Abstract We consider the algorithmic problem of generating each subset of [ n ] : = 1 , 2 , … , n whose size is in some interval [ k , l ] , 0 ≤ k ≤ l ≤ n , exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k = 0 and l = n this is the classical problem of generating all 2 n subsets of [ n ] by element additions removals, and for k = l this is the classical problem of generating all ( n k ) subsets of [ n ] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n , k , and l , improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant O ( 1 ) time per generated set and O ( n ) space. Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n -dimensional cube Q n induced by all levels [ k , l ] . We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q 2 k + 1 induced by all levels [ k − c , k + 1 + c ] , c ∈ 0 , 1 , … , k , has a Hamilton cycle. We also prove an approximate version of this generalized conjecture, showing that this graph has a cycle that visits a ( 1 − o ( 1 ) ) -fraction of all vertices.
In @cite_0 the authors present algorithms for enumerating all vertices of @math for an arbitrary interval @math such that any two consecutive bitstrings have Levenshtein distance at most 2 and Hamming distance at most 4. The Levenshtein distance is the minimum number of bit insertions, deletions, and bitflips necessary to transform one bitstring into another. Again, these enumerations are not cycles in @math and they are not tight. However, the corresponding generation algorithms are loopless, so they are very simple and fast. Improving on this, as a byproduct of the results mentioned in the previous section we obtain a simple loopless algorithm to generate all vertices of @math for an arbitrary interval @math such that any two consecutive bitstrings have Hamming distance and Levenshtein distance at most 2.
@cite_0
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We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size @math that halts in time @math with probability @math , each message containing @math bit: thus its bit complexity per channel is @math (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) for general graphs which is @math per channel (it halts in time @math and the size of each message is @math ). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices. We provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby's algorithm presented by Lynch and by Wattenhofer. This question remains open for the variant given by Peleg. The sequence reconstruction problem , first proposed by Levenshtein, models the setup in which a sequence from some set is transmitted over several channels, and the decoder receives the outputs from every channel. The channels are almost independent as it is only required that all outputs are different from each other. The main problem of interest is to determine the minimum number of channels required to reconstruct the transmitted sequence. In the combinatorial context, the problem is equivalent to finding the maximum intersection between two balls of radius @math , where the distance between their centers is at least @math . The setup of this problem was studied before for several error metrics such as the Hamming metric, the Kendall-tau metric, and the Johnson metric. In this paper, we extend the study initiated by Levenshtein for reconstructing sequences over the deletion channel. While he solved the case where the transmitted sequence can be arbitrary, we study the setup, where the transmitted sequence belongs to a single-deletion-correcting code and there are @math deletions in every channel. Under this paradigm, we study the minimum number of different channel outputs in order to construct a successful decoder. We prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. To the best of our knowledge, this is the first computational setting in which the complexity of estimating the edit distance is provably larger than that of Hamming distance. Our lower bound exhibits a trade-off between approximation and communication, asserting, for example, that protocols with @math bits of communication can obtain only approximation @math , where @math is the length of the input strings. This case of @math communication is of particular importance since it captures constant-size sketches as well as embeddings into spaces like @math and squared- @math , two prevailing algorithmic approaches for dealing with edit distance. Indeed, the known nontrivial communication upper bounds are all derived from embeddings into @math . By excluding low-communication protocols for edit distance, we rule out a strictly richer class of algorithms than previous results. Furthermore, our lower bound holds not only for strings over a binary alphabet but also for strings that are permutations (aka the Ulam metric). For this case, our bound nearly matches an upper bound known via embedding the Ulam metric into @math . Our proof uses a new technique that relies on Fourier analysis in a rather elementary way. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure.