A data structure for dynamic trees
Journal of Computer and System Sciences
The cell probe complexity of dynamic data structures
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Maintenance of a minimum spanning forest in a dynamic plane graph
Journal of Algorithms
A new data structure for cumulative frequency tables
Software—Practice & Experience
Complexity models for incremental computation
Theoretical Computer Science - Special issue on dynamic and on-line algorithms
Optimal Biweighted Binary Trees and the Complexity of Maintaining Partial Sums
SIAM Journal on Computing
Randomized fully dynamic graph algorithms with polylogarithmic time per operation
Journal of the ACM (JACM)
Journal of the ACM (JACM)
The Complexity of Maintaining an Array and Computing Its Partial Sums
Journal of the ACM (JACM)
Near-optimal fully-dynamic graph connectivity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Optimal bounds for the predecessor problem and related problems
Journal of Computer and System Sciences - STOC 1999
Optimal Algorithms for List Indexing and Subset Rank
WADS '89 Proceedings of the Workshop on Algorithms and Data Structures
New Lower Bound Techniques for Dynamic Partial Sums and Related Problems
SIAM Journal on Computing
Tight bounds for the partial-sums problem
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Design of data structures for mergeable trees
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
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We prove an Ω(lg Erik n) cell-probe lower bound on maintaining connectivity in dynamic graphs, as well as a more general trade-off between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight for these restricted cases, proving optimality of these data structures (e. g., Sleator and Tarjan's dynamic trees). Our trade-off is known to be tight for trees, and the best two data structures for dynamic connectivity in general graphs are points on our trade-off curve. In this sense these two data structures are optimal, and this tightness serves as strong evidence that our lower bounds are the best possible. From a more theoretical perspective, our result is the first logarithmic cell-probe lower bound for any problem in the natural class of dynamic language membership problems, breaking the long standing record of Ω(lg n / lg lg n). In this sense, our result is the first data-structure lower bound that is "truly" logarithmic, i. e., logarithmic in the problem size counted in bits. Obtaining such a bound is listed as one of three major challenges for future research by Miltersen [13] (the other two challenges remain unsolved). Our techniques form a general framework for proving cell-probe lower bounds on dynamic data structures. We show how our framework also applies to the partial-sums problem to obtain a nearly complete understanding of the problem in cell-probe and algebraic models, solving several previously posed open problems.