We generalize the classical nuts and bolts problem to a setting where the input is a collection of n nuts and m bolts, and there is no promise of any matching pairs. It is not allowed to compare a nut directly with a nut or a bolt directly with a bolt, and the goal is to perform the fewest nut-bolt comparisons to discover the partial order between the nuts and bolts. We term this problem bipartite sorting. We show that instances of bipartite sorting of the same size exhibit a wide range of complexity, and propose to perform a fine-grained analysis for this problem. We rule out straightforward notions of instance-optimality as being too stringent, and adopt a neighborhood-based definition. Our definition may be of independent interest as a unifying lens for instance-optimal algorithms for other static problems existing in literature. This includes problems like sorting (Estivill-Castro and Woods, ACM Comput. Surv. 1992), convex hull (Afshani, Barbay and Chan, JACM 2017), adaptive joins (Demaine, López-Ortiz and Munro, SODA 2000), and the recent concept of universal optimality for graphs (Haeupler, Hladík, Rozhoň, Tarjan and Tětek, 2023). As our main result on bipartite sorting, we give a randomized algorithm that is within a factor of O(log³(n+m)) of being instance-optimal w.h.p., with respect to the neighborhood-based definition. As our second contribution, we generalize bipartite sorting to DAG sorting, when the underlying DAG is not necessarily bipartite. As an unexpected consequence of a simple algorithm for DAG sorting, we rule out a potential lower bound on the widely-studied problem of sorting with priced information, posed by (Charikar, Fagin, Guruswami, Kleinberg, Raghavan and Sahai, STOC 2000). In this problem, comparing keys i and j has a known cost c_{ij} ∈ ℝ^+ ∪ {∞}, and the goal is to sort the keys in an instance-optimal way, by keeping the total cost of an algorithm as close as possible to ∑_{i=1}^{n-1} c_{x(i)x(i+1)}. Here x(1) < ⋯ < x(n) is the sorted order. While several special cases of cost functions have received a lot of attention in the community, no progress on the general version with arbitrary costs has been reported so far. One reason for this lack of progress seems to be a widely-cited Ω(n) lower bound on the competitive ratio for finding the maximum. This Ω(n) lower bound by (Gupta and Kumar, FOCS 2000) uses costs in {0,1,n, ∞}, and although not extended to sorting, this barrier seems to have stalled any progress on the general cost case. We rule out such a potential lower bound by showing the existence of an algorithm with a Õ(n^{3/4}) competitive ratio for the {0,1,n,∞} cost version. This generalizes the setting of generalized sorting proposed by (Huang, Kannan and Khanna, FOCS 2011), where the costs are either 1 or infinity, and the cost of the cheapest proof is always n-1.

A \emph{saddlepoint} of an $n \times n$ matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the \emph{value} of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. For finding a \emph{strict saddlepoint} (an entry that is the strict maximum of its row and the strict minimum of its column) we recently gave an $O({n\log^*{n}})$-time algorithm, improving the $O({n\log{n}})$ bounds from 1991 of Bienstock, Chung, Fredman, Schäffer, Shor, Suri and of Byrne and Vaserstein. In this paper we present an optimal $O({n})$-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial $O(n^2)$ runtime cannot be improved even with the use of randomness.

The problem of sorting with priced information was introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to find a sorting algorithm with small competitive ratio, defined as the (worst-case) ratio of the algorithm's cost to the cost of the cheapest proof of the sorted order. The simple case of bichromatic sorting posed by [CFGKRS] remains open: We are given two sets $A$ and $B$ of total size $N$, and the cost of an $A-A$ comparison or a $B-B$ comparison is higher than an $A-B$ comparison. The goal is to sort $A \cup B$. An $Ω(\log N)$ lower bound on competitive ratio follows from unit-cost sorting. Note that this is a generalization of the famous nuts and bolts problem, where $A-A$ and $B-B$ comparisons have infinite cost, and elements of $A$ and $B$ are guaranteed to alternate in the final sorted order. In this paper we give a randomized algorithm InversionSort with an almost-optimal w.h.p. competitive ratio of $O(\log^{3} N)$. This is the first algorithm for bichromatic sorting with a $o(N)$ competitive ratio.

