POJ - 1521 Entropy
来源:互联网 发布:神机妙算软件介绍 编辑:程序博客网 时间:2024/04/30 07:42
Entropy
Time Limit: 1000MS Memory Limit: 10000KB 64bit IO Format: %I64d & %I64u
Description
An entropy encoder is a data encoding method that achieves lossless data compression by encoding a message with "wasted" or "extra" information removed. In other words, entropy encoding removes information that was not necessary in the first place to accurately encode the message. A high degree of entropy implies a message with a great deal of wasted information; english text encoded in ASCII is an example of a message type that has very high entropy. Already compressed messages, such as JPEG graphics or ZIP archives, have very little entropy and do not benefit from further attempts at entropy encoding.
English text encoded in ASCII has a high degree of entropy because all characters are encoded using the same number of bits, eight. It is a known fact that the letters E, L, N, R, S and T occur at a considerably higher frequency than do most other letters in english text. If a way could be found to encode just these letters with four bits, then the new encoding would be smaller, would contain all the original information, and would have less entropy. ASCII uses a fixed number of bits for a reason, however: it’s easy, since one is always dealing with a fixed number of bits to represent each possible glyph or character. How would an encoding scheme that used four bits for the above letters be able to distinguish between the four-bit codes and eight-bit codes? This seemingly difficult problem is solved using what is known as a "prefix-free variable-length" encoding.
In such an encoding, any number of bits can be used to represent any glyph, and glyphs not present in the message are simply not encoded. However, in order to be able to recover the information, no bit pattern that encodes a glyph is allowed to be the prefix of any other encoding bit pattern. This allows the encoded bitstream to be read bit by bit, and whenever a set of bits is encountered that represents a glyph, that glyph can be decoded. If the prefix-free constraint was not enforced, then such a decoding would be impossible.
Consider the text "AAAAABCD". Using ASCII, encoding this would require 64 bits. If, instead, we encode "A" with the bit pattern "00", "B" with "01", "C" with "10", and "D" with "11" then we can encode this text in only 16 bits; the resulting bit pattern would be "0000000000011011". This is still a fixed-length encoding, however; we’re using two bits per glyph instead of eight. Since the glyph "A" occurs with greater frequency, could we do better by encoding it with fewer bits? In fact we can, but in order to maintain a prefix-free encoding, some of the other bit patterns will become longer than two bits. An optimal encoding is to encode "A" with "0", "B" with "10", "C" with "110", and "D" with "111". (This is clearly not the only optimal encoding, as it is obvious that the encodings for B, C and D could be interchanged freely for any given encoding without increasing the size of the final encoded message.) Using this encoding, the message encodes in only 13 bits to "0000010110111", a compression ratio of 4.9 to 1 (that is, each bit in the final encoded message represents as much information as did 4.9 bits in the original encoding). Read through this bit pattern from left to right and you’ll see that the prefix-free encoding makes it simple to decode this into the original text even though the codes have varying bit lengths.
As a second example, consider the text "THE CAT IN THE HAT". In this text, the letter "T" and the space character both occur with the highest frequency, so they will clearly have the shortest encoding bit patterns in an optimal encoding. The letters "C", "I’ and "N" only occur once, however, so they will have the longest codes.
There are many possible sets of prefix-free variable-length bit patterns that would yield the optimal encoding, that is, that would allow the text to be encoded in the fewest number of bits. One such optimal encoding is to encode spaces with "00", "A" with "100", "C" with "1110", "E" with "1111", "H" with "110", "I" with "1010", "N" with "1011" and "T" with "01". The optimal encoding therefore requires only 51 bits compared to the 144 that would be necessary to encode the message with 8-bit ASCII encoding, a compression ratio of 2.8 to 1.
English text encoded in ASCII has a high degree of entropy because all characters are encoded using the same number of bits, eight. It is a known fact that the letters E, L, N, R, S and T occur at a considerably higher frequency than do most other letters in english text. If a way could be found to encode just these letters with four bits, then the new encoding would be smaller, would contain all the original information, and would have less entropy. ASCII uses a fixed number of bits for a reason, however: it’s easy, since one is always dealing with a fixed number of bits to represent each possible glyph or character. How would an encoding scheme that used four bits for the above letters be able to distinguish between the four-bit codes and eight-bit codes? This seemingly difficult problem is solved using what is known as a "prefix-free variable-length" encoding.
In such an encoding, any number of bits can be used to represent any glyph, and glyphs not present in the message are simply not encoded. However, in order to be able to recover the information, no bit pattern that encodes a glyph is allowed to be the prefix of any other encoding bit pattern. This allows the encoded bitstream to be read bit by bit, and whenever a set of bits is encountered that represents a glyph, that glyph can be decoded. If the prefix-free constraint was not enforced, then such a decoding would be impossible.
