We study greedy routing over @math nodes placed in a ring, with the between two nodes defined to be the clockwise or the absolute distance between them along the ring. Such graphs arise in the context of modeling social networks and in routing networks for peer-to-peer systems. We construct the first network over @math nodes in which greedy routing takes @math hops in the worst-case, with @math out-going links per node. Our result has the first asymptotically optimal greedy routing complexity. Previous constructions required @math hops.
Xu al @cite_16 provide a partial answer to the above question by studying routing with distance function @math over graph topologies. A graph over @math nodes placed in a circle is said to be uniform if the set of clockwise offsets of out-going links is identical for all nodes. Chord is an example of a uniform graph. Xu al show that for any uniform graph with @math links per node, routing with distance function @math necessitates @math hops in the worst-case.
@cite_16
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We consider a system of @math servers inter-connected by some underlying graph topology @math . Tasks arrive at the various servers as independent Poisson processes of rate @math . Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in @math . Tasks have unit-mean exponential service times and leave the system upon service completion. The above model has been extensively investigated in the case @math is a clique. Since the servers are exchangeable in that case, the queue length process is quite tractable, and it has been proved that for any @math , the fraction of servers with two or more tasks vanishes in the limit as @math . For an arbitrary graph @math , the lack of exchangeability severely complicates the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph @math is said to be @math -optimal or @math -optimal when the occupancy process on @math is equivalent to that on a clique on an @math -scale or @math -scale, respectively. We prove that if @math is an Erd o s-R 'enyi random graph with average degree @math , then it is with high probability @math -optimal and @math -optimal if @math and @math as @math , respectively. This demonstrates that optimality can be maintained at @math -scale and @math -scale while reducing the number of connections by nearly a factor @math and @math compared to a clique, provided the topology is suitably random. It is further shown that if @math contains @math bounded-degree nodes, then it cannot be @math -optimal. In addition, we establish that an arbitrary graph @math is @math -optimal when its minimum degree is @math , and may not be @math -optimal even when its minimum degree is @math for any @math . We introduce a family of directed geometric graphs, denoted @math , that depend on two parameters @math and @math . For @math and @math , the @math graph is a strong @math -spanner, with @math . The out-degree of a node in the @math graph is at most @math . Moreover, we show that routing can be achieved locally on @math . Next, we show that all strong @math -spanners are also @math -spanners of the unit disk graph. Simulations for various values of the parameters @math and @math indicate that for random point sets, the spanning ratio of @math is better than the proven theoretical bounds. In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result and by prior work, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α We show that if a connected graph with @math nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(φ-1 • log n), many rounds with high probability, regardless of the source, by using the PUSH-PULL strategy. The O(••) notation hides a polylog φ-1 factor. This result is almost tight since there exists graph of n nodes, and conductance φ, with diameter Ω(φ-1 • log n). If, in addition, the network satisfies some kind of uniformity condition on the degrees, our analysis implies that both both PUSH and PULL, by themselves, successfully broadcast the message to every node in the same number of rounds.