数据结构(陈越) 作业题 第三周

Posted 2020-06-14

03-1. 二分法求多项式单根

时间限制

400 ms

内存限制

65536 kB

代码长度限制

8000 B

判题程序

Standard

作者

杨起帆（浙江大学城市学院）

二分法求函数根的原理为：如果连续函数f(x)在区间[a, b]的两个端点取值异号，即f(a)f(b)<0，则它在这个区间内至少存在1个根r，即f(r)=0。

二分法的步骤为：

 

• 检查区间长度，如果小于给定阈值，则停止，输出区间中点(a+b)/2；否则
• 如果f(a)f(b)<0，则计算中点的值f((a+b)/2)；
• 如果f((a+b)/2)正好为0，则(a+b)/2就是要求的根；否则
• 如果f((a+b)/2)与f(a)同号，则说明根在区间[(a+b)/2, b]，令a=(a+b)/2，重复循环；
• 如果f((a+b)/2)与f(b)同号，则说明根在区间[a, (a+b)/2]，令b=(a+b)/2，重复循环；

   

  本题目要求编写程序，计算给定3阶多项式f(x)=a₃x³+a₂x²+a₁x+a₀在给定区间[a, b]内的根。

  输入格式：

  输入在第1行中顺序给出多项式的4个系数a₃、a₂、a₁、a₀，在第2行中顺序给出区间端点a和b。题目保证多项式在给定区间内存在唯一单根。

  输出格式：

  在一行中输出该多项式在该区间内的根，精确到小数点后2位。

  输入样例：

  3 -1 -3 1
  -0.5 0.5
  

  输出样例：

  0.33

此题比较简单，有两个点要注意。

1.输入完之后，应该先判断两个端点是否是根，如果是直接输出，否则再用二分法来查找

2.输出时精确到小数点后2位，所以应该保证精确解和近似解相差不超0.001，所以循环终止条件为找到根，或者|b-a|<0.001

 1 //一是如何判断精度，二是如何设置输出的小数点后位数
 2 #include<iostream>
 3 #include<iomanip>
 4 using namespace std;
 5 
 6 double a[4];
 7 double f(double);
 8 
 9 int main()
10 {
11     cin >> a[0] >> a[1] >> a[2] >> a[3];
12 
13     double a,b;
14     cin >> a >> b;
15 
16     double left=a,right=b,mid=(a+b)/2.0;
17 
18     if (f(left)==0.)
19         cout << fixed << setprecision(2) << left << endl;
20     else if (f(right)==0.)
21         cout << fixed << setprecision(2) << right << endl;
22     else
23     {
24         while (  f(mid) && (right-left)>0.001 )
25         {
26             if ( f(left)*f(mid) < 0 )
27                 right = mid;
28             else
29                 left = mid;
30             mid = (left+right)/2.0;
31         }
32         cout << fixed << setprecision(2) << mid << endl;
33     }
34 
35     return 0;
36 }
37 
38 double f (double x)
39 {
40     double results=a[0]*x+a[1];
41 
42     for (int i=2;i<4;i++)
43     {
44         results = results*x+a[i];
45     }
46     return results;
47 }

 

03-2. List Leaves

时间限制

400 ms

内存限制

65536 kB

代码长度限制

8000 B

判题程序

Standard

作者

CHEN, Yue

Given a tree, you are supposed to list all the leaves in the order of top down, and left to right.

Input Specification:

Each input file contains one test case. For each case, the first line gives a positive integer N (<=10) which is the total number of nodes in the tree -- and hence the nodes are numbered from 0 to N-1. Then N lines follow, each corresponds to a node, and gives the indices of the left and right children of the node. If the child does not exist, a "-" will be put at the position. Any pair of children are separated by a space.

Output Specification:

For each test case, print in one line all the leaves‘ indices in the order of top down, and left to right. There must be exactly one space between any adjacent numbers, and no extra space at the end of the line.

Sample Input:

8
1 -
- -
0 -
2 7
- -
- -
5 -
4 6

Sample Output:

4 1 5

本题的第一个关键点是，如何找出树的根结点。很简单，输入的每一行分别是每个结点的左右孩子，因此输入中不出现的那个结点就是根结点，很容易理解。输入完之后构建树，构建完之后用队列进行层序遍历，输出的条件是判断结点是叶结点。

 1 #include<iostream>
 2 #include<fstream>
 3 #include<queue>
 4 using namespace std;
 5 
 6 struct node
 7 {
 8     int data;
 9     node* left;
10     node* right;
11 };
12 
13 int main()
14 {
15     int N;
16     char left,right;
17     cin >> N;
18 
19     node **tree = new node* [N];
20     int *flag = new int [N];
21 
22     for (int i=0;i<N;i++)
23     {
24         tree[i] = new node;
25         tree[i]->data = i;
26         flag[i]=1;
27     }
28 
29     for (int i=0;i<N;i++)
30     {
31         cin >> left;
32         if (left == ‘-‘)
33             tree[i]->left = nullptr;
34         else
35         {
36             tree[i]->left = tree[left-‘0‘];
37             flag[left-‘0‘]=0;
38         }
39 
40         cin >> right;
41         if (right == ‘-‘)
42             tree[i]->right = nullptr;
43         else
44         {
45             tree[i]->right = tree[right-‘0‘];
46             flag[right-‘0‘]=0;
47         }
48     }
49 
50     int head=0;
51     for (int i=0;i<N;i++)
52         if (flag[i] == 1)
53             head = i;
54 
55     //开始层序遍历输出叶节点
56     queue<node*> Q;
57     Q.push(tree[head]);
58     node *temp_node;
59     int output_flag=0;
60 
61     while (!Q.empty())
62     {
63         temp_node = Q.front();
64         Q.pop();
65         if (!temp_node->left && !temp_node->right )
66         {
67             if (output_flag)
68                 cout << ‘ ‘ << temp_node->data;
69             else
70             {
71                 cout << temp_node->data;
72                 output_flag=1;
73             }
74         }
75 
76         if (temp_node->left)
77             Q.push(temp_node->left);
78         if (temp_node->right)
79             Q.push(temp_node->right);
80     }
81 
82     return 0;
83 }

