Compact Rich-Functional Binary Relation Representations

Abstract

Abstract. Binary relations are an important abstraction arising in a number of data representation problems. Each existing data structure specializes in the few basic operations required by one single application, and takes only limited advantage of the inherent redundancy of binary relations. We show how to support more general operations efficiently, while taking better advantage of some forms of redundancy in practical instances. As a basis for a more general discussion on binary relation data structures, we list the operations of potential interest for practical applications, and give reductions between operations. We identify a set of operations that yield the support of all others. As a first contribution to the discussion, we present two data structures for binary relations, each of which achieves a distinct tradeoff between the space used to store and index the relation, the set of operations supported in sublinear time, and the time in which those operations are supported. The experimental performance of our data structures shows that they not only offer good time complexities to carry out many operations, but also take advantage of regularities that arise in practical instances in order to reduce space usage.

Entropy-bounded representation of point grids

“…This time is O(1) in our "compressed" solution, and in our "fully compressed" solution it is O lg n 2 m in the range m = O n 2 lg 1/4 n . This is never worse than the previous result [2], and is strictly better if m = n 2−o(1) . For report and select we are faster or slower depending on the case.…”

“…Table 1 compares the previous and new complexities achieved for our operations. The previous compressed representation [2] achieves H + o(H) bits only if ω( n lg lg n lg 2 n ) = m = o(n 2 ) (note our m is their t). Also, it always supports rank in time O(lg n).…”

“…2), which does not fit in o(H) whenever m = O n lg lg n lg 2 n . This is because Barbay et al [2] chose Raman et al's representation [19] to achieve constant-time rank and select on a bit vector. By using instead a binary searchable representation [11], the o(n) term becomes o(H) + O(m + lg n) and the O(lg σ) time becomes O(lg σ + lg m).…”

“…That space is not the best possible when m is larger than n. A (worst-case) lower bound on the number of bits needed to represent a grid is the logarithm of the number of possible grids, called the "entropy" H = lg n 2 m = m lg n 2 m + O(m + lg n). To the best of our knowledge, the only previous work achieving this "compressed" space is by Barbay et al [2]. They propose a data structure using…”

Compressed Dynamic Binary Relations

“…On large binary relations, reducing space while retaining functionality is crucial in order to operate efficiently in main memory. There has been work on compressing general binary relations [2], as well as specific ones such as Web graphs [3].…”