# David G. Harris's research while affiliated with University of Maryland, College Park and other places

## Publications (77)

Preprint
We describe a new algorithm for vertex cover with runtime $O^*(1.25400^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential functio...
Article
Full-text available
A classic conjecture of Füredi, Kahn, and Seymour (1993) states that any hypergraph with non‐negative edge weights w(e)$$w(e)$$ has a matching M$$M$$ such that ∑e∈M(|e|−1+1/|e|)w(e)≥w∗$${\sum}_{e\in M}\left(|e|-1+1/|e|\right)\kern0.3em w(e)\ge {w}^{\ast }$$, where w∗$${w}^{\ast }$$ is the value of an optimum fractional matching. We show the...
Article
In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent resul...
Article
An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existenc...
Preprint
Karppa & Kaski (2019) proposed a novel type of "broken" or "opportunistic" multiplication algorithm, based on a variant of Strassen's alkgorithm, and used this to develop new algorithms for Boolean matrix multiplication, among other tasks. For instance, their algorithm can compute Boolean matrix multiplication in $O(n^{\log_2(6 + 6/7)} \log n) = O(... Article Full-text available We consider column‐sparse covering integer programs, a generalization of set cover. We develop a new rounding scheme based on the partial resampling variant of the Lovász Local Lemma developed by Harris and Srinivasan. This achieves an approximation ratio of , where amin is the minimum covering constraint and is the maximum ℓ1‐norm of any column of... Preprint A classic conjecture of F\"{u}redi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights$w(e)$, there exists a matching$M$such that$\sum_{e \in M} (|e|-1+1/|e|)\, w(e) \geq w^*$, where$w^*$is the value of an optimum fractional matching. We show the conjecture is true for rank-3 hypergraphs, and is achieved b... Preprint Given a graph$G=(V,E)$with arboricity$\alpha$, we study the problem of decomposing the edges of$G$into$(1+\epsilon)\alpha$disjoint forests in the distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm that computes a$(2+\epsilon)\alpha$-forest decomposition using$O(\frac{\log n}{\epsilon})$rounds. Ghaffari and Su [... Preprint The Lov\'{a}sz Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such... Preprint Full-text available The main focus of this paper is radius-based clustering problems in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsac... Article Full-text available The Lovász local lemma (LLL) is a probabilistic tool to generate combinatorial structures with good “local” properties. The “LLL‐distribution” further shows that these structures have good global properties in expectation. The seminal algorithm of Moser and Tardos turned the simplest, variable‐based form of the LLL into an efficient algorithm; this... Preprint We consider \emph{Gibbs distributions}, which are families of probability distributions over a discrete space$\Omega$with probability mass function of the form$\mu^\Omega_\beta(x) \propto e^{\beta H(x)}$for$\beta$in an interval$[\beta_{\min}, \beta_{\max}]$and$H(x) \in \{0 \} \cup [1, n]$. The \emph{partition function} is the normalization... Preprint In the$k$-cut problem, we want to find the smallest set of edges whose deletion breaks a given (multi)graph into$k$connected components. Algorithms of Karger & Stein and Thorup showed how to find such a minimum$k$-cut in time approximately$O(n^{2k})$. The best lower bounds come from conjectures about the solvability of the$k$-clique problem a... Article The Lovász Local Lemma (LLL) shows that, for a collection of “bad” events B in a probability space that are not too likely and not too interdependent, there is a positive probability that no events in B occur. Moser and Tardos (2010) gave sequential and parallel algorithms that transformed most applications of the variable-assignment LLL into effic... Article Full-text available Preprint The Lovasz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection$\mathcal B$of "bad" events which are mostly independent and have low probability. In its simplest "symmetric" form, it asserts that whenever a bad-event has probability$p$and affects at most$d$bad-e... Article The resampling algorithm of Moser and Tardos is a powerful approach to develop constructive versions of the Lovász Local Lemma. We generalize this to partial resampling: When a bad event holds, we resample an appropriately random subset of the variables that define this event rather than the entire set, as in Moser and Tardos. This is particularly... Article Full-text available Motivated by the Erdos-Faber-Lovasz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizing's Theorem for graphs. Article A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a long-standing algorithmic challenge, dating back nearly 30 years to a survey of Karp and Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame and Luby (1990) and Kelsen (1992), running in time roughly (log n... Preprint An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph with a given vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT... Preprint We consider the family of \emph{Gibbs distributions}, which are probability distributions over a discrete space$\Omega$given by$\mu^\Omega_\beta(x)=\frac{e^{\beta H(x)}}{Z(\beta)}$. Here$H:\Omega\rightarrow \{0,1,\ldots,n\}$is a fixed function (called a {\em Hamiltonian}),$\beta$is the parameter of the distribution, and$Z(\beta)=\sum_{x\in\...
