The Complexity of Two - DimensionalCompressed Pattern -

We consider the complexity of problems for highly compressed 2-dimensional texts: compressed pattern-matching (when the pattern is not compressed and the text is compressed) and fully compressed pattern-matching (when also the pattern is compressed). First we consider 2-dimensional compression in terms of straight-line programs, see 9]. It is a natural way for representing very highly compressed images, by describing larger parts in terms of smaller (earlier described) ones. For 1-dimensional strings there exist polynomial-time deterministic algorithms for similar types of compression 2, 6, 8, 9]. We show that the complexity dramatically increases in a 2-dimensional setting: Compressed matching for two dimensional compressed images is NP-complete. Fully compressed matching for two dimensional compressed images is 2 P-complete. Testing a given occurrence of a two dimensional compressed pattern is co-NP-complete. On the other hand we show eecient algorithms for some related problems: Testing equality of two compressed two dimensional patterns (an application of algebraic techniques). Testing a given occurrence of an uncompressed pattern in a two dimensional compressed image. We also show the surprising fact that the compressed size of a subrectangle of a compressed two dimensional array can grow exponentially, unlike the one dimensional case.