Although many authors have considered how many ternary comparisons it takes to sort a multiset $S$ of size $n$, the best known upper and lower bounds still differ by a term linear in $n$. In this paper we restrict our attention to online stable sorting and prove upper and lower bounds that are within (o (n)) not only of each other but also of the best known upper bound for offline sorting. Specifically, we first prove that if the number of distinct elements (σ= o (n / \log n)), then ((H + 1) n + o (n)) comparisons are sufficient, where $H$ is the entropy of the distribution of the elements in $S$. We then give a simple proof that ((H + 1) n - o (n)) comparisons are necessary in the worst case.
Given a text $T$ of length $n$, we propose a deterministic online algorithm computing the sparse suffix array and the sparse longest common prefix array of $T$ in $O(c \sqrt{\lg n} + m \lg m \lg n \lg^* n)$ time with $O(m)$ words of space under the premise that the space of $T$ is rewritable, where $m \le n$ is the number of suffixes to be sorted (provided online and arbitrarily), and $c$ is the number of characters with $m \le c \le n$ that must be compared for distinguishing the designated suffixes.
In this paper, we present a framework based on a simple data structure and parameterized algorithms for the problems of finding items in an unsorted list of linearly ordered items based on their rank (selection) or value (search). As a side-effect of answering these online selection and search queries, we progressively sort the list. Our algorithms are based on Hoare's Quickselect, and are parameterized based on the pivot selection method. For example, if we choose the pivot as the last item in a subinterval, our framework yields algorithms that will answer q<=n unique selection and/or search …
The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $π$ and $π$ is in $S_n$. $T$ is the operation and the edge set $E_T$ denotes the corresponding generator set. The goal is to sort a given permutation $π$ with $T$. The number of generators of $E_T$ that suffices to sort any $π\in S_n$ constitutes an upper bound. It is an upper bound, on the diameter of the corresponding Cayley graph $Γ$ i.e. $diam(Γ)$. A precise u…
Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most $3 ω(G) - 3$, where $ω(G)$ is the size of the maximum clique in a given graph $G$, and it is the best possible.
Sorting extremely large datasets is a frequently occuring task in practice. These datasets are usually much larger than the computer's main memory; thus external memory sorting algorithms, first introduced by Aggarwal and Vitter (1988), are often used. The complexity of comparison based external memory sorting has been understood for decades by now, however the situation remains elusive if we assume the keys to be sorted are integers. In internal memory, one can sort a set of $n$ integer keys of $Θ(\lg n)$ bits each in $O(n)$ time using the classic Radix Sort algorithm, however in external mem…
Huang et al.~(STOC 2018) introduced the fully online matching problem, a generalization of the classic online bipartite matching problem in that it allows all vertices to arrive online and considers general graphs. They showed that the ranking algorithm by Karp et al.~(STOC 1990) is strictly better than $0.5$-competitive and the problem is strictly harder than the online bipartite matching problem in that no algorithms can be $(1-1/e)$-competitive. This paper pins down two tight competitive ratios of classic algorithms for the fully online matching problem. For the fractional version of the pr…
The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an $n$-vertex spanning tree $T$ and an additional set $L$ of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links $F$ such that every terminal pair that has arrived so far is $2$-edge-connected in $T \cup F$. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network desig…
We present a polynomial time algorithm for online maximization of $k$-submodular maximization. For online (nonmonotone) $k$-submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone $k$-submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as $k$ tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.
