Final answer:
The question involves implementing Huffman Coding in Java and analyzing its time complexity, which is O(n log n) in both best and worst cases. Huffman Coding is a method for data compression based on the frequency of characters in the input.
Step-by-step explanation:
The subject of the question is Huffman Coding, which is a widely used algorithm for data compression. The student is asking for a dynamic approach to solving Huffman Codes, its implementation in Java, and analysis of the algorithm's time complexity in terms of T(n) for both best and worst cases.
Huffman Coding Implementation in Java
Implementing Huffman Coding dynamically in Java involves creating a priority queue to store frequency of characters, building a Huffman tree, and generating codes based on the tree structure. Here is a simplified version of such an implementation:
import java.util.*;
class HuffmanNode {
int frequency;
char character;
HuffmanNode left, right;
}
// Comparator class used for priority queue
class MyComparator implements Comparator {
public int compare(HuffmanNode x, HuffmanNode y) {
return x.frequency - y.frequency;
}
}
public class HuffmanCode {
public static void main(String[] args) {
// Assume we have an input string to encode
String input = "example string to encode";
// A method here would calculate frequency of each character and create nodes
// Those nodes would be added to a priority queue
// A Huffman tree would be built from the queue
// Another method would traverse the tree and generate codes
}
// Additional methods for building tree and generating codes would be here
}
Time Complexity Analysis
In the best case scenario, the time complexity of building the Huffman tree is O(n log n), where n is the number of unique characters. This occurs when the priority queue operations (insertions and extractions) take logarithmic time, which is the case with balanced data.
Conversely, in the worst-case scenario, the complexity is also O(n log n), since each insertion and extraction operation on the priority queue is based on the height of the tree, which is bound by log n due to the binary nature of the Huffman tree construction.