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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

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int i, j; char c; float r; int i1[10], j1[10]; float r1[10]; i1[0]=20; i2[1]=90; …

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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

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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

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Example of a stack and a queue EDCBAEDCBA top stack A B C D E front rear queue

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Representation of a stack: *a one-dimensional array: stack[0:n-1] *a variable: top *linked list *node: data, link *a variable: top E DCBA0 top data link

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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 BCD data link front rear 0

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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] ‧ ‧ ‧ J4J3 J2 J1 Circular queue of capacity n-1 containing four elements J1, J2, J3, and J4

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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): T 1, T 2, …, T n n 0 * 節點 (node) * 分支 (branch, edge) * 樹根 (root) * 子點 (children) * 父點 (parent) * 終端節點 (leaf, terminal nodes)

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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

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a Sample Tree A C G DB F E KL HJI M level

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Trees * 高度 (height, depth): the maximum level of any node in the tree *A forest is a set of n 0 disjoint trees * 若一樹有 n 個 nodes ，則必有且唯有 n-1 個 edges *A graph without a cycle

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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 A0 BF0CG0DIJ0 EKL0HM0 The tag field of a node is one if it has a down-pointing arrow; otherwise it is zero.

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Binary Trees : *Any node can have at most two children * 二元樹上每個節點的 degree 2 * 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 2 i-1 *The maximum number of nodes in a binary tree of depth k is 2 k -1, k>0. *For any non-empty binary tree, if n 0 is the number of terminal nodes and n 2 is the number of nodes of degree 2, then n 0 =n 2 +1.

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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 2 k -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.

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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 i 1. If i=1, i is the root and has no parent. 2. lchild(i) is at 2i if 2i n. If 2i>n, i has no left child. 3. rchild(i) is at 2i+1 if 2i+1 n. 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

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Binary Trees A C G B FE HI B D E level 5 D C A (a) (b)

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Sequential Representation Sequential representation of the binary trees AB-C---D-…E ABCDEFGHI (1)(2)(3)(4)(5)(6)(7)(8)(9)…(16) Tree

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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。

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Linked Representations A0B0C0D0E00ABCE00F00DG00H00I00 (a) (b) Linked representation of the binary trees tree

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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

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Binary trees (a) (b)(c)

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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

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Representation of a Priority Queue: *Any data structure that supports the operations of search max, insert, and delete max is called a priority queue.

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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)

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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

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Insertion into a Heap Action of Inset inserting 90 into an existing heap

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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)

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

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Forming a Heap (a)(b)(c) (d) (e) (f)(g) Forming a heap from the set{40,80,35,90,45,50,70}

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

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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)

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Representations of Union Possible representations of S 1 ∪ S S1S1 S2S S1∪S2S1∪S2 5 2 or S1∪S2S1∪S2

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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.

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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.

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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].

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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

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有序樹 (ordered tree)

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Sample Graphs Three sample graphs (a) G 1 (b) G 2 (c) G 3

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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

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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

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Connected Components A graph with two connected components G4G4 H1H1 H2H

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Subgraphs Some subgraphs (i) (b) Some of the subgraphs of G (ii) (a) Some of the subgraphs of G 1 (iii) (iv) (i)(ii)(iii)(iv)

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Graphlike Structures Examples of graphlike structures (a) Graph with a self edge (b) Multigraph

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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 )

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Directed Graph(Digraph, 有方向圖形 ): * 假如 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 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

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Directed Graph(Digraph, 有方向圖形 ): *Directed Path: 假如,,..., 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

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Directed Graph(Digraph, 有方向圖形 ): *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)

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Strongly Connected Components A graph and its strongly connected components (a)(b)

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Graph Representations: * 相鄰矩陣 (adjacency matrix) * 相鄰串列 (adjacency list) * 相鄰多元串列 (adjacency multilist)

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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(n 2 ) *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.

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Adjacency Matrices (a) G 1 (b) G 3 (c) G

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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.

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(a) G 1 (b) G [1] [2] [3] [1] [2] [3] 4 [4] head nodes vertex link

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(c) G [1] [2] [3] 10 [4] head nodes vertex link [5] [6] [7] [8]

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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.

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Inverse Adjacency Lists Inverse adjacency lists for G [1] [2] [3]

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

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Sequential representation Sequential representation of graph G 4 ： Array node[1 ： n + 2e + 1]

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Orthogonal list Each node has four fields and represents one edge. Node structure: tailheadcolumn link for headrow link for tail

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Orthogonal list representation Orthogonal list representation for G Head nodes (shown twice)

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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 mvertex1vertext2list1list2

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Adjacency Multilist Adjacency multilists for G 1 N1 12N2N4edge (1,2) N2 13N3N4edge (1,3) [1] [2] [3] [4] N3 140N5edge (1,4) N4 23N5N6edge (2,3) head nodes N5 240N6edge (2,4) N6 3400edge (3,4) The lists are vertex 1: N1 N2 N3 vertex 2: N1 N4 N5 vertex 3: N2 N4 N6 vertex 4: N3 N5 N6

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