A strong lower bound for approximate nearest neighbor searching

We prove a lower bound of d^1−o(1) on the query time for any deterministic algorithms that solve approximate nearest neighbor searching in Yao's cell probe model. Our result greatly improves the best previous lower bound for this problem, which is [A. Chakrabarti et al., in: Proc. 31st Ann. ACM Symp. Theory of Computing, 1999, pp. 305-311]. Our proof is also much simpler than the proof of A. Chakrabarti et al.     

On the false-positive rate of Bloom filters

Bloom filters are a randomized data structure for membership queries dating back to 1970. Bloom filters sometimes give erroneous answers to queries, called false positives. Bloom analyzed the probability of such erroneous answers, called the false-positive rate, and Bloom's analysis has appeared in many publications throughout the years. We show that Bloom's analysis is incorrect and give a correct analysis.     

A new analysis of the false positive rate of a Bloom filter

A Bloom filter is a space-efficient data structure used for probabilistic set membership testing. When testing an object for set membership, a Bloom filter may give a false positive. The analysis of the false positive rate is a key to understanding the Bloom filter and applications that use it. We show experimentally that the classic analysis for false positive rate is wrong. We formally derive a correct formula using a balls-and-bins model and show how to numerically compute the new, correct formula in a stable manner. We also prove that the new formula always results in a predicted greater false positive rate than the classic formula. This correct formula is numerically compared to the classic formula for relative error - for a small Bloom filter the prediction of false positive rate will be in error when the classic formula is used.     

Canonical density control

The Sparse Table is a data structure for controlling density in an array which was first proposed in 1981 and has recently reappeared as a component of cache-oblivious data structures. All existing variants of the Sparse Table divide the array into blocks that have a calibrator tree above them. We show that the same amortized complexity can be achieved without this auxiliary structure, obtaining a canonical data structure that can be updated by conceptually simpler algorithms.     

Using spine decompositions to efficiently solve the length-constrained heaviest path problem for trees

The length-constrained heaviest path (LCHP) in a weighted tree T, where each edge is assigned a weight and a length, is the path P in T with maximum total path weight and total path length bounded by a given value B. This paper presents an O(nlogn) time LCHP algorithm which utilizes a data structure constructed from the spine decomposition of the input tree. This is an improvement over the existing algorithm by Wu et al. (1999), which runs in O(nlog²n) time. Our method also improves on a previous O(nlogn) time algorithm by Kim (2005) for the special case of finding a longest nonnegative path in a constant degree tree in that we can handle trees of arbitrary degree within the same time bounds.     

A note on the online First-Fit algorithm for coloring k-inductive graphs

In a FOCS 1990 paper, S. Irani proved that the First-Fit online algorithm for coloring a graph uses at most O(klogn) colors for k-inductive graphs. In this note we provide a very short proof of this fact.     

Tuning the boyer-moore-horspool string searching algorithm

Substring search is a common activity in computing. The fastest known search method is that of Boyer and Moore with the improvements introduced by Horspool. This paper presents a new implementation which takes advantage of the dependencies between the characters. The resulting code runs 25 per cent faster than the best currently-known routine.     

An improved lower bound on approximation algorithms for the Closest Substring problem

The Closest Substring problem (the CSP problem) is a basic NP-hard problem in the study of computational biology. It is known that the problem has polynomial time approximation schemes. In this paper, we prove that unless the Exponential Time Hypothesis fails, the CSP problem has no polynomial time approximation schemes of running time f(1/ε)n^o(1/ε) for any function f. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound ε. As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the problem for small error bound ε.     

Double hashing with passbits

Double hashing with bucket capacity one is augmented with multiple passbits to obtain significant reduction to unsuccessful search lengths. This improves the analysis of Martini et al. [P.M. Martini, W.A. Burkhard, Double hashing with multiple passbits, Internat. J. Found. Theoret. Comput. Sci. 14 (6) (2003) 1165-1188] by providing a closed form expression for the expected unsuccessful search lengths.     

A note on constructing binary heaps with periodic networks

We consider the problem of constructing binary heaps on constant degree networks performing compare-exchange operations only. The heap data structure, introduced by William and Williams [Comm. ACM 7 (6) (1964) 347-348], has many applications and, therefore, has been intensively studied in sequential and parallel context. In particular, Brodal and Pinotti [Theoret. Comput. Sci. 250 (2001) 235-245] have recently presented two families of comparator networks: the first of depth 4logN and the second of size O(NloglogN) for constructing binary heaps of size N. In this note, we give an new construction of such a network with the running time improved to 3logN. Moreover, the network has a novel property of being 3-periodic, that is, for each unit of time i the same sets of operations are performed in units i and i+3. Then we argue that our construction is optimal with respect to the length of the period, that is, we prove that there is no 2-periodic network that is able to build a binary heap in sublinear time. Finally, we show that our construction can be used to decrease also the depth of the networks with O(NloglogN) size.     

