An Approximation Algorithm for Path Computation and Function Placement in SDNs

Abstract

We consider the task of embedding multiple service requests in Software-Defined Networks (SDNs), i.e. computing (combined) mappings of network functions on physical nodes and finding routes to connect the mapped network functions. A single service request may either be fully embedded or be rejected. The objective is to maximize the sum of benefits of the served requests, while the solution must abide node and edge capacities.

We follow the framework suggested by Even et al. [5] for the specification of the network functions and routing of requests via processing-and-routing graphs (PR-graphs): a request is represented as a directed acyclic graph with the nodes representing network functions. Additionally, a unique source and a unique sink node are given for each request, such that any source-sink path represents a feasible chain of network functions to realize the service. This allows for example to choose between different realizations of the same network function. Requests are attributed with a global demand (e.g. specified in terms of bandwidth) and a benefit.

Our main result is a randomized approximation algorithm for path computation and function placement with the following guarantee. Let m denote the number of links in the substrate network, $\varepsilon$ denote a parameter such that $0< \varepsilon <1$, and $\mathsf {opt}^*$ denote the maximum benefit that can be attained by a fractional solution (one in which requests may be partly served and flow may be split along multiple paths). Let $c_{\min }$ denote the minimum edge capacity, let $d_{\max }$ denote the maximum demand, and let $b_{\max }$ denote the maximum benefit of a request. Let $\varDelta _{\max }$ denote an upper bound on the number of processing stages a request undergoes. If $c_{\min }/(\varDelta _{\max }\cdot d_{\max })=\varOmega ((\log m)/\varepsilon ^2)$, then with probability at least $1-\frac{1}{m}-\textit{exp}(-\varOmega (\varepsilon ^2\cdot \mathsf {opt}^*/(b_{\max }\cdot d_{\max })))$, the algorithm computes a $(1-\varepsilon )$-approximate solution.

Appendices

A Multi-Commodity Flows

Consider a directed graph $G=(V,E)$. Assume that edges have non-negative capacities c(e). For a vertex $u\in V$, let $\mathsf {out}(u)$ denote the outward neighbors, namely the set $\{y \in V \mid (u,y)\in E\}$. Similarly, $\mathsf {in}(u) \triangleq \{ x\in V \mid (x,u)\in E\}$. Consider two vertices s and t in V (called the source and destination vertices, respectively). A flow from s to t is a function $f:E \rightarrow \mathbb {R}^{\ge 0}$ that satisfies the following conditions:

• (i) Capacity constraints: for every edge $(u,v)\in E$, $0 \le f(u,v)\le c(u,v)$.

• (ii) Flow conservation: for every vertex $u\in V\setminus \{s,t\}$

  $$\begin{aligned} \sum _{x\in \mathsf {in}(u)} f(x,u)&= \sum _{y\in \mathsf {out}(u)} f(u,y). \end{aligned}$$

The amount of flow delivered by the flow f is defined by

$$\begin{aligned} |f|&\triangleq \sum _{y\in \mathsf {out}(s)} f(s,y) - \sum _{x\in \mathsf {in}(s)} f(x,s). \end{aligned}$$

Consider a set ordered pairs of vertices $\{(s_i,t_i)\}_{i\in I}$. An element $i\in I$ is called a commodity as it denotes a request to deliver flow from $s_i$ to $t_i$. Let $F\triangleq \{f_i\}_{i\in I}$ denote a set of flows, where each flow $f_i$ is a flow from the source vertex $s_i$ to the destination vertex $t_i$. We abuse notation, and let F denote the sum of the flows, namely $F(e)\triangleq \sum _{i\in I} f_i (e)$, for every edge e. Such a sequence is a multi-commodity flow if, in addition it satisfies cumulative capacity constraints defined by:

$$\begin{aligned} \hbox {for every edge}~(u,v)\in E:&~~~F(u,v) \le c(u,v). \end{aligned}$$

Demands are used to limit the amount of flow per commodity. Formally, let $\{d_i\}_{i\in I}$ denote a sequence of positive real numbers. We say that $d_i$ is the demand of flow $f_i$ if we impose the constraint that $|f_i|\le d_i$. Namely, one can deliver at most $d_i$ amount of flow for commodity i.

