Adaptive Succinctness

Representing a static set of integers S , |S| = n from a finite universe U = [1..u] is a fundamental task in computer science. Our concern is to represent S in small space while supporting the operations of 𝗋𝖺𝗇𝗄 and 𝗌𝖾𝗅𝖾𝖼𝗍 on S ; if S is viewed as its characteristic vector, the problem becomes that of representing a bit-vector, which is arguably the most fundamental building block of succinct data structures. Although there is an information-theoretic lower bound of ℬ(n, u)= u ()n bits on the space needed to represent S , this applies to worst-case (random) sets S , and sets found in practical applications are compressible. We focus on the case where elements of S contain runs of | ℓ >1 consecutive elements, one that occurs in many practical situations. Let 𝒞^ (n ) denote the class of u ()n distinct sets of n elements over the universe [1..u] . Let also 𝒞^ (n )_g⊂𝒞^ (n ) contain the sets whose n elements are arranged in g ≤ n runs of ℓ _i ≥ 1 consecutive elements from U for i=1,… , g , and let 𝒞^ (n )_g,r⊂𝒞^ (n )_g contain all sets that consist of g runs, such that r ≤ g of them have at least 2 elements. This paper yields the following insights and contributions related to 𝗋𝖺𝗇𝗄 / 𝗌𝖾𝗅𝖾𝖼𝗍 succinct data structures: We introduce new compressibility measures for sets, including: ℬ_1(g,n,u)= |𝒞^ (n )_g| = u-n+1 ()g + n-1 ()g-1 , and ℬ_2(r, g, n,u)= |𝒞^ (n )_g,r| =u-n+1 ()g + n-g-1 ()r-1 + g ()r , such that ℬ_2(r, g, n,u)≤ℬ_1(g,n,u)≤ℬ(n, u) . We give data structures that use space close to bounds ℬ_1(g,n,u) and ℬ_2(r, g, n,u) and support 𝗋𝖺𝗇𝗄 and 𝗌𝖾𝗅𝖾𝖼𝗍 in O(1) time. We provide additional measures involving entropy-coding run lengths and gaps between items, and data structures to support 𝗋𝖺𝗇𝗄 and 𝗌𝖾𝗅𝖾𝖼𝗍 using space close to these measures.