Question

A prefix-free encoding of a finite alphabet Γ assigns each symbol in Γ a binary codeword, such that no codeword is a prefix of another codeword. A prefix-free encoding is minimal if it is not possible to arrive at another prefix-free encoding (of the same symbols) by contracting some of the keywords. For instance, the encoding $\left\{0,101\right\}$ is not minimal since the codeword 101 can be contracted to 1 while still maintaining the prefix-free property.

Show that a minimal prefix-free encoding can be represented by a full binary tree in which each leaf corresponds to a unique element of Γ, whose codeword is generated by the path from the root to that leaf (interpreting a left branch as 0 and a right branch as 1 ).

Short answer

Every symbol in a binary codeword is assigned a finite alphabet in a prefix free encoding. The encoding $\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{10}\mathrm{\}}$ for example, is not a tiny codeword and has been reduced to 1 while retaining the prefix base characteristic.

Step-by-step solution

Step 1: Prefix-free encoding

• This code is prefix-free encoding, which implies that there are no codewords that are prefixes to other codewords.

• Its prefix-free encode with finite alphabet " $\Gamma$" allots as well as every symbol$\Gamma$ in a binary codeword that represents the entire binary tree and constructs the route from the root to the leaf node.

o There seem to be based on two different nodes: left leaf & right leaf, with the path from the tree's root to the left leaf node indicating " 0 " and the path from the tree's root to the right leaf node indicating " 1 ."

Step 2: Proof of minimal prefix-free encoding

Consider the whole binary tree with both the string " s " and a length of " k " for all strings.

• This binary tree has two branches that are labelled "left" and "right."

o Its left branching is labelled " 0 ," while the right branch is labelled " 1 ."

• To achieve the maximum string length, all binary strings are encoded.

o To put it another way, only utilise the total number of levels in all strings.

• Also, encode all of the path's intermediary nodes.

o That is, the path through root node towards intermediate node " a " must be prefix-free, as well as the strings of intermediate node " a " must be encoded as left node with codeword " s0 ."

• Every binary tree currently has to be a full binary tree. Assume that perhaps the intermediate node "b" corresponds to the string " s ." However, the string " s " only has one node, such as " a," which is the value of the codeword " s0."

• The intermediate node " b " is a subtree of the prefix-free string " s " with the codeword " s0."

o To get the superior encoding scheme, change the codeword " s0" with " s " for intermediate node " b."

o As a result, the prefix-free encoding is kept to a minimum.

As a result of this contradiction, the minimum prefix-free encoding is established.

Conclusion

Binary tree has to be perhaps the intermediate node “ b ” corresponds to string “ S ”. Base on this we can get result prefix – free encoding is kept to a minimum. As a result, we can get contradiction, the minimum prefix encoding is established.