Analyzing a Greedy Approximation of an MDL Summarization

Many OLAP (On-line Analytical Processing) applications have produced data cubes that summarize and aggregate details of data queries. These data cubes are multi-dimensional matrices where each cell that satisfies a specific property or trait is represented as a 1, notated as a 1-cell in this report. A cell that does not satisfy that specific property is represented as a 0, notated as a 0-cell. in this report In order to compress the amount of space required to represent this matrix completely, others have used MDL (Minimum Description Length) Summarization, including the MDL Summarization with Holes. While it is NP-Hard to compute the optimal MDL Summarization with Holes for a data matrix of 2 or more dimensions (Proven by Bu et al. [1]), there exists a greedy algorithm to approximate the MDL Summarization with Holes, proven to give an answer that within a factor of lm ∗ log(M) of the optimal solution (Proven by Guha and Tan [3]), where M is a factor dependent size of the data matrix. See the Technical Approach section of this report for a definition of lm. However, Guha and Tan in [3] mention that this bound has not been proven tight. I studied this for 2-dimensional matrices where the algorithm can only compress by covering rows and columns (here lm = 2). Currently, I have a proof that the greedy algorithm is a 4-approximation algorithm in this special 2-dimensional case and a constant-factor (2 ∗ (κ − 2))-approximation algorithm in the general case. Furthermore, I have written a program that uses the greedy approximation to MDL Summarize with Holes an arbitrary n-by-n 2-dimensional matrix of 1’s and 0’s.