In this paper, we present a probabilistic self-balancing dictionary data structure for massive data sets, and prove expected amortized I/O-optimal bounds on the dictionary operations. We show how to use the structure as an I/O-optimal priority queue. The data structure, which we call as the random buffer tree, abstracts the properties of the random treap and the buffer tree and has the same expected I/O-bounds as the buffer tree.
In the bounded delay buffer management problem unit size packets arrive online to be sent over a network link. The objective is to maximize the total weight of packets sent before their deadline. In this paper we are interested in the two-valued variant of the problem, where every packet has either low (1) or high priority weight ($α$ > 1). We show that its randomized competitive ratio against an oblivious adversary is 1 + ($α$ -- 1)/($α$ 2 + $α$).
We give a memoryless scale-invariant randomized algorithm for the Buffer Management with Bounded Delay problem that is e/(e-1)-competitive against an adaptive adversary, together with better performance guarantees for many restricted variants, including the s-bounded instances. In particular, our algorithm attains the optimum competitive ratio of 4/3 on 2-bounded instances. Both the algorithm and its analysis are applicable to a more general problem, called Collecting Items, in which only the relative order between packets' deadlines is known. Our algorithm is the optimal randomized memoryless…
We give an $O(\log\log k)$-competitive randomized online algorithm for reordering buffer management, where $k$ is the buffer size. Our bound matches the lower bound of Adamaszek et al. (STOC 2011). Our algorithm has two stages which are executed online in parallel. The first stage computes deterministically a feasible fractional solution to an LP relaxation for reordering buffer management. The second stage "rounds" using randomness the fractional solution. The first stage is based on the online primal-dual schema, combined with a dual fitting argument. As multiplicative weights steps and dual…
We present deterministic and randomized algorithms for the problem of online packet routing in grids in the competitive network throughput model \cite{AKOR}. In this model the network has nodes with bounded buffers and bounded link capacities. The goal in this model is to maximize the throughput, i.e., the number of delivered packets. Our deterministic algorithm is the first online algorithm with an $O\left(\log^{O(1)}(n)\right)$ competitive ratio for uni-directional grids (where $n$ denotes the size of the network). The deterministic online algorithm is centralized and handles packets with de…
In runtime verification, the central problem is to decide if a given program execution violates a given property. In online runtime verification, a monitor observes a program's execution as it happens. If the program being observed has hard real-time constraints, then the monitor inherits them. In the presence of hard real-time constraints it becomes a challenge to maintain enough information to produce error traces, should a property violation be observed. In this paper we introduce a data structure, called tree buffer, that solves this problem in the context of automata-based monitors: If th…
We study the problem of buffer management in QoS-enabled network switches in the bounded delay model where each packet is associated with a weight and a deadline. We consider the more realistic situation where the network switch has a finite buffer size. A 9.82-competitive algorithm is known for the case of multiple buffers (Azar and Levy, SWAT'06). Recently, for the case of a single buffer, a 3-competitive deterministic algorithm and a 2.618-competitive randomized algorithm was known (Li, INFOCOM'09). In this paper we give a simple deterministic 2-competitive algorithm for the case of a singl…
In the online packet buffering problem (also known as the unweighted FIFO variant of buffer management), we focus on a single network packet switching device with several input ports and one output port. This device forwards unit-size, unit-value packets from input ports to the output port. Buffers attached to input ports may accumulate incoming packets for later transmission; if they cannot accommodate all incoming packets, their excess is lost. A packet buffering algorithm has to choose from which buffers to transmit packets in order to minimize the number of lost packets and thus maximize t…
We consider algorithms to schedule packets with values and deadlines in a size-bounded buffer. At any time, the buffer can store at most B packets. Packets arrive over time. Each packet has a non-negative value and an integer deadline. In each time step, at most one packet can be sent. Packets can be dropped at any time before they are sent. The objective is to maximize the total value gained by delivering packets no later than their respective deadlines. This model generalizes the well-studied bounded-delay model (Hajek. CISS 2001. Kesselman et al. STOC 2001). We first provide an optimal offl…
This paper presents a new kind of self-balancing ternary search trie that uses a randomized balancing strategy adapted from Aragon and Seidel's randomized binary search trees ("treaps"). After any sequence of insertions and deletions of strings, the tree looks like a ternary search trie built by inserting strings in random order. As a result, the time cost of searching, inserting, or deleting a string of length k in a tree with n strings is at most O(k + log n) with high probability.