We present comparison-based parallel algorithms for sorting n comparable items subject to comparison errors. We consider errors to occur according to a well-studied framework, where the comparison of two elements returns the wrong answer with a fixed probability. In the persistent model, the result of the comparison of two given elements, x and y, always has the same result, and is independent of all other pairs of elements. In the non-persistent model, the result of the comparison of each pair of elements, x and y, is independent of all prior comparisons, including for x and y. It is not possible to always correctly sort a given input set in the persistent model, so we study algorithms that achieve a small maximum dislocation and small total dislocation of the elements in the output permutation. In this paper, we provide parallel algorithms for sorting with comparison errors in the persistent and non-persistent models. Our algorithms are asymptotically optimal in terms of their span, work, and, in the case of persistent errors, maximum and total dislocation. The main results are algorithms for the binary-forking parallel model with atomics, but we also provide algorithms for the CREW PRAM model. Our algorithms include a number of novel techniques and analysis tools, including a PRAM-to-binary-forking-model simulation result, and are the first optimal parallel algorithms for the persistent model and the non-persistent model in the binary-forking parallel model with atomics. In particular, our algorithms have O(log n) span, O(n log n) work, and, in the case of the persistent model, O(log n) maximum dislocation and O(n) total dislocation, with high probability. We achieve similar results for the CREW PRAM model, which are the first optimal methods for the persistent model and the first optimal results for the non-persistent model with reasonable constant factors in the performance bounds.

The fragile complexity of a comparison-based algorithm is $f(n)$ if each input element participates in $O(f(n))$ comparisons. In this paper, we explore the fragile complexity of algorithms adaptive to various restrictions on the input, i.e., algorithms with a fragile complexity parameterized by a quantity other than the input size~$n$. We show that searching for the predecessor in a sorted array has fragile complexity $\Theta(\log k)$, where $k$ is the rank of the query element, both in a randomized and a deterministic setting. For predecessor searches, we also show how to optimally reduce the amortized fragile complexity of the elements in the array. We also prove the following results: Selecting the $k$th smallest element has expected fragile complexity $O(\log\log k)$ for the element selected. Deterministically finding the minimum element has fragile complexity $\Theta(\log(\INV))$ and $\Theta(\log(\RUNS))$, where $\INV$ is the number of inversions in a sequence and $\RUNS$ is the number of increasing runs in a sequence. Deterministically finding the median has fragile complexity $O(\log(\RUNS) + \log\log n)$ and $\Theta(\log (\INV))$. Deterministic sorting has fragile complexity $\Theta(\log (\INV))$ but it has fragile complexity $\Theta(\log n)$ regardless of the number of runs.

We prove an Ω (log n log log n) lower bound for the span of implementing the n input, log n-depth FFT circuit (also known as butterfly network) in the nonatomic binary fork-join model. In this model, memory-access synchronizations occur only through fork operations, which spawn two child threads, and join operations, which resume a parent thread when its child threads terminate. Our bound is asymptotically tight for the nonatomic binary fork-join model, which has been of interest of late, due to its conceptual elegance and ability to capture asynchrony. Our bound implies super-logarithmic lower bound in the nonatomic binary fork-join model for implementing the butterfly merging networks used, e.g., in Batcher's bitonic and odd-even mergesort networks. This lower bound also implies an asymptotic separation result for the atomic and nonatomic versions of the fork-join model, since, as we point out, FFT circuits can be implemented in the atomic binary fork-join model with span equal to their circuit depth.

We consider static, external memory indexes for exact and approximate versions of the k-nearest neighbor (k-NN) problem, and show new lower bounds under a standard indivisibility assumption: Polynomial space indexing schemes for high-dimensional k-NN in Hamming space cannot take advantage of block transfers: í(k) block reads are needed to to answer a query. For the l∞ metric the lower bound holds even if we allow c-appoximate nearest neighbors to be returned, for c ∈ (1, 3). The restriction to c < 3 is necessary: For every metric there exists an indexing scheme in the indexability model of Hellerstein et al. using space O(kn), where n is the number of points, that can retrieve k 3-approximate nearest neighbors using optimal ⌈k/B⌉ I/Os, where B is the block size. For specific metrics, data structures with better approximation factors are possible. For k-NN in Hamming space and every approximation factor c>1 there exists a polynomial space data structure that returns k c-approximate nearest neighbors in ⌈k/B⌉ I/Os. To show these lower bounds we develop two new techniques: First, to handle that approximation algorithms have more freedom in deciding which result set to return we develop a relaxed version of the λ-set workload technique of Hellerstein et al. This technique allows us to show lower bounds that hold in d ≥ n dimensions. To extend the lower bounds down to d = O(k log(n/k)) dimensions, we develop a new deterministic dimension reduction technique that may be of independent interest.

We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible.

