Lower bounds for union-split-find related problems on random access machines (original) (raw)

Worst-case and amortised optimality in union-find (extended abstract)

Proceedings of the thirty-first annual ACM symposium on Theory of Computing, 1999

We study the interplay between worst-case and amortised time bounds for the classic Disjoint Set Union problem (Union-Find). We ask whether it is possible to achieve optimal worst-case and amortised bounds simultaneously. Furthermore we would like to allow a tradeoff between the worst-case time for a query and for an update. We answer this question by first providing lower bounds for the possible worst-case time tradeoffs, as well as lower bounds which show where in this tradeoff range optimal amortised time is achievable. We then give an algorithm which tightly matches both lower bounds simultaneously. The lower bounds are provided in the cell-probe model as well as in the algebraic real-number RAM, and the upper bounds hold for a RAM with logarithmic word size and a modest instruction set. Our lower bounds show that for worst-case query and update time t, and t, respectively, one must have t, = a(log n/ log t,), and only for t, 2 cx(m, n) can this tradeoff be achieved simultaneously with the optimal amortised time of o(c~(m,n)). Our 'DIK", Camp. sci. Dept., ""ivorsity of Copenhagen, Denmark. E-mail: slephe"mdik".dk. Part of thin work W&S done while "isiting BRIGS and Lund University.

A Space Lower Bound for Dynamic Approximate Membership Data Structures

An approximate membership data structure is a randomized data structure for representing a set which supports membership queries. It allows for a small false positive error rate but has no false negative errors. Such data structures were first introduced by Bloom in the 1970's, and have since had numerous applications, mainly in distributed systems, database systems, and networks.

Succinct Dynamic Ordered Sets with Random Access

ArXiv, 2020

The representation of a dynamic ordered set of nnn integer keys drawn from a universe of size mmm is a fundamental data structuring problem. Many solutions to this problem achieve optimal time but take polynomial space, therefore preserving time optimality in the \emph{compressed} space regime is the problem we address in this work. For a polynomial universe m=nTheta(1)m = n^{\Theta(1)}m=nTheta(1), we give a solution that takes textsfEF(n,m)+o(n)\textsf{EF}(n,m) + o(n)textsfEF(n,m)+o(n) bits, where textsfEF(n,m)leqnlceillog_2(m/n)rceil+2n\textsf{EF}(n,m) \leq n\lceil \log_2(m/n)\rceil + 2ntextsfEF(n,m)leqnlceillog_2(m/n)rceil+2n is the cost in bits of the \emph{Elias-Fano} representation of the set, and supports random access to the iii-th smallest element in O(logn/loglogn)O(\log n/ \log\log n)O(logn/loglogn) time, updates and predecessor search in O(loglogn)O(\log\log n)O(loglogn) time. These time bounds are optimal.

A lower bound for set intersection queries

We consider the following set intersection reporting problem. We have a collection of initially empty sets and would like to process an intermixed sequence of n updates (insertions into and deletions from individual sets) and q queries (reporting the intersection of two sets). We cast this problem in the arithmetic model of computation ofFredman [Fre81] and Yao [Yao85] and show that any algorithm that fits in this model must take time O(q+nytq) to process a sequence of n updates and q queries, ignoring factors that are polynomial in log n.We also show that this bound is tight in this model of computation, agam to within a polynomial in log n factbr, improving upon a result of Yellin (YeI92]. Furthermore we consider the case q = O(n) with an additional space restriction. We only allow to use m memory locations, where m :5 n 3 / 2 • We show• a tight bound of 0(n 2 /m 1 / 3) for a sequence of O(n) operations, agam ignoring polynomial in logn factors.

Linear expected time of a simple union-find algorithm

Information Processing Letters, 1976

The average time required by this algorithm will be computed by averaging over all legal sequences of n in. structions. Such sequences of instructions are of the form where for each k, 1 kn,'k is either (a) Find (i) for some i, I is, or (b) Union (i, I) for some i and!, 1 i* js, sub ...