1、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 i110, j110; float r110; i10=20; i21=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(Se
2、quential 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,Representation of a stack: *a one-d
3、imensional array: stack0:n-1 *a variable: top *linked list *node: data, link *a variable: top,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)
4、*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 anode *樹的分支度(degree of a tree): the maximum degree of the nodes in the tree *階度(level): initially let the
5、 root be at level one,Trees *高度(height, depth): the maximum level of anynode 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 toa list rat
6、her than a data item,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 binarytree of depth k is 2k-1, k0. *For any non-em
7、pty binary tree, if n0 is the number ofterminal 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 tr
8、ee) A binary tree with n nodes and depth k is complete iffits 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 twoadjacent levels.,Representation of a Binary Tree: *a one-dimensional array: tree1:n the node
9、numbered i is stored in treei1. 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 2in, i has no left child. 3. rchild(i) is at 2i+1 if 2i+1n. If 2i+1n, 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
10、Insertion or deletion,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。,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 thanthe key in th
11、e root.3. The keys in the right subtree are larger thanthe key in the root. 4. The left and right subtrees are also binarysearch trees. *Search by key value *Node: lchild, rchild and data,Binary Search Trees *Search by rank(find the kth-smallest element) *Node: lchild, rchild, data and leftsize *Lef
12、tsize: one plus the number of elements inthe left subtree of the node,Representation of a Priority Queue: *Any data structure that supports the operationsof search max, insert, and delete max is called a priority queue.,Priority Queues *unordered linear list insert time: (1) delete time: (n) n-eleme
13、nt 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 asthe value at its children(任何一父點的值必大於 *The largest number is in root node
14、*A max heap can be implemented using an array a. *Insertion into a heap *Deletion from a 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),Sets and Disjoint Set Union Set: *The sets are assumed to be pairwise disjoint
15、. *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 unionMake one of the trees a subtree of the other *Find(i),Weighting rule for Union(i, j): If the number of nodes in the tree with root i is
16、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 anegative 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.
17、 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 pirooti, then set pj to rooti.,Graph A graph G(V,E) is a structure which consis
18、ts 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,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 edgesincident
19、to that vertex G has n vertices and e edges, *A subgraph of G is a graph G(V,E)，VV 且 EE *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,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
20、 且(u,v)是incident on u和v *In G, two vertices u and v are said to be connectediff 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)
21、graph. *Self-edge(self-loop): an edge from a vertex back to itself *Multigraph: have multiple occurrences of the same edge,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),Directe
22、d 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 isthe tail.*Subgraph:G=(V, E)， V V 且 E E,Directed Graph(Digrap
23、h,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,Directed Graph(Digraph,有方向圖形): *Strongly connected graph: for every pair of distinct vertices u and v in V(G
24、), 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),Graph Representations: *相鄰矩陣(adjacency matrix) *相鄰串列(adjacency list) *相鄰多元串列(adjacency multilist)
25、,Adjacency Matrix G(V, E): a graph with n vertices A two-dimensional n*n array: a1:n, 1:n *ai, j=1 iff (i, j)E(G) *Space: O(n2) *The adjacency matrix for an undirected graphis symmetric.Use the upper or lower triangle of the matrix *For an undirected graph the degree of i is its row sum. *For a dire
26、cted graph the row sum is the out-degree,and the column sum is the in-degree.,Adjacency List G(V, E): a graph with n vertices and e edges *the n rows of the adjacency matrix are representedas n linked lists *There is one list for each vertex in G node: vertex, link *Each list has a head node *Space:
27、 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 num
28、ber of nodes in its adjacency list.,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.,Array node1: n+2e+1 to represent a graph G(V, E) with n vertices and e edges - eliminate
29、 the use of pointers *The nodei gives the starting point of the list for vertex i, 1in *noden+1:=n+2e+2,Orthogonal list Each node has four fields and represents one edge. Node structure: tailheadcolumn link for headrow link for tail,Adjacency Multilist *One list for each vertex *node can be shared among several lists *for each edge there is exactly one node, butthe node is in two lists *node strucure,有序樹(ordered tree),
