Demand-aware network designs of bounded degree

Abstract

Traditionally, networks such as datacenter interconnects are designed to optimize worst-case performance under arbitrary traffic patterns. Such network designs can however be far from optimal when considering the actual workloads and traffic patterns which they serve. This insight led to the development of demand-aware datacenter interconnects which can be reconfigured depending on the workload. Motivated by these trends, this paper initiates the algorithmic study of demand-aware networks, and in particular the design of bounded-degree networks. The inputs to the network design problem are a discrete communication request distribution, ${{\mathcal {D}}}$, defined over communicating pairs from the node set V, and a bound, ${\varDelta }$, on the maximum degree. In turn, our objective is to design an (undirected) demand-aware network $N=(V,E)$ of bounded-degree ${\varDelta }$, which provides short routing paths between frequently communicating nodes distributed across $N$. In particular, the designed network should minimize the expected path length on $N$ (with respect to ${{\mathcal {D}}}$), which is a basic measure of the efficiency of the network. We derive a general lower bound based on the entropy of the communication pattern ${{\mathcal {D}}}$, and present asymptotically optimal demand-aware network design algorithms for important distribution families, such as sparse distributions and distributions of locally bounded doubling dimensions.

Appendices

Appendix

We first discuss different types of distortions on spanners and then show that actually Theorem 5 requires a weaker condition than having a constant sparse spanner.

Notions of distortion

In the spanner problem, the goal is to find a sparse subgraph $S=(V,E')$ of G, i.e., $E'\subseteq E$ with $|E'|\le O(n)$ which approximately preserves the distances of G despite having less edges. Usually, the following notion of average distortion [9] is considered and referred to as the all-pairs distortion:

Definition 4

(All-pairs distortion ($\mathrm {APD}$)) The average all-pairs distortion on a spanner $S$ of a graph G is

$$\begin{aligned} \mathrm {APD}(G,S) =\frac{1}{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } \sum _{\{u,v\}\in \left( {\begin{array}{c}V\\ 2\end{array}}\right) } \frac{d_{S}(u,v)}{d_G(u,v)}. \end{aligned}$$

We in this paper are only interested in preserving distances between communicating neighbors in G, henceforth defined as the neighborhood distortion:

Definition 5

(Neighborhood distortion ($\mathrm {ND}$)) The average neighborhood distortion on a spanner $S$ of a graph G (with m edges) is,

$$\begin{aligned} \mathrm {ND}(G,S)= & {} \frac{1}{|E(G) |} \sum _{\{u,v\}\in E(G)} \frac{d_{S}(u,v)}{d_G(u,v)} \\= & {} \frac{1}{m} \sum _{\{u,v\}\in E(G)} d_{S}(u,v). \end{aligned}$$

Next we claim that these two notions of distortion are indeed different, that is, low all-pairs distortion does not imply a low neighborhood distortion; and vice versa.

Claim

There is a family of graphs $G_n$ and a corresponding family of spanners $S_n$ (where n is the size of the graph and $S_n$ is a spanner of $G_n$) where

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\mathrm {APD}(G_n, S_n)}{\mathrm {ND}(G_n, S_n)} = \infty . \end{aligned}$$

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Claim

There is a family of graphs $G_n$ and a corresponding family of spanners $S_n$ (where n is the size of the graph and $S_n$ is a spanner of $G_n$) where

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\mathrm {ND}(G_n, S_n)}{\mathrm {APD}(G_n, S_n)} = \infty . \end{aligned}$$

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Fig. 3

a Different distortions on tree spanner w.r.t. different definitions, b different distortion on tree network design (with auxiliary edges) w.r.t. different definitions

We will show this by examples. First consider Fig. 3a. There is a ${\varTheta }(\sqrt{n})$-sized clique in the center, and each of those clique nodes is associated with a line containing ${\varTheta }(\sqrt{n})$ nodes. To compute the optimal tree spanner with maximum degree ${\varDelta }$, we turn the clique nodes into a ${\varDelta }$-regular tree of diameter ${\varTheta }(\log _{\varDelta }{\sqrt{n}})=O(\log _{\varDelta }{n})$. The nodes remain connected with the corresponding lines. The asymptotic distortion w.r.t. Definition 5 is:

$$\begin{aligned} \frac{n \cdot \log _{\varDelta }{n}+n \cdot 1}{n}={\varTheta }(\log _{\varDelta }{n}). \end{aligned}$$

Now we discuss all-pair distortion on the same spanner for this graph. Consider any two nodes which belong to different lines, but are also a member of the clique. Their distance in the spanner may become $\log \sqrt{n}$. So, according to Definition 4, $d_{S}(u,v)/d_G(u,v)$ is equal to $\frac{1}{2}\log n$. Now we provide an upper bound on the number of such pairs $\varphi$ whose distance can be up to $O(\log n)$ times their earlier distance. Consider all the nodes on all the lines which are within distance $\log n$ from the corresponding clique node. On the original graph, distances between any two such nodes were in the range $[1,2\log n +1]$. The number of such node pairs is $n \log ^2 n$. Clearly, $\varphi < n\log ^2 n$. Now consider any node on a line which is at least at a distance $(1+\log n)$ from the corresponding clique node on the line. The distance from this node to any other node on any other line was at least $(2+\log n)$. On the spanner, this distance can be at most $1+2\log n$. So, for all such node pairs, $d_{S}(u,v)/d_G(u,v)<2$. Hence, according to Definition 4, all pair distortion becomes constant, as stated in the following expression.

$$\begin{aligned} \frac{n \log ^2 n\cdot \log n + (n^2-n \log ^2 n)\cdot 2}{n^2}={\varTheta }(1). \end{aligned}$$

Now look at Fig. 3b. There is a star of size $n/\log {n}$ in the center, and each of the $n/\log {n}$ nodes is associated with a clique of size $\log {n}$. Thus, in total, there are $n\log {n}$ edges. To compute a tree spanner of degree ${\varDelta }=\log {n}$, we replace the cliques consisting of $\log {n}$ nodes with stars of size $\log {n}$ nodes; the star of $n/\log {n}$ nodes in the center is transformed into a ${\varDelta }$-regular tree whose diameter is ${\varTheta }(\log {n}/\log {\log {n}})$. Then each tree node is associated with the root of the star corresponding to its associated clique. This tree spanner contains auxiliary edges too. Then, the asymptotic distortion w.r.t. Definition 5 is:

$$\begin{aligned} \frac{\frac{n}{\log n}\log ^2n+\frac{n}{\log n}\cdot \frac{\log n}{\log \log n}}{n\log n}=O(1). \end{aligned}$$

In contrast, the distortion w.r.t. Definition 4 is ${\varOmega }(\log n/\log \log n)$ since all pairs from the two different cliques now suffer a distortion of ${\varTheta }(\log n/\log \log n)$, and there are $O(n^2)$ such pairs.

Corollary 2

Theorem 5 holds even if there exists a sparse spanner $S$ with constant neighborhood distortion instead of having a constant spanner.

Proof

If the request distribution ${{\mathcal {D}}}$ is uniform, i.e., $p(i,j)=1/m$ for all the m non-zero entries of the matrix $M_{{{\mathcal {D}}}}~$, then from Definition 5 and from our objective function,

$$\begin{aligned} \mathrm {EPL}({{\mathcal {D}}}, S) = \mathrm {ND}(G, S). \end{aligned}$$

Hence $\mathrm {ND}(G,S)$ is constant, which implies that $\mathrm {EPL}({{\mathcal {D}}}, S)$ is also constant i.e., Eq. 16 holds if $\mathrm {ND}(G,S)$ is constant. $\square$