D E

front rear

queue

Representation of a stack:

*a one-dimensional array: stack[0:n-1]

*a variable: top

*linked list

*node: data, link

*a variable: top

E D C B A 0

top

data link

Representation of a queue:

*a one-dimensional array: q[0:n-1]

*two variables: front & rear

*linked list

*node: data, link

* two variables: front & rear

A B C D

data link

front rear

0

Circular Queue

front = 0 ; rear = 4

[n-1][n-2]

[n-3]

[n-4]

[0]

[1]

[2]

[3]

[4] ‧ ‧‧J4

J3

J2

J1

front = n-4 ; rear = 0

[n-1][n-2]

[n-3]

[n-4]

[0]

[1]

[2]

[3]

[4] ‧ ‧‧

J4 J3J2

J1

Circular queue of capacity n-1 containing four elements J1, J2, J3, and J4

Trees *A tree is a finite set of one or more nodes * 樹是一個或多個節點 (node) 所組成的有限集合* 有一個特殊的節點稱為樹根 (root)* 每一個節點底下有零個或一個以上的子樹 (subtree): T1, T2, …, Tn n0* 節點 (node) * 分支 (branch, edge) * 樹根 (root) * 子點 (children) * 父點 (parent) * 終端節點 (leaf, terminal nodes)

Trees

* 非終端節點 (nonterminal nodes)

* 兄弟 (siblings)

* 祖先 (ancestor of a node): all the nodes along

the path from the root to that node

* 分支度 (degree): the number of subtrees of a

node

* 樹的分支度 (degree of a tree): the maximum

degree of the nodes in the tree

* 階度 (level): initially let the root be at level one

a Sample Tree

A

C

G

DB

FE

K L

H JI

M

level

1

2

3

4

Trees

* 高度 (height, depth): the maximum level of any

node in the tree

*A forest is a set of n0 disjoint trees

* 若一樹有 n 個 nodes ，則必有且唯有 n-1 個 edges

*A graph without a cycle

Representation of a Tree:

*linked list

*node: data, link, tag

*when tag=1, data field contains a pointer to

a list rather than a data item

A 0

B F 0 C G 0 D I J 0

E K L 0 H M 0

The tag field of a node is one if it has a down-pointing arrow;otherwise it is zero.

Binary Trees :

*Any node can have at most two children

* 二元樹上每個節點的 degree2

* 左子樹 (left subtree) & 右子樹 (right subtree)

* 二元樹可以是空集合 (empty)

*The maximum number of nodes on leve i of a

binary tree is 2i-1

*The maximum number of nodes in a binary

tree of depth k is 2k-1, k>0.

*For any non-empty binary tree, if n0 is the number of

terminal nodes and n2 is the number of nodes of

degree 2, then n0=n2+1.

二元樹的特例 :

* 歪斜樹 (skewed binary tree)

left-skewed binary tree, right-skewed binary tree

* 全滿二元樹 (full binary tree)

The binary tree of depth k that has exactly 2k-1 nodes

* 完整二元樹 (complete binary tree)

A binary tree with n nodes and depth k is complete iff

its nodes corresponds to the nodes that are

numbered one to n in the full binary tree of depth k.

In a complete tree, leaf nodes occur on at most two

adjacent levels.

Representation of a Binary Tree: *a one-dimensional array: tree[1:n] the node numbered i is stored in tree[i] 1. parent(i) is at i/2 if i1. If i=1, i is the root and has no parent. 2. lchild(i) is at 2i if 2in. If 2i>n, i has no left child. 3. rchild(i) is at 2i+1 if 2i+1n. If 2i+1>n, i has no right child. * 優點 : 處理簡單，若為 full binary tree ，則相當節省 空間。 * 缺點 : 若為 skewed binary tree ，則相當浪費空間。 不容易處理 Insertion or deletion

Binary Trees

A

C

G

B

FE

H I

1

2

3

4

B

D

E

level

5

D

C

A

(a) (b)