We present a randomized algorithm for reconstructing directed rooted trees of $n$ nodes and node degree at most $d$, by asking at most $O(dn\log^2 n)$ path queries. Each path query takes as input an origin node and a target node, and answers whether there is a directed path from the origin to the target. Regarding lower bounds, we show that any randomized algorithm requires at least $Ω(n\log n)$ queries, while any deterministic algorithm requires at least $Ω(dn)$ queries. Additionally, we present a $O(dn\log^3 n)$ randomized algorithm for noisy queries, and a $O(dn\log^2 n)$ randomized algorit…
Radboud University Nijmegen（拉德堡德大学（奈梅亨））；University of Copenhagen（哥本哈根大学）
A buffer k-d tree is a k-d tree variant for massively-parallel nearest neighbor search. While providing valuable speed-ups on modern many-core devices in case both a large number of reference and query points are given, buffer k-d trees are limited by the amount of points that can fit on a single device. In this work, we show how to modify the original data structure and the associated workflow to make the overall approach capable of dealing with massive data sets. We further provide a simple yet efficient way of using multiple devices given in a single workstation. The applicability of the mo…
The k-d tree was one of the first spatial data structures proposed for nearest neighbor search. Its efficacy is diminished in high-dimensional spaces, but several variants, with randomization and overlapping cells, have proved to be successful in practice. We analyze three such schemes. We show that the probability that they fail to find the nearest neighbor, for any data set and any query point, is directly related to a simple potential function that captures the difficulty of the point configuration. We then bound this potential function in two situations of interest: the first, when data co…
Many network design problems deal with the design of low-cost networks that are resilient to the failure of their elements, such as nodes or links. One such problem is Connectivity Augmentation, where the goal is to cheaply increase the connectivity of a network from a value k to k+1. The most studied setting focuses on edge-connectivity, which reduces to k=2, called Cactus Augmentation. From an approximation perspective, Byrka, Grandoni, and Jabal Ameli (2020) were the first to break the 2-approximation barrier for this problem, by exploiting a connection to the Steiner Tree problem, and by t…
We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of $(1/2-δ) \cdot 2.57143^h$ for the two-sided-error randomized decision tree complexity of evaluating height $h$ formulae with error $δ\in [0,1/2)$. This improves the lower bound of $(1-2δ)(7/3)^h$ given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of $(1-2δ) \cdot 2.55^h$ given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most $(1.007) \cdot 2.64944^h$. The previous best k…
LSM-tree has been widely used in cloud computing systems by Google, Facebook, and Amazon, to achieve high performance for write-intensive workloads. However, in LSM-tree, random key-value queries can experience long latency and low throughput due to the interference from the compaction, a basic operation in the algorithm, to caching. LSM-tree relies on frequent compaction operations to merge data into a sorted structure. After a compaction, the original data are reorganized and written to other locations on the disk. As a result, the cached data are invalidated since their referencing addresse…
We introduce a new metric of match, called Cartesian tree matching, which means that two strings match if they have the same Cartesian trees. Based on Cartesian tree matching, we define single pattern matching for a text of length n and a pattern of length m, and multiple pattern matching for a text of length n and k patterns of total length m. We present an O(n+m) time algorithm for single pattern matching, and an O((n+m) log k) deterministic time or O(n+m) randomized time algorithm for multiple pattern matching. We also define an index data structure called Cartesian suffix tree, and present…
We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models. In the {\em Online Preemptive} model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algo…
In this paper, we settle the randomized $k$-sever conjecture for the following metric spaces: line, circle, Hierarchically well-separated tree (HST). Specially, we show that there are $O(\log k)$-competitive randomized $k$-sever algorithms for above metric spaces. For any general metric space with $n$ points, we show that there is an $O( \log k \log n)$-competitive randomized $k$-sever algorithm.
