【2020-MOOC-浙江大学-陈越、何钦铭-数据结构】树和堆（第五周的笔记和编程作业）

布满荆棘的人生 2023-07-17 15:56 12阅读 0赞

### 文章目录 ###

*  〇、前言
 *  一、堆
 *  二、堆的基本操作
 *  三、哈夫曼树
 *  四、集合及运算
 *  五、堆中的路径
 *  六、课后题
 *   *   *  1、05-树7 堆中的路径 (25分)
         *  2、05-树8 File Transfer (25分)
         *  3、05-树9 Huffman Codes (30分)
 *  总结

# 〇、前言 #

这两周开始跟着【MOOC-浙江大学-陈越、何钦铭-数据结构】进行数据结构与算法的学习，特此记录复习一下，虽然记不住，但是一直记一直记一直记，成为复读机就好了。  
![在这里插入图片描述][20200314162146148.png_pic_center]

# 一、堆 #

优先队列（Priority Queue）：特殊的“队列”，取出元素的顺序是依照元素的优先权（关键字）大小，而不是元素进入队列的先后顺序。  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70]

# 二、堆的基本操作 #

![在这里插入图片描述][2020032310125742.png]

*  MaxHeap Create( int MaxSize )：创建一个空的最大堆。
 *  Boolean IsFull( MaxHeap H )：判断最大堆H是否已满。
 *  Insert( MaxHeap H, ElementType item )：将元素item插入最大堆H。
 *  Boolean IsEmpty( MaxHeap H )：判断最大堆H是否为空。
 *  ElementType DeleteMax( MaxHeap H )：返回H中最大元素(高优先级)。

# 三、哈夫曼树 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 1]

# 四、集合及运算 #

*  集合运算：交、并、补、差，判定一个元素是否属于某一集合
 *  并查集：集合并、查某元素属于什么集合
 *  并查集问题中集合存储如何实现？
    
     *  可以用树结构表示集合，树的每个结点代表一个集合元素

# 五、堆中的路径 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 2]

# 六、课后题 #

![在这里插入图片描述][20200323101653973.png]

### 1、05-树7 堆中的路径 (25分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 3]  
**输入样例:**

5 3  
46 23 26 24 10  
5 4 3

**输出样例:**

24 23 10  
46 23 10  
26 10

--------------------

#include <stdio.h> 
    
    #define MAXN 1001
    #define MINH -10001
    
    int H[MAXN], size;
    void Create ()
    { 
    	size = 0;
    	H[0] = MINH;
    	/*设置“岗哨”*/
    }
    
    void Insert ( int X )
    { 
    	/* 将X插入H。这里省略检查堆是否已满的代码 */
    	int i;
    	for (i=++size; H[i/2] > X; i/=2)
    		H[i] = H[i/2];
    	H[i] = X;
    }
    
    int main()
    {  
    	int n, m, x, i, j;
    	scanf("%d %d", &n, &m);
    	Create(); /* 堆初始化 */
    	for (i=0; i<n; i++) {  /*以逐个插入方式建堆 */
    		scanf("%d", &x);
    		Insert(x);
    	}
    	for (i=0; i<m; i++) { 
    		scanf("%d", &j);
    		printf("%d", H[j]);
    		while (j>1) {  /*沿根方向输出各结点*/
    			j /= 2;
    			printf(" %d", H[j]);
    		}
    		printf("\n");
    	}
    	return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 4]

--------------------

### 2、05-树8 File Transfer (25分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 5]  
![在这里插入图片描述][20200323101903871.png]  
**Sample Input 1:**

5  
C 3 2  
I 3 2  
C 1 5  
I 4 5  
I 2 4  
C 3 5  
S

**Sample Output 1:**

no  
no  
yes  
There are 2 components.

**Sample Input 2:**

5  
C 3 2  
I 3 2  
C 1 5  
I 4 5  
I 2 4  
C 3 5  
I 1 3  
C 1 5  
S

**Sample Output 2:**

no  
no  
yes  
yes  
The network is connected.