 

03-3. Tree Traversals Again

时间限制

200 ms

内存限制

65536 kB

代码长度限制

8000 B

判题程序

Standard

作者

CHEN, Yue

An inorder binary tree traversal can be implemented in a non-recursive way with a stack. For example, suppose that when a 6-node binary tree (with the keys numbered from 1 to 6) is traversed, the stack operations are: push(1); push(2); push(3); pop(); pop(); push(4); pop(); pop(); push(5); push(6); pop(); pop(). Then a unique binary tree (shown in Figure 1) can be generated from this sequence of operations. Your task is to give the postorder traversal sequence of this tree.

Figure 1

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N (<=30) which is the total number of nodes in a tree (and hence the nodes are numbered from 1 to N). Then 2N lines follow, each describes a stack operation in the format: "Push X" where X is the index of the node being pushed onto the stack; or "Pop" meaning to pop one node from the stack.

Output Specification:

For each test case, print the postorder traversal sequence of the corresponding tree in one line. A solution is guaranteed to exist. All the numbers must be separated by exactly one space, and there must be no extra space at the end of the line.

Sample Input:

6
Push 1
Push 2
Push 3
Pop
Pop
Push 4
Pop
Pop
Push 5
Push 6
Pop
Pop

Sample Output:

3 4 2 6 5 1

这题我的做法是，根据输入中的操作和序号，来构建树。从第一个操作开始，相邻的两个操作有4种可能，push-push,push-pop,pop-pop,pop-push,根据这四种可能来构建树。构建完之后用后序遍历输出即可。值得注意的是，后序遍历可以递归，可以非递归，建议大家两种都写一下，让自己更熟悉。我自己第一遍写后序遍历的非递归程序的过程就比较麻烦了。

  1 #include<string>
  2 #include<stack>
  3 #include<iostream>
  4 #include<fstream>
  5 using namespace std;
  6 
  7 struct node
  8 {
  9     int data;
 10     node *left;
 11     node *right;
 12     int flag;
 13 };
 14 
 15 int main()
 16 {    
 17     int N;
 18     cin >> N;
 19 
 20     string *operation = new string [N*2];
 21     int *num = new int [N*2];
 22     string temp_op;
 23 
 24     for (int i=0;i<2*N;i++)
 25     {
 26         cin >> temp_op;
 27         if (temp_op == "Push")
 28         {
 29             operation[i] = "Push";
 30             cin >> num[i];
 31         }
 32         else
 33         {
 34             operation[i] = "Pop";
 35             num[i]=-1;
 36         }
 37     }
 38 
 39     node **tree = new node* [N+1];
 40     for (int i=1;i<N+1;i++)
 41     {
 42         tree[i] = new node;
 43         tree[i]->data = i;
 44         tree[i]->left = nullptr;
 45         tree[i]->right = nullptr;
 46         tree[i]->flag=0;
 47     }
 48 
 49     int head=num[0];
 50     stack<int> sta;
 51     int temp=0;
 52     for (int i=0;i<2*N-1;i++)
 53     {
 54         if (operation[i] == "Push")
 55             sta.push(num[i]);
 56         else
 57         {
 58             temp = sta.top();
 59             sta.pop();
 60         }
 61 
 62         if (operation[i] == "Push" && operation[i+1] == "Push" )
 63         {
 64             tree[num[i]]->left = tree[num[i+1]];
 65         }
 66         else if (operation[i] == "Push" && operation[i+1] == "Pop" )
 67         {
 68             tree[num[i]]->left = nullptr;
 69         }
 70         else if (operation[i] == "Pop" && operation[i+1] == "Push" )
 71         {
 72             tree[temp]->right = tree[num[i+1]];
 73         }
 74         else //pop pop
 75             tree[temp]->right = nullptr;
 76     }
 77         
 78     //开始后序输出
 79     stack<node*> out;
 80     node *T=tree[head];
 81 
 82     while (T || !out.empty())
 83     {
 84         while (T)
 85         {
 86             out.push(T);
 87             T = T->left;
 88         }
 89 
 90         if (!out.empty())
 91         {
 92             T = out.top();
 93 
 94             if (T->flag)
 95             {
 96                 cout << T->data << ‘ ‘;
 97                 out.pop();
 98                 T=out.top();
 99             }
100             else
101             {
102                 T->flag = 1;
103                 T=T->right;
104             }
105         }
106     }
107     
108     return 0;
109 }