The dispersion problem has received much attention recently in the distributed computing literature. In this problem, k ≤ n agents placed initially arbitrarily on the nodes of an n-node, m-edge anonymous graph of max...
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Consider a simple finite graph and its nodes to represent identical water barrels (containing different amounts of water) on a level plane. Each edge corresponds to a (locked, water-filled) pipe connecting two barrels...
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We consider two variants, (s,z,)-Temporal Separator and (s,z,)-Temporal Cut, respectively, of the vertex separator and the edge cut problem in temporal graphs. The goal is to remove the minimum number of vertices (tem...
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The problems of detecting and recovering planted structures/subgraphs in ErdősRényi random graphs, have received significant attention over the past three decades, leading to many exciting results and mathematic...
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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
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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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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