We consider the following fundamental routing problem. An adversary inputs packets arbitrarily at sources, each packet with an arbitrary destination. Traffic is constrained by link capacities and buffer sizes, and packets may be dropped at any time. The goal of the routing algorithm is to maximize throughput, i.e., route as many packets as possible to their destination. Our main result is an $O\left(\log n\right)$-competitive deterministic algorithm for an $n$-node line network (i.e., $1$-dimensional grid), requiring only that buffers can store at least $5$ packets, and that links can deliver …
We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in $O(m\logα(m,n))$ time, where $α$ is the inverse-Ackermann function. This improves upon a long standing bound of $O(mα(m,n))$ established by Tarjan. Our algorithms are based on an efficient split-findmin data structure, which maintains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our split-findmin algorithm is the first with superlinear but sub-inverse-Ackermann complexity. We also give a reduction fro…
We show an algorithm that, given an $n$-vertex graph $G$ and a parameter $k$, in time $2^{O(k \log k)} n^{O(1)}$ finds a tree decomposition of $G$ with the following properties: * every adhesion of the tree decomposition is of size at most $k$, and * every bag of the tree decomposition is $(i,i)$-unbreakable in $G$ for every $1 \leq i \leq k$. Here, a set $X \subseteq V(G)$ is $(a,b)$-unbreakable in $G$ if for every separation $(A,B)$ of order at most $b$ in $G$, we have $|A \cap X| \leq a$ or $|B \cap X| \leq a$. The resulting tree decomposition has arguably best possible adhesion size bounds…
We consider network design problems with deadline or delay. All previous results for these models are based on randomized embedding of the graph into a tree (HST) and then solving the problem on this tree. We show that this is not necessary. In particular, we design a deterministic framework for these problems which is not based on embedding. This enables us to provide deterministic $\text{poly-log}(n)$-competitive algorithms for Steiner tree, generalized Steiner tree, node weighted Steiner tree, (non-uniform) facility location and directed Steiner tree with deadlines or with delay (where $n$ …
We consider parametrized versions of metrical task systems and metrical service systems, two fundamental models of online computing, where the constrained parameter is the number of possible distinct requests $m$. Such parametrization occurs naturally in a wide range of applications. Striking examples are certain power management problems, which are modeled as metrical task systems with $m=2$. We characterize the competitive ratio in terms of the parameter $m$ for both deterministic and randomized algorithms on hierarchically separated trees. Our findings uncover a rich and unexpected picture …
We revisit the method of Kirschenhofer, Prodinger and Tichy to calculate the moments of comparisons used by the quick sort algorithm. We reemphasize that this approach helps in calculating these quantities with less computation. We also point out that as observed by Knuth this method also gives moments for total path length of a binary search tree built over a random set of n keys.
The $k$-Steiner-2NCS problem is as follows: Given a constant $k$, and an undirected connected graph $G = (V,E)$, non-negative costs $c$ on $E$, and a partition $(T, V-T)$ of $V$ into a set of terminals, $T$, and a set of non-terminals (or, Steiner nodes), where $|T|=k$, find a minimum-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized PTAS for the weighted problem. We obtain similar results for the $k$-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a…
Uniquely represented data structures represent each logical state with a unique storage state. We study the problem of maintaining a dynamic set of $n$ keys from a totally ordered universe in this context. We introduce a two-layer data structure called $(α,\varepsilon)$-Randomized Block Search Tree (RBST) that is uniquely represented and suitable for external memory. Though RBSTs naturally generalize the well-known binary Treaps, several new ideas are needed to analyze the {\em expected} search, update, and storage, efficiency in terms of block-reads, block-writes, and blocks stored. We prove …
We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, $k$, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider $k=2$, and for trees, we consider any $k \geq 2$. We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms, on paths as well as trees…
We extend randomized jumplists introduced by Brönnimann et al. (STACS 2003) to choose jump-pointer targets as median of a small sample for better search costs, and present randomized algorithms with expected $O(\log n)$ time complexity that maintain the probability distribution of jump pointers upon insertions and deletions. We analyze the expected costs to search, insert and delete a random element, and we show that omitting jump pointers in small sublists hardly affects search costs, but significantly reduces the memory consumption. We use a bijection between jumplists and "dangling-min BSTs…