--------------------

#include <stdio.h> 
    
    #define MaxSize 10001
    
    typedef int ElementType; /*默认元素可以用非负整数表示*/
    typedef int SetName; /*默认用根结点的下标作为集合名称*/
    typedef ElementType SetType[MaxSize];
    
    SetName Find( SetType S, ElementType X )
    {  
    	/* 默认集合元素全部初始化为-1 */
    	for ( ; S[X]>=0; X=S[X] ) ;
    	return X;
    }
    
    void Union( SetType S, SetName Root1, SetName Root2 )
    {  
    	if ( S[Root2]<S[Root1] ){ 
    		S[Root2] += S[Root1];
    		S[Root1] = Root2;
    	}
    	else { 
    		S[Root1] += S[Root2];
    		S[Root2] = Root1;
    	}
    }
    
    void Input_connection( SetType S )
    {  
    	ElementType u, v;
    	SetName Root1, Root2;
    	scanf("%d %d\n", &u, &v);
    	Root1 = Find(S, u-1);
    	Root2 = Find(S, v-1);
    	if ( Root1 != Root2 )
    		Union( S, Root1, Root2 );
    }
    
    void Check_connection( SetType S )
    {  
    	ElementType u, v;
    	SetName Root1, Root2;
    	scanf("%d %d\n", &u, &v);
    	Root1 = Find(S, u-1);
    	Root2 = Find(S, v-1);
    	if ( Root1 == Root2 )
    		printf("yes\n");
    	else printf("no\n");
    }
    
    void Check_network( SetType S, int n )
    {  
    	int i, counter = 0;
    	for (i=0; i<n; i++) { 
    		if ( S[i] < 0 ) counter++;
    	}
    	if ( counter == 1 )
    		printf("The network is connected.\n");
    	else
    		printf("There are %d components.\n", counter);
    }
    Initialization( SetType S, int n ){ 
    	int i;
    	for(i = 0; i < n; i++) S[i] = -1;
    }
    
    int main()
    {  
    	SetType S;
    	int n;
    	char in;
    	scanf("%d\n", &n);
    	Initialization( S, n );
    	do { 
    		scanf("%c", &in);
    		switch (in) { 
    		case 'I': Input_connection( S ); break;
    		case 'C': Check_connection( S ); break;
    		case 'S': Check_network( S, n ); break;
    		}
    	} while ( in != 'S');
    	return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 6]

--------------------

### 3、05-树9 Huffman Codes (30分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 7]  
![在这里插入图片描述][20200323102052560.png]  
**Sample Input:**

7  
A 1 B 1 C 1 D 3 E 3 F 6 G 6  
4  
A 00000  
B 00001  
C 0001  
D 001  
E 01  
F 10  
G 11  
A 01010  
B 01011  
C 0100  
D 011  
E 10  
F 11  
G 00  
A 000  
B 001  
C 010  
D 011  
E 100  
F 101  
G 110  
A 00000  
B 00001  
C 0001  
D 001  
E 00  
F 10  
G 11

**Sample Output:**

Yes  
Yes  
No  
No

--------------------

#include <stdio.h>
    #include <stdlib.h>
    #include <string.h>
    #define Maxn 64
     
    int N, w[Maxn];
    char ch[Maxn];
    int codelen, cnt1, cnt2;
     
    typedef struct TreeNode* Tree;
    struct TreeNode { 
        int Weight;
        Tree Left, Right;
    };
     
    typedef struct HeapNode* Heap;
    struct HeapNode { 
        struct TreeNode Data[Maxn];
        int Size;
    };
     
    Tree CreatTree() { 
        Tree T;
        T = (Tree)malloc(sizeof(struct TreeNode));
        T->Left = T->Right = NULL;
        T->Weight = 0;
        return T;
    }
     