Article
Full-text available
Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low independence. A series of papers, beginning with work by Luby (1988), showed that in many cases these techniques can be combined to give deterministic parallel (NC) algorithms for a variety of combinatorial o...
Preprint
The study of graph problems in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Czumaj et al. [STOC'18], Assadi et al. [SODA'19], and Ghaffari et al. [PODC'18], gave algorithms for finding a $1+\varepsilon$ approximate maximum matching in $O(\log \log n)$ rounds using $\widetilde{O}(n)$ memory per machine....
Preprint
We consider stochastic settings for clustering, and develop provably-good (approximation) algorithms for a number of these notions. These algorithms allow one to obtain better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including providing fairer clustering and cluster...
Preprint
We describe randomized and deterministic approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in hypergraphs. For a rank-$r$ hypergraph, our algorithm generates a matching within an $O(r)$ factor of the maximum weight matching. The runtime is $\tilde O(\log r \log \Delta)$ for the random...
Preprint
The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ other bad-eve...
Preprint
Full-text available
A $k$-truss is a relaxation of a $k$-clique developed by Cohen (2005), specifically a connected graph in which every edge is incident to at least $k$ triangles. The $k$-truss has proved to be a useful tool in identifying cohesive networks in real-world graphs such as social networks. Despite its simplicity and its utility, the combinatorial and alg...
Article
Full-text available
We give a new randomized distributed algorithm for (Δ +1)-coloring in the LOCAL model, running in O(&sqrt; log Δ)+ 2O(&sqrt;log log n) rounds in a graph of maximum degree Δ. This implies that the (Δ +1)-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of Ω(min(&sqrt;&fra...
Article
While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this b...
Article
The Lovász Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser and Tardos (2010) provides an efficient randomized algorithm to implement it. This can be parallelized to give an algorithm that uses polynomially many processors and runs in O(log³n) time on an EREW PRAM, stemming...
Article
The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing randomized LOCAL algorithms and transforming them into efficient deterministic LOCAL algorithm....
Article
Full-text available
In this paper, we give tight approximation algorithms for the $k$-center and matroid center problems with outliers. Unfairness arises naturally in this setting: certain clients could always be considered as outliers. To address this issue, we introduce a lottery model in which each client $j$ is allowed to submit a parameter $p_j \in [0,1]$ and we...
Article
Full-text available
Dependent rounding is a popular technique in designing approximation algorithms. It allows us to randomly round a fractional solution $x$ to an integral vector $X$ such that $E[X] = x$, all $X_i$'s are ("cylindrically") negatively correlated, and the cardinality constraint $\sum_i X_i = \sum_i x_i$ holds with probability one. One can also essential...
Article
Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial approximation scheme to estimate the probability that a graph G becomes disconnected, given that its edges are removed independently with probability p. This algorithm runs in n5+o(1)ϵ−3 time to obtain an estimate within relative error ϵ. We improve this run-time through...
Article
We consider an issue of much current concern: could fairness, an issue that is already difficult to guarantee, worsen when algorithms run much of our lives? We consider this in the context of resource-allocation problems; we show that algorithms can guarantee certain types of fairness in a verifiable way. Our conceptual contribution is a simple app...
