We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures.

Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of 1.883 when the table was full. Furthermore, by using a non-standard mean-centred search algorithm, this would imply that searches could be performed in expected constant time even in a full table.

In spite of the time elapsed since these observations were made, no progress has been made in proving them. In this paper we introduce a technique to work around the intractability of the recurrence equation by solving instead an associated differential equation. While this does not provide an exact solution, it is sufficiently powerful to prove a bound of π2/3 for the variance, and thus obtain a proof that the variance of Robin Hood is bounded by a small constant for load factors arbitrarily close to 1. As a corollary, this proves that the mean-centred search algorithm runs in expected constant time.

We also use this technique to study the performance of Robin Hood hash tables under a long sequence of insertions and deletions, where deletions are implemented by marking elements as deleted. We prove that, in this case, the variance is bounded by 1/(1−α), where α is the load factor.

To model the behaviour of these hash tables, we use a unified approach that we apply also to study the First-Come-First-Served and Last-Come-First-Served collision resolution disciplines, both with and without deletions.

We present an average-case analysis of a variant of dual-pivot quicksort. We show that the algorithmic partitioning strategy used is optimal, that is, it minimizes the expected number of key comparisons. For the analysis, we calculate the expected number of comparisons exactly as well as asymptotically; in particular, we provide exact expressions for the linear, logarithmic and constant terms.

An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proved.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

Let k ⩾ 2 be an integer. We show that if s = 2 and t ⩾ 2, or s = t = 3, then the maximum possible number of edges in a C2k+1-free graph containing no induced copy of Ks,t is asymptotically equal to (t − s + 1)1/s(n/2)2−1/s except when k = s = t = 2.

This strengthens a result of Allen, Keevash, Sudakov and Verstraëte [1], and answers a question of Loh, Tait, Timmons and Zhou [14].

In 1960 Rényi, in his Michigan State University lectures, asked for the number of random queries necessary to recover a hidden bijective labelling of n distinct objects. In each query one selects a random subset of labels and asks, which objects have these labels? We consider here an asymmetric version of the problem in which in every query an object is chosen with probability p > 1/2 and we ignore ‘inconclusive’ queries. We study the number of queries needed to recover the labelling in its entirety (Hn), before at least one element is recovered (Fn), and to recover a randomly chosen element (Dn). This problem exhibits several remarkable behaviours: Dn converges in probability but not almost surely; Hn and Fn exhibit phase transitions with respect to p in the second term. We prove that for p > 1/2 with high probability we need

$$H_n=\log_{1/p} n +{\tfrac{1}{2}} \log_{p/(1-p)}\log n +o(\log \log n)$$

$$ H_n=\log_{2} n +\sqrt{2 \log_{2} n} +o(\sqrt{\log n})$$

We prove that the number of multigraphs with vertex set {1, . . ., n} such that every four vertices span at most nine edges is an2+o(n2) where a is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.

Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

Let Φ be a random k-SAT formula in which every variable occurs precisely d times positively and d times negatively. Assuming that k is sufficiently large and that d is slightly below the critical degree where the formula becomes unsatisfiable with high probability, we determine the limiting distribution of the number of satisfying assignments.

One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a function f: {0,1}n → ℝ. The algorithm starts with a random search point ξ ∈ {0,1}n, and in each round it flips each bit of ξ with probability c/n independently at random, where c > 0 is a fixed constant. The thus created offspring ξ' replaces ξ if and only if f(ξ') ≥ f(ξ). The analysis of the runtime of this simple algorithm for monotone and for linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbours coloured by 1, and locally stable functions, in which every vertex has a constant fraction of neighbours coloured by its own colour. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for ℤn and other graphs, and offer several follow-up questions.