Sequential Representation

Sequential representation of the binary trees

A B - C - - - D - … E

A B C D E F G H I

(1) (2) (3) (4) (5) (6) (7) (8) (9) …(16)

Tree

Tree

Representation of a Binary Tree:

*linked list

node: lchild, data, rchild

a variable: tree

* 優點 : 插入與刪除一個節點相當容易。 * 缺點 : 缺點 : 很難找到該節點的父點。

Linked Representations

A 0

B 0

C 0

D 0

E 00

A

B C

E 00 F 00D G 00

H 00 I 00

(a) (b)

Linked representation of the binary trees

treetree

Binary Search Trees

*A binary search tree is a binary tree.

1. All the keys are distinct.

2. The keys in the left subtree are smaller than

the key in the root.

3. The keys in the right subtree are larger than

the key in the root.

4. The left and right subtrees are also binary

search trees.

*Search by key value

*Node: lchild, rchild and data

Binary trees

20

15 25

12 10 22

30

5 40

2

60

80

70

65

(a) (b) (c)

Binary Search Trees

*Search by rank(find the kth-smallest element)

*Node: lchild, rchild, data and leftsize

*Leftsize: one plus the number of elements in

the left subtree of the node

Representation of a Priority Queue:

*Any data structure that supports the operations

of search max, insert, and delete max is

called a priority queue.

Priority Queues

*unordered linear list

insert time: (1)

delete time: (n) n-element unordered list

*ordered linear list

insert time: O(n)

delete time: (1) n-element ordered list

*heap

insert time: O(log n)

delete time: O(log n)

MAX Heap

*a complete binary tree

*the value at each node is at least as large as

the value at its children( 任何一父點的值必大於*The largest number is in root node

*A max heap can be implemented using an

array a[].

*Insertion into a heap

*Deletion from a heap

Insertion into a Heap

80

45 70

40 35 9050

80

45 90

40 35 7050

90

45 80

40 35 7050

Action of Inset inserting 90 into an existing heap

Heap sort

1) 建立一 binary tree

2) 將 binary tree 轉成 Heap (Heapify)

3)Output: Removing the largest number and

restoring the heap (Adjust)

100,119,118,171,112,151,32,40,80,35,90,45,50,70

Forming a Heap

40

80

40 80

40

80

40 35

90

80 35

40 45

90

80 35

40 45 50

90

80 50

40 45 35

80

40 35

90

90

80 35

40

(a) (b) (c)

(d) (e)

90

80 50

40 45 35 70

90

80 70

40 45 35 50

(f) (g)

Forming a heap from the set{40,80,35,90,45,50,70}

Heapify

100

119 118

171 112 132151

Action of Heapify(a,7) on the data (100, 119, 118, 171, 112, 151, and 132)

(a)

100

119 151

171 112 132118

(b)

100

171 151

119 112 132118

(c)

171

119 151

100 112 132118

(d)

Sets and Disjoint Set Union Set: *The sets are assumed to be pairwise disjoint. *Each set is represented as a tree. *Link the nodes from the children to the parent Ex: S1={1, 7, 8, 9}, S2={2, 5, 10}, S3={3, 4, 6}

Set Operations: *Disjoint set union Make one of the trees a subtree of the other*Find(i)

Representations of Union

Possible representations of S1∪S2

9

1 5

7 8 102

S1 S2

9

1

7 8

S1 S∪ 2

5

102

or

10

5

1 2

97 8

S1 S∪ 2

Weighting rule for Union(i, j):

If the number of nodes in the tree with root i is less than the number in the tree with root j, then make j the parent of i; otherwise make i the parent of j.

*Maintain a count in the p field of the roots as a

negative number.

*Count field: the number of nodes in that tree.

Lemma 2.3

Assume that we start with a forest of trees, each having one node. Let T be a tree with m nodes created as a result of a sequence of unions each performed using WeightedUnion. The height of T is no greater than log m+1.

Collapsing rule:

If j is a node on the path from i to its root and p[i]root[i], then set p[j] to root[i].

