Bit vector encoding via decomposition

The decomposition encoding of an n-bit vector V is an approach to the problem of how best to encode a bit vector under the constraints that this vector be encoded into blocks of t bits, and that access time for each bit of the original vector be constant, i.e. "random access." This method involves encoding the vector as two separate matrices F1 and F2. Essential to the method is the decomposition of N--the indexing set for V--by means of an injection (h1,h2): N --> R1 x R2 : m │--> (h1(m),h2(m)). Decoding involves some simply computed function f(F1[h1(m);],F2[h2(m);]). The emphasis in this paper is upon compact encodings for static vectors. For one choice of f--XOR decomposition encoding--it is shown by linear algebraic techniques that a decomposition encoding requiring the minimal t for a given (h1,h2) pair can be determined. Of more practical interest is minimization of the total number of bits in the encoding for all possible injections (h1,h2). Experimental results using a number of easily computed (h1,h2) pairs show that compactions to one half of the ordinary packed representation can be achieved via decomposition encoding for relatively sparse bit vectors.