Sorting and Searching in Multisets

Dynamic Shannon coding

“…We used this algorithm in a later paper [7] as a dynamic protocol for communication over channels with asymmetric bandwidth. We can also use it to sort S in (H + O(1))m binary comparisons; previous algorithms used ternary comparisons [22] or had a bound of O((H + 1)m) comparisons. When used for sorting, our algorithm queries a partial-sum data structure once for each comparison; with an augmented AVL-tree implementation, for example, it takes a total of O((H + 1)m log n) time.…”

Range majority in constant time and linear space

“…The decision problem of determining whether the frequency m of the mode exceeds one reduces to the element uniqueness problem, resulting in a lower bound of Ω(n lg n) time in the algebraic decision tree model [5]. Better bounds have been obtained by parameterizing in terms of m: Munro and Spira [21] and Dobkin and Munro [11] described an O (n lg( n m )) time algorithm and corresponding lower bound of Ω(n lg( n m )) time. Misra and Gries [20] gave O (n) and O (n lg( 1 α )) time algorithms for computing an α-majority when α 1 2 and α < 1 2 , respectively.…”

“…Numerous results established bounds on the number of comparisons required for computing a majority, α-majority, mode, or plurality (e.g., [1,2,11,21]). …”

Tight bounds for online stable sorting

“…By a ternary comparison, we mean one that can return <, = or >; we count only comparisons between elements of the multiset, not those between data generated by the algorithm. Over thirty years ago, Munro and Spira [11] proved distribution-sensitive upper and lower bounds that differ by O(n log log σ ), where σ is the number of distinct elements in S. Their bounds have been improved in a series of papers -summarized in Table 1 -and now the best known upper and lower bounds differ by a term linear in n (about (1 + log e)n ≈ 2.44n when σ = o(n)). In this paper we restrict our attention to online stable sorting and prove distribution-sensitive upper and lower bounds that are within o(n) not only of each other but also of the best known upper bound for offline sorting.…”