We study the online facility location problem with uniform facility costs in the random-order model. Meyerson's algorithm [FOCS'01] is arguably the most natural and simple online algorithm for the problem with several advantages and appealing properties. Its analysis in the random-order model is one of the cornerstones of random-order analysis beyond the secretary problem. Meyerson's algorithm was shown to be (asymptotically) optimal in the standard worst-case adversarial-order model and $8$-competitive in the random order model. While this bound in the random-order model is the long-standing …
We study the online bipartite matching problem, introduced by Karp, Vazirani and Vazirani [1990]. For bipartite graphs with matchings of size $n$, it is known that the Ranking randomized algorithm matches at least $(1 - \frac{1}{e})n$ edges in expectation. It is also known that no online algorithm matches more than $(1 - \frac{1}{e})n + O(1)$ edges in expectation, when the input is chosen from a certain distribution that we refer to as $D_n$. This upper bound also applies to fractional matchings. We review the known proofs for this last statement. In passing we observe that the $O(1)$ additive…
The online Dial-a-Ride problem is a fundamental online problem in a metric space, where transportation requests appear over time and may be served in any order by a single server with unit speed. Restricted to the real line, online Dial-a-Ride captures natural problems like controlling a personal elevator. Tight results in terms of competitive ratios are known for the general setting and for online TSP on the line (where source and target of each request coincide). In contrast, online Dial-a-Ride on the line has resisted tight analysis so far, even though it is a very natural online problem. W…
We study the $b$-matching problem in bipartite graphs $G=(S,R,E)$. Each vertex $s\in S$ is a server with individual capacity $b_s$. The vertices $r\in R$ are requests that arrive online and must be assigned instantly to an eligible server. The goal is to maximize the size of the constructed matching. We assume that $G$ is a $(k,d)$-graph~\cite{NW}, where $k$ specifies a lower bound on the degree of each server and $d$ is an upper bound on the degree of each request. This setting models matching problems in timely applications. We present tight upper and lower bounds on the performance of deter…
We show tight bounds for online Hamming distance computation in the cell-probe model with word size w. The task is to output the Hamming distance between a fixed string of length n and the last n symbols of a stream. We give a lower bound of Omega((d/w)*log n) time on average per output, where d is the number of bits needed to represent an input symbol. We argue that this bound is tight within the model. The lower bound holds under randomisation and amortisation.
Vizing's celebrated theorem asserts that any graph of maximum degree $Δ$ admits an edge coloring using at most $Δ+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2Δ-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $Δ=O(\log n)$, and they conjectured the existence of online algorithms using $Δ(1+o(1))$ colors for $Δ=ω(\log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS'03 a…
Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and $L_r$ norms of loads are among the most popular objectives considered. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J of Comp. 2006), vector scheduling is a generali…
Communication is a major factor determining the performance of algorithms on current computing systems; it is therefore valuable to provide tight lower bounds on the communication complexity of computations. This paper presents a lower bound technique for the communication complexity in the bulk-synchronous parallel (BSP) model of a given class of DAG computations. The derived bound is expressed in terms of the switching potential of a DAG, that is, the number of permutations that the DAG can realize when viewed as a switching network. The proposed technique yields tight lower bounds for the f…
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. …
We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $\ell$ servers, where servers have capacity $k$ and we assume that the graph has $\ell \cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $\operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $\varepsilon k$ for constant $\varepsilon>0$. Whenever $\op…
Motivated by the dynamic assortment offerings and item pricings occurring in e-commerce, we study a general problem of allocating finite inventories to heterogeneous customers arriving sequentially. We analyze this problem under the framework of competitive analysis, where the sequence of customers is unknown and does not necessarily follow any pattern. Previous work in this area, studying online matching, advertising, and assortment problems, has focused on the case where each item can only be sold at a single price, resulting in algorithms which achieve the best-possible competitive ratio of…
The following online bin packing problem is considered: Items with integer sizes are given and variable sized bins arrive online. A bin must be used if there is still an item remaining which fits in it when the bin arrives. The goal is to minimize the total size of all the bins used. Previously, a lower bound of 5/4 on the competitive ratio of this problem was achieved using jobs of size S and 2S-1. For these item sizes and maximum bin size 4S-3, we obtain asymptotically matching upper and lower bounds, which vary depending on the ratio of the number of small jobs to the number of large jobs.
Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy Sorting problem, where each comparison result is flipped independently with probability $p$ for some fixed $p\in (0, \frac 12)$. As our main result, we show that $$(1\pm o(1)) \left( \frac{1}{I(p)} + \frac{1}{(1-2p) \log_2 \left(\frac{1-p}p\right)} \right) n\log_2 n$$ noisy comparisons are both necessary and sufficient to sort $n$ elements with error probabili…
The car-sharing problem, proposed by Luo, Erlebach and Xu in 2018, mainly focuses on an online model in which there are two locations: 0 and 1, and $k$ total cars. Each request which specifies its pick-up time and pick-up location (among 0 and 1, and the other is the drop-off location) is released in each stage a fixed amount of time before its specified start (i.e. pick-up) time. The time between the booking (i.e. released) time and the start time is enough to move empty cars between 0 and 1 for relocation if they are not used in that stage. The model, called $k$S2L-F, assumes that requests i…
We show tight bounds for both online integer multiplication and convolution in the cell-probe model with word size w. For the multiplication problem, one pair of digits, each from one of two n digit numbers that are to be multiplied, is given as input at step i. The online algorithm outputs a single new digit from the product of the numbers before step i+1. We give a Theta((d/w)*log n) bound on average per output digit for this problem where 2^d is the maximum value of a digit. In the convolution problem, we are given a fixed vector V of length n and we consider a stream in which numbers arriv…
In the unit-cost comparison model, a black box takes an input two items and outputs the result of the comparison. Problems like sorting and searching have been studied in this model, and it has been generalized to include the concept of priced information, where different pairs of items (say database records) have different comparison costs. These comparison costs can be arbitrary (in which case no algorithm can be close to optimal (Charikar et al. STOC 2000)), structured (for example, the comparison cost may depend on the length of the databases (Gupta et al. FOCS 2001)), or stochastic (Angel…
We consider the file maintenance problem (also called the online labeling problem) in which n integer items from the set {1,...,r} are to be stored in an array of size m >= n. The items are presented sequentially in an arbitrary order, and must be stored in the array in sorted order (but not necessarily in consecutive locations in the array). Each new item must be stored in the array before the next item is received. If r<=m then we can simply store item j in location j but if r>m then we may have to shift the location of stored items to make space for a newly arrived item. The algorithm is ch…
In this paper we study the mincut problem in the online setting. We consider two distinct models: A) competitive analysis and B) regret analysis. In the competitive setting we consider the vertex arrival model; whenever a new vertex arrives it's neighborhood with respect to the set of known vertices is revealed. An online algorithm must make an irrevocable decision to determine the side of the cut that the vertex must belong to in order to minimize the size of the final cut. Various models are considered. 1) For classical and advice models we give tight bounds on the competitive ratio of deter…
In this paper, we exploit linear programming duality in the online setting (i.e., where input arrives on the fly) from the unique perspective of designing lower bounds on the competitive ratio. In particular, we provide a general technique for obtaining online deterministic and randomized lower bounds (i.e., hardness results) on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight lower bounds for three diverse online problems. The three problems we show tight lower bounds for are the Vector Bin Packing problem, Ad-auctions (and v…
A permutation $π$ over alphabet $Σ= {1,2,3,\ldots,n}$, is a sequence where every element $x$ in $Σ$ occurs exactly once. $S_n$ is the symmetric group consisting of all permutations of length $n$ defined over $Σ$. $I_n$ = $(1, 2, 3,\ldots, n)$ and $R_n =(n, n-1, n-2,\ldots, 2, 1)$ are identity (i.e. sorted) and reverse permutations respectively. An operation, that we call as an $LRE$ operation, has been defined in OEIS with identity A186752. This operation is constituted by three generators: left-rotation, right-rotation and transposition(1,2). We call transposition(1,2) that swaps the two left…
In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piece-by-piece, our model presents one of the players, Bob, his input piece-by-piece, and has the players Alice and Bob cooperate to compute a result each time before the next piece is revealed to Bob. This model has a closer and more natural correspondence to dynamic data structures than classic communication models do, and hence presents a new perspective on data structures. We first present a tight lower bound for the online set intersection problem in the online c…
