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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Abstract-It is well known that load balancing and low delivery communication cost are two critical issues in mapping requests to servers in Content Delivery Networks (CDNs). However, the trade-off between these two performance metrics has not been yet quantitatively investigated in designing efficient request mapping schemes. In this work, we formalize this trade-off through a stochastic optimization problem. While the solutions to the problem in the extreme cases of minimum communication cost and optimum load balancing can be derived in closed form, finding the general solution is hard to derive. Thus we propose three heuristic mapping schemes and compare the trade-off performance of them through extensive simulations.Our simulation results show that at the expense of high query cost, we can achieve a good trade-off curve. Moreover, by benefiting from the power of multiple choices phenomenon, we can achieve almost the same performance with much less query cost. Finally, we can handle requests with different delay requirements at the cost of degrading network performance.Index Terms-Content delivery networks, distributed load balancing, power of multiple choices, query cost.

We propose algorithms for allocating n sequential balls into n bins that are interconnected as a -regular n-vertex graph G, where ⩾ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ⩽ t ⩽ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of , with high probability.KEYWORDS balls-into-bins models, balanced allocation, nonbacktracking random walks 1 * A preliminary version of this paper appeared in the proceedings of the 22nd International Computing and Combinatorics Conference (COCOON'16).Random Struct Alg. 2019;55:980-1009.wileyonlinelibrary.com/journal/rsa

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