Low time complexity algorithms for path computation in Cayley Graphs

Abstract

We study the problem of path computation in Cayley Graphs (CG) from an approach of word processing in groups. This approach consists in encoding the topological structure of CG in an automaton called Diff , then techniques of word processing are applied for computing the shortest paths. We present algorithms for computing the K-shortest paths, the shortest disjoint paths and the shortest path avoiding a set of nodes and edges. For any CG with diameter D, the time complexity of the proposed algorithms is O(KD|Diff |), where |Diff | denotes the size of Diff . We show that our proposal outperforms the state of art of topology-agnostic algorithms for disjoint shortest paths and stays competitive with respect to proposals for specific families of CG. Therefore, the proposed algorithms set a base in the design of adaptive and low-complexity routing schemes for networks whose interconnections are defined by CG.

Predefined Location Formation: Keeping Control for UAV Clusters Based on Monte Carlo Strategy

“…The computational complexity of the algorithm is an important indicator of the efficiency of the algorithm [25]. The computational complexity of the method provided in this study is O(n*m) for each UAV in the cluster using the asymptotic expression, where m is the number of UAVs in the cluster and n is similar to the number of populations in a genetic algorithm.…”

Using semidirect products of groups to build classes of interconnection networks

“…As might be expected from our discussion above, Cayley graphs have featured heavily in interconnection network design, and continue to do so, for the algebra provides a means for succinct description and any Cayley graph is vertex-transitive (though not necessarily edge-transitive) which, as we hinted earlier, has many benefits in an interconnection network context. We refer the reader to the review papers [18,20] for more on Cayley graphs and their relevance as interconnection networks and also note some more recent examples of research involving Cayley graphs in relation to interconnection network design: in a consideration of the structural properties of data centre interconnection networks with reference to universal routing schemes, it was shown in [10] that every Cayley graph has large hyperbolicity (hyperbolicity is a parameter that, intuitively, compares the metric space of a graph with the metric space of a tree); in [1], the computation of various paths in Cayley graphs was undertaken using automata theory (in the style of automatic groups [15]); and in [41], the generalized 3-connectivity of various classes of Cayley graphs was considered (generalized k-connectivity is a refined graph connectivity measure). These examples have been chosen as they involve path computation in Cayley graphs, as does our research; however, the general area encompassed by Cayley graphs and interconnection networks is thriving.…”

“…Consider the D-walk π 1 = π 1 0 π 1 1 , where π 1 0 ∈ ∆ B and π 1 1 is a shortest path of row-edges from (x 0 , 1) to (x 0 , y 0 ), in comparison with the D-walk defined as the shortest Type A D-walk from (0, 0) to (x 0 , c − 1) (built as in Lemma 7 and Lemma 9) extended with the additional edges ((x 0 , c − 1), (x 0 , 0)) and ((x 0 , 0), (x 0 , 1)), and then the path π 1 1 . The length of the former D-walk is r + x 0 + |π 1 1 | whereas the length of the latter is at most max{r − x 0 , 2µ − r + x 0 } + 2 + |π 1 1 |. Hence, as x 0 > 0, we can dispense with the D-walk π…”