A note on compressed sensing and the complexity of matrix multiplication

We consider the conjectured O(N2+?) time complexity of multiplying any two N×N matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A⋅B provided that the resulting matrix product is sparse/compressible. As a consequence of our main result we increase the class of matrices A, for any given N×N matrix B, which allows the exact computation of A⋅B to be carried out using the conjectured O(N2+?) operations. Additionally, in the process of developing our matrix multiplication procedure, we present a modified version of Indyk's recently proposed extractor-based CS algorithm [P. Indyk, Explicit constructions for compressed sensing of sparse signals, in: SODA, 2008] which is resilient to noise.     

A note on searching line arrangements and applications

We study the problem of searching for a vertex with a desired property in the arrangement of a set of lines in the plane. We show that this problem can be solved efficiently by modifying (and simplifying) two slope selection algorithms without using parametric search. We apply this result to a points approximation problem and obtain an optimal solution for it without using parametric search. Since this line arrangement searching problem is quite natural, our result may find other applications as well.     

A lower bound for the time to assure interactive consistency

A fast algorithm for composition of sparse maps (i.e.,maps which act as the identity on most of the elements of their domain) is presented.     

Lower bounds on the OBDD size of two fundamental functions' graphs

Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which supports many useful operations. Among others it finds applications in CAD, model checking, and symbolic graph algorithms. Nevertheless, many simple functions are known to have exponential OBDD size with respect to their number of variables. In order to investigate the limits of symbolic graph algorithms which work on OBDD-represented graph instances, it is useful to have simply-structured graphs whose OBDD representation has exponential size. Therefore, we consider two fundamental functions with exponential lower bounds on their OBDD size and transfer these results to their corresponding graphs. Concretely, we consider the Indirect Storage Access function and the Hidden Weighted Bit function.     

A note on the query complexity of the Condorcet winner problem

Given an unknown tournament over {1,…,n}, we show that the query complexity of the question “Is there a vertex with outdegree n−1?” (known as a Condorcet winner in social choice theory) is exactly 2n−⌊log(n)⌋−2. This stands in stark contrast to the evasiveness of this property in general digraphs.     

A note on maximum independent set and related problems on box graphs

A box graph is the intersection graph of orthogonal rectangles in the plane. We show that maximum independent set and minimum vertex cover on box graphs can be solved in subexponential time, more precisely, in time , by applying Miller's simple cycle planar separator theorem [J. Comput. System Sci. 32 (1986) 265-279] (in spite of the fact that the input box graph might be strongly non-planar).     

Trading uninitialized space for time

The design of efficient graph algorithms usually precludes the test of edge existence, because an efficient support of that operation already requires time for the initialization of an adjacency-matrix representation. We describe an alternative representation of static directed graphs taking Θ(n+m) initialization time and using Θ(n²) space, which supports the efficient implementation of all usual operations on static graphs. The sparse graph representation allows the design of efficient graph algorithms using both iteration over all vertices adjacent with a given vertex and edge-existence operations, although at the expense of additional (uninitialized) space which may, nevertheless, be used for other purposes. To the best of our knowledge, the representation leads to the first graph algorithms with the disconcerting property that the time complexity is better than the space complexity.     

A note on finding all homogeneous set sandwiches

A module is a set of vertices H of a graph G=(V,E) such that each vertex of V?H is either adjacent to all vertices of H or to none of them. A homogeneous set is a nontrivial module. A graph G_s=(V,E_s) is a sandwich for a pair of graphs G_t=(V,E_t) and G=(V,E) if E_t⊆E_s⊆E. In a recent paper, Tang et al. [Inform. Process. Lett. 77 (2001) 17-22] described an O(Δn²) algorithm for testing the existence of a homogeneous set in sandwich graphs of G_t=(V,E_t) and G=(V,E) and then extended it to an enumerative algorithm computing all these possible homogeneous sets. In this paper, we invalidate this latter algorithm by proving there are possibly exponentially many such sets, even if we restrict our attention to strong modules. We then give a correct characterization of a homogeneous set of a sandwich graph.