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25
Logarithmic lower bounds in the cellprobe model
 SIAM Journal on Computing
"... Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the ..."
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Cited by 34 (4 self)
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Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the data structure. We use our technique to prove better bounds for the partialsums problem, dynamic connectivity and (by reductions) other dynamic graph problems. Our proofs are surprisingly simple and clean. The bounds we obtain are often optimal, and lead to a nearly complete understanding of the problems. We also present new matching upper bounds for the partialsums problem. Key words. cellprobe complexity, lower bounds, data structures, dynamic graph problems, partialsums problem AMS subject classification. 68Q17
Interaction in Quantum Communication and the Complexity of Set Disjointness
, 2001
"... One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible ..."
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Cited by 34 (7 self)
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One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structurethey involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a \simpler" quantum protocolone that has similar eciency, but uses fewer message exchanges.
Distance oracles for sparse graphs
 In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS
"... Abstract — Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn 1+1/k) that can answe ..."
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Cited by 18 (3 self)
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Abstract — Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn 1+1/k) that can answer queries in time O(k) with a distance estimate that is at most α = 2k − 1 times larger than the actual shortest distance (α is called the stretch). They proved that, under a combinatorial conjecture, their data structure is optimal in terms of space: if a stretch of at most 2k−1 is desired, then the space complexity is at least n 1+1/k. Their proof holds even if infinite query time is allowed: it is essentially an “incompressibility ” result. Also, the proof only holds for dense graphs, and the best bound it can prove only implies that the size of the data structure is lower bounded by the number of edges of the graph. Naturally, the following question arises: what happens for sparse graphs? In this paper we give a new lower bound for approximate distance oracles in the cellprobe model. This lower bound holds even for sparse (polylog(n)degree) graphs, and it is not an “incompressibility ” bound: we prove a threeway tradeoff between space, stretch and query time. We show that, when the query time is t, and the stretch is α, then the space S must be S ≥ n 1+Ω(1/tα) / lg n. (1) This lower bound follows by a reduction from lopsided set disjointness to distance oracles, based on and motivated by recent work of Pǎtras¸cu. Our results in fact show that for any highgirth regular graph, an approximate distance oracle that supports efficient queries for all subgraphs of G must obey Eq. (1). We also prove some lemmas that count sets of paths in highgirth regular graphs and highgirth regular expanders, which might be of independent interest. Keywordsdistance oracle; data structures; lower bounds; cellprobe model; lopsided set disjointness 1.
Higher lower bounds for nearneighbor and further rich problems
 in Proc. 47th IEEE Symposium on Foundations of Computer Science (FOCS
"... We convert cellprobe lower bounds for polynomial space into stronger lower bounds for nearlinear space. Our technique applies to any lower bound proved through the richness method. For example, it applies to partial match, and to nearneighbor problems, either for randomized exact search, or for d ..."
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Cited by 16 (2 self)
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We convert cellprobe lower bounds for polynomial space into stronger lower bounds for nearlinear space. Our technique applies to any lower bound proved through the richness method. For example, it applies to partial match, and to nearneighbor problems, either for randomized exact search, or for deterministic approximate search (which are thought to exhibit the curse of dimensionality). These problems are motivated by search in large databases, so nearlinear space is the most relevant regime. Typically, richness has been used to imply Ω(d / lg n) lower bounds for polynomialspace data structures, where d is the number of bits of a query. This is the highest lower bound provable through the classic reduction to communication complexity. However, for space n lg O(1) n, we now obtain bounds of Ω(d / lg d). This is a significant improvement for natural values of d, such as lg O(1) n. In the most important case of d = Θ(lg n), we have the first superconstant lower bound. From a complexity theoretic perspective, our lower bounds are the highest known for any static data structure problem, significantly improving on previous records. 1
Tight lower bounds for selection in randomly ordered streams
 In SODA
, 2008
"... We show that any algorithm computing the median of a stream presented in random order, using polylog(n) space, requires an optimal Ω(log log n) passes, resolving an open question from the seminal paper on streaming by Munro and Paterson, from FOCS 1978. 1 ..."
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Cited by 15 (3 self)
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We show that any algorithm computing the median of a stream presented in random order, using polylog(n) space, requires an optimal Ω(log log n) passes, resolving an open question from the seminal paper on streaming by Munro and Paterson, from FOCS 1978. 1
Property Testing Lower Bounds Via Communication Complexity
, 2011
"... We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexit ..."
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Cited by 12 (3 self)
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We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is klinear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with twosided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as kjuntas. For some classes, such as the class of monotone functions and the class of ssparse GF(2) polynomials, we significantly strengthen the best known bounds.
UNIFYING THE LANDSCAPE OF CELLPROBE LOWER BOUNDS
, 2008
"... We show that a large fraction of the datastructure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • highdimensional problems, where the goal is to show large space lower bounds. • co ..."
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Cited by 11 (1 self)
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We show that a large fraction of the datastructure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • highdimensional problems, where the goal is to show large space lower bounds. • constantdimensional geometric problems, where the goal is to bound the query time for space O(n·polylogn). • dynamic problems, where we are looking for a tradeoff between query and update time. (In this case, our bounds are slightly weaker than the originals, losing a lglgn factor.) Our reductions also imply the following new results: • an Ω(lgn/lglgn) bound for 4dimensional range reporting, given space O(n · polylogn). This is quite timely, since a recent result [39] solved 3D reporting in O(lg 2 lgn) time, raising the prospect that higher dimensions could also be easy. • a tight space lower bound for the partial match problem, for constant query time. • the first lower bound for reachability oracles. In the process, we prove optimal randomized lower bounds for lopsided set disjointness.
Lower Bounds for Dynamic Transitive Closure, Planar Point Location, and Parentheses Matching
 Nordic Journal of Computing
, 1996
"... 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 operations. ..."
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Cited by 11 (4 self)
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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 operations.
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 ..."
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Cited by 10 (1 self)
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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