The cup game on $n$ cups is a multi-step game with two players, a filler and an emptier. At each step, the filler distributes $1$ unit of water among the cups, and then the emptier selects a single cup to remove (up to) $1$ unit of water from. There are several objective functions that the emptier might wish to minimize. One of the strongest guarantees would be to minimize tail size, which is defined to be the number of cups with fill $2$ or greater. A simple lower-bound construction shows that the optimal tail size for deterministic emptying algorithms is $Θ(n)$, however. We present a simple …
We define a search problem on trees that closely captures the backtracking behavior of all current practical graph isomorphism algorithms. Given two trees with colored leaves, the goal is to find two leaves of matching color, one in each of the trees. The trees are subject to an invariance property which promises that for every pair of leaves of equal color there must be a symmetry (or an isomorphism) that maps one leaf to the other. We describe a randomized algorithm with errors for which the number of visited leaves is quasilinear in the square root of the size of the smaller of the two tree…
We investigate the power of randomized algorithms for the maximum cardinality matching (MCM) and the maximum weight matching (MWM) problems in the online preemptive model. In this model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded. The complexity of the problem is settled for deterministic algorithms. Almost nothing is known for randomized algorithms. A lower bound of $1.693$ is known for MCM with a triv…
In this paper we improve the approximation ratio for the problem of scheduling packets on line networks with bounded buffers, where the aim is that of maximizing the throughput. Each node in the network has a local buffer of bounded size $B$, and each edge (or link) can transmit a limited number, $c$, of packets in every time unit. The input to the problem consists of a set of packet requests, each defined by a source node, a destination node, and a release time. We denote by $n$ the size of the network. A solution for this problem is a schedule that delivers (some of the) packets to their des…
We consider the vehicle routing problem with stochastic demands (VRPSD) on tree structured networks with a single depot. The problem we are concerned with in this paper is to find a set of tours for the vehicle with minimum expected length. Every tour begins at the depot, visits a subset of customers and returns to the depot without violating the capacity constraint. Randomized approximation algorithm achieving approximation guarantees of 2 for split-delivery VRPSD, and 3 for un-split delivery VRPSD are obtained.
We consider metrical task systems on tree metrics, and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O…
In this paper, we revisit the classic problem of run generation. Run generation is the first phase of external-memory sorting, where the objective is to scan through the data, reorder elements using a small buffer of size M , and output runs (contiguously sorted chunks of elements) that are as long as possible. We develop algorithms for minimizing the total number of runs (or equivalently, maximizing the average run length) when the runs are allowed to be sorted or reverse sorted. We study the problem in the online setting, both with and without resource augmentation, and in the offline settin…
An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with $k$ vertices. The algorithms are of runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for deciding whether an input digraph contains a spanning out-tree with at least $k$ internal vertices. This answers i…
This paper studies a fundamental algorithmic problem related to the design of demand-aware networks: networks whose topologies adjust toward the traffic patterns they serve, in an online manner. The goal is to strike a tradeoff between the benefits of such adjustments (shorter routes) and their costs (reconfigurations). In particular, we consider the problem of designing a self-adjusting tree network which serves single-source, multi-destination communication. The problem has interesting connections to self-adjusting datastructures. We present two constant-competitive online algorithms for thi…
Tree path minimum query problem is a fundamental problem while processing trees, and is used widely in minimum spanning tree verification and randomized minimum spanning tree algorithms. In this paper, we study the possibility of building an oracle in advance, which is able to answer the queries efficiently. We present an algorithm based on Boruvka trees. Our algorithm is the first to achieve a near-optimal bound on query time, while matching the currently optimal trade-off between construction time and the number of comparisons required at query. Particularly, in order to answer each query wi…
Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast $Δ$-coloring of trees requires random bits: Building on the recent lower b…
The Gomory-Hu tree or cut tree (Gomory and Hu, 1961) is a classic data structure for reporting $(s,t)$ mincuts (and by duality, the values of $(s,t)$ maxflows) for all pairs of vertices $s$ and $t$ in an undirected graph. Gomory and Hu showed that it can be computed using $n-1$ exact maxflow computations. Surprisingly, this remains the best algorithm for Gomory-Hu trees more than 50 years later, *even for approximate mincuts*. In this paper, we break this longstanding barrier and give an algorithm for computing a $(1+ε)$-approximate Gomory-Hu tree using $\text{polylog}(n)$ maxflow computations…
