This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel. We introduce our new model on Dyck steps with the nondeterministic step set {{--1}, {1}, {--1, 1}} and Motzkin steps with the nondeterministic step set {{--1}, {0}, {1}, {--1, 0}, {--1, 1}, {0, 1}, {--1, 0, 1}}. For general lists of step sets and a given length, we express the generating function of nondeterministic walks where at least one of the walks explored in parallel is a bridge (ends at the origin). In the particular cases of Dyck and Motzkin steps, we also compute the asymptotic probability that at least one of those parallel walks is a meander (stays nonnegative) or an excursion (stays nonnegative and ends at the origin). This research is motivated by the study of networks involving encapsulations and decapsulations of protocols. Our results are obtained using generating functions and analytic combinatorics.
In many data structure settings, it has been shown that using "double hashing" in place of standard hashing, by which we mean choosing multiple hash values according to an arithmetic progression instead of choosing each hash value independently, has asymptotically negligible difference in performance. We attempt to extend these ideas beyond data structure settings by considering how threshold arguments based on second moment methods can be generalized to "arithmetic progression" versions of problems. With this motivation, we define a novel "quasi-random" hypergraph model, random arithmetic progression (AP) hypergraphs, which is based on edges that form arithmetic progressions and unifies many previous problems. Our main result is to show that second moment arguments for 3-NAE-SAT and 2-coloring of 3-regular hypergraphs extend to the double hashing setting. We leave several open problems related to these quasi-random hypergraphs and the thresholds of associated problem variations.
Because of unmatched improvements in CPU performance, memory transfers have become a bottleneck of program execution. As discovered in recent years, this also affects sorting in internal memory. Since partitioning around several pivots reduces overall memory transfers, we have seen renewed interest in multiway Quicksort. Here, we analyze in how far multiway partitioning helps in Quickselect.We compute the expected number of comparisons and scanned elements (approximating memory transfers) for a generic class of (non-adaptive) multiway Quickselect and show that three or more pivots are not helpful, but two pivots are. Moreover, we consider "adaptive" variants which choose partitioning and pivot-selection methods in each recursive step from a finite set of alternatives depending on the current (relative) sought rank. We show that "Sesquickselect", a new Quickselect variant that uses either one or two pivots, makes better use of small samples w.r.t. memory transfers than other Quickselect variants.
Hillclimbing is an essential part of optimization. An important benchmark for hillclimbing algorithms on functions f : {0, 1}(n) -> R are (strictly) montone functions, on which a surprising number of hillclimbers fail to be efficient. For example, the (1+1)-Evolutionary Algorithm is a standard hillclimber which flips each bit independently with probability c/n in each round. Perhaps surprisingly, this algorithm shows a phase transition: it optimizes any such monotone function in quasilinear time if c < 1, but there are monotone functions for which the algorithm needs exponential time if c > 2.2. But so far it was unclear whether the threshold is at c = 1. In this paper we show that there exists a c(0) > 1 such that for all 0 < c <= c(0) the (1 + 1)-Evolutionary Algorithm with rate c/n finds the optimum in O(n log(2) n) steps in expectation. Our proof is an adaptation of Moser's entropy compression argument. That is, we show that a long runtime would allow us to encode the random steps of the algorithm with less bits than their entropy.
We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a discrete-time version of the first-named author's ``Meeting time lemma~ that bounds the probability of random walk hitting a deterministic trajectory in terms of hitting times of static vertices. The meeting time result is then used to bound the expected full coalescence time of multiple random walks over a graph. This last theorem is a discrete-time version of a result by the first-named author, which had been previously conjectured by Aldous and Fill. Our bounds improve on recent results by Lyons and Oveis-Gharan; Kanade et al; and (in certain regimes) Cooper et al.
We analyze a selection procedure introduced by Krieger, Pollak, and Samuel-Cahn. It retains an item if it is among the top 100p percent, as compared to the items that have been accepted so far. Gaither and Ward analyzed the average behavior of the number of items selected. We present the asymptotic properties of the higher moments of the number of items retained by the selection procedure. We derive a general formula for the moments. To demonstrate the complexity of these moments, we present the exact first-order asymptotic growth of some of these moments, for various rational values of p.