The maximum benefit optimization problem associated with multi-commodity flow is formulated as follows. The input consists of a (directed) graph $G=(V,E)$, edge capacities c(e), a sequence source-destination pairs for commodities $\{(s_i,t_i)\}_{i\in I}$. Each commodity has a nonnegative demand $d_i$ and benefit $b_i$. The goal is to find a multi-commodity flow that maximizes the objective $\sum _{(u,v)\in E} b_i \cdot |f_i|$. We often refer to this objective as the benefit of the multi-commodity flow. When the demands are identical and the benefits are identical, the maximum benefit problem reduces to a maximum throughput problem.

A multi-commodity flow is all-or-nothing if $|f_i|\in \{0,d_i\}$, for every commodity $i\in I$. A multi-commodity flow is unsplittable if the support of each flow is a simple path. (The support of a flow $f_i$ is the set of edges (u, v) such that $f_i(u,v)>0$.) We often emphasize the fact that a multi-commodity flow is not all-or-nothing or not unsplittable by saying that it fractional.

B Randomized Rounding Procedure

In this section we overview the randomized rounding procedure. The presentation is based on [11]. Given an instance $F=\{f_i\}_{i\in I}$ of a fractional multi-commodity flow with demands and benefits, we are interested in finding an all-or-nothing unsplittable multi-commodity flow $F'=\{f'_i\}_{i\in I}$ such that the benefit of $F'$ is as close to the benefit of F as possible.

Observation 1

As flows along cycles are easy to eliminate, we assume that the support of every flow $f_i\in F$ is acyclic.

We employ a randomized procedure, called randomized rounding, to obtain $F'$ from F. We emphasize that all the random variables used in the procedure are independent. The procedure is divided into two parts. First, we flip random independent coins to decide which commodities are supplied. Next, we perform a random walk along the support of the supplied commodities. Each such walk is a simple path along which the supplied commodity is delivered. We describe the two parts in detail below.

Deciding which commodities are supplied. For each commodity, we first decide if $|f'_i|=d_i$ or $|f'_i|=0$. This decision is made by tossing a biased coin $\textit{bit}_i\in \{0,1\}$ such that

$$\begin{aligned} \mathbf {Pr} \left[ {\textit{bit}_i=1} \right]&\triangleq \frac{|f_i|}{d_i}. \end{aligned}$$

If $\textit{bit}_i=1$, then we decide that $|f'_i|=d_i$ (i.e., commodity i is fully supplied). Otherwise, if $\textit{bit}_i=0$, then we decide that $|f'_i|=0$ (i.e., commodity i is not supplied at all).

Assigning paths to the supplied commodities. For each commodity i that we decided to fully supply (i.e., $\textit{bit}_i=1$), we assign a simple path $P_i$ from its source $s_i$ to its destination $t_i$ by following a random walk along the support of $f_i$. At each node, the random walk proceeds by rolling a dice. The probabilities of the sides of the dice are proportional to the flow amounts. A detailed description of the computation of the path $P_i$ is given in Algorithm 1.

Definition of $F'$ . Each flow $f'_i\in F'$ is defined as follows. If $\textit{bit}_i=0$, then $f'_i$ is identically zero. If $\textit{bit}_i=1$, then $f'_i$ is defined by

$$\begin{aligned} f'_i(u,v)&\triangleq {\left\{ \begin{array}{ll} d_i &{} \text {if } (u,v)\in P_i,\\ 0&{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Hence, $F'$ is an all-or-nothing unsplittable flow, as required.

C Analysis of Randomized Rounding - Expected Flow per Edge

The presentation in this section is based on [11].

