Range Majorities and Minorities in Arrays

Compressed Dynamic Range Majority and Minority Data Structures

“…Durocher et al [10] presented the first solution that achieves optimal O(1/α) query time, and their structure also occupies O(n/α) words. Subsequent researchers have worked to make the space usage independent of α [15,8,5] and even to achieve compression [15,5]. Among all these works, the most recent one is that of Belazzougui et al [3,5], who showed how to represent S using (1 + ǫ)nH 0 + o(n) bits for any constant ǫ > 0 to answer range α-majority queries in O(1/α) time, where H 0 is the 0-th order empirical entropy of S. When more compression is desired, they also showed how to represent S in nH 0 + o(n)(H 0 + 1) bits to support range α-majority in O(f (n)/α) time, for any f (n) = ω (1).…”

“…Subsequent researchers have worked to make the space usage independent of α [15,8,5] and even to achieve compression [15,5]. Among all these works, the most recent one is that of Belazzougui et al [3,5], who showed how to represent S using (1 + ǫ)nH 0 + o(n) bits for any constant ǫ > 0 to answer range α-majority queries in O(1/α) time, where H 0 is the 0-th order empirical entropy of S. When more compression is desired, they also showed how to represent S in nH 0 + o(n)(H 0 + 1) bits to support range α-majority in O(f (n)/α) time, for any f (n) = ω (1). Their solutions work for variable α, that is, α is not known at construction time; the value of α is given together with the range [i, j] in each query.…”

“…Their solutions work for variable α, that is, α is not known at construction time; the value of α is given together with the range [i, j] in each query. We refer readers to their most recent paper [5] for a more thorough survey.…”

“…Chan et al [8] studied this problem and designed an O(n)-word data structure that answers range α-minority queries in O(1/α) time. Belazzougui et al [3,5] further designed succinct data structures for range α-minority. They again presented two tradeoffs: they either represent S using (1 + ǫ)nH 0 + o(n) bits for any constant ǫ > 0 to answer range α-minority queries in O(1/α) time, or use nH 0 + o(n)(H 0 + 1) bits and support range α-minority queries in O(f (n)/α) time, for any f (n) = ω (1).…”

“…They again presented two tradeoffs: they either represent S using (1 + ǫ)nH 0 + o(n) bits for any constant ǫ > 0 to answer range α-minority queries in O(1/α) time, or use nH 0 + o(n)(H 0 + 1) bits and support range α-minority queries in O(f (n)/α) time, for any f (n) = ω (1). The solutions of both Chan et al [8] and Belazzougui et al [3,5] work for variable α. No work has been done for dynamic range α-minority queries.…”