PDLB: Path Diversity-aware Load Balancing with Adaptive Granularity in Data Center Network

Modern data center topologies often take the form of a multi-rooted tree with rich parallel paths to provide high bandwidth. However, various path diversities caused by traffic dynamics, link failures and heterogeneous switching equipments widely exist in production data center network. Therefore, the multi-path load balancer in data center should be robust to these diversities. Although prior fine-grained schemes such as RPS and Presto make full use of available paths, they are prone to experi-ence packet reordering problem under asymmetric topology. The coarse-grained solutions such as ECMP and LetFlow effectively avoid packet reordering, but easily lead to under-utilization of multiple paths. To cope with these inefficiencies, we propose a load balancing mechanism called PDLB, which adaptively adjusts flowcell granularity according to path diversity. PDLB increases flowcell granularity to alleviate packet reordering under large degrees of topology asymmetry, while reducing flowcell granularity to obtain high link utilization under small degrees of topology asymmetry. PDLB is only deployed on the sender without any modification on switch. We evaluate PDLB through large-scale NS2 simulations. The experimental results show that PDLB reduces the average flow completion time by up to ∼11-53% over the state-of-the-art load balancing schemes.

On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

Oddlab: fault-tolerant aware load-balancing framework for data center networks

Data center networks (DCNs) act as critical infrastructures for emerging technologies. In general, a DCN involves a multi-rooted tree with various shortest paths of equal length from end to end. The DCN fabric must be maintained and monitored to guarantee high availability and better QoS. Traditional traffic engineering (TE) methods frequently reroute large flows based on the shortest and least-congested paths to maintain high service availability. This procedure results in a weak link utilization with frequent packet reordering. Moreover, DCN link failures are typical problems. State-of-the-art approaches address such challenges by modifying the network components (switches or hosts) to discover and avoid broken connections. This study proposes Oddlab (Odds labels), a novel deployable TE method to guarantee the QoS of multi-rooted data center (DC) traffic in symmetric and asymmetric modes. Oddlab creatively builds a heuristic model for efficient flow scheduling and faulty link detection by exclusively using the gathered statistics from the DCN data plane, such as residual bandwidth and the number of installed elephant flows. Besides, the proposed method is implemented in an SDN-based DCN without altering the network components. Our findings indicate that Oddlab can minimize the flow completion time, maximize bisection bandwidth, improve network utilization, and recognize faulty links with sufficient accuracy to improve DC productivity.

Constructing uncountably many groups with the same profinite completion

Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.

2-Distance Strong b-coloring of Perfect 𝜟-ary Tree

A 2-distance 𝑏-coloring is a 2-distance coloring in which every color class contains a vertex which has a neighbor in every other color class. A 2-distance strong 𝑏-coloring (2𝑠𝑏- coloring) is a 2-distance coloring in which every color class contains a vertex 𝑢 such that there is a vertex 𝑣 in every other color class satisfying the condition that the distance between 𝑢 and 𝑣 is at most 2. The 2-distance 𝑏-chromatic number 𝜒2𝑏(𝐺) (2𝑏-number) is the largest integer 𝑘 such that 𝐺 admits a 2-distance 𝑏-coloring with 𝑘 colors and the 2-distance strong bchromatic number 𝜒2𝑠𝑏(𝐺) (2𝑠𝑏-number) is the maximum k such that 𝐺 admits a 2𝑠𝑏-coloring with 𝑘 colors. A tree with a special vertex called the root is called a rooted tree. A perfect 𝛥- ary tree, is a rooted tree in which all internal vertices are of degree 𝛥 and all pendant vertices are at the same level. In this paper, the exact bound of the 2𝑠𝑏-number of perfect 𝛥-ary tree are obtained.

The Human Ecology: An Inverted Asswattha Plant (Ficus religiosa), A Symbolic Presentation by Sri Krishna in The Bhagavad Gita–An Analytical Review

ABSTRACT SriKrishna has presented human ecology (Manusya Loka) as an imperishable banyan tree in Bhagavad Gita that has its roots upward and branches down. The leaves of the plant are the Vedic hymns (human knowledge) and the twigs and buds are the objects of senses. The tree has also roots growing down (towards the branches), which are bound to the fruitive actions of human society. The plant is nourished by the three modes (Gunas) of material nature. The real form of this tree cannot be perceived in this world. No one can understand its foundation, beginning and end. The workculture of detachment is the weapon to cut down this strongly rooted tree, to make oneself free from the cycle of birth and death.