Cell Probe Model Research Articles

Optimal Bounds for the Predecessor Problem and Related Problems

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.

Tighter Lower Bounds for Nearest Neighbor Search and Related Problems in the Cell Probe Model

We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n,d) cells, each containing poly(d, logn) bits. We get a lower bound of Ω(d/logn) on the search time, a significant improvement over the recent bound of Ω(logd) of Borodin et al. This should be contrasted with the upper bound of O(loglogd) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.

Lower Bounds for Dynamic Data Structures on Algebraic RAMs

In a seminal paper of 1989, Fredman and Saks proved lower bounds for some important data-structure problems in the cell probe model. This model assumes that data structures are stored in memory with a known word length. In this paper we consider random access machines (RAMs) that can add, subtract, compare, multiply and divide integer or real numbers, with no size limitation. These are referred to as algebraic RAMs . We prove new lower bounds for two important data-structure problems, union-findand dynamic prefix sums . To this end we apply the generalized Fredman—Saks techniqueintroduced by the authors in a previous paper. The generalized technique relies on a certain well-defined function, Output Variability , that characterizes in some sense the power of the computational model. Fredman and Saks' work implied bounds on output variability for the cell probe model; in this paper we provide the first bounds for algebraic RAMs, and show that they suffice for proving tight lower bounds for useful problems. An important feature of the problems we consider is that in a data structure of size n , the data stored are members of {0,\ldots,n} . This makes the derivation of lower bounds for such problems on a RAM without word-size limitations a particular challenge; previous RAM lower bounds we are aware of depend on the fact that the data for the computation can vary over a large domain.

A Generalization of a Lower Bound Technique due to Fredman and Saks

{In a seminal paper of 1989, Fredman and Saks proved lower bounds for some important data-structure problems in the cell probe model. In particular, lower bounds were established on worst-case and amortized operation cost for the union-find problem and the prefix sum problem. The goal of this paper is to turn their proof technique into a general tool that can be applied to different problems and computational models. To this end we define two quantities: Output Variability depends only on the model of computation. It indicates how much variation can be found in the results of a program with certain resource bounds. This measures in some sense the power of a model. Problem Variability characterizes in a similar sense the difficulty of the problem.

A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs

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 O(log n) worst-case time per edge insertion or deletion. We prove a close lower bound of $\Omega(\log n/(\log\log n+\log b))$ 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. As a byproduct of our algorithm, we solve in O(log n) worst-case time the problem of maintaining connectivity in a dynamically changing proper interval graph.

Improved bounds for dictionary look-up with one error

Given a dictionary S of n binary strings each of length m , we consider the problem of designing a data structure for S that supports d - queries; given a binary query string q of length m , a d -query reports if there exists a string in S within Hamming distance d of q . We construct a data structure for the case d=1 , that requires space O(n logm) and has query time O(1) in a cell probe model with word size m . This generalizes and improves the previous bounds of Yao and Yao for the problem in the bit probe model. The data structure can be constructed in randomized expected time O(nm) .

Lower Bounds for Fully Dynamic Connectivity Problems in Graphs

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 $\Omega (\log n / {k (\log \log n} + \log b))$ 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 $\Omega( \log n /(\log \log n + \log b))$ per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.

Marked Ancestor Problems

Full text not available.<br /> [Extended abstract appears in Motwani, editor, <em>39th Annual Symposium on Foundations of Computer Science</em>, FOCS '98 Proceedings, 1998, pages 534-543.]

On Data Structures and Asymmetric Communication Complexity

In this paper we consider two-party communication complexity, the “asymmetric case”, when the input sizes of the two players differ significantly. Most of previous work on communication complexity only considers the total number of bits sent, but we study trade-offs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our “round elimination” lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the “round reduction” techniques used in many previous lower bound proofs.

Marked Ancestor Problems (Preliminary Version)

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.<br /> We show tight upper and lower bounds for this problem. The lower bounds are proved in the cell probe model, the upper bounds run on a unit-cost RAM.<br /> As easy corollaries we prove (often optimal) lower bounds on a number of problems. These include planar range searching, including the existential or emptiness problem, priority search trees, static tree union-find, and several problems from dynamic computational geometry, including intersection problems, proximity problems, and ray shooting. Our upper bounds improve a number of algorithms from various fields, including dynamic dictionary matching and coloured ancestor problems.

Hardness Results for Dynamic Problems by Extensions of Fredman and Saks’ Chronogram Method

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). <br />From these results we easily derive a large number of lower bounds of order Omega(log n/log log n) for conventional dynamic models like the random access machine. We prove lower bounds for dynamic algorithms for reachability in directed graphs, planarity testing, planar point location, incremental parsing, fundamental data structure problems like maintaining the majority of the prefixes of a string of bits and range queries. We characterise the complexity of maintaining the value of any symmetric function on the prefixes of a bit string.

Dynamic word problems

Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector 2 = (21,22,. . .,z,,) E initially (1,1,. . ., 1) wifh two kinds of operations, for each i E { 1,. . . , n}, a E M, an operation changei,a which changes xi to a and a single operation product returning n:==, xi. This i s the dynamic word problem. If we in addition for each j E { 1, . . . , n} have an operation pref ixj returning ni=, xi, we talk about the dynamic prefix problem. We analyze the complexity of these problems in the cell probe or decision assignment tree model for two natural cell sizes, 1 bit and logn bits. We obtain a classification of the Complexity based on algebraic properties of M.

