The 3/4 Conjecture for Fix-Free Codes With at Most Three Distinct Codeword Lengths

Abstract

A property of prefix codes called strong monotonicity is introduced. Then it is proven that for a prefix code C for a given probability distribution, the following are equivalent: (i) C is expected length minimal; (ii) C is length equivalent to a Huffman code; and (iii) C is complete and strongly monotone. Also, three relations are introduced between prefix code trees called same-parent, same-row, and same-probability swap equivalence, and it is shown that for a given source, all Huffman codes are same-parent, same-probability swap equivalent, and all expected length minimal prefix codes are same-row, same-probability swap equivalent.