We present priority queues in the external memory model with block size B and main memory size M that support on N elements, operation Update (a combination of operations Insert and DecreaseKey) in O(1/Blog_{M/B} N/B) amortized I/Os and operations ExtractMin and Delete in O(ceil[(M^epsilon)/B log_{M/B} N/B] log_{M/B} N/B) amortized I/Os, for any real epsilon in (0,1), using O(N/Blog_{M/B} N/B) blocks. Previous I/O-efficient priority queues either support these operations in O(1/Blog_2 N/B) amortized I/Os [Kumar and Schwabe, SPDP '96] or support only operations Insert, Delete and ExtractMin in optimal O(1/Blog_{M/B} N/B) amortized I/Os, however without supporting DecreaseKey [Fadel et al., TCS '99]. We also present buffered repository trees that support on a multi-set of N elements, operation Insert in O(1/Blog_M/B N/B) I/Os and operation Extract on K extracted elements in O(M^{epsilon} log_M/B N/B + K/B) amortized I/Os, using O(N/B) blocks. Previous results achieve O(1/Blog_2 N/B) I/Os and O(log_2 N/B + K/B) I/Os, respectively [Buchsbaum et al., SODA '00]. Our results imply improved O(E/Blog_{M/B} E/B) I/Os for single-source shortest paths, depth-first search and breadth-first search algorithms on massive directed dense graphs (V,E) with E = Omega (V^(1+epsilon)), epsilon > 0 and V = Omega (M), which is equal to the I/O-optimal bound for sorting E values in external memory.

An oblivious data structure is a data structure where the memory access patterns reveals no information about the operations performed on it. Such data structures were introduced by Wang et al. [ACM SIGSAC’14] and are intended for situations where one wishes to store the data structure at an untrusted server. One way to obtain an oblivious data structure is simply to run a classic data structure on an oblivious RAM (ORAM). Until very recently, this resulted in an overhead of ω(lg n) for the most natural setting of parameters. Moreover, a recent lower bound for ORAMs by Larsen and Nielsen [CRYPTO’18] show that they always incur an overhead of at least Ω(lg n) if used in a black box manner. To circumvent the ω(lg n) overhead, researchers have instead studied classic data structure problems more directly and have obtained efficient solutions for many such problems such as stacks, queues, deques, priority queues and search trees. However, none of these data structures process operations faster than Θ(lg n), leaving open the question of whether even faster solutions exist. In this paper, we rule out this possibility by proving Ω(lg n) lower bounds for oblivious stacks, queues, deques, priority queues and search trees.

Motivated by the asymmetric read and write costs of emerging non-volatile memory technologies, we study lower bounds for the problems of sorting, permuting and multiplying a sparse matrix by a dense vector in the asymmetric external memory model (AEM). Given an AEM with internal (symmetric) memory of size M, transfers between symmetric and asymmetric memory in blocks of size B and the ratio ω between write and read costs, we show Ω(min (N, ωN/B logω M/B N/B) lower bound for the cost of permuting N input elements. This lower bound also applies to the problem of sorting N elements. This proves that the existing sorting algorithms in the AEM model are optimal to within a constant factor for reasonable ranges of parameters N, M, B, and ω. We also show a lower bound of Ω(min {H, ω H/B logω M/B N/ max{δ ,M}}) for the cost of multiplying an N x N matrix with at most H= δ N non-empty entries by a vector with N elements.

Peer-to-peer systems rely on a scalable overlay network that enables efficient routing between its members. Hypercubic topologies facilitate such operations while each node only needs to connect to a small number of other nodes. In contrast to static communication networks, peer-to-peer networks allow nodes to adapt their neighbor set over time in order to react to join and leave events and failures. This article shows how to maintain such networks in a robust manner. Concretely, we present a distributed and self-stabilizing algorithm that constructs a (slightly extended) skip graph, SKIP + , in polylogarithmic time from any given initial state in which the overlay network is still weakly connected. This is an exponential improvement compared to previously known self-stabilizing algorithms for overlay networks. In addition, our algorithm handles individual joins and leaves locally and efficiently.

The name sparse grids denotes a highly space-efficient, grid-based numerical technique to approximate high-dimensional functions. Although employed in a broad spectrum of applications from different fields, there have only been few tries to use it in real time visualization (e.g. [1]), due to complex data structures and long algorithm runtime. In this work we present a novel approach inspired by principles of I/0-efficient algorithms. Locally applied coefficient permutations lead to improved cache performance and facilitate the use of vector registers for our sparse grid benchmark problem hierarchization. Based on the compact data structure proposed for regular sparse grids in [2], we developed a new algorithm that outperforms existing implementations on modern multi-core systems by a factor of 37 for a grid size of 127 million points. For larger problems the speedup is even increasing, and with execution times below 1 s, sparse grids are well-suited for visualization applications. Furthermore, we point out how a broad class of sparse grid algorithms can benefit from our approach.

