We prove lower bounds on the complexity of maintaining fully dynamic k-edge or k-vertex connectivity in plane graphs and in (k − 1)-vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, where b is the word size of the machine and n is the number of vertices in G. We also show an amortized lower bound of �(log n/(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.

We develop a new technique for proving cell-probe lower bounds for static data structures. Previous lower bounds used a reduction to communication games, which was known not to be tight by counting arguments. We give the first lower bound for an explicit problem which breaks this communication complexity barrier. In addition, our bounds give the first separation between polynomial and near linear space. Such a separation is inherently impossible by communication complexity. Using our lower bound technique and new upper bound constructions, we obtain tight bounds for searching predecessors among a static set of integers. Given a set Y of n integers of ℓ bits each, the goal is to efficiently find predecessor(x) = max {y ∈ Y | y ≤ x}. For this purpose, we represent Y on a RAM with word length w using S words of space. Defining a = lg S n +lg w, we show that the optimal search time is, up to constant factors: logw n lg min ℓ−lg n

The choice of data structure for tour representation plays a critical role in the efficiency of local improvement heuristics for the Traveling Salesman Problem. The tour data structure must permit queries about the relative order of cities in the current tour and must allow sections of the tour to be reversed. The traditional array-based representation of a tour permits the relative order of cities to be determined in small constant time, but requires worst-case W(N) time (where N is the number of cities) to implement a reversal, which renders it impractical for large instances. This paper considers alternative tour data structures, examining them from both a theoretical and experimental point of view. The first alternative we consider is a data structure based on splay trees, where all queries and updates take amortized time O(logN). We show that this is close to the best possible, because in the cell probe model of computation any data structure must take worst-case amortized time W(...

We develop succinct data structures to represent (i) a sequence of values to support partial sum and select queries and update (changing values) and (ii) a dynamic array consisting of a sequence of elements which supports insertion, deletion and access of an element at any given index. For the partial sums problem...

Abstract. We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized Ω(lg n) lower bound per operation for several data structural problems on n elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of Ω(lg n/lg lg n) for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan’s dynamic trees. We also prove the first Ω(log B n) lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update. Key words. Cell-probe complexity, lower bounds, data structures, dynamic graph problems, partial-sums problem AMS subject classification. 68Q17 1. Introduction. The

A static dictionary is a data structure for storing subsets of a nite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size (log jU j), and show that for n-element subsets, constant worst case query time can be obtained using B +O(log log jU j) + o(n) bits of storage, where B = dlog 2 jUj n e is the minimum number of bits needed to represent all such subsets. For jU j = n log O(1) n the dictionary supports constant time rank queries.

Certification trails are a recently introduced and promising approach to faultdetection and fault-tolerance [19]. In this paper, we significantly generalize the applicability of the certification trail technique. Previously, certification trails had to be customized to each algorithm application, but here we develop trails appropriate to wide classes of algorithms. These certification trails are based on common data-structure operations such as those carried out using balanced binary trees and heaps. Any algorithm using these sets of operations can therefore employ the certification trail method to achieve software fault tolerance. To exemplify the scope of the generalization of the certification trail technique provided in this paper, constructions of trails for abstract data types such as priority queues and union-find structures will be given. These trails are applicable to any data-structure implementation of the abstract data type. It will also be shown that these ideas lead natur...

In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusion-free family of intervals. This problem has important applications in physical mapping of DNA. We give a near-optimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with word-size b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time.

. We investigate the problem of storing a subset of the elements of a boundeduniverse so that searches canbe performed in constant time and the space used is within a constant factor of the minimum required. Initially we focus on the static version of this problem and conclude with an enhancement that permits insertions and deletions. 1 Introduction Given a universal set M = f0; : : : ; M \Gamma 1g and any subset N = fe 1 ; : : : ; e N g the membership problem is to determine whether given query element in M is an element of N . There are two standard approaches to solve this problem: to list all elements of N (e.g. in a hash table) or to list all the answers (e.g. a bit map of size M ). When N is small the former approach comes close to the information theoretic lower bound on the number of bits needed to represent an arbitrary subset of the given size (i.e. a function of both N and M , l lg \Gamma M N \Delta m ). Similarly, when N is large (say ffM ) the later approach is near...

.<F3.849e+05> Let<F3.187e+05> A<F3.849e+05> be an array. The<F3.262e+05> partial sum problem<F3.849e+05> concerns the design of a data structure for implementing the following operations. The operation<F3.187e+05> update(j,<F3.849e+05> x) has the e#ect<F3.187e+05><F3.849e+05><F3.187e+05><F3.849e+05> A[j]<F5.57e+05> #<F3.187e+05><F3.849e+05><F3.187e+05><F3.849e+05><F3.187e+05> A[j]+x<F3.849e+05> , and the query operation<F3.187e+05><F3.849e+05> sum(j) returns the partial sum<F7.1e+05> #<F1.882e+05> j<F2.831e+05> i=1<F3.187e+05><F3.849e+05><F3.187e+05><F3.849e+05> A[i] . Our interest centers upon the optimal e#ciency with which sequences of such operations can be performed, and we derive new upper and lower bounds in the semigroup model of computation. Our analysis relates the optimal complexity of the partial sum problem to optimal binary trees relative to a type of weighting scheme that defines the notion of<F3.262e+05> biweighted<F3.849e+05> binary tree.<F4.005e+05> Key words.<F3...