Article
Moser and Tardos have developed a powerful algorithmic approach (henceforth MT) to the Lovász Local Lemma (LLL); the basic operation done in MT and its variants is a search for “bad” events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables that arise during i...
Article
The Lov\'{a}sz Local Lemma (LLL) shows that, if a set of collection of "bad" events $\mathcal B$ in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in $\mathcal B$ occur. Moser \& Tardos (2010) gave sequential and parallel algorithms which transformed most applications of th...
Article
The Lopsided Lovász Local Lemma (LLLL) is a powerful probabilistic principle that has been used in a variety of combinatorial constructions. While this principle began as a general statement about probability spaces, it has recently been transformed into a variety of polynomial-time algorithms. The resampling algorithm of Moser and Tardos [2010] is...
Article
While there has been significant progress on algorithmic aspects of the Lov\'{a}sz Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve th...
Article
Set Cover is a classic NP-hard problem; as shown by Slav\'{i}k (1997) the greedy algorithm gives an approximation ratio of $\ln n - \ln \ln n + \Theta(1)$. A series of works by Lund \& Yannakakis (1994), Feige (1998), Moshkovitz (2015) have shown that, under the assumption $P \neq NP$, it is impossible to obtain a polynomial-time approximation rati...
Article
The Lovasz Local Lemma (LLL) is a probabilistic principle which has been used in a variety of combinatorial constructions to show the existence of structures that have good "local" properties. Using the "LLL-distribution", one can show that the resulting combinatorial structures have good global properties. Nearly all applications of the LLL in com...
Article
Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low (almost-) independence. A series of papers, beginning with work by Luby (1988) and continuing with Berger & Rompel (1991) and Chari et al. (1994), showed that these techniques can be combined to give determin...
Article
The Lopsided Lov\'{a}sz Local Lemma (LLLL) is a probabilistic tool which is a cornerstone of the probabilistic method of combinatorics, which shows that it is possible to avoid a collection of "bad" events as long as their probabilities and interdependencies are sufficiently small. The strongest possible criterion that can be stated in these terms...
Article
Finding a maximal independent set in hypergraphs has been a long-standing algorithmic challenge. The best parallel algorithm for hypergraphs of rank $r$ was developed by Beame \& Luby (1990) and Kelsen (1992), running in time roughly $(\log n)^{r!}$. This is in RNC for fixed $r$, but is still quite expensive. We improve on the analysis of Kelsen to...
Article
Whether or not the problem of finding maximal independent sets (MIS) in hypergraphs is in (R)NC is one of the fundamental problems in the theory of parallel computing. Essentially, the challenge is to design (randomized) algorithms in which the number of processors used is polynomial and the (expected) runtime is polylogarithmic in the size of the...
Conference Paper
Full-text available
The (∆+1)-coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for (∆+1)-coloring running in O(√log ∆)+ 2^O(√log log n) rounds with probability 1-1/n^Ω(1) in a graph with n nodes and maximum degree ∆. This implies that the (∆+1)-coloring problem is easier than the maximal...
Article
Full-text available
Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is \emph{isolating} if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating w...
Article
A variety of powerful results have been shown for the chromatic number of triangle-free graphs; typically, this can be bounded much more strongly than is possible for arbitrary graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Nilli, Gimbel \& Thomassen, and Johansson. There have been compara...
Article
Basic graph structures such as maximal independent sets (MIS's) have spurred much theoretical research in randomized and distributed algorithms, and have several applications in networking and distributed computing as well. However, the extant (distributed) algorithms for these problems do not necessarily guarantee fault-tolerance or load-balance p...
Article
Full-text available
The $(\Delta+1)$-coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for $(\Delta+1)$-coloring running in $O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})}$ rounds with probability $1-1/n^{\Omega(1)}$ in a graph with $n$ nodes and maximum degree $\Delta$. This implies tha...
Article
The Lopsided Lovász Local Lemma (LLLL) is a powerful probabilistic principle which has been used in a variety of combinatorial constructions. While this principle began as a general statement about probability spaces, it has recently been transformed into a variety of polynomial-time algorithms. The resampling algorithm of Moser & Tardos is the mos...