Graph

A graph G(V,E) is a structure which consists of

1. a finite nonempty set V(G) of vertices

(points, nodes)

2. a (possible empty) set E(G) of edges (lines)

V(G): vertex set of G

E(G): edge set of G

有序樹 (ordered tree)

Sample Graphs

1

4

2 3

1

2 3

4 5 6 7

1

2

3

Three sample graphs

(a) G1 (b) G2 (c) G3

Undirected Graph( 無方向圖形 ): (u,v)=(v,u) * 假如 (u,v)E(G) ，則 u 和 v 是 adjacent vertices ， 且 (u,v) 是 incident on u 和 v *Degree of a vertex: the number of edges incident to that vertex G has n vertices and e edges,

*A subgraph of G is a graph G'(V',E') ， V'V 且 E'E *Path: 假如 (u,i1), (i1,i2), ..., (ik,v)E ，則 u 與 v 有一條 Path( 路徑 ) 存在。 *Path Length: the number of edges on the path *Simple path: 路徑上除了起點和終點可能相同外，其它 的頂點都是不同的

2/)(1

n

iide

Undirected Graph( 無方向圖形 ): (u,v)=(v,u) *Cycle: a simple path 且此路徑上的起點和終點相同 且 (u,v) 是 incident on u 和 v *In G, two vertices u and v are said to be connected iff there is a path in G from u to v. *Connected graph: 圖上每個頂點都有路徑通到其它 頂點 *Connected component: a maximal connected subgraph ( 圖上相連在一起的最大子圖 ) *Complete graph: if |V|=n then |E|=n(n-1)/2 *A tree is a connected acyclic(no cycles) graph.*Self-edge(self-loop): an edge from a vertex back to itself *Multigraph: have multiple occurrences of the same edge

Connected Components

A graph with two connected components

G4

H1 H2

1

4

3 2

5

8

6 7

Subgraphs

1

2

3

Some subgraphs

(i)

(b) Some of the subgraphs of G3

1 1

2 3 4

2 31

4

2 3

(ii)

1 1

2

3

(a) Some of the subgraphs of G1

(iii)

1

2

3

(iv)

(i) (ii) (iii) (iv)

Graphlike Structures

Examples of graphlike structures

(a) Graph with a self edge

1

3

2

1

3

2 4

(b) Multigraph

Undirected Graph( 無方向圖形 ): (u,v)=(v,u)

*Eulerian walk(cycle): 從任何一個頂點開始， 經過每個邊 一次，再回到原出發點 ( 每個頂點的 degree 必須是偶數 )

*Eulerian chain: 從任一頂點開始，經過每個邊 一次，不一定要回到原點 ( 只有兩個頂點的 degree 是奇數，其它必須是 偶數 )

Directed Graph(Digraph, 有方向圖形 ): <u,v><v,u>

* 假如 <u,v>E(G) ，則稱 u is the tail and v the

head of the edge

u 是 adjacent to v ，而 v 是 adjacent from v

<u,v> 是 incident to u 和 v

*in-degree of v: the number of edges for which v is

the head

*out-degree of v: the number of edges for which v is

the tail.

*Subgraph:G'=(V', E') ， V' V 且 E' E

Directed Graph(Digraph, 有方向圖形 ): <u,v><v,u>

*Directed Path: 假如 <u,i1>, <i1,i2>, ..., <ik,v>E ， 則 u 與 v 有一條 Path( 路徑 ) 存在。*Path Length: 路徑上所包含的有向邊的數目 *Simple directed path: 路徑上除了起點和終點可能 相同外，其它的頂點都是不同的 *Directed Cycle(Circuit): A simple directed path

且路徑上的起點和終點相同

Directed Graph(Digraph, 有方向圖形 ): <u,v><v,u>

*Strongly connected graph: for every pair of distinct vertices u and v in V(G), there is a directed path from u to v and also from v to u.

