Results 1  10
of
25
TimeSpace Tradeoff Lower Bounds for Randomized Computation of Decision Problems
 In Proc. of 41st FOCS
, 2000
"... We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. ..."
Abstract

Cited by 35 (5 self)
 Add to MetaCart
We prove the first timespace lower bound tradeoffs for randomized computation of decision problems.
A geometric approach to lower bounds for approximate nearneighbor search and partial match
 In Proc. 49th IEEE Symposium on Foundations of Computer Science (FOCS
, 2008
"... This work investigates a geometric approach to proving cell probe lower bounds for data structure problems. We consider the approximate nearest neighbor search problem on the Boolean hypercube ({0, 1} d, ‖ · ‖1) with d = Θ(log n). We show that any (randomized) data structure for the problem that a ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
This work investigates a geometric approach to proving cell probe lower bounds for data structure problems. We consider the approximate nearest neighbor search problem on the Boolean hypercube ({0, 1} d, ‖ · ‖1) with d = Θ(log n). We show that any (randomized) data structure for the problem that answers capproximate nearest neighbor search queries using t probes must use space at least n1+Ω(1/ct). In particular, our bound implies that any data structure that uses space Õ(n) with polylogarithmic word size, and with constant probability gives a constant approximation to nearest neighbor search queries must be probed Ω(log n / log log n) times. This improves on the lower bound of Ω(log log d / log log log d) probes shown by Chakrabarti and Regev [8] for any polynomial space data structure, and the Ω(log log d) lower bound in Pătras¸cu and Thorup [26] for linear space data structures. Our lower bound holds for the near neighbor problem, where the algorithm knows in advance a good approximation to the distance to the nearest neighbor. Additionally, it is an average case lower bound for the natural distribution for the problem. Our approach also gives the same bound for (2 − 1)approximation to the farthest neighbor problem. c For the case of nonadaptive algorithms we can improve the bound slightly and show a Ω(log n) lower bound on the time complexity of data structures with O(n) space and logarithmic word size. We also show similar lower bounds for the partial match problem: any randomized tprobe data structure that solves the partial match problem on {0, 1, ⋆} d for d = Θ(log n) must use space n1+Ω(1/t). This implies an Ω(log n / log log n) lower bound for time complexity of near linear space data structures, slightly improving the Ω(log n/(log log n) 2) lower bound from [25],[16] for this range of d. Recently and independently Pătras¸cu achieved similar bounds [24]. Our results also generalize to approximate partial match, improving on the bounds of [4, 25]. 1 1
Lower bounds on near neighbor search via metric expansion
 CoRR
"... In this paper we show how the complexity of performing nearest neighbor (NNS) search on a metric space is related to the expansion of the metric space. Given a metric space we look at the graph obtained by connecting every pair of points within a certain distance r. We then look at various notions o ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
In this paper we show how the complexity of performing nearest neighbor (NNS) search on a metric space is related to the expansion of the metric space. Given a metric space we look at the graph obtained by connecting every pair of points within a certain distance r. We then look at various notions of expansion in this graph relating them to the cell probe complexity of NNS for randomized and deterministic, exact and approximate algorithms. For example if the graph has node expansion Φ then we show that any deterministic tprobe data structure for n points must use space S where (St/n)t> Φ. We show similar results for randomized algorithms as well. These relationships can be used to derive most of the known lower bounds in the well known metric spaces such as l1, l2, l ∞ by simply computing their expansion. In the process, we strengthen and generalize our previous results [19]. Additionally, we unify the approach in [19] and the communication complexity based approach. Our work reduces the problem of proving cell probe lower bounds of near neighbor search to computing the appropriate expansion parameter. In our results, as in all previous results, the dependence on t is weak; that is, the bound drops exponentially in t. We show a much stronger (tight) timespace tradeoff for the class of dynamic low contention data structures. These are data structures that supports updates in the data set and that do not look up any single cell too often. 1 1
CellProbe Lower Bounds for the Partial Match Problem
, 2003
"... Given a database of n points in f0; 1g , the partial match problem is: In response to a query x in f0; 1; g database point y such that for every i whenever x i 6= , we have x i = y i . In this paper we show randomized lower bounds in the cellprobe model for this wellstudied problem [18, 11, ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
Given a database of n points in f0; 1g , the partial match problem is: In response to a query x in f0; 1; g database point y such that for every i whenever x i 6= , we have x i = y i . In this paper we show randomized lower bounds in the cellprobe model for this wellstudied problem [18, 11, 19, 16, 4, 6]. Our lower
