A local coloring of a graph G is a function c : V (G) → N having the property that for each set S ⊆ V (G) with 2 ≤ |S| ≤ 3, there exist vertices u, v ∈ S such that |c(u) − c(v)| ≥ m S , where m S is the number of edges of the induced subgraph S . The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ (c). The local chromatic number of G is χ (G) = min{χ (c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K (n, k) for some values of n and k is determined.
We consider the random phone call model introduced by Demers et al., which is a well-studied model for information dissemination on networks. One basic protocol in this model is the so-called Push protocol which proceeds in synchronous rounds. Starting with a single node which knows of a rumor, every informed node calls in each round a random neighbor and informs it of the rumor. The Push-Pull protocol works similarly, but additionally every uninformed node calls a random neighbor and may learn the rumor from it.It is well-known that both protocols need $\Theta(\log n)$ rounds to spread a rumor on a complete network with $n$ nodes. Here we are interested in how much the spread can be speeded by enabling nodes to make more than one call in each round. We propose a new model where the number of calls of a node is chosen independently according to a probability distribution $R$. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of $R$ such as the mean or the variance (if they exist). In particular, if $R$ follows a power law distribution with exponent $\beta \in (2,3)$, we show that the Push-Pull protocol spreads a rumor in $\Theta(\log \log n)$ rounds. Moreover when $\beta=3$, the Push-Pull protocol spreads a rumor in $\Theta(\frac{ \log n}{\log\log n})$ rounds.
We consider load balancing in a network of caching servers delivering contents to end users. Randomized load balancing via the so-called power of two choices is a wellknown approach in parallel and distributed systems that reduces network imbalance. In this paper, we propose a randomized load balancing scheme which simultaneously considers cache size limitation and proximity in the server redirection process.Since the memory limitation and the proximity constraint cause correlation in the server selection process, we may not benefit from the power of two choices in general. However, we prove that in certain regimes, in terms of memory limitation and proximity constraint, our scheme results in the maximum load of order Θ(log log n) (here n is the number of servers and requests), and at the same time, leads to a low communication cost. This is an exponential improvement in the maximum load compared to the scheme which assigns each request to the nearest available replica. Finally, we investigate our scheme performance by extensive simulations.
Abstract-We consider load balancing in a network of caching servers delivering contents to end users. Randomized load balancing via the so-called power of two choices is a wellknown approach in parallel and distributed systems. In this framework, we investigate the tension between storage resources, communication cost, and load balancing performance. To this end, we propose a randomized load balancing scheme which simultaneously considers cache size limitation and proximity in the server redirection process.In contrast to the classical power of two choices setup, since the memory limitation and the proximity constraint cause correlation in the server selection process, we may not benefit from the power of two choices. However, we prove that in certain regimes of problem parameters, our scheme results in the maximum load of order Θ(log log n) (here n is the network size). This is an exponential improvement compared to the scheme which assigns each request to the nearest available replica. Interestingly, the extra communication cost incurred by our proposed scheme, compared to the nearest replica strategy, is small. Furthermore, our extensive simulations show that the trade-off trend does not depend on the network topology and library popularity profile details.
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