Summary

We investigate the Lempel-Ziv '77 data compression algorithm by considering an analogous algorithm for efficiently embedding strings in binary trees. This project includes a discussion of this comparison with two optional addenda on error correction and decompression, followed by exercises and solutions.

Notes for the instructor

Students in discrete mathematics often have a dual interest in computer science. This project succinctly combines these two areas. Data compression can be viewed as a discrete mathematics topic with many ramifications for computer scientists. Students who have completed one or two semesters of computer science (in particular, who are familiar with trees) may be eager to implement the algorithms discussed in C++, Java, or another object-oriented programming language.

The Lempel-Ziv '77 data compression algorithm was introduced in [1]. Analysis of the multiplicity matching parameter of suffix trees was presented in the present author's Ph.D. thesis; an abridged journal version with many more references to the literature can be found in [3]. An error correcting version of LZ'77 is outlined in [2].

Bibliography

[1] Lempel, A. and J. Ziv. “A universal algorithm for sequential data compression,” IEEE Transactions on Information Theory 23 (1977) 337–343.

[2] Lonardi, S., W. Szpankowski, and M. D. Ward. “Error resilient LZ'77 data compression: algorithms, analysis, and experiments,” IEEE Transactions on Information Theory 53 (2007) 1799–1813.