Quantum Branching Programs and SpaceBounded Nonuniform Quantum Complexity
, 2005
"... In this paper, the space complexity of nonuniform quantum algorithms is investigated using the model of quantum branching programs (QBPs). In order to clarify the relationship between QBPs and nonuniform quantum Turing machines, simulations between these two models are presented which allow to trans ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
In this paper, the space complexity of nonuniform quantum algorithms is investigated using the model of quantum branching programs (QBPs). In order to clarify the relationship between QBPs and nonuniform quantum Turing machines, simulations between these two models are presented which allow to transfer upper and lower bound results. Exploiting additional insights about the connection between the running time and the precision of amplitudes, it is shown that nonuniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Furthermore, quantum ordered binary decision diagrams (QOBDDs) are considered, which are restricted QBPs that can be regarded as a nonuniform analog of oneway quantum finite automata. Upper and lower bounds are proved that allow a classification of the computational power of QOBDDs in comparison to usual deterministic and randomized variants of the model. Finally, an extension of QBPs is proposed where the performed unitary operation may depend on the result of a previous measurement. A simulation of randomized BPs by this generalized QBP model as well as exponential lower bounds for its ordered variant are presented.
Parity graphdriven readonce branching programs and an exponential lower bound for integer multiplication
 In Proc. of 2nd TCS
, 2002
"... Abstract Branching programs are a wellestablished computation model for boolean functions, especially readonce branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic readonce branching programs are known for a long time. On the other hand ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
Abstract Branching programs are a wellestablished computation model for boolean functions, especially readonce branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic readonce branching programs are known for a long time. On the other hand, the problem of proving superpolynomial lower bounds for parity readonce branching programs is still open. In this paper restricted parity readonce branching programs are considered and an exponential lower bound on the size of wellstructured parity graphdriven readonce branching programs for integer multiplication is proven. This is the first strongly exponential lower bound on the size of a nonoblivious parity readonce branching program model for an explicitly defined boolean function. In addition, more insight into the structure of integer multiplication is yielded.
Timespace tradeoff lower bounds for integer . . . (Extended Abstract)
, 2003
"... We prove exponential size lower bounds for nondeterministic and randomized readk BPs as well as a timespace tradeoff lower bound for unrestricted, deterministic multiway BPs computing the middle bit of integer multiplication. The lower bound for randomized readk BPs is superpolynomial as long ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We prove exponential size lower bounds for nondeterministic and randomized readk BPs as well as a timespace tradeoff lower bound for unrestricted, deterministic multiway BPs computing the middle bit of integer multiplication. The lower bound for randomized readk BPs is superpolynomial as long as the error probability is superpolynomially small. For polynomially small error, we have a polynomial upper bound on the size of approximating readonce BPs for this function. The lower bounds follow from a more general result for the graphs of universal hash classes that is applicable to the graphs of arithmetic functions such as integer multiplication, convolution, and finite field multiplication.
The multiparty communication complexity of ExactT: Improved bounds and new problems
 In Proc. of 31st MFCS
, 2006
"... Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 case, and a lower bound of!(1) for k> = 3 when T = \Theta (2n). We obtain(1) for general k> = 3 an upper bound of k + O(n1/(k1)), (2) for k = 3, T = \Theta (2n), a lowerbound of \Omega (log log n), (3) a generalization of the protocol to abelian groups, (4) lower boundson the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case. 1 Introduction Multiparty communication complexity was first defined by Chandra, Furst, and Lipton [8] and used to obtain lower bounds on branching programs. Since then it has been used to get additional lower bounds and tradeoffs for branching programs [1, 5], lower bounds on problems in data structures [5], timespace tradeoffs for restricted Turing machines [1], and unconditional pseudorandom generators for logspace [1]. Def 1.1 Let f: {{0, 1}n}k! {0, 1}. Assume, for 1 < = i < = k, Pi has all of the inputs except xi. Let d(f) be the total number of bits broadcast in the optimal deterministic protocol for f. This is called the multiparty communication complexity of f. The scenario is called the forehead model.