We extend randomized jumplists introduced by Bronnimann, Cazals, and Durand [2] 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", a variant of (fringe-balanced) binary search trees for the analysis. Despite their similarities, some standard analysis techniques for search trees fail for dangling-min trees (and hence for jumplists).
We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $\log[1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(\log x)^2$; the corresponding order for the Janson (2015) bound is the lead order, $x \log x$. Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).
We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using an additive tree parameter model combined with a recursive characterization. Our results include asymptotic formulas for mean and variance of this quantity as well as a central limit theorem.
When asymptotically analysing the summatory function of a q-regular sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of matrices of the linear representation of the sequence determine the "shape" (in particular the growth) of the asymptotic formula. Existing general results for determining the precise behavior (including the Fourier coefficients of the appearing fluctuations) have previously been restricted by a technical condition on these eigenvalues. The aim of this work is to lift these restrictions by providing an insightful proof based on generating functions for the main pseudo Tauberian theorem for all cases simultaneously. (This theorem is the key ingredient for overcoming convergence problems in Mellin-Perron summation in the asymptotic analysis.) One example is discussed in more detail: A precise asymptotic formula for the amount of esthetic numbers in the first N natural numbers is presented. Prior to this only the asymptotic amount of these numbers with a given digit-length was known.
One of the central topics in the theory of random graphs deals with the phase transition in the order of the largest components. In the binomial random graph $\mathcal{G}(n,p)$, the threshold for t...
A hooking network is grown from a set of graphs called blocks, each block with a labelled vertex called a hook. At each step in the growth of the network, a vertex called a latch is chosen from the hooking network, and a block is attached by joining the hook of the block with the latch. These graphs generalize trees, which are hooking networks grown from a single edge as the only block. Using Polya urns, we show multivariate normal limit laws for the degree distributions of hooking networks. We extend previous results by allowing for more than one block in the growth of the network and by studying arbitrarily large degrees.
The protection number of a tree is the minimal distance from its root to a leaf. In this paper we are interested in the protection number of a uniformly chosen random recursive tree of size n. Due to different construction of plane oriented and non-plane recursive trees we consider them separately. We use the singularity analysis of derived generating functions to find out the number of relevant trees, which leads us to the probability distribution of the protection number. Our results are also compared to outcomes of computer simulations.
In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schr\"oder tree model is not adapted to take into account the chronology between the branching nodes. In particular, it does not answer the question: how many different phylogenetic stories lead to the creation of n species and what is the average time to get there? In this paper, we enrich this model in two distinct ways in order to obtain two ranked tree models for phylogenetics, i.e. models coding chronology. For that purpose, we first develop a model of (strongly) increasing Schr\"oder trees, symbolically described in the classical context of increasing labeling. Then we introduce a generalization for the labeling with some unusual order constraint in Analytic Combinatorics (namely the weakly increasing trees). Although these models are direct extensions of the Schr\"oder tree model, it appears that they are also in one-to-one correspondence with several classical combinatorial objects. Through the paper, we present these links, exhibit some parameters in typical large trees and conclude the studies with efficient uniform samplers.
Deletions in open addressing tables have often been seen as problematic. The usual solution is to use a special mark ’deleted’ so that probe sequences continue past deleted slots, as if there was an element still sitting there. Such a solution, notwithstanding is wide applicability, may involve performance degradation. In the first part of this paper we review a practical implementation of the often overlooked deletion algorithm for linear probing hash tables, analyze its properties and performance, and provide several strong arguments in favor of the Robin Hood variant. In particular, we show how a small variation can yield substantial improvements for unsuccessful search. In the second part we propose an algorithm for true deletion in open addressing hashing with secondary clustering, like quadratic hashing. As far as we know, this is the first time that such an algorithm appears in the literature. Moreover, for tables built using the Robin Hood variant the deletion algorithm strongly preserves randomness (the resulting table is identical to the table that would result if the item were not inserted at all). Although it involves some extra memory for bookkeeping, the algorithm is comparatively easy and efficient, and it might be of some practical value, besides its theoretical interest.
