Centralized approaches for virtual network function placement in SDNenabled networks
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Abstract
Softwaredefined networking (SDN) and network function virtualization (NFV) represent significant changes to the architecture of data networks. SDN provides clean separation of the control plane from the data plane while NFV helps virtualize functions typically implemented using middleboxes into virtual network functions (VNFs). The network function placement (NFP) problem involves placing VNFs to satisfy the service function chaining (SFC) requests of the flows in the network. Current solutions to this problem are slow and cannot handle realtime requests. In this paper, a static NFP solution that uses a divideandconquer approach is discussed first, with complexity similar to that of existing solutions. It is shown that the solution is complete and sound. Next, customization of this solution to obtain an agile version that trades off precision for time complexity is discussed. A combination of this divideandconquer solution with a modified version of Dijkstra’s algorithm is used to solve the problem for dynamic requests. Finally, a centralized, online SDNenabled approach to solving the problem is proposed. The proposed architecture and algorithms are simulated and analyzed with various system parameters and are shown to scale to large data center network (DCN) topologies.
Keywords
VNF NFV SDN NFP DivideandConquer MANO DCN1 Introduction
Current data networks are built using massive infrastructure and serve diverse functions in commerce, education, research, and social networks. The network itself is built in layers in accordance with the open systems interconnection (OSI) model. The layers can be broadly classified into the following: (i) L1–L3 that handle the physical and network layers responsible for connectivity and routing and (ii) L4–L7 that handle transport over the network, security and applications. Traditionally, L1–L3 (lower layers) were implemented within the network, while L4–L7 (upper layers) were implemented at the edge of the network. Over time, data networks implemented service functions such as firewalls, intrusion detection, proxy services, and video compression. These service functions primarily operate at the upper layers and have been termed as L4–L7 services. These services are complex and have been encapsulated in specialized hardware known as middleboxes. The middleboxes are usually proprietary with a closed architecture. Thus, they impede the cycle of innovation and entail expensive upgrade cycles when the network as a whole must be upgraded to a larger scale. In recent times, middleboxes have been virtualized and replaced by virtual network functions (VNFs) that are more flexible and can scale on demand.
The chain of VNFs that are required to process a flow is called its service function chain (SFC). We formalize NFP problem as the problem of finding the optimal placement of VNFs for a set of flows subject to their SFC requirements, bandwidth, and resource constraints with the objective of minimizing the overall use of the resources. The NFP problem can be handled either statically or dynamically.
The static NFP problem deals with the problem of servicing a static set of incoming flows, each with an SFC requirement, where the routes are provided by an oracle. This reduces the NFP problem to that of allocating the VNFs appropriately along the routes to optimize the use of available resources. Even this static version of the NFP problem has been proven to be NPcomplete [1]. The dynamic NFP problem involves handling dynamic requests for flows with SFC requirements, where even the routes need to be decided by us.

Routing: This component deals with choosing an appropriate route through the network topology to reach the destination considering some path computation objective such as latency, prior placement of VNFs and the bandwidth constraints. Thus, depending on the previous set of VNF allocations and available bandwidth, we need to update the routing metrics to minimize congestion and resource usage in the network. We assume that this component is solved for the static NFP by an oracle.

