In 1953, David A. Huffman published his paper "A Method for the Construction of Minimum-Redundancy Codes", and hence printed his name in the history of computer science. As a professor who gives the final exam problem on Huffman codes, I am encountering a big problem: the Huffman codes are NOT unique. For example, given a string "aaaxuaxz", we can observe that the frequencies of the characters 'a', 'x', 'u' and 'z' are 4, 2, 1 and 1, respectively. We may either encode the symbols as {'a'=0, 'x'=10, 'u'=110, 'z'=111}, or in another way as {'a'=1, 'x'=01, 'u'=001, 'z'=000}, both compress the string into 14 bits. Another set of code can be given as {'a'=0, 'x'=11, 'u'=100, 'z'=101}, but {'a'=0, 'x'=01, 'u'=011, 'z'=001} is NOT correct since "aaaxuaxz" and "aazuaxax" can both be decoded from the code 00001011001001. The students are submitting all kinds of codes, and I need a computer program to help me determine which ones are correct and which ones are not.

Input Specification:

Each input file contains one test case. For each case, the first line gives an integer N (2), then followed by a line that contains all the N distinct characters and their frequencies in the following format:

c[1] f[1] c[2] f[2] ... c[N] f[N]

where c[i] is a character chosen from {'0' - '9', 'a' - 'z', 'A' - 'Z', '_'}, and f[i] is the frequency of c[i] and is an integer no more than 1000. The next line gives a positive integer M (≤), then followed by M student submissions. Each student submission consists of N lines, each in the format:

c[i] code[i]

where c[i] is the i-th character and code[i] is an non-empty string of no more than 63 '0's and '1's.

Output Specification:

For each test case, print in each line either "Yes" if the student's submission is correct, or "No" if not.

Note: The optimal solution is not necessarily generated by Huffman algorithm. Any prefix code with code length being optimal is considered correct.

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<iostream>
#include<cstring>
using namespace std;
#define maxn 70
int N,codelen,cnt1,cnt2,w[maxn];
char ch[maxn];
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 = new struct TreeNode;
T->weight = ;
T->Left = T->Right = NULL;
return T;
} Heap creatHeap(){
Heap H;
H = new struct HeapNode;
H->Data[].weight = -;
H->size = ;
return H;
} void Insert(Heap H,struct TreeNode T){
int i = ++H->size;
for(; T.weight < H->Data[i/].weight; i /= )
H->Data[i] = H->Data[i/];
H->Data[i] = T;
} Tree Delete(Heap H){
int child,parent;
struct TreeNode Temp = H->Data[H->size--];
Tree T = creatTree();
*T = H->Data[];
for(parent = ; * parent <= H->size; parent = child){
child = * parent;
if(child < H->size && H->Data[child].weight > H->Data[child+].weight)
child++;
if(H->Data[child].weight > Temp.weight) break;
H->Data[parent] = H->Data[child];
}
H->Data[parent] = Temp;
return T;
} Tree Huffman(Heap H){
Tree T = creatTree();
while(H->size != ){
T->Left = Delete(H);
T->Right = Delete(H);
T->weight = T->Right->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+)+WPL(T->Right,depth+);
} void JudgeTree(Tree T){
if(T){
if(T->Right && T->Left) cnt2++;
else if(!T->Left && !T->Right) cnt1++;
else cnt1 = ;
JudgeTree(T->Left);
JudgeTree(T->Right);
}
} int Judge(){
int i,j,wgh,flag = ;;
char s1[maxn],s2[maxn];
Tree T = creatTree(), pt = NULL;
for(i = ; i < N; i++){
cin >> s1 >> s2;
if(strlen(s2) > N) return ;
for(j = ; s1[] != ch[j]; j++); wgh = w[j];
pt = T;
for(j = ; s2[j] ; j++){
if(s2[j] == ''){
if(!pt->Left) pt->Left = creatTree();
pt = pt->Left;
}
if(s2[j] == ''){
if(!pt->Right) pt->Right = creatTree();
pt = pt->Right;
}
if(pt->weight) flag = ;
if(!s2[j+]){
if(pt->Left || pt->Right) flag = ;
else pt->weight = wgh;
}
}
}
if(flag == ) return ;
cnt1 = cnt2 = ;
JudgeTree(T);
if(cnt1 != cnt2 + ) return ;
if(codelen == WPL(T,)) return ;
else return ;
} int main(){
int i,n;
Tree T;
Heap H;
T = creatTree();
H = creatHeap();
cin >> N;
for(i = ; i < N; i++){
getchar();
cin >> 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);
codelen = WPL(T,);
cin >> n;
while(n--){
if(Judge()) cout<< "Yes" << endl;
else cout << "No" << endl;
}
return ;
}

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