A (molecular) graph in which all vertices have the same degree is known as a regular graph. According to Gutman, Hansen, and Mélot [J. Chem. Inf. Model. 45 (2005) 222–230], it is of interest to measure the irreg...
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The Beeping Network (BN) model captures important properties of biological processes, for instance, when the beeping entity, called node, models a cell. Perhaps paradoxically, even the fact that the communication capa...
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The Beeping Network (BN) model captures important properties of biological processes, for instance, when the beeping entity, called node, models a cell. Perhaps paradoxically, even the fact that the communication capabilities of such nodes are extremely limited has helped BN become one of the fundamental models for networks where nodes’ transmissions interfere with each other. Since in each round, a node may transmit at most one bit, it is useful to treat the communications in the network as distributed coding and design it to overcome the interference. We study both non-adaptive and adaptive codes. Some communication and graph problems already studied in the Beeping Networks admit fast (i.e., polylogarithmic in the network size n) randomized algorithms. On the other hand, all known deterministic algorithms for non-trivial problems have time complexity (i.e., the number of beeping rounds, corresponding to the length of the used codes) at least polynomial in the maximum node-degree ∆. We improve known results for deterministic algorithms by first showing that this polynomial can be as low as Õ(∆2). More precisely, we prove that beeping out a single round of any CONGEST algorithm in any network of maximum node-degree ∆ can be done in O(∆2 polylog n) beeping rounds, each accommodating at most one beep per node, even if the nodes intend to send different messages to different neighbors. This upper bound reduces polynomially the time for a deterministic simulation of CONGEST in a Beeping network, comparing to the best known algorithms, and nearly matches the time obtained recently using randomization (up to a poly-logarithmic factor). Our simulator allows us to implement any efficient algorithm designed for the CONGEST networks in the Beeping Networks, with O(∆2 polylog n) overhead. This O(∆2 polylog n) implementation results in a polynomial improvement upon the best-to-date Θ(∆3)-round beeping MIS algorithm (and of related tasks). Using a more specialized (and thus, mor
As large graph datasets become increasingly common across many fields, sampling is often needed to reduce the graphs into manageable sizes. This procedure raises critical questions about representativeness as no sampl...
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Listing k-cliques plays a fundamental role in various data mining tasks, such as community detection and mining of cohesive substructures. Existing algorithms for the k-clique listing problem are built upon a general ...
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We review the problem of finding paths in Cayley graphs of groups and group actions, using the Rubik’s cube as an example, and we list several more examples of significant mathematical interest. We then show how to f...
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The main result of this paper utilizes the representation graph of a group G, R(V, G), and gives a general construction of a diagrammatic category DgramsR(V,G). The proof of the main theorem shows that, given explicit...
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In the estimation of causal effects, one common method for removing the influence of confounders is to adjust the variables that satisfy the back-door criterion. However, it is not always possible to uniquely determin...
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The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of ...
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This paper presents a novel multicriteria shortest path search algorithm called Hierarchical MLS. The distinguishing feature of the algorithm is the multilayered structure of compressed k-Path-Cover graphs it operates...
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A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increa...
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