ืืืืืขืจ ืื ืึธื ืืืื ืคืื ืืขื ืงืืจืก ืืืจ ืืึธืื ืฆืืืขืืจืืื ืึท ืืืืขืจืืขืฆืื ื ืคืื ืื ืื ืืขืจ ื ืืฆืืง ืืึทืืขืจืืึทื ืคึฟืึทืจ ืืืจ.
Huffman ืงืึธืืืจืื ื ืืื ืึท ืืึทืื ืงืึทืืคึผืจืขืฉืึทื ืึทืืืขืจืืืึทื ืืืึธืก ืคืึธืจืืืืืจื ืื ืืจืื ื ืืขืืึทื ืง ืคืื ืืขืงืข ืงืึทืืคึผืจืขืฉืึทื. ืืื ืืขื ืึทืจืืืงื ืืืจ ืืืขืื ืจืขืื ืืืขืื ืคืึทืจืคืขืกืืืงื ืืื ืืืขืจืืึทืืึทื ืืขื ื ืงืึธืืืจืื ื, ืืื ืืงืื ืืืงืึธืืืึทืืึทื ืงืึธืืื, ืคึผืจืขืคืืงืก ืึผืืืื ืืื Huffman ืืืื ืงืึทื ืกืืจืึทืงืฉืึทื.
ืืืจ ืืืืกื ืึทื ืืขืืขืจ ืืึทืจืึทืงืืขืจ ืืื ืกืืึธืจื ืืื ืึท ืกืืงืืืึทื ืก ืคืื 0 ืก ืืื 1 ืก ืืื ืึทืงืืึทืคึผืืื 8 ืืืื. ืืึธืก ืืื ืืขืจืืคื ืคืึทืจืคืขืกืืืงื-ืืขื ื ืงืึธืืืจืื ื ืืืืึทื ืืขืืขืจ ืืึทืจืึทืงืืขืจ ื ืืฆื ืื ืืขืืืข ืคืึทืจืคืขืกืืืงื ื ืืืขืจ ืคืื ืกืืึธืจืืืืฉ ืืืื.
ืืื ืก ืืึธืื ืื ืืขืงืกื ืืื ืืขืืขืื. ืืื ืงืขื ืขื ืืืจ ืจืขืืืฆืืจื ืื ืกืืืข ืคืื โโืคึผืืึทืฅ ืคืืจืืื ืื ืฆื ืงืจืึธื ืืืื ืืึทืจืึทืงืืขืจ?
ืื ืืจืื ื ืืขืืึทื ืง ืืื ืงืึธืืืจืื ื ืคืื ืืืขืจืืึทืืึทื ืืขื ื. ืืืจ ืงืขื ืขื ื ืืฆื ืืขื ืคืึทืงื ืึทื ืขืืืขืืข ืืืชืืืช ืืขืจืฉืืึทื ืขื ืืขืจ ืึธืคื ืืื ืืขืงืกื ืืื ืื ืืขืจืข () ืฆื ืึทื ืืืืืงืืขื ืึทื ืึทืืืขืจืืืึทื ืืืึธืก ืืืขื ืคืึธืจืฉืืขืื ืื ืืขืืืข ืกืืงืืืึทื ืก ืคืื ืืืชืืืช ืืื ืืืืื ืืงืขืจืข ืืืื. ืืื ืืืขืจืืึทืืึทื ืืขื ื ืงืึธืืืจืื ื, ืืืจ ืืึทืฉืืืืขื ืึท ืืืึทืืขืืืืืง ื ืืืขืจ ืคืื ืืืื ืฆื ืืืชืืืช ืืืคึผืขื ืืื ื ืืืืฃ ืื ืึธืคืืงืืึทื ืคืื ืืืืขืจ ืคึผืึทืกืืจืื ื ืืื ืึท ืืขืืขืื ืืขืงืกื. ืืขืกืึธืฃ, ืขืืืขืืข ืืืชืืืช ืงืขื ื ืขืืขื ืืืืื 1 ืืืกื, ืืฉืขืช ืื ืืขืจืข ืงืขื ื ืขืืขื 2 ืืืื, 3 ืึธืืขืจ ืืขืจ. ืื ืคึผืจืึธืืืขื ืืื ืื ืงืึธืืืจืื ื ืคืื ืืืขืจืืึทืืึทื ืืขื ื ืืื ืืืืื ืื ืกืึทืืกืึทืงืืืึทื ื ืืืงืึธืืืื ื ืคืื ืื ืกืืงืืืึทื ืก.
ืืื, ืืืืกื ืึท ืกืืงืืืึทื ืก ืคืื ืืืื, ืงืขื ืขื ืืืจ ืืขืงืึธืืข ืขืก ืึทื ืึทืืืืืืืึทืกืื?
ืืึทืืจืึทืืื ืื ืฉืืจื "ืึทืึทืืึทืงืืึทื". ืขืก ืืื 8 ืืืชืืืช, ืืื ืืื ืคืึทืจืคืขืกืืืงื ืืขื ื ืงืึธืืืจืื ื ืขืก ืืืขื ืืึทืจืคื 64 ืืืฅ ืฆื ืงืจืึธื ืขืก. ืืึทืืขืจืงืื ื ืึทื ืื ืกืืืืึธื ืึธืคืืงืืึทื "ืึท", "ื", "ืง" ะธ "ื" ืืงืืืึทืื 4, 2, 1, 1 ืจืืกืคึผืขืงืืืืืื. ืืื ืก ืคึผืจืืืืจื ืฆื ืืืึทืืืฉืึทื "ืึทืึทืืึทืงืืึทื" ืืื ืืืืื ืืงืขืจืข ืืืื, ื ืืฆื ืื ืคืึทืงื ืึทื "ืฆื" ืึทืงืขืจื ืืขืจ ืึธืคื ืืื "ื"ืืื "ื" ืึทืงืขืจื ืืขืจ ืึธืคื ืืื "ื" ะธ "ื". ืืืจ ืืืขืื ืึธื ืืืืื ืืื ืงืึธืืืจืื ื "ืฆื" ื ืืฆื ืืืื ืืืกื ืืืืึทื ืฆื 0, "ื" ืืืจ ืืืขืื ืืึทืฉืืืืขื ืึท ืฆืืืื-ืืืกื ืงืึธื ืฆื 11, ืืื ื ืืฆื ืืจืื ืืืื 100 ืืื 011, ืืืจ ืขื ืงืึธืื "ื" ะธ "ื".
ืืื ืึท ืจืขืืืืืึทื, ืืืจ ืืืขืื ืืึทืงืืืขื:
a
0
b
11
c
100
d
011
ืึทืืื ืื ืฉืืจื "ืึทืึทืืึทืงืืึทื" ืืืจ ืืืขื ืงืึธื ืขืก ืืื 00110100011011 (0|0|11|0|100|011|0|11)ื ืืฆื ืื ืงืึธืืื ืฆืืืขืฉืืขืื ืืืืื. ืึธืืขืจ, ืืขืจ ืืืืคึผื ืคึผืจืึธืืืขื ืืืขื ืืืื ืืื ืืืงืึธืืืื ื. ืืืขื ืืืจ ืคึผืจืืืืจื ืฆื ืืขืงืึธืืข ืื ืฉืืจืืงื 00110100011011, ืืืจ ืืืขืื ืืึทืงืืืขื ืึทื ืึทืืืืืืืึทืก ืจืขืืืืืึทื, ืืืืึทื ืขืก ืงืขื ืขื ืืืื ืจืขืคึผืจืืืขื ืืื ืืื:
0|011|0|100|011|0|11 adacdab
0|0|11|0|100|0|11|011 aabacabd
0|011|0|100|0|11|0|11 adacabab
...
ืืื"ื ื
ืฆื ืืืกืืืืื ืืขื ืึทืืืืืืืึทืื, ืืืจ ืืืื ืขื ืฉืืจ ืึทื ืืื ืืืขืจ ืงืึธืืืจืื ื ืกืึทืืืกืคืืื ืึท ืืึทืืจืืฃ ืึทืืึท ืืื ืคึผืจืขืคืืงืก ืืขืจืฉื, ืืืึธืก ืืื ืงืขืจ ืืืคึผืืืื ืึทื ืื ืงืึธืืื ืงืขื ืขื ืืืื ืืืงืึธืืืื ืืื ืืืืื ืืืื ืืื ืฆืืง ืืืขื. ืื ืคึผืจืขืคืืงืก ืืขืจืฉื ืื ืฉืืจื ืึทื ืงืืื ืงืึธื ืืื ืึท ืคึผืจืขืคืืงืก ืคืื ืื ืืขืจื. ืืื ืงืึธื ืืืจ ืืืื ืขื ืื ืืืื ืืขื ืืฆื ืฆื ืคืึธืจืฉืืขืื ืึท ืืึทืืื ืืขืจ ืืึทืจืึทืงืืขืจ. ืืื ืื ืืืืื ืืืึทืฉืคึผืื 0 ืืื ืึท ืคึผืจืขืคืืงืก 011, ืืืืก ืคืืจืืขืฆื ืื ืคืจืขืคืืงืก ืืขืจืฉื. ืึทืืื, ืืืื ืืื ืืืขืจ ืงืึธืืื ืืึทืคืจืืืืงื ืื ืคึผืจืขืคืืงืก ืืขืจืฉื, ืืืจ ืงืขื ืขื ืึทื ืึทืืืืืืืึทืกืื ืืขืงืึธืืข (ืืื ืืืืฆืข ืืืขืจืกืึท).
ืืื ืก ืจืืืืืื ืื ืืืึทืฉืคึผืื ืืืืื. ืืึธืก ืืึธื ืืืจ ืืืขืื ืืึทืฉืืืืขื ืคึฟืึทืจ ืกืืืืึธืืก "ืึท", "ื", "ืง" ะธ "ื" ืงืึธืืื ืืืึธืก ืืึทืคืจืืืืงื ืื ืคึผืจืขืคืืงืก ืืขืจืฉื.
a
0
b
10
c
110
d
111
ื ืืฆื ืืขื ืงืึธืืืจืื ื, ืื ืฉืืจืืงื "ืึทืึทืืึทืงืืึทื" ืืืขื ืืืื ืขื ืงืึธืืืื ืืื 00100100011010 (0|0|10|0|100|011|0|10). ืืื ืืึธ 00100100011010 ืืืจ ืงืขื ืขื ืืืฆื ืึทื ืึทืืืืืืืึทืกืื ืืขืงืึธืืข ืืื ืฆืืจืืงืงืืืขื ืฆื ืืื ืืืขืจ ืึธืจืืืื ืขื ืฉืืจืืงื "ืึทืึทืืึทืงืืึทื".
ืืืคืืึทื ืงืึธืืืจืื ื
ืืืฆื ืึทื ืืืจ ืคึฟืึทืจืฉืืืื ืื ืงืึธืืืจืื ื ืคืื ืืืขืจืืึทืืึทื ืืขื ื ืืื ืื ืคึผืจืขืคืืงืก ืืขืจืฉื, ืืึธืื ืืื ืื ืจืขืื ืืืขืื Huffman ืงืึธืืืจืื ื.
ืืขืจ ืืืคึฟื ืืื ืืืืืจื ืืืืฃ ืงืจืืืืืื ื ืืืื ืขืจื ืืืืืขืจ. ืืื ืขืก, ืึท ื ืึธืืข ืงืขื ืขื ืืืื ืึธืืขืจ ืืขืฆื ืึธืืขืจ ืื ืขืจืืขื. ืืืืืขืก, ืึทืืข ื ืึธืืื ืืขื ืขื ืืขืืืืื ืืืขืืขืจ (ืขื ืื), ืืืึธืก ืคืึธืจืฉืืขืื ืื ืกืืืืึธื ืืื ืืื ืืืึทื ืืืึธื (ืืึธืก ืืื ืื ืึธืคืืงืืึทื ืคืื ืคึผืึทืกืืจืื ื). ืื ืื ืขืจืืขื ื ืึธืืื ืึทื ืืืึทืืื ืื ืืืึธื ืคืื ืื ืืึทืจืึทืงืืขืจ ืืื ืึธืคึผืฉืืงื ืฆื ืฆืืืื ืกืึทืงืกืขืกืขืจ ื ืึธืืื. ืืืื ืึทืืืขืืืื ืืขืกืงืขื, ืืืกื ยซ0ยป ืจืขืคึผืจืึทืืขื ืฅ ืืืืึทืืขืจืืืง ืื ืืื ืงืก ืฆืืืืึทื, ืืื ยซ1ยป - ืืืืฃ ืื ืจืขืืืข ืืืื. ืืื ืคืื ืืืื N ืืืขืืขืจ ืืื N-ืงืกื ืืืงืก ืื ืขืจืืขื ื ืึธืืื. ืขืก ืืื ืจืขืงืึทืืขื ืืื ืึทื ืืืขื ืงืึทื ืกืืจืึทืงืืื ื ืึท Huffman ืืืื, ืึทื ืืืื ืกืืืืึธืืก ืืขื ืขื ืืืกืงืึทืจืืื ืฆื ืืึทืงืืืขื ืงืึธืืื ืคืื ืึธืคึผืืืืึทื ืืขื ื.