    Heap CreatHeap() { 
        Heap H;
        H = (Heap)malloc(sizeof(struct HeapNode));
        H->Size = 0;
        H->Data[0].Weight = -1;
        return H;
    }
     
    void Insert(Heap H, struct TreeNode T) { 
        int i = ++H->Size;
        for( ; T.Weight < H->Data[i/2].Weight; i /= 2)
            H->Data[i] = H->Data[i/2];
        H->Data[i] = T;
    }
     
    Tree Delete(Heap H) { 
        int parent, child;
        struct TreeNode Temp = H->Data[H->Size--];
        Tree T = CreatTree();
        *T = H->Data[1];
        for(parent = 1; 2*parent <= H->Size; parent = child) { 
            child = 2 * parent;
            if(child != H->Size && H->Data[child].Weight > H->Data[child+1].Weight) child++;
            if(Temp.Weight < H->Data[child].Weight) break;
            H->Data[parent] = H->Data[child];
        }
        H->Data[parent] = Temp;
        return T;
    }
     
    Tree Huffman(Heap H) { 
        Tree T = CreatTree(); //分配空间
        while(H->Size != 1) { 
            T->Left = Delete(H);
            T->Right = Delete(H);
            T->Weight = T->Left->Weight + T->Right->Weight;
            Insert(H, *T);
        }
        T = Delete(H);
        return T;
    }
     
    int WPL(Tree T, int Depth) { 
        if(!T->Left && !T->Right) {  return Depth*T->Weight; }
        else return WPL(T->Left, Depth+1) + WPL(T->Right, Depth+1);
    }
     
    void JudgeTree(Tree T) { 
        if(T) { 
            if(T->Left && T->Right) cnt2++;
            else if(!T->Left && !T->Right) cnt1++;
            else cnt1 = 0;
            JudgeTree(T->Left);
            JudgeTree(T->Right);
        }
    }
     
    int Judge() { 
        int i, j, wgh, flag = 1;
        char s1[Maxn], s2[Maxn];
        Tree T = CreatTree(), pt = NULL;
        for(i = 0; i < N; i++) { 
            scanf("%s%s", s1, s2);
            if(strlen(s2) > N) return 0;
            for(j = 0; s1[0] != ch[j]; j++); wgh = w[j];
            pt = T;//每次建树前先将指针移动到根节点上；
            for(j = 0; s2[j]; j++) { 
                if(s2[j] == '0') { 
                    if(!pt->Left) pt->Left = CreatTree();
                    pt = pt->Left;
                }
                if(s2[j] == '1') { 
                    if(!pt->Right) pt->Right = CreatTree();
                    pt = pt->Right;
                }
                if(pt->Weight) flag = 0;
                if(!s2[j+1]) { 
                    if(pt->Left || pt->Right) flag = 0;//判断是否为前缀
                    pt->Weight = wgh;
                }
            }
        }
        if(!flag) return 0;
        cnt1 = cnt2 = 0;
        JudgeTree(T);//判断是否不存在度数1的节点
        if(cnt1 != cnt2 + 1) return 0;
        if(codelen == WPL(T, 0)) return 1;
        else return 0;
    }
     
    int main() { 
        int i, n;
        Heap H;
        Tree T;
        H = CreatHeap();
        T = CreatTree();
        scanf("%d", &N);
        for(i = 0; i < N; i++) { 
            getchar();
            scanf("%c %d", &ch[i], &w[i]);
            H->Data[H->Size].Left = H->Data[H->Size].Right = NULL;
            T->Weight = w[i];
            Insert(H, *T);
        }
        T = Huffman(H);     //PreTravel(T);
        codelen = WPL(T, 0);
        scanf("%d", &n);
        while(n--) { 
            if(Judge()) printf("Yes\n");
            else printf("No\n");
        }
        return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 8]

# 总结 #

简单总结下这周的学习内容，我哭了，笔断了，电脑死机了，

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70_pic_center]  
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