Embracing the giant component

Consider a game in which edges of a graph are provided a pair at a time, and the player selects one edge from each pair, attempting to construct a graph with a component as large as possible. This game is in the spirit of recent papers on avoiding a giant component, but here we embrace it. We analyze this game in the offline and online setting, for arbitrary and random instances, which provides for interesting comparisons. For arbitrary instances, we find that the competitive ratio (the best possible solution value divided by best possible online solution value) is large. For “sparse” random instances the competitive ratio is also large, with high probability (whp); If the instance has ¼(1 + ε)n random edge pairs, with 0 < ε ≤ 0.003, then any online algorithm generates a component of size O((log n)^3/2) whp , while the optimal offline solution contains a component of size Ω(n) whp . For “dense” random instances, the average‐case competitive ratio is much smaller. If the instance has ½(1 ? ε)n random edge pairs, with 0 < ε ≤ 0.015, we give an online algorithm which finds a component of size Ω(n) whp . © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Nash equilibria in random games

We consider Nash equilibria in 2‐player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m × n payoff matrices, it runs in time O(m²nloglog n + n²mloglog m) with high probability. Our result follows from showing that a 2‐player random game has a Nash equilibrium with supports of size two with high probability, at least 1 − O(1/log n). Our main tool is a polytope formulation of equilibria. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Phase transitions of the Moran process and algorithmic consequences

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ?>0, we show that the expected absorption time on an n‐vertex graph is o(n^3+?). Specifically, it is at most , and there is a family of graphs where it is Ω(n³). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.     

Random channel assignment in the plane

In the model for random radio channel assignment considered here, points corresponding to transmitters are thrown down independently at random in the plane, and we must assign a radio channel to each point but avoid interference. In the most basic version of the model, we assume that there is a threshold d such that, in order to avoid interference, points within distance less than d must be assigned distinct channels. Thus we wish to color the nodes of a corresponding scaled unit disk graph. We consider the first n random points, and we are interested in particular in the behavior of the ratio of the chromatic number to the clique number. We show that, as n → ∞, in probability this ratio tends to 1 in the “sparse” case [when d = d(n) is such that the average degree grows more slowly than ln n] and tends to ～ 1.103 in the “dense” case (when the average degree grows faster than ln n). We also consider related graph invariants, and the more general channel assignment model when assignments must satisfy “frequency‐distance” constraints. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 187–212, 2003     

Covering minimum spanning trees of random subgraphs

We consider the problem of finding a sparse set of edges containing the minimum spanning tree (MST) of a random subgraph of G with high probability. The two random models that we consider are subgraphs induced by a random subset of vertices, each vertex included independently with probability p, and subgraphs generated as a random subset of edges, each edge with probability p. Let n denote the number of vertices, choose p ∈ (0, 1) possibly depending on n, and let b = 1/(1 ? p). We show that in both random models, for any weighted graph G, there is a set of edges Q of cardinality O(n log_bn) that contains the minimum spanning tree of a random subgraph of G with high probability. This result is asymptotically optimal. As a consequence, we also give a bound of O(kn) on the size of the union of all minimum spanning trees of G with some k vertices (or edges) removed. More generally, we show a bound of O(n log_bn) on the size of a covering set in a matroid of rank n, which contains the minimum‐weight basis of a random subset with high probability. Also, we give a randomized algorithm that calls an MST subroutine only a polylogarithmic number of times and finds the covering set with high probability. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

A spectral technique for random satisfiable 3CNF formulas

Let I be a random 3CNF formula generated by choosing a truth assignment ? for variables x₁, x_n uniformly at random and including every clause with i literals set true by ? with probability p_i, independently. We show that for any constants 0 ≤ η₂,η₃ ≤ 1 there is a constant d_min so that for all d ≥ d_min a spectral algorithm similar to the graph coloring algorithm of Alon and Kahale will find a satisfying assignment with high probability for p₁ = d/n², p₂ = η₂d/n², and p₃ = η₃d/n². Appropriately setting the η_i's yields natural distributions on satisfiable 3CNFs, not‐all‐equal‐sat 3CNFs, and exactly‐one‐sat 3CNFs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On extractors and exposure‐resilient functions for sublogarithmic entropy

We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n ‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n ‐ k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure‐resilient functions achieving these parameters are known. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2‐connected (simple) graphs on n vertices and m edges, provided that m ‐ n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2‐edge‐connectedness is treated similarly. We also obtain formulae for the number of 2‐connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

The diameter of randomly perturbed digraphs and some applications

The central observation of this paper is that if εn random arcs are added to any n‐node strongly connected digraph with bounded degree then the resulting graph has diameter 𝒪(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL‐complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ∼ 𝔻_n,ε/n we obtain a “smoothed” instance. We show that, with high probability, a log‐space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL ⫅̸ almost‐L then no heuristic can recognize similarly perturbed instances of (s,t)‐connectivity. Property Testing: A digraph is called k‐linked if, for every choice of 2k distinct vertices s₁,…,s_k,t₁,…,t_k, the graph contains k vertex disjoint paths joining s_r to t_r for r = 1,…,k. Recognizing k‐linked digraphs is NP‐complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k‐linked graphs with high probability, and rejects all graphs that are at least εn arcs away from being k‐linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The emergence of a giant component in random subgraphs of pseudo‐random graphs

