11 research outputs found

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

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    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 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

    Marked Ancestor Problems (Preliminary Version)

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    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. 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. 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

    Adapt Or Die: Polynomial Lower Bounds For Non-Adaptive Dynamic Data Structures

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    In this paper, we study the role non-adaptivity plays in maintaining dynamic data structures. Roughly speaking, a data structure is non-adaptive if the memory locations it reads and/or writes when processing a query or update depend only on the query or update and not on the contents of previously read cells. We study such non-adaptive data structures in the cell probe model. The cell probe model is one of the least restrictive lower bound models and in particular, cell probe lower bounds apply to data structures developed in the popular word-RAM model. Unfortunately, this generality comes at a high cost: the highest lower bound proved for any data structure problem is only polylogarithmic (if allowed adaptivity). Our main result is to demonstrate that one can in fact obtain polynomial cell probe lower bounds for non-adaptive data structures. To shed more light on the seemingly inherent polylogarithmic lower bound barrier, we study several different notions of non-adaptivity and identify key properties that must be dealt with if we are to prove polynomial lower bounds without restrictions on the data structures. Finally, our results also unveil an interesting connection between data structures and depth-2 circuits. This allows us to translate conjectured hard data structure problems into good candidates for high circuit lower bounds; in particular, in the area of linear circuits for linear operators. Building on lower bound proofs for data structures in slightly more restrictive models, we also present a number of properties of linear operators which we believe are worth investigating in the realm of circuit lower bounds

    Static Data Structure for Discrete Advance Bandwidth Reservations on the Internet

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    In this paper we present a discrete data structure for reservations of limited resources. A reservation is defined as a tuple consisting of the time interval of when the resource should be reserved, IRI_R, and the amount of the resource that is reserved, BRB_R, formally R={IR,BR}R=\{I_R,B_R\}. The data structure is similar to a segment tree. The maximum spanning interval of the data structure is fixed and defined in advance. The granularity and thereby the size of the intervals of the leaves is also defined in advance. The data structure is built only once. Neither nodes nor leaves are ever inserted, deleted or moved. Hence, the running time of the operations does not depend on the number of reservations previously made. The running time does not depend on the size of the interval of the reservation either. Let nn be the number of leaves in the data structure. In the worst case, the number of touched (i.e. traversed) nodes is in any operation O(logn)O(\log n), hence the running time of any operation is also O(logn)O(\log n)

    Data Structuring Problems in the Bit Probe Model

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    We study two data structuring problems under the bit probe model: the dynamic predecessor problem and integer representation in a manner supporting basic updates in as few bit operations as possible. The model of computation considered in this paper is the bit probe model. In this model, the complexity measure counts only the bitwise accesses to the data structure. The model ignores the cost of computation. As a result, the bit probe complexity of a data structuring problem can be considered as a fundamental measure of the problem. Lower bounds derived by this model are valid as lower bounds for any realistic, sequential model of computation. Furthermore, some of the problems are more suitable for study in this model as they can be solved using less than ww bit probes where ww is the size of a computer word. The predecessor problem is one of the fundamental problems in computer science with numerous applications and has been studied for several decades. We study the colored predecessor problem, a variation of the predecessor problem, in which each element is associated with a symbol from a finite alphabet or color. The problem is to store a subset SS of size n,n, from a finite universe UU so that to support efficient insertion, deletion and queries to determine the color of the largest value in SS which is not larger than x,x, for a given xU.x \in U. We present a data structure for the problem that requires O(klogUloglogUk)O(k \sqrt[k]{{\log U} \over {\log \log U}}) bit probes for the query and O(k2logUloglogU)O(k^2 {{\log U} \over {\log \log U}}) bit probes for the update operations, where UU is the universe size and kk is positive constant. We also show that the results on the colored predecessor problem can be used to solve some other related problems such as existential range query, dynamic prefix sum, segment representative, connectivity problems, etc. The second structure considered is for integer representation. We examine the problem of integer representation in a nearly minimal number of bits so that increment and decrement (and indeed addition and subtraction) can be performed using few bit inspections and fewer bit changes. In particular, we prove a new lower bound of Ω(n)\Omega(\sqrt{n}) for the increment and decrement operation, where nn is the minimum number of bits required to represent the number. We present several efficient data structures to represent integers that use a logarithmic number of bit inspections and a constant number of bit changes per operation

    Lower Bounds for Encrypted Multi-Maps and Searchable Encryption in the Leakage Cell Probe Model

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    Encrypted multi-maps (EMMs) enable clients to outsource the storage of a multi-map to a potentially untrusted server while maintaining the ability to perform operations in a privacy-preserving manner. EMMs are an important primitive as they are an integral building block for many practical applications such as searchable encryption and encrypted databases. In this work, we formally examine the tradeoffs between privacy and efficiency for EMMs. Currently, all known dynamic EMMs with constant overhead reveal if two operations are performed on the same key or not that we denote as the global key-equality pattern\mathit{global\ key\text{-}equality\ pattern}. In our main result, we present strong evidence that the leakage of the global key-equality pattern is inherent for any dynamic EMM construction with O(1)O(1) efficiency. In particular, we consider the slightly smaller leakage of decoupled key-equality pattern\mathit{decoupled\ key\text{-}equality\ pattern} where leakage of key-equality between update and query operations is decoupled and the adversary only learns whether two operations of the same type\mathit{same\ type} are performed on the same key or not. We show that any EMM with at most decoupled key-equality pattern leakage incurs Ω(logn)\Omega(\log n) overhead in the leakage cell probe model\mathit{leakage\ cell\ probe\ model}. This is tight as there exist ORAM-based constructions of EMMs with logarithmic slowdown that leak no more than the decoupled key-equality pattern (and actually, much less). Furthermore, we present stronger lower bounds that encrypted multi-maps leaking at most the decoupled key-equality pattern but are able to perform one of either the update or query operations in the plaintext still require Ω(logn)\Omega(\log n) overhead. Finally, we extend our lower bounds to show that dynamic, response-hiding\mathit{response\text{-}hiding} searchable encryption schemes must also incur Ω(logn)\Omega(\log n) overhead even when one of either the document updates or searches may be performed in the plaintext

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

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

    Get PDF
    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..
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