Abstract In this article, we study the train classification problem. Train classification basically is the process of rearranging the cars of a train in a specified order, which can be regarded as a special sorting problem. This sorting is done in a special railway installation called a classification yard, and a classification process is described by a classification schedule. In this article, we develop a novel encoding of classification schedules, which allows characterizing train classification methods simply as classes of schedules. Applying this efficient encoding, we achieve a simpler, more precise analysis of well‐known classification methods. Furthermore, we elaborate a valuable optimality condition inherent in our encoding, which we succesfully apply to obtain tight lower bounds for the length of schedules in general and to develop new classification methods. Finally, we present complexity results and algorithms to derive optimal schedules for several real‐world settings. Together, our theoretical results provide a solid foundation for improving train classification in practice. © 2010 Wiley Periodicals, Inc. NETWORKS, 2011

Peer-to-peer systems rely on scalable overlay networks that enable efficient routing between its members. Hypercubic topologies facilitate such operations while each node only needs to connect to a small number of other nodes. In contrast to static communication networks, peer-to-peer networks allow nodes to adapt their neighbor set over time in order to react to join and leave events and failures. This paper shows how to maintain such networks in a robust manner. Concretely, we present a distributed and self-stabilizing algorithm that constructs a (variant of the) skip graph in polylogarithmic time from any initial state in which the overlay network is still weakly connected. This is an exponential improvement compared to previously known self-stabilizing algorithms for overlay networks. In addition, individual joins and leaves are handled locally and require little work.

We analyze the problem of sparse-matrix dense-vector multiplication (SpMV) in the I/O-model. The task of SpMV is to compute y := Ax, where A is a sparse N x N matrix and x and y are vectors. Here, sparsity is expressed by the parameter k that states that A has a total of at most kN nonzeros, i.e., an average number of k nonzeros per column. The extreme choices for parameter k are well studied special cases, namely for k=1 permuting and for k=N dense matrix-vector multiplication.

Abstract Motivation: Tandem mass spectrometry allows for high-throughput identification of complex protein samples. Searching tandem mass spectra against sequence databases is the main analysis method nowadays. Since many peptide variations are possible, including them in the search space seems only logical. However, the search space usually grows exponentially with the number of independent variations and may therefore overwhelm computational resources. Results: We provide fast, cache-efficient search algorithms to screen large peptide search spaces including non-tryptic peptides, whole genomes, dozens of posttranslational modifications, unannotated point mutations and even unannotated splice sites. All these search spaces can be screened simultaneously. By optimizing the cache usage, we achieve a calculation speed that closely approaches the limits of the hardware. At the same time, we control the size of the overall search space by limiting the combinations of variations that can co-occur on the same peptide. Using a hypergeometric scoring scheme, we applied these algorithms to a dataset of 1 420 632 spectra. We were able to identify a considerable number of peptide variations within a modest amount of computing time on standard desktop computers. Availability: PepSplice is available as a C++ application for Linux, Windows and OSX at www.ti.inf.ethz.ch/pw/software/pepsplice/. It is open source under the revised BSD license. Contact: [email protected] or [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.

In this paper we determine the computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.

Given an alphabet $\Sigma$, a (directed) graph G whose edges are weighted and $\Sigma$-labeled, and a formal language $L\subseteq\Sigma^*$, the formal-language-constrained shortest/simple path problem consists of finding a shortest (simple) path p in G complying with the additional constraint that l(p) \in L$. Here l(p) denotes the unique word obtained by concatenating the $\Sigma$-labels of the edges along the path p. The main contributions of this paper include the following: We show that the formal-language-constrained shortest path problem is solvable efficiently in polynomial time when L is restricted to be a context-free language (CFL). When L is specified as a regular language we provide algorithms with improved space and time bounds. In contrast, we show that the problem of finding a simple path between a source and a given destination is NP-hard, even when L is restricted to fixed simple regular languages and to very simple classes of graphs (e.g., complete grids). For the class of treewidth-bounded graphs, we show that (i) the problem of finding a regular-language-constrained simple path between source and destination is solvable in polynomial time and (ii) the extension to finding CFL-constrained simple paths is NP-complete. Our results extend the previous results in [SIAM J. Comput., 24 (1995), pp. 1235--1258; Proceedings of the 76th Annual Meeting of the Transportation Research Board, 1997; and Proceedings of the 9th ACM SIGACT-SIGMOD-SIGART Symposium on Database Systems, 1990, pp. 230--242]. Several additional extensions and applications of our results in the context of transportation problems are presented. For instance, as a corollary of our results, we obtain a polynomial-time algorithm for the best k-similar path problem studied in Proceedings of the 76th Annual Meeting of the Transportation Reasearch Board, 1997]. The previous best algorithm was given by [ Proceedings of the 76th Annual Meeting of the Transportation Research Board, 1997] and takes exponential time in the worst case.