 Shannon-Fano-Elias code

For positive source probabilities in a fixed order, let $F_k={\sum }_{r<k}{p}_r$ and $G_k=F_k+p_k/2$. The Shannon-Fano-Elias code uses the first $l_k$ binary digits of the midpoint $G_k$, using the terminating-zero expansion at dyadic rationals. Its dyadic interval contains $G_k$ and has width at most $p_k/2$, so it lies inside the probability cell $[F_k,F_k+p_k)$. These cells are disjoint, whereas a codeword that is a prefix of another has a dyadic interval containing the other interval. The code is therefore a prefix code. Since $l_k<-{\log }_2{p}_k+2$, averaging proves $\mathbb{E}L<H(X)+2$. This is a direct cumulative construction and does not require sorting the probabilities or constructing a Huffman tree.