Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the well-known predecessor problem, where the tree is a path.

The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1 Introduction 1.1 The 'Were-you-last?' game A Dream Team, consisting of m players, is held captive in the dungeon of their adversary, Hannibal. He now makes them play his favourite game, Were-you-last?. Before the game starts the players of the Team are allowed to meet to discuss a strategy (obviously, Hannibal has the room bugged and is listening in). After the discussion they are led to separate waiting rooms. Then Hannibal leads each of the players of the team, one by one, to the playing field. The players do not know the order in which they are led to the field and they spend their time there alone. The playing field is a room, containing an infinite number of boxes, labelled 0, 1, 2, 3, . . . . Inside each box is a switch that can be ...

We study the complexity of the dynamic partial sum problem in the cell-probe model. We give the model access to nondeterministic queries and prove that the problem remains hard. We give the model access to the right answer as an oracle and prove that the problem remains hard. This suggests which kind of information is hard to maintain. From these results, we derive a number of lower bounds for dynamic algorithms and data structures: We prove lower bounds for dynamic algorithms for existential range queries, reachability in directed graphs, planarity testing, planar point location, incremental parsing, and fundamental data structure problems like maintaining the majority of the prefixes of a string of bits. We prove a lower bound for reachability in grid graphs in terms of the graph's width. We characterize the complexity of maintaining the value of any symmetric function on the prefixes of a bit string. Keywords. cell-probe model, partial sum, dynamic algorithm, data structure AMS subject classifications. 68Q17, 68Q10, 68Q05, 68P05

We introduce new models for dynamic computation based on the cell probe model of Fredman and Yao. We give these models access to nondeterministic queries or the right answer ±1 as an oracle. We prove that for the dynamic partial sum problem, these new powers do not help, the problem retains its lower bound of Omega (log n/ log log n). From...

Pattern matching is the problem of nding all occurrences of a pattern in a text. In a dynamic setting the problem is to support pattern matching in a text which can be manipulated on-line, i.e., the usual situation in text editing. We present a data structure that supports insertions and deletions of characters and movements of arbitrary large blocks within a text in O(log 2 n log log n log n) time per operation. Furthermore a search for a pattern P in the text is supported in time O(log n log log n + occ + jP j), where occ is the number of occurrences to be reported. An ingredient in our solution to the above main result is a data structure for the dynamic string equality problem introduced by Mehlhorn, Sundar and Uhrig. As a secondary result we give almost quadratic better time bounds for this problem which in addition to keeping polylogarithmic factors low for our main result also improves the complexity for several other problems.

We consider the problem of maintaining a string of n brackets ‘(’ or ‘)’ under the operation reverse(i) that changes the ith bracket from ‘(’ to ‘)’ or vice versa, and returns ‘yes’ if and only if the resulting string is properly balanced. We show that this problem can be solved on the RAM in time O(log n/loglog n) per operation using linear space and preprocessing. Moreover, we show that this is optimal in the sense that every data structure supportingreverse (no matter its space and preprocessing complexity) needs time Omega(log n/loglog n) per operation in the cell probe model.

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.

In this thesis, we investigate various interpretations of the Union-Split-Find problem, an extension of the classic Union-Find problem. In the Union-Split-Find problem, we maintain disjoint sets of ordered elements subject to the operations of constructing singleton sets, merging two sets together, splitting a set by partitioning it around a specified value, and finding the set that contains a given element. The different interpretations of this problem arise from the different assumptions made regarding when sets can be merged and any special properties the sets may have. We define and analyze the Interval, Cyclic, Ordered, and General Union-Split-Find problems. Previous work implies optimal solutions to the Interval and Ordered Union-Split-Find problems and an Ω(log n / log log n) lower bound for the Cyclic Union-Split-Find problem in the cell-probe model. We present a new data

ABSTRACT. Pattern matching is the problem of finding all occurrences of a pattern in a text. For a long period of time significant progress has been made in solving increasingly more generalized and dynamic versions of this problem. In this paper we introduce a fully dynamic generalization of the pattern matching problem. We show how to maintain a family of strings under split and concatenation operations. Given a string in the family, all occurrences of it in the family are reported within time occ time, where is the total size of the strings and occ is the number of occurrences. Updates are performed in time. These bounds are competitive or improve former results for less generalized versions of the problem. As an intermediate result of independent interest, we provide an almost quadratic improvement of the time bounds for the dynamic string equality problem due to Mehlhorn, Sundar and Uhrig.