Article
The Lovasz Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provided an efficient randomized algorithm to implement it. This algorithm could be parallelized to give an algorithm that uses polynomially many processors and $O(\log^3 n)$ time, stemming from $O(\... Article We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variab... Article Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables which arise during... Article Full-text available Species in the genus Plasmodium cause malaria in humans and infect a variety of mammals and other vertebrates. Currently, estimated ages for several mammalian Plasmodium parasites differ by as much as one order of magnitude, an inaccuracy that frustrates reliable estimation of evolutionary rates of disease-related traits. We developed a novel stati... Article As shown in the original work on the Lovász Local Lemma due to Erdős & Lovász (Infinite and Finite Sets, 1975), a basic application of the Local Lemma answers an infinitary coloring question of Strauss, showing that given any integer set S, the integers may be k-colored so that S and all its translates meet every color. The quantitative bounds here... Article Whether or not the problem of finding maximal independent sets (MIS) in hypergraphs is in (R)NC is one of the fundamental problems in the theory of parallel computing. Unlike the well-understood case of MIS in graphs, for the hypergraph problem, our knowledge is quite limited despite considerable work. It is known that the problem is in \emph{RNC}... Article The reliability polynomial of a graph counts its connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph’s reliability polynomial. We develop an improved SIS algorithm for estimating the reliability polynomial. The new algorithm runs in expected time O(mlogn α(m,n)) at wo... Article Full-text available Suppose that$a$and$d$are positive integers with$a \geq 2$. Let$h_{a,d}(n)$be the largest integer$t$such that any set of$n$points in$\mathbb{R}^d$contains a subset of$t$points for which all the non-zero volumes of the${t \choose a}$subsets of order$a$are distinct. Beginning with Erd\H{o}s in 1957, the function$h_{2,d}(n)\$ has bee...
Article
Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial approximation scheme to estimate the probability that a graph G becomes disconnected, given that its edges are removed independently with probability p. This algorithm runs in O(n5+o(1)ε-3) time to obtain an estimate within relative error e. We improve this runtime in two...
Article
While there has been significant progress on algorithmic aspects of the Lovasz Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations: The "lopsided" version of the LLL is usually at play here, and we do not yet have subexponential-time algorithms. We resolve this by developing a rand...
Conference Paper
The resampling algorithm of Moser & Tardos is a powerful approach to develop versions of the Lovasz Local Lemma. We develop a partial resampling approach motivated by this methodology: when a bad event holds, we resample an appropriately-random subset of the set of variables that define this event, rather than the entire set as in Moser & Tardos. T...
Conference Paper
Basic graph structures such as maximal independent sets (MIS's) have spurred much theoretical research in distributed algorithms, and have several applications in networking and distributed computing as well. However, the extant (distributed) algorithms for these problems do not necessarily guarantee fault-tolerance or load-balance properties: For...
Conference Paper
Constraint-satisfaction problems (CSPs) form a basic family of NP-hard optimization problems that includes satisfiability. Motivated by the sufficient condition for the satisfiability of SAT formulae that is offered by the Lovasz Local Lemma, we seek such sufficient conditions for arbitrary CSPs. To this end, we identify a variable-covering radius-...
Article
In a matroid secretary problem, one is presented with a sequence of objects of various weights in a random order, and must choose irrevocably to accept or reject each item. There is a further constraint that the set of items selected must form an independent set of an associated matroid. Constant-competitive algorithms (algorithms whose expected so...
Article
The reliability polynomial of a graph gives the probability that a graph is connected as a function of the probability that each edge is connected. The coefficients of the reliability polynomial count the number of connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph's...
Article
In a related-key attack, an attacker seeks to discover the secret key by requesting encryptions under keys related to the secret key in a manner chosen by the attacker. We describe a new related-key attack against generic ciphers, requiring just O(1) work to distinguish a cipher from random, and O(key length) to completely recover the secret key. T...