Let G be a d‐regular graph G on n vertices. Suppose that the adjacency matrix of G is such that the eigenvalue λ which is second largest in absolute value satisfies λ = o(d). Let G_p with p = α/d be obtained from G by including each edge of G independently with probability p. We show that if α < 1, then whp the maximum component size of G_p is O(log n) and if α > 1, then G_p contains a unique giant component of size Ω(n), with all other components of size O(log n). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Randomly coloring constant degree graphs

We study a simple Markov chain, known as the Glauber dynamics, for generating a random k ‐coloring of an n ‐vertex graph with maximum degree Δ. We prove that, for every ε > 0, the dynamics converges to a random coloring within O(nlog n) steps assuming k ≥ k₀(ε) and either: (i) k/Δ > α* + ε where α*≈? 1.763 and the girth g ≥ 5, or (ii) k/Δ >β * + ε where β*≈? 1.489 and the girth g ≥ 7. Our work improves upon, and builds on, previous results which have similar restrictions on k/Δ and the minimum girth but also required Δ = Ω (log n). The best known result for general graphs is O(nlog n) mixing time when k/Δ > 2 and O(n²) mixing time when k/Δ > 11/6. Related results of Goldberg et al apply when k/Δ > α* for all Δ ≥ 3 on triangle‐free “neighborhood‐amenable” graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Counting the onion

Iteratively computing and discarding a set of convex hulls creates a structure known as an “onion.” In this paper, we show that the expected number of layers of a convex hull onion for n uniformly and independently distributed points in a disk is Θ(n^2/3). Additionally, we show that in general the bound is Θ(n^2/(d+1)) for points distributed in a d‐dimensional ball. Further, we show that this bound holds more generally for any fixed, bounded, full‐dimensional shape with a nonempty interior. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

The scaling window for a random graph with a given degree sequence

We consider a random graph on a given degree sequence D, satisfying certain conditions. Molloy and Reed defined a parameter Q = Q(D) and proved that Q = 0 is the threshold for the random graph to have a giant component. We introduce a new parameter R = R( \begin{align*}\mathcal {D}\end{align*}) and prove that if |Q| = O(n^‐1/3R^2/3) then, with high probability, the size of the largest component of the random graph will be of order Θ(n^2/3R^‐1/3). If |Q| is asymptotically larger than n^‐1/3R^2/3 then the size of the largest component is asymptotically smaller or larger than n^2/3R^‐1/3. Thus, we establish that the scaling window is |Q| = O(n^‐1/3R^2/3). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A new implementation of an algorithm for the optimal assignment problem: An improved version of Munkres' algorithm

Existing implementations of Munkres' algorithm for the optimal assignment problem are shown to requireO(n ⁴) time in the worstn×n case. A new implementation is presented which runs in worst-case timeO(n ³) and compares favorably in performance with the algorithm of Edmonds and Karp for this problem.The results of this paper were obtained by the author while at the Department of Computer Science, Cornell University. This work was supported in part by a Vanderbilt University Research Council Grant.     

Dynamic point location in arrangements of hyperplanes

We present algorithms for maintaining data structures supporting fast (polylogarithmic) point-location and ray-shooting queries in arrangements of hyperplanes. This data structure allows for deletion and insertion of hyperplanes. Our algorithms use random bits in the construction of the data structure but do not make any assumptions about the update sequence or the hyperplanes in the input. The query bound for our data structure isÕ(polylog(n)), wheren is the number of hyperplanes at any given time, and theÕ notation indicates that the bound holds with high probability, where the probability is solely with respect to randomization in the data structure. By high probability we mean that the probability of error is inversely proportional to a large degree polynomial inn. The space requirement isÕ(n ^d). The cost of update isÕ(n ^d?1 logn. The expected cost of update isO(n ^d?1); the expectation is again solely with respect to randomization in the data structure. Our algorithm is extremely simple. We also give a related algorithm with optimalÕ(logn) query time, expectedO(n ^d) space requirement, and amortizedO(n ^d?1) expected cost of update. Moreover, our approach has a versatile quality which is likely to have further applications to other dynamic algorithms. Ford=2, 3 we also show how to obtain polylogarithmic update time in the CRCW PRAM model so that the processor-time product matches (within a polylogarithmic factor) the sequential update time.     

Shrinkage of de Morgan formulae under restriction

It is shown that a random restriction leaving only a fraction ? of the input variables unassigned reduces the expected de Morgan formula size of the induced function by a factor of O(E^(5?√2)/2) = O(?^1.63). (A de Morgan, or unate, formula is a formula over the basis {∧, ∨, ¬}.) This improves a long-standing result of O(?^1.5) by Subbotovskaya and a recent improvement to O(?^(21?√73)/8) = O(?^1.55) by Nisan and Impagliazzo. The New exponent yields an increased lower bound of n^{(7?√3)/2?o(1)} = Ω(n^2.63) for the de Morgan formula size of a function in P defined by Andreev. This is the largest formula size lower bound known, even for functions in NP. © 1993 John Wiley & Sons, Inc.