Consider the text "AAAAABCD". Using ASCII, encoding this would require 64 bits. If, instead, we encode "A" with the bit pattern "00", "B" with "01", "C" with "10", and "D" with "11" then we can encode this text in only 16 bits; the resulting bit pattern would be "0000000000011011". This is still a fixed-length encoding, however; we’re using two bits per glyph instead of eight. Since the glyph "A" occurs with greater frequency, could we do better by encoding it with fewer bits? In fact we can, but in order to maintain a prefix-free encoding, some of the other bit patterns will become longer than two bits. An optimal encoding is to encode "A" with "0", "B" with "10", "C" with "110", and "D" with "111". (This is clearly not the only optimal encoding, as it is obvious that the encodings for B, C and D could be interchanged freely for any given encoding without increasing the size of the final encoded message.) Using this encoding, the message encodes in only 13 bits to "0000010110111", a compression ratio of 4.9 to 1 (that is, each bit in the final encoded message represents as much information as did 4.9 bits in the original encoding). Read through this bit pattern from left to right and you’ll see that the prefix-free encoding makes it simple to decode this into the original text even though the codes have varying bit lengths.
As a second example, consider the text "THE CAT IN THE HAT". In this text, the letter "T" and the space character both occur with the highest frequency, so they will clearly have the shortest encoding bit patterns in an optimal encoding. The letters "C", "I’ and "N" only occur once, however, so they will have the longest codes.
There are many possible sets of prefix-free variable-length bit patterns that would yield the optimal encoding, that is, that would allow the text to be encoded in the fewest number of bits. One such optimal encoding is to encode spaces with "00", "A" with "100", "C" with "1110", "E" with "1111", "H" with "110", "I" with "1010", "N" with "1011" and "T" with "01". The optimal encoding therefore requires only 51 bits compared to the 144 that would be necessary to encode the message with 8-bit ASCII encoding, a compression ratio of 2.8 to 1.
Input
The input file will contain a list of text strings, one per line. The text strings will consist only of uppercase alphanumeric characters and underscores (which are used in place of spaces). The end of the input will be signalled by a line containing only the word “END” as the text string. This line should not be processed.
Output
For each text string in the input, output the length in bits of the 8-bit ASCII encoding, the length in bits of an optimal prefix-free variable-length encoding, and the compression ratio accurate to one decimal point.
Sample Input
AAAAABCDTHE_CAT_IN_THE_HATEND
Sample Output
64 13 4.9144 51 2.8
哈弗曼树
#include<iostream>#include<string>#include<cstring>#include<iomanip>#include<cstdio>#include<queue> //哈夫曼树,用优先队列实现using namespace std;int main(){string s;while (cin >> s){if (s == "END") break;int len = s.size();int date[30] = { 0 }; //date数组记录text中各个字符的频数priority_queue<int>q;for (int i = 0; i < len; i++){if (s[i] == '_') date[0]++;else date[s[i] - 'A' + 1]++;}for (int i = 0; i < 27; i++){if (date[i] != 0) q.push(-date[i]); //只把不同字符的频数加入优先队列,字符本身与题目要求无关} //处理使小的数据的优先级别高int ans = 0;int tem;while (!q.empty()){tem = -q.top(); //取出最小的两个数,相加累计到ans中,并加入队列,一直处理到队列中没有数q.pop();if (!q.empty()){tem = tem - q.top();q.pop();}ans = ans + tem;if (!q.empty())q.push(-tem); //若队列已没有数据,则不添加(上面已经取出最后两个,或一个),若没有这一步,上面whlie的判断不成立。}int ans8 = len << 3;double bi = (double)ans8 / ans;printf("%d %d %.1lf\n", ans8, ans, bi);}}
0 0
- POJ 1521 Entropy
- poj 1521 Entropy
- poj 1521 Entropy
- POJ 1521 Entropy
- POJ 1521 Entropy
- POJ - 1521 Entropy
- poj 1521 Entropy
- POJ 1521 , Entropy , Huffman
- POJ 1521-Entropy 贪心问题
- POJ 1521 Entropy(哈夫曼树)
- poj 1521(ENTROPY) huffman 编码
- POJ-1521/ZOJ-1117/Entropy
- 贪心1003 POJ 1521-Entropy
- hdu-1053-Entropy && poj-1521-Entropy (哈夫曼编码)
- POJ 1521 Entropy(哈夫曼编码)
- poj 1521 Entropy huffman(哈夫曼)编码
- poj 1521 Entropy(优先队列)
- POJ 1521 Entropy 优先队列/multiset
- 终于找到了-----------让工资再涨的编程方式
- 将ppt文件如何转换成pdf文件
- Redirecting携带数据
- 优化实例内存
- Android Studio 1.1.0 配置androidannotations框架
- POJ - 1521 Entropy
- android获取在线视频略缩图
- Discover Feature Engineering, How to Engineer Features and How to Get Good at It
- mysql命令行修改字符编码
- C++中vector用法
- PHP5.6新特性介绍
- Contiki的Event_Driven 模型的整理与思考
- 谈谈redis与memcached(一)
- 第三周项目5.3 5.4 5.5 文件输入工资 通过多文件组织