Claim

For every commodity i and every edge $(u,v)\in E$:

$$\begin{aligned} \mathbf {Pr} \left[ {(u,v)\in P_i} \right]&= \frac{f_i(u,v)}{d_i},\\ \mathbf {E} \left[ {f'_i(u,v)} \right]&= f_i(u,v). \end{aligned}$$

Proof

Since

$$\begin{aligned} \mathbf {E} \left[ {f'_i(u,v)} \right]&= d_i \cdot \mathbf {Pr} \left[ {(u,v)\in P_i} \right] , \end{aligned}$$

it suffices to prove the first part.

An edge (u, v) can belong to the path $P_i$ only if $f_i(u,v)>0$. We now focus on edges in the support of $f_i$. By Observation 1, the support is acyclic, hence we can sort the support in topological ordering. The claim is proved by induction on the position of an edge in this topological ordering.

The induction basis, for edges $(s_i,y)\in \mathsf {out}(s_i)$, is proved as follows. Since the support of $f_i$ is acyclic, it follows that $f_i(x,s_i)=0$ for every $(x,s_i)\in \mathsf {in}(s_i)$. Hence $|f_i|=\sum _{y \in \mathsf {out}(s_i,f_i)}f_i(s_i,y)$. Hence,

$$\begin{aligned} \mathbf {Pr} \left[ {(s_i,y)\in P_i} \right]&= \mathbf {Pr} \left[ {\textit{bit}_i =1} \right] \cdot \mathbf {Pr} \left[ {\text {dice } C(s_i,f_i)\text { selects } (s_i,y) \mid \textit{bit}_i=1} \right] \\&= \frac{|f_i|}{d_i} \cdot \frac{f_i(s_i,y)}{\sum _{y \in \mathsf {out}(s_i,f_i)}f_i(s_i,y)} \\&=\frac{f_i(s_i,y)}{d_i}, \end{aligned}$$

and the induction basis follows.

The induction step, for an edge (u, v) in the support of $f_i$ such that $u\ne s_i$, is proved as follows. Vertex u is in $P_i$ if and only if $P_i$ contains an edge whose head is u. We apply the induction hypothesis to these incoming edges, and use flow conservation to obtain

$$\begin{aligned} \mathbf {Pr} \left[ {u\in P_i} \right]&= \mathbf {Pr} \left[ {\bigcup _{x\in \mathsf {in}(u)} (x,u)\in P_i} \right] \\&= \frac{1}{d_i} \cdot \sum _{x\in \mathsf {in}(u)} f_i(x,u)\\&= \frac{1}{d_i} \cdot \left( \sum _{y\in \mathsf {out}(u)} f_i(u,y) \right) . \end{aligned}$$

Now,

$$\begin{aligned} \mathbf {Pr} \left[ {(u,v)\in P_i} \right]&=\mathbf {Pr} \left[ {u\in P_i} \right] \cdot \mathbf {Pr} \left[ {\text {dice }C(u,f_i)\text { selects }(u,v)\mid u\in P_i} \right] \\&= \frac{1}{d_i} \cdot \left( \sum _{y\in \mathsf {out}(u)} f_i(u,y)\right) \cdot \frac{f_i(u,v)}{\sum _{y\in \mathsf {out}(u)} f_i(u,y)}\\&=\frac{f_i(u,v)}{d_i}, \end{aligned}$$

and the claim follows.    $\square$

By linearity of expectation, we obtain the following corollary.

Corollary 1

$\mathbf {E} \left[ {|f'_i|} \right] = |f_i|$.

D Chernoff Bounds

In this section we present material from Raghavan [15] and Young [20] about the Chernoff bounds used in the analysis of randomized rounding.

Fact 1

$e^x \ge 1+x$ and $x\ge \ln (1+x)$ for $x>-1$.

Fact 2

$(1+\alpha )^x \le 1+\alpha \cdot x$, for $0\le x \le 1$ and $\alpha \ge -1$.