We revisit self-adjusting external memory tree data structures, which combine the optimal (and practical) worst-case I/O performances of B-trees, while adapting to the online distribution of queries. Our approach is analogous to undergoing efforts in the BST model, where Tango Trees (Demaine et al. 2007) were shown to be $O(\log\log N)$-competitive with the runtime of the best offline binary search tree on every sequence of searches. Here we formalize the B-Tree model as a natural generalization of the BST model. We prove lower bounds for the B-Tree model, and introduce a B-Tree model data str…
We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. …
Recently Kubica et al. (Inf. Process. Let., 2013) and Kim et al. (submitted to Theor. Comp. Sci.) introduced order-preserving pattern matching. In this problem we are looking for consecutive substrings of the text that have the same "shape" as a given pattern. These results include a linear-time order-preserving pattern matching algorithm for polynomially-bounded alphabet and an extension of this result to pattern matching with multiple patterns. We make one step forward in the analysis and give an $O(\frac{n\log{n}}{\log\log{n}})$ time randomized algorithm constructing suffix trees in the ord…
We consider the model of random trees introduced by Devroye [SIAM J. Comput. 28 (1999) 409-432]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths of the items) is a natural measure of the efficiency of the algorithm/data structure. Using renewal theory, we prove convergence in distribution of the total path length toward a distribution characterized uniquely by a fixed point equation. Our result covers, using a unified approach, many data str…
This paper analyzes the cache miss cost of algorithms when scheduled using randomized work stealing (RWS) in a parallel environment, taking into account the effects of false sharing. First, prior analyses (due to Acar et al.) are extended to incorporate false sharing. However, to control the possible delays due to false sharing, some restrictions on the algorithms seem necessary. Accordingly, the class of Hierarchical Tree algorithms is introduced and their performance analyzed. In addition, the paper analyzes the performance of a subclass of the Hierarchical Tree Algorithms, called HBP algori…
The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special c…
This paper presents a randomized Las Vegas distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in $\tilde{O}(D + \sqrt{n})$ time and exchanges $\tilde{O}(m)$ messages (both with high probability), where $n$ is the number of nodes of the network, $D$ is the diameter, and $m$ is the number of edges. This is the first distributed MST algorithm that matches \emph{simultaneously} the time lower bound of $\tildeΩ(D + \sqrt{n})$ [Elkin, SIAM J. Comput. 2006] and the messa…
We propose a novel randomized linear programming algorithm for approximating the optimal policy of the discounted Markov decision problem. By leveraging the value-policy duality and binary-tree data structures, the algorithm adaptively samples state-action-state transitions and makes exponentiated primal-dual updates. We show that it finds an $ε$-optimal policy using nearly-linear run time in the worst case. When the Markov decision process is ergodic and specified in some special data formats, the algorithm finds an $ε$-optimal policy using run time linear in the total number of state-action …
Kvasir Ltd.（Kvasir有限公司）；University of Helsinki（赫尔辛基大学）；Helsinki Institute for Information Technology (HIIT)（赫尔辛基信息技术研究所（HIIT））
Approximate nearest neighbor algorithms are used to speed up nearest neighbor search in a wide array of applications. However, current indexing methods feature several hyperparameters that need to be tuned to reach an acceptable accuracy--speed trade-off. A grid search in the parameter space is often impractically slow due to a time-consuming index-building procedure. Therefore, we propose an algorithm for automatically tuning the hyperparameters of indexing methods based on randomized space-partitioning trees. In particular, we present results using randomized k-d trees, random projection tre…
This article studies the Minimum Spanning Tree Problem under Explorable Uncertainty as well as a related vertex uncertainty version of the problem. We particularly consider special instance types, including cactus graphs, for which we provide randomized algorithms. We introduce the problem of finding a minimum weight spanning star under uncertainty for which we show that no algorithm can achieve constant competitive ratio.
We present a uniform self-stabilizing algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of an arbitrary positively real-weighted graph. Our algorithm consists in two stages of stabilizing protocols. The first stage is a uniform randomized stabilizing {\em unique naming} protocol, and the second stage is a stabilizing {\em MDST} protocol, designed as a {\em fair composition} of Merlin--Segall's stabilizing protocol and a distributed deterministic stabilizing protocol solving the (MDST) problem. The resulting randomized distributed algorithm presented…