## Citations

... For k > 2, we note that fast algorithms for Graph-k-Cut have been of interest since they help in generating cutting planes while solving TSP [5,20]. A recent series of works aimed towards improving the bounds on the number of optimum solutions for Graph-k-Cut culminated in a drastic improvement in the run-time to solve Graph-k-Cut [28,31,32]. Given the status of counting and enumeration results for k-partitioning in graphs and their algorithmic and representation implications that were discovered subsequently, we believe that a similar understanding in hypergraphs could serve as an important ingredient in the algorithmic and representation theory of hypergraphs. ...
... In the proof of Theorem 22 in [13], an algorithmic version of the Local Lemma is used to obtain a randomized algorithm for finding the desired coloring for the hypergraph under consideration. There are deterministic algorithms known for the Local Lemma [18,19] which can be used in place of the randomized algorithm used in [13]. By applying Theorem 1.1 (1) from [19], we get a deterministic polynomial time algorithm to find the colorings C i (1 ≤ i ≤ r ). ...
... Hence, the logarithmic barrier for graphs of bounded arboricity seems to be inevitable. There is a progress in providing a decomposition with as fewest forest as possible [15,16], but we are not aware of any work breaking the logarithmic round complexity barrier without sacrificing the constant approximation on the number of forests. Working on a non-constant number of forests, happens to be useful in designing distributed coloring algorithms, however for approximation algorithms, in particular the case of the dominating set problem, it blows up the approximation guarantee. ...
... In this paper, we propose a framework for tackling the above challenge that we call the rainbow matching technique. Our technique capitalizes on the deep combinatorial results of Graf and Haxell in [26] (see also [25]). In fact, in Subsection 2.2, we derive the following "rainbow-matching" outcome of the main result of [26] asserting that there exists a polynomial time algorithm that, given an edge-multicolored graph G (by p colors), either outputs a matching of G carrying all p colors or outputs a non-empty set of colors C ⊆ {1, . . . ...
... The k-truss is a maximal subgraph in which each edge is incident to k triangles. See [18] for a more complete analysis of graph trusses. The importance of triangles was recognized earlier in 1998 when Watts and Strogatz [70] found that triangles were integral to the property of real-world networks, and introduced the clustering coefficient as a measure of how likely a pair of neighbors of a vertex may themselves be directly connected. ...
... This leads to one of the most powerful and general bounds for the MT-distribution [16]: ...
... A high level pseudocode of the main algorithm is given in Algorithm 1. return Min Cut of calculated on a single machine 3 end 4 Let 1 , . . ., be copies of with assigned random weight on edges (independently for each copy); 5 In parallel for all ∈ [ ], ← MinSingletonCut( ); 6 In parallel for all ∈ [ ], ← copy of after first contractions; 7 In parallel, ← AMPC-MinCut( ); 8 return min( 1 , . . . , , 1 , . . . ...
... While special graph classes are out of the scope of this paper, we mention the extensively studied case of distributed edge coloring. Here, poly log n-round algorithms were designed for progressively improving number of colors, from (2 + ε)Δ [28,31] to (2Δ − 1) [22,32], then to (1 + ε)Δ [30,32,48]. In a very recent manuscript, Bernshteyn gives a deterministic (Δ + 1)-edge coloring algorithm with runtime polynomial in Δ and log n [15]. ...
Reference: Linial for lists
... Aharoni, Berger & Ziv [1] showed that when b ≥ 2∆, there exists an IT I with w(I) ≥ w(V )/b. The work [11], building on [10], gives a nearly-matching randomized algorithm under the condition b ≥ (2 + ε)∆ for constant ε > 0. Similarly, [20] shows that, when w(v) ≥ 0 for all v (we say in this case that w is non-negative) and that b ≥ (4 + ε)∆, then the randomized MT algorithm directly gives an IT I with w(I) ≥ Ω(w(V )/b). ...
... Implemented in the sequential RAM model, it uses O (log 2 n) bits of space. For the more general problem of finding maximal independent sets in hypergraphs, the recent PRAM algorithm of Harris [30] yields a polylogarithmicspace algorithm for hypergraphs with edges of fixed size. ...