Fact 3

(Markov Inequality). For a non-negative random variable X and $\alpha >0$, $\mathbf {Pr} \left[ {X\ge \alpha } \right] \le \frac{\mathbf {E} \left[ {X} \right] }{\alpha }$.

Definition 3

The function $\beta :(-1,\infty ) \rightarrow \mathbb {R}$ is defined by $\beta (\varepsilon )\triangleq (1+\varepsilon )\ln (1+\varepsilon ) - \varepsilon$.

Fact 4

For $\varepsilon$ such that $-1<\varepsilon <1$ we have $\beta (-\varepsilon ) \ge \frac{\varepsilon ^2}{2} \ge \beta (\varepsilon ) \ge \frac{2\varepsilon ^2}{4.2\,+\,\varepsilon }$. Hence, $\beta (-\varepsilon )=\varOmega (\varepsilon ^2)$ and $\beta (\varepsilon )=\varTheta (\varepsilon ^2)$.

Theorem 2

(Chernoff Bound). Let $\{X_i\}_i$ denote a sequence of independent random variables attaining values in [0, 1]. Assume that $\mathbf {E} \left[ {X_i} \right] \le \mu _i$. Let $X\triangleq \sum _i X_i$ and $\mu \triangleq \sum _i \mu _i$. Then, for $\varepsilon >0$,

$$\begin{aligned} \mathbf {Pr} \left[ {X\ge (1+\varepsilon )\cdot \mu } \right]&\le e^{-\beta (\varepsilon )\cdot \mu }. \end{aligned}$$

Proof

Let A denote the event that $X\ge (1+\varepsilon )\cdot \mu$. Let $f(x)\triangleq (1+\varepsilon )^x$. Let B denote the event that

$$\begin{aligned} \frac{f(X)}{f((1+\varepsilon )\cdot \mu )}&\ge 1. \end{aligned}$$

Because $f(x)>0$ and f(x) is monotonously increasing, it follows that $\mathbf {Pr} \left[ {A} \right] =\mathbf {Pr} \left[ {B} \right]$. By Markov’s Inequality,

$$\begin{aligned} \mathbf {Pr} \left[ {B} \right]&\le \frac{\mathbf {E} \left[ {f(X)} \right] }{f((1+\varepsilon )\cdot \mu )}. \end{aligned}$$

Since $X=\sum _i X_i$ is the sum of independent random variables,

$$\begin{aligned} \mathbf {E} \left[ {f(X)} \right]&=\prod _i \mathbf {E} \left[ {(1+\varepsilon )^{X_i}} \right] \\&\le \prod _i \mathbf {E} \left[ {1+\varepsilon \cdot X_i} \right] \quad&(by~Fact~2)\\&\le \prod _i (1+\varepsilon \cdot \mu _i) \\&\le \prod _i e^{\varepsilon \cdot \mu _i} \qquad \qquad&(by~Fact~1)\\&= e^{\varepsilon \cdot \mu } \end{aligned}$$

We conclude that

$$\begin{aligned} \mathbf {Pr} \left[ {A} \right]&\le \frac{e^{\varepsilon \cdot \mu } }{f((1+\varepsilon )\cdot \mu )}\\&=e^{-\beta (\varepsilon )\cdot \mu }, \end{aligned}$$

and the theorem follows.   $\square$

Using the same tools, the following theorem can be obtained for bounding the probability of the event that X is much smaller than $\mu$ (see [6] for the proof).

Theorem 3

(Chernoff Bound). Under the same premises as in Theorem 2 except that $\mathbf {E} \left[ {X_i} \right] \ge \mu _i$, it holds that, for $1> \varepsilon \ge 0$,

$$\begin{aligned} \mathbf {Pr} \left[ {X\le (1-\varepsilon )\cdot \mu } \right]&\le e^{-\beta (-\varepsilon )\cdot \mu }. \end{aligned}$$