ืืืจ ืืืขืื ื ืืฆื ืึท ืืืืืขืจืงืืึทื ืจืื ืฆื ืืืืขื ืึท ืืืคืืึทื ืืืื, ืืื ืื ื ืึธืืข ืืื ืื ืืึธืืึทืกื ืึธืคืืงืืึทื ืืืขื ืืึทืงืืืขื ืื ืืขืืกืื ืืืืืขืจืงืืึทื. ืื ืงืึทื ืกืืจืึทืงืฉืึทื ืกืืขืคึผืก ืืขื ืขื ืืืกืงืจืืืื ืืื ืื:
- ืฉืึทืคึฟื ืึท ืืืึทื ื ืึธืืข ืคึฟืึทืจ ืืขืืขืจ ืกืืืืึธื ืืื ืืืืื ืืื ืฆื ืื ืืืืืขืจืงืืึทื ืจืื.
- ืืฉืขืช ืขืก ืืื ืืขืจ ืืื ืืืื ืืืึทื ืืื ืืขืจ ืจืื, ืืึธื ืื ืคืืืืขื ืืข:
- ืึทืจืึธืคึผื ืขืืขื ืื ืฆืืืื ืืขืืกืื ืืืืืขืจืงืืึทื (ืืึธืืึทืกื ืึธืคืืงืืึทื) ื ืึธืืื ืคืื ืื ืจืื;
- ืฉืึทืคึฟื ืึท ื ืืึทืข ืื ืขืจืืขื ื ืึธืืข, ืืื ืื ืฆืืืื ื ืึธืืื ืืืขื ืืืื ืื ืกืึทืงืกืขืกืขืจื, ืืื ืื ืึธืคืืงืืึทื ืคืื ืคึผืึทืกืืจืื ื ืืืขื ืืืื ืืืืึทื ืฆื ืื ืกืึทืืึทืงื ืคืื ืื ืคืจืืงืืืึทื ืกืื ืคืื ืื ืฆืืืื ื ืึธืืื.
- ืืืื ืึท ื ืืึทืข ื ืึธืืข ืฆื ืื ืืืืืขืจืงืืึทื ืจืื.
- ืืขืจ ืืืืื ืจืืขื ื ืึธืืข ืืืขื ืืืื ืืขืจ ืืืึธืจืฆื ื ืึธืืข, ืืื ืืึธืก ืืืขื ืคืึทืจืขื ืืืงื ืื ืงืึทื ืกืืจืึทืงืฉืึทื ืคืื ืืขื ืืืื.
ืืืืืจ ืืื ืคืืจืฉืืขืื ืื ืืืจ ืืืื ืขืคืขืก ื ืืขืงืกื ืืืืก ืืืฉืืืื ืืืืื ืคืื ืืืชืืืช "ื ื ื ื" ะธ "ืืื", ืืื ืืืืขืจ ืึธืคืืงืืึทื ืคืื ืคึผืึทืกืืจืื ื ืืขื ืขื 15, 7, 6, 6 ืืื 5, ืจืืกืคึผืขืงืืืืืื. ืื ืืขืจ ืืขื ืขื ืืืืืกืืจืืฆืืขืก ืืืึธืก ืคืึทืจืืจืึทืืื ืืื ืื ืกืืขืคึผืก ืคืื ืื ืึทืืืขืจืืืึทื.
ืืขืจ ืืจื ืคืื ืืขืจ ืืืึธืจืฆื ืฆื ืงืืื ืืืึทื ื ืึธืืข ืืืขื ืงืจืึธื ืื ืึธืคึผืืืืึทื ืคึผืจืขืคืืงืก ืงืึธื (ืืืื ืืืงืื ื ืืื ืึท Huffman ืงืึธื) ืงืึธืจืึทืกืคึผืึทื ืืื ื ืฆื ืื ืืึทืจืึทืงืืขืจ ืคืืจืืื ืื ืืื ืืขื ืืืึทื ื ืึธืืข.