Lower bounds for dynamic transitive closure, planar point location, and parantheses matching

We give a number of new lower bounds in the cell probe model with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operati...

Lower Bounds for Dynamic Transitive Closure, Planar Point Location, and Parentheses Matching

We give a number of new lower bounds in the cell probe model<br />with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations. We study the signed prefix sum problem: given a string of length n of zeroes and signed ones, compute the sum of its ith prefix during updates. We show a<br />lower bound of Omega(log n/log log n) time per operations, even if the prefix sums are bounded by log n/log log n during all updates. We also show that if the update time is bounded by the product of the worst-case update time and the<br />answer to the query, then the update time must be Omega(sqrt(log n/ log log n)).<br /> These results allow us to prove lower bounds for a variety of seemingly unrelated<br />dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of <br />Omega(log n/ log log n) per operation. We give<br />a lower bound for the dynamic transitive closure problem on upward planar graphs with one source and one sink of <br />Omega(log n/(log logn)^2) per operation. We give a lower bound of Omega(sqrt(log n/log log n)) for the dynamic membership problem of any Dyck language with two or more letters. This implies the same<br />lower bound for the dynamic word problem for the free group with k generators. We also give lower bounds for the dynamic prefix majority and prefix equality problems.

Data Structures for Traveling Salesmen

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 Ω(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(log N). 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 Ω(log N/log log N) per operation. Empirically (for random Euclidean instances), splay trees overcome their large constant-factor overhead and catch up to arrays by N = 10,000, pulling ahead by a factor of 4-10 (depending on machine) when N = 100,000. Two alternative tree-based data structures do even better in this range, however. Although both are asymptotically inferior to the splay tree representations, the latter does not appear to pull even with them until N ∼ to 1,000,000.

On Data Structures and Asymmetric Communication Complexity

In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the ``asymmetric'' case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds, and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our ``round elimination'' lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the ``round reduction'' techniques used in many previous lower bound proofs.<em> </em>

Fully Dynamic Transitive Closure in Plane Dags with one Source and one Sink

We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from u to v ?' for arbitrary vertices u and v can be answered in logarithmic time. The size of the data structure and the initialisation time are linear in the number of edges.<br /> <br />The result enlarges the class of graphs for which a logarithmic (or even polylogarithmic) time dynamic transitive closure algorithm exists. Previously, the only algorithms within the stated resource bounds put restrictions on the topology of the graph or on the delete operation. To obtain our result, we use a new characterisation of the transitive closure in plane graphs with one source and one sink and introduce new techniques to exploit this characterisation.<br /> <br />We also give a lower bound of Omega(log n/log log n) on the amortised complexity of the problem in the cell probe model with logarithmic word size. This is the first dynamic directed graph problem with almost matching lower and upper bounds.

The Set Union Problem with Unlimited Backtracking

An extension of the disjoint set union problem is considered, where the extra primitive backtrack($i$) can undo the last $i$ unions not yet undone. Let $n$ be the total number of elements in all the sets. A data structure is presented that supports each union and find in $O(\log n/\log \log n)$ worst-case time and each backtrack($i$) in $O(1)$ worst-case time, irrespective of $i$. The total space required by the data structure is $O(n)$. A byproduct of this construction is a partially persistent data structure for the standard set union problem, capable of supporting union, find, and find-in-the-past operations, each in $O(\log n/\log \log n)$ worst-case time. All these upper bounds are tight for the class of separable-pointer algorithms as well as in the cell probe model of computation.

Dynamic Word problems

Let <em>M</em> be a fixed finite monoid. We consider the problem of implementing a data type containing a vector x=(x_1, x_2, ..., x_n) in M^n, initially (1, 1, ..., 1) with two kinds of operations, for each i in {1, ..., n}, a in M, an operation <code>change</code>_{i,a} which changes x_i to a and a single operation <code> product</code> returning ½_{i=1}^j x_i. This is the dynamic <em>word</em> problem. If we in addition for each j in {1, ..., n} have an operation <code>prefix_j</code> returning ½_{i=1}^j x_i, we talk about the dynamic <em>prefix</em> problem. We analyze the complexity of these problems in the <em>cell probe</em> or <em>decision assignment tree</em> model for two natural cell sizes, 1 bit and log n bits. We obtain a classification of the complexity based on algebraic properties of <em>M</em>. f

A lower bound for finding predecessors in Yao's cell probe model

LetL be the set consisting of the firstq positive integers. We prove in this paper that there does not exist a data structure for representing an arbitrary subsetA ofL which uses poly (¦A¦) cells of memory (where each cell holdsc logq bits of information) and which the predecessor inA of an arbitraryx≦q can be determined by probing only a constant (independent ofq) number of cells. Actually our proof gives more: the theorem remains valid if this number is less thane log logq, that is D. E. Willard's algorithm [2] for finding the predecessor inO(log logq) time is optimal up to a constant factor.