In this paper, we build on recent results by Chauve et al. (2014) and Bahrani and Lumbroso (2017), which combined the split-decomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs---in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper, can be thought of as trees with their edges replaced by cycles (of arbitrary lengths). Their split-decomposition trees contain prime nodes that are cycles, making them ideal to study. We derive a characterization for the split-decomposition trees of cactus graphs, produce a general template of symbolic grammars for cactus graphs, and implement random generation for these graphs, building on work by Iriza (2015).
Beaconless geocast protocols are routing protocols used to send messages in mobile ad-hoc wireless networks, in which the only information available to each node is its own location. Messages get routed in a distributed manner: each node uses local decision rules based on the message source and destination, and its own location. In this paper we analyze six different beaconless geocast protocols, focusing on two relevant 1D scenarios. The selection of protocols reflects the most relevant types of protocols proposed in the literature, including those evaluated in previous computer simulations. We present a formal and structured analysis of the maximum number of messages that a node can receive, for each protocol, in each of the two scenarios. This is a measure of the network load incurred by each protocol. Our analysis, that for some of the protocols requires an involved probabilistic analysis, confirms behaviors that had been observed only through simulations before.
We study a new way of measuring the expected performance of various binary search tree algorithms that is between the worst and the average case. Our starting point is the correspondence between binary search trees and insertion sequences, and we will measure the difficulty of such sequences based on inversion counts. This measure naturally interpolates between random and sequential insertion orders. We show that if the tree is randomly picked from all trees built upon insertion length n permutations with t inversions, the height of the tree can be bounded by .
In this paper, building on previous work by Nakano et al. [23], we develop an alternate technique which almost automatically translates (existing) vertex incremental characterizations of graph classes into asymptotics of that class. Specifically, we construct trees corresponding to the sequences of vertex incremental operations which characterize a graph class, and then use analytic combinatorics to enumerate the trees, giving an upper bound on the graph class. This technique is applicable to a wider set of graph classes compared to the tree decompositions, and we show that this technique produces accurate upper bounds.We first validate our method by applying it to the case of distance-hereditary graphs, and comparing the bound obtained by our method with that obtained by Nakano et al. [23], and the exact enumeration obtained by Chauve et al. [7, 8]. We then illustrate its use by applying it to switch cographs, for which there are few known results: our method provide a bound of ∼6.301n, to be compared with the precise exponential growth, ∼6.159n, which we obtained independently through the relationship between switch cographs and bicolored cographs, first introduced by Hertz [19].We believe the popularity of vertex incremental characterizations might mean this may prove a fairly convenient to tool for future exploration of graph classes.
The concentration of the sizes of largest induced paths and cycles (holes) is studied in ErdoH \s--Re'\nyi random graphs. A 2-point concentration is proved for the size of the largest induced path and cycle for all p = p(n) satisfying p \geq n -1/2(ln n)2 and p \leq 1 -\varepsilon , where \varepsilon > 0 is any constant. No such tight concentration (within two consecutive values) was previously known for induced paths and cycles. As a corollary, a significant additive improvement is obtained over a 40-year-old result of ErdoH \s and Palka [Discrete Math., 46 (1983), pp. 145--150] concerning the size of the largest induced tree in a dense random graph. Further, the induced path decomposition number and induced tree decomposi-tion number, i.e., the smallest number of parts into which the vertex set of a graph can be partitioned such that every part induces a (i) path or (ii) tree, respectively, are studied for \scrG (n, p). The arguments involve the second moment method together with an adaptation of a martingale-based technique of Krivelevich et al. [Random Structures Algorithms, 22 (2003), pp. 1--14] for monotone high-degree polynomial random variables to the nonmonotone setting. A lower bound is proved showing the tightness of the application of the inequality up to logarithmic factors in the exponent. The modified inequality is then stated and proved in a general setting, which may be of independent interest.
