A Polynomial Time Algorithm for Constructing Optimal Binary AIFV-2 Codes

Huffman Codes are optimal Instantaneous Fixed-to-Variable (FV) codes in which every source symbol can only be encoded by one codeword. Relaxing these constraints permits constructing better FV codes. More specifically, recent work has shown that AIFV-
$m$
codes can beat Huffman coding. AIFV-
$m$
codes construct an
$m$
-tuple of different coding trees between which the code alternates and are only almost instantaneous (AI). This means that decoding a word might require a delay of a finite number of bits. Current algorithms for constructing optimal AIFV-
$m$
codes are iterative processes that construct progressively “better sets” of code trees. The processes have been proven to finitely converge to the optimal code but with no known bounds on the convergence rate. This paper derives a geometric interpretation of the space of binary AIFV-2 codes, permitting the development of the first polynomially time-bounded procedure for constructing optimal AIFV codes. This binary-search like procedure will run in
$O(n^{3} b)$
time, where
$n$
is the number of symbols in the source alphabet and
$b$
is the maximum number of bits used to encode any one input probability.