Succincter

We can represent an array of n values from {0,1,2} using ceil(n log_2 3) bits (arithmetic coding), but then we cannot retrieve a single element efficiently. Instead, we can encode every block of t elements using ceil(t log_2 3)  bits, and bound the retrieval time by t. This gives a linear trade-off between the redundancy of the representation and the query time.In fact, this type of linear trade-off is ubiquitous in known succinct data structures, and in data compression. The folk wisdom is that if we want to waste one bit per block, the encoding is so constrained that it cannot help the query in any way. Thus, the only thing a query can do is to read the entire block and unpack it.We break this limitation and show how to use recursion to improve redundancy. It turns out that if a block is encoded with two (!) bits of redundancy, we can decode a single element, and answer many other interesting queries, in time logarithmic in the block size.Our technique allows us to revisit classic problems in succinct data structures, and give surprising new upper bounds. We also construct a locally-decodable version of arithmetic coding.