ืืืคืืื ืืืื
ืื ืืขืจ ืืืจ ืืืขื ืืขืคึฟืื ืขื ืึทื ืืืคึผืืึทืืขื ืืืืฉืึทื ืคืื ืื Huffman ืงืึทืืคึผืจืขืฉืึทื ืึทืืืขืจืืืึทื ืืื C ++ ืืื Java:
#include <iostream>
#include <string>
#include <queue>
#include <unordered_map>
using namespace std;
// A Tree node
struct Node
{
char ch;
int freq;
Node *left, *right;
};
// Function to allocate a new tree node
Node* getNode(char ch, int freq, Node* left, Node* right)
{
Node* node = new Node();
node->ch = ch;
node->freq = freq;
node->left = left;
node->right = right;
return node;
}
// Comparison object to be used to order the heap
struct comp
{
bool operator()(Node* l, Node* r)
{
// highest priority item has lowest frequency
return l->freq > r->freq;
}
};
// traverse the Huffman Tree and store Huffman Codes
// in a map.
void encode(Node* root, string str,
unordered_map<char, string> &huffmanCode)
{
if (root == nullptr)
return;
// found a leaf node
if (!root->left && !root->right) {
huffmanCode[root->ch] = str;
}
encode(root->left, str + "0", huffmanCode);
encode(root->right, str + "1", huffmanCode);
}
// traverse the Huffman Tree and decode the encoded string
void decode(Node* root, int &index, string str)
{
if (root == nullptr) {
return;
}
// found a leaf node
if (!root->left && !root->right)
{
cout << root->ch;
return;
}
index++;
if (str[index] =='0')
decode(root->left, index, str);
else
decode(root->right, index, str);
}
// Builds Huffman Tree and decode given input text
void buildHuffmanTree(string text)
{
// count frequency of appearance of each character
// and store it in a map
unordered_map<char, int> freq;
for (char ch: text) {
freq[ch]++;
}
// Create a priority queue to store live nodes of
// Huffman tree;
priority_queue<Node*, vector<Node*>, comp> pq;
// Create a leaf node for each character and add it
// to the priority queue.
for (auto pair: freq) {
pq.push(getNode(pair.first, pair.second, nullptr, nullptr));
}
// do till there is more than one node in the queue
while (pq.size() != 1)
{
// Remove the two nodes of highest priority
// (lowest frequency) from the queue
Node *left = pq.top(); pq.pop();
Node *right = pq.top(); pq.pop();
// Create a new internal node with these two nodes
// as children and with frequency equal to the sum
// of the two nodes' frequencies. Add the new node
// to the priority queue.
int sum = left->freq + right->freq;
pq.push(getNode('', sum, left, right));
}
// root stores pointer to root of Huffman Tree
Node* root = pq.top();
// traverse the Huffman Tree and store Huffman Codes
// in a map. Also prints them
unordered_map<char, string> huffmanCode;
encode(root, "", huffmanCode);
cout << "Huffman Codes are :n" << 'n';
for (auto pair: huffmanCode) {
cout << pair.first << " " << pair.second << 'n';
}
cout << "nOriginal string was :n" << text << 'n';
// print encoded string
string str = "";
for (char ch: text) {
str += huffmanCode[ch];
}
cout << "nEncoded string is :n" << str << 'n';
// traverse the Huffman Tree again and this time
// decode the encoded string
int index = -1;
cout << "nDecoded string is: n";
while (index < (int)str.size() - 2) {
decode(root, index, str);
}
}
// Huffman coding algorithm
int main()
{
string text = "Huffman coding is a data compression algorithm.";
buildHuffmanTree(text);
return 0;
}import java.util.HashMap;
import java.util.Map;
import java.util.PriorityQueue;
// A Tree node
class Node
{
char ch;
int freq;
Node left = null, right = null;
Node(char ch, int freq)
{
this.ch = ch;
this.freq = freq;
}
public Node(char ch, int freq, Node left, Node right) {
this.ch = ch;
this.freq = freq;
this.left = left;
this.right = right;
}
};
class Huffman
{
// traverse the Huffman Tree and store Huffman Codes
// in a map.
