Imagine you have several wood pieces of different lengths—say,
1, 4, 9, and 11. You want to combine
them into one long stick with the least total merging cost. Each merge
costs the sum of the two sticks being combined. If you merge larger
sticks too early, the large values get added repeatedly, increasing
the overall cost.
Example 1:
Merge 4 and 9 first: cost = 4 + 9 = 13
Then merge 13 (from previous merge) with 1: cost = 13 + 1 = 14
Finally, merge the resulting stick (length 14) with 11: cost = 14 +
11 = 25
Total Cost: 13 + 14 + 25 = 52
Example 2 (Better Approach):
Merge the two smallest sticks first: merge 1 and 4 to get 5; cost =
1 + 4 = 5
Then merge the next smallest: merge 5 (result) and 9 to get 14; cost
= 5 + 9 = 14
Finally, merge 14 with 11; cost = 14 + 11 = 25
Total Cost: 5 + 14 + 25 = 44
Key Insight: By always merging the two smallest
sticks available, you minimize the immediate cost at each step. This
prevents larger numbers from being added repeatedly later in the
process.
Connection to Huffman Coding
Huffman Coding uses a very similar strategy for data compression.
Instead of wood pieces, you start with characters and their
frequencies. The goal is to build a binary tree that minimizes the
total weighted path length, which directly relates to the length of
the final encoded string.
Main Points:
Frequency as Length: Think of the frequency of each
character as the “length” of a wood piece. More frequent characters
have smaller “lengths” (or weights) to keep the overall cost low.
Greedy Strategy: At each step, choose the two
characters (or nodes) with the smallest frequencies. Merge them to
create a new node with a frequency equal to the sum of the two.
Building the Tree: This process is repeated until
all characters are merged into a single tree. The tree structure
ensures that no character's bit representation is a prefix of any
other, which is crucial for decoding the compressed data uniquely.
Resulting Code: The path from the root of the tree
to each leaf (character) gives its unique binary code. Characters
with higher frequencies tend to have shorter codes, reducing the
overall size of the encoded message.
Why This Works
Efficiency: Merging the smallest elements first
minimizes the repeated addition of large numbers (or frequencies).
This approach is optimal for both stick merging and Huffman Coding.
Non-Prefix Property: In the Huffman tree, no code
is a prefix of another. This is because each merge creates a new
internal node, ensuring that every final code (assigned to a leaf)
can be uniquely identified when reading the encoded string.
Summary
Huffman Coding is a practical application of a greedy algorithm. It:
Begins by considering each character’s frequency.
Repeatedly merges the two smallest elements.
Builds a binary tree that assigns shorter codes to more frequent
characters.
Guarantees that no code is a prefix of another, enabling unambiguous
decoding.
This method, analogous to merging the smallest sticks first to reduce
overall merging cost, ensures that the final compressed representation
of data is as efficient as possible.
