We tackle the problem of graph transformation with a particular focus on node cloning. We propose a new approach to graph rewriting where nodes can be cloned zero, one or more times. A node can be cloned together with all its incident edges, with only its outgoing edges, with only its incoming edges or with none of its incident edges. We thus subsume previous works such as the sesqui-pushout, the heterogeneous pushout and the adaptive star grammars approaches. A rewrite rule is defined as a span where the right-hand and left-hand sides are graphs while the interface is a polarized graph. A polarized graph is a graph endowed with some annotations on nodes. The way a node is cloned is indicated by its polarization annotation. We use these annotations for designing graph transformation with polarized cloning. We show how a clone of a node can be built according to the different possible polarizations and define a rewrite step as a final pullback complement followed by a pushout. This is called the polarized sesqui-pushout approach. We also provide an algorithmic presentation of the proposed graph transformation with polarized cloning.
Cloning has also been subject of interest in @cite_8 . The authors considered rewrite rules of the form @math where @math is a star, i.e., @math is a (nonterminal) node surrounded by its adjacent nodes together with the edges that connect them. Rewrite rules which perform the cloning of a node have been given in [Def. 6] drewes06 . These rules show how a star can be removed, kept identical to itself or copied (cloned) more than once. Here again, unlike our framework, the cloning does not care about the arity of the nodes and, as in the case of the sesqui-pushout approach, a node is copied together with all its incoming and outgoing edges.
@cite_8
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We address the problem of replicating a Voronoi diagram @math of a planar point set @math by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in @math ; 2. the distance to and label(s) of the nearest site(s) in @math ; 3. a unique label for every nearest site in @math . We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of @math with @math queries and @math processing time. We also prove that queries of Type 3 can never exactly clone @math , but we show that with @math queries we can construct an @math -approximate cloning of @math . In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures. We propose an extension of node and hyperedge replacement grammars, called adaptive star grammars, and study their basic properties. A rule in an adaptive star grammar is actually a rule schema which, via the so-called cloning operation, yields a potentially infinite number of concrete rules. Adaptive star grammars are motivated by application areas such as modeling and refactoring object-oriented programs. We prove that cloning can be applied lazily. Unrestricted adaptive star grammars are shown to be capable of generating every type-0 string language. However, we identify a reasonably large subclass for which the membership problem is decidable. Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an @math - @math MSR code @math over @math stores a file @math consisting of @math symbols over @math among @math nodes, each storing @math symbols, in such a way that: 1) the file @math can be recovered by downloading the content of any @math of the @math nodes and 2) the content of any failed node can be reconstructed by accessing any @math of the remaining @math nodes and downloading @math symbols from each of these nodes. In practice, the file @math is typically available in uncoded form on some @math of the @math nodes, known as systematic nodes , and the defining node-repair condition above can be relaxed to requiring the optimal repair bandwidth for systematic nodes only . Such codes are called systematic–repair MSR codes . Unfortunately, finite– @math constructions of @math MSR codes are known only for certain special cases: either low rate, namely @math , or high repair connectivity, namely @math . Our main result in this paper is a finite– @math construction of systematic-repair @math MSR codes for all possible values of parameters @math . We also introduce a generalized construction for @math MSR codes to achieve the optimal repair bandwidth for all values of @math simultaneously. We propose a generic transformation that can convert any nonbinary @math maximum distance separable (MDS) code into another @math MDS code over the same field such that: 1) some arbitrarily chosen @math nodes have the optimal repair bandwidth and the optimal rebuilding access; 2) for the remaining @math nodes, the normalized repair bandwidth and the normalized rebuilding access (over the file size) are preserved; and 3) the sub-packetization level is increased only by a factor of @math . Two immediate applications of this generic transformation are then presented. The first application is that we can transform any nonbinary MDS code with the optimal repair bandwidth or the optimal rebuilding access for the systematic nodes only, into a new MDS code which possesses the corresponding repair optimality for all nodes. The second application is that by applying the transformation multiple times, any nonbinary @math scalar MDS code can be converted into an @math MDS code with the optimal repair bandwidth and the optimal rebuilding access for all nodes, or only a subset of nodes, whose sub-packetization level is also optimal.