public static void encode(Node root, String str,
Map<Character, String> huffmanCode)
{
if (root == null)
return;
// found a leaf node
if (root.left == null && root.right == null) {
huffmanCode.put(root.ch, str);
}
encode(root.left, str + "0", huffmanCode);
encode(root.right, str + "1", huffmanCode);
}
// traverse the Huffman Tree and decode the encoded string
public static int decode(Node root, int index, StringBuilder sb)
{
if (root == null)
return index;
// found a leaf node
if (root.left == null && root.right == null)
{
System.out.print(root.ch);
return index;
}
index++;
if (sb.charAt(index) == '0')
index = decode(root.left, index, sb);
else
index = decode(root.right, index, sb);
return index;
}
// Builds Huffman Tree and huffmanCode and decode given input text
public static void buildHuffmanTree(String text)
{
// count frequency of appearance of each character
// and store it in a map
Map<Character, Integer> freq = new HashMap<>();
for (int i = 0 ; i < text.length(); i++) {
if (!freq.containsKey(text.charAt(i))) {
freq.put(text.charAt(i), 0);
}
freq.put(text.charAt(i), freq.get(text.charAt(i)) + 1);
}
// Create a priority queue to store live nodes of Huffman tree
// Notice that highest priority item has lowest frequency
PriorityQueue<Node> pq = new PriorityQueue<>(
(l, r) -> l.freq - r.freq);
// Create a leaf node for each character and add it
// to the priority queue.
for (Map.Entry<Character, Integer> entry : freq.entrySet()) {
pq.add(new Node(entry.getKey(), entry.getValue()));
}
// do till there is more than one node in the queue
while (pq.size() != 1)
{
// Remove the two nodes of highest priority
// (lowest frequency) from the queue
Node left = pq.poll();
Node right = pq.poll();
// Create a new internal node with these two nodes as children
// and with frequency equal to the sum of the two nodes
// frequencies. Add the new node to the priority queue.
int sum = left.freq + right.freq;
pq.add(new Node('', sum, left, right));
}
// root stores pointer to root of Huffman Tree
Node root = pq.peek();
// traverse the Huffman tree and store the Huffman codes in a map
Map<Character, String> huffmanCode = new HashMap<>();
encode(root, "", huffmanCode);
// print the Huffman codes
System.out.println("Huffman Codes are :n");
for (Map.Entry<Character, String> entry : huffmanCode.entrySet()) {
System.out.println(entry.getKey() + " " + entry.getValue());
}
System.out.println("nOriginal string was :n" + text);
// print encoded string
StringBuilder sb = new StringBuilder();
for (int i = 0 ; i < text.length(); i++) {
sb.append(huffmanCode.get(text.charAt(i)));
}
System.out.println("nEncoded string is :n" + sb);
// traverse the Huffman Tree again and this time
// decode the encoded string
int index = -1;
System.out.println("nDecoded string is: n");
while (index < sb.length() - 2) {
index = decode(root, index, sb);
}
}
public static void main(String[] args)
{
String text = "Huffman coding is a data compression algorithm.";
buildHuffmanTree(text);
}
}ืืึทืืขืจืงืื ื: ืืขืจ ืืึผืจืื ืืขื ืืฆื ืืืจื ืื ืึทืจืืึทื ืฉืจืืึทื ืฉืืจืืงื ืืื 47 * 8 = 376 ืืืฅ ืืื ืื ืขื ืงืึธืืืื ืฉืืจืืงื ื ืขืื ืืืืื 194 ืืืฅ ื.ื. ืื ืืึทืื ืืขื ืขื ืงืึทืืคึผืจืขืกื ืืื ืืขืขืจืขื 48%. ืืื ืื C ++ ืคึผืจืึธืืจืึทื ืืืืื, ืืืจ ื ืืฆื ืื ืฉืืจืืงื ืงืืึทืก ืฆื ืงืจืึธื ืึท ืขื ืงืึธืืืื ืฉืืจืืงื ืฆื ืืึทืื ืื ืคึผืจืึธืืจืึทื ืืืื ืขืืืืืง.
ืืืืึทื ืขืคืขืงืืืื ืืืืืขืจืงืืึทื ืจืื ืืึทืื ืกืืจืึทืงืืฉืขืจื ืืึทืจืคื ืคึผืขืจ ืื ืกืขืจืฉืึทื ืึธ(ืืึธื(ืขื)) ืฆืืึทื, ืืื ืืื ืึท ืืึทื ืฅ ืืืื ืขืจื ืืืื ืืื N ืืืขืืขืจ ืคืึธืจืฉืืขืื ืงืกื ืืืงืกื-ืงืกื ืืืงืก ื ืึธืืื, ืืื ืื Huffman ืืืื ืืื ืึท ืืึทื ืฅ ืืืื ืขืจื ืืืื, ืืขืจ ืึทืืืขืจืืืึทื ืึทืจืืขื ืืื O(Nlog(N)) ืฆืืื, ืืื N - ืืืชืืืช.
ืงืืืืื:
ืืงืืจ: www.habr.com