Compression of low entropy strings with Lempel-Ziv algorithms

We compare the compression ratio of the Lempel-Ziv algorithms with the empirical entropy of the input string, We show that although these algorithms are optimal according to the generally accepted definition, we can find families of low entropy strings which are not compressed optimally. More precisely, we show that the compression ratio achieved by LZ78 (resp. LZ77) can be much higher than the zeroth order entropy H ₀ (resp. the first order entropy H₁). We present a compression algorithm which combines LZ78 with run length encoding, and we show that for any string s the new algorithm achieves a compression ratio bounded by 3H₀(s). Finally, we prove that for any string s the compression ratio achieved by LZ77 is bounded by 8H₀ (s)