We can represent an array of n values from {0, 1, 2} using ⌈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 ⌈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.

Minimal perfect hash functions have been shown to be useful to compress data in several data management tasks. In particular, order-preserving minimal perfect hash functions [12] have been used to retrieve the position of a key in a given list of keys: however, the ability to preserve any given order leads to an unavoidable.n log n / lower bound on the number of bits required to store the function. Recently, it was observed [1] that very frequently the keys to be hashed are sorted in their intrinsic (i.e., lexicographical) order. This is typically the case of dictionaries of search engines, list of URLs of web graphs, etc. We refer to this restricted version of the problem as monotone minimal perfect hashing. We analyse experimentally the data structures proposed in [1], and along our way we propose some new methods that, albeit asymptotically equivalent or worse, perform very well in practice, and provide a balance between access speed, ease of construction, and space usage. 1

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