*Strongly connected component: a maximal subgraph that is strongly connected

( 有向圖上緊密相連在一起的最大有向子圖 )

*Complete digraph: if |V|=n then |E|=n(n-1)

Strongly Connected Components

A graph and its strongly connected components

(a) (b)

1

2

3

1

2

3

Graph Representations:

* 相鄰矩陣 (adjacency matrix)

* 相鄰串列 (adjacency list)

* 相鄰多元串列 (adjacency multilist)

Adjacency Matrix

G(V, E): a graph with n vertices

A two-dimensional n*n array: a[1:n, 1:n]

*a[i, j]=1 iff (i, j)E(G)

*Space: O(n2)

*The adjacency matrix for an undirected graph

is symmetric.

Use the upper or lower triangle of the matrix

*For an undirected graph the degree of i is its

row sum.

*For a directed graph the row sum is the out-degree,

and the column sum is the in-degree.

n

j

jia1

],[

Adjacency Matrices

1

4

2 3

1

2

3

(a) G1 (b) G3 (c) G4

1

4

3 2

5

8

6 7

0111

1011

1101

1110

4

3

2

1

4321

000

101

010

3

2

1

321

01000000

10100000

01010000

00100000

00000110

00001001

00001001

00000110

8

7

6

5

4

3

2

1

87654321

Adjacency List G(V, E): a graph with n vertices and e edges *the n rows of the adjacency matrix are represented as n linked lists *There is one list for each vertex in G node: vertex, link *Each list has a head node *Space: n head nodes and 2e nodes for an undirected graph n head nodes and e nodes for a directed graph Use the upper or lower triangle of the matrix *For an undirected graph the degree of i is to count the number of nodes in its adjacency list. *For a directed graph the out-degree of i is to count the number of nodes in its adjacency list.

1

4

2 3

1

2

3

(a) G1

(b) G3

0

2 0

3

[1]

[2]

[3]

1 0

4

3

[1]

[2]

[3]

4

[4]

2 4

1 2

2 3 0

1 0

1 0

3 0

head nodes

head nodes vertex link

vertex link

(c) G4

1

4

3 2

5

8

6 7

3

4

[1]

[2]

[3]

1 0

[4]

1 4 0

2 3 0

2 0

head nodes vertex link

[5]

[6]

[7]

[8]

6 0

7 5 0

6 8 0

7 0

Adjacency List

Inverse adjacency lists – to count the in-degree easily

*One list for each vertex

*Each list contains a node for each vertex

adjacent to the vertex it represents.

Inverse Adjacency Lists

1

2

3

Inverse adjacency lists for G3

2 0

1 0

2 0

[1]

[2]

[3]

Array node[1: n+2e+1] to represent a graph G(V, E) with n vertices and e edges - eliminate the use of pointers

*The node[i] gives the starting point of the list

for vertex i, 1in

*node[n+1]:=n+2e+2

Sequential representation

Sequential representation of graph G4 ： Array node[1 ： n + 2e + 1]

1

4

3 2

5

8

6 7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

10 12 14 16 18 19 21 23 24 3 2 4 1 1 4 2 3 6 7 5 6 8 7

Orthogonal list

Each node has four fields and represents one edge.

Node structure:

tail head column link for head row link for tail

Orthogonal list representation

Orthogonal list representation for G3

1

2

3

1 2 3

1 1 2 0 0

2 2 1 0 2 3 0 0

3 0

Head nodes(shown twice)

Adjacency Multilist

*One list for each vertex

*node can be shared among several lists

*for each edge there is exactly one node, but

the node is in two lists

*node strucure

m vertex1 vertext2 list1 list2

Adjacency Multilist

Adjacency multilists for G1

N1 1 2 N2 N4 edge (1,2)

N2 1 3 N3 N4 edge (1,3)

[1]

[2]

[3]

[4]

N3 1 4 0 N5 edge (1,4)

N4 2 3 N5 N6 edge (2,3)

head nodes

N5 2 4 0 N6 edge (2,4)

N6 3 4 0 0 edge (3,4)

1

4

2 3

The lists are vertex 1: N1N2N3vertex 2: N1N4N5vertex 3: N2N4N6vertex 4: N3N5N6