Allocation: This component deals with allocating the optimal set of resources for the flows over the paths selected by the routing component to provision the SFCs of the flows, considering the resource availability and previous placements. Given the set of routes provided by the routing component, VNFs are allocated on the network elements or switches along the routes to fulfill the SFC requirements while aiming to optimize the use of the compute resources available in the network.
Even though attempts [2, 3] have been made to develop fast and optimal solutions for both of the above components individually, they remain impractical for realworld implementation. Also, optimality of any one of the above components does not guarantee the optimality of the overall solution since the output of one feeds into the other and thus constraining the overall solution. Instead, we aim to solve this problem considering both components simultaneously, for developing realworld online NFP solvers. Additionally, we propose a consolidated SDNenabled architecture and protocol to perform realtime solutions for SFC requests.
The main contributions in this work are the following: (i) A new divideandconquer approach for the problem of static NFP which is theoretically proven to be sound and complete; (ii) A heuristic version of the above solution which allows users to trade off accuracy for fast computation times; (iii) An extended version of this heuristic for dynamic NFP along with a complementary and compatible routing algorithm; (iv) A framework for using the above heuristic for realtime online service of SFC requests; (v) A detailed theoretical and empirical analysis of time complexity and the tradeoff between the quality of solution and time complexity of the algorithm.
The rest of this paper is organized as follows: first, the previous work done by researchers on this problem and the need for better algorithms are described. Next, the proposed algorithm for solving static NFP is described. In the subsequent section, the above solution is extended with a complementary routing algorithm for dynamically servicing SFC requests. Later, a framework is proposed to harness the power of SDN to use the above algorithm for realtime service of such requests. Simulation results with appropriate inferences and analysis to highlight the major advantages are then discussed. Finally, in conclusion, the proposed protocol, empirical results, and future problems to be solved are summarized.
2 Related work
In this section, brief background information on NFV and SDN are presented and the previous approaches to solving the NFP problem are discussed.
2.1 Network function virtualization (NFV)
NFV architecture aims to harness the power of stateoftheart virtualization technologies to virtualize the compute, network, and storage resources of commodity network elements to host VNFs that perform the middlebox functionality. This has the advantage of avoiding investments in specialized hardware, while allowing elastic scaling of the capacity of the functions [4]. Moreover, switches and routers can use their spare computational resources to implement these VNFs [5]. The VNFs perform the same functions as the conventional middleboxes except for the fact that these functions are now implemented in software. This architecture accelerates innovation of the capabilities of the network functions and supports realtime scaling of the virtualized resources as the number of data flows through the network and their service requirements change dynamically [6].
2.2 Softwaredefined networking (SDN)
Another nextgeneration network technology, softwaredefined networking (SDN) [7], is a paradigm of networking that separates the control and the data planes. In this architecture, the data plane is primarily responsible for forwarding the data packets through the network based on flow rules. A centralized controller (or a set of synchronized centralized controllers) is responsible for computing the flow rules based on the state of the network to be pushed to the data plane. A protocol is required for communication between the centralized controller and the switches that implement the forwarding, with OpenFlow [8] being a leading example. SDN allows commodity switches in the data plane to be controlled by control software through standard interfaces. Moreover, in this architecture, the controllers can be modified and upgraded easily to effect changes to the behavior of the network as a whole [9].
2.3 Network function placement (NFP)
The NFV architecture gives rise to many interesting problems. One such problem is about the number of instances of the various VNFs required and their placement in the network [2, 3]. Each flow in the network requires a chain (either a strict order or a partial order) of functions that must process the flow according to some defined network policy. Such a chain is termed as a service function chain (SFC) [11]. Each instance of a function can be applied to several flows and takes up resources in the network that are constrained in their availability. Thus the placement of these functions is a complex optimization problem that takes into account the requirements and the constraints and the overall objective of minimizing the consumption of resources [12].
The problem has been widely studied and an extensive survey of the solutions is provided in [2, 3]. The placement problem is known to be NPComplete [1] and has primarily been solved by treating it as a constraint satisfaction problem.
The work presented in [12] attempts to solve path computation (routing) and function allocation as a unified problem. It combines the path computation constraints and the network function placement constraints into a single instance of a mixed integer quadratically constrained program (MIQCP) and uses a SAT solver to find solutions. Similarly, the work discussed in [13] attempts a coordinated solution that solves both the composition of the network functions in a chain as well as embedding them in the substrate network. The approach described in [1] also integrates the solution for VNF allocation and endtoend demand realization. Some researchers have also attempted to treat VNFs as being capable of getting broken into relatively simple functions that can be distributed and implemented among the switches themselves by using commodity compute, network, and storage resources. The work in [14] proposes a programming model which allows a centralized stateful program to be distributed and deployed using state variables implemented at various switches in the network.
The routing aspect of the NFP problem involves finding the kshortest paths through the network. This problem has been widely studied, and D. Eppstein’s solution [15] is the best known solution that allows looping paths. However, if we consider only loopless paths, then Yen’s algorithm [16, 17] gives the solution in pseudopolynomial time. Two variants of these previous algorithms are proposed in this paper as part of the dynamic NFP solution.
The consolidated approach in this paper for dynamic NFP is in contrast to the above solutions, since we propose an architecture and a protocol which are consistent with current stateoftheart NFV and SDN architectures. The paper addresses the interaction of the two technologies with each other and builds on the prior research work on this topic by the authors [18, 19].
3 Static network function placement
In this section, we formally describe the problem statement for static NFP. Then, we present our sound and complete algorithm to solve the problem. A customizable, agile heuristic version of the algorithm that is exponentially faster than the stateoftheart solutions is proposed. We define completeness of an NFP algorithm as its ability to consider all possible outcomes and generate a solution if one exists and soundness as the property of the output being the optimal solution. Also, a solution is considered more agile than another solution if the algorithm has lower time complexity and customizable if there are parameters that network administrators can tweak to improve the performance according to their needs or available resources.
3.1 Problem definition
As described in previous sections, for the static NFP problem, the paths of the flows are provided by an oracle. The computation of such paths is left for the dynamic NFP solution. Given the paths, we attempt to address the problem of placement of network functions, subject to the various constraints such as the SFC requirement of each flow and the resources available at each switch to deploy the functions. We consider the chain of functions to be of linear order. That is, functions that split or merge the flows are not considered. Similarly, We restrict our discussion to network functions that can be deployed on the network switches [5]. We call these nodes as NFVenabled nodes. Given the flows through the network and their SFC requirements (SFC _{k}), we need to place function instances at the switches such that the overall resource cost is minimized.

Instance cost (I _{j}): This is the resource cost to create an instance of the function on any switch.

Service cost (S _{j}): This is the dynamic resource cost of the function that a switch requires to service one unit of flow data rate.
Modern stateoftheart switches usually have three types of resources—network, computational, and storage. To simplify our discussion, we will capture the capacity of a switch to host network functions as a single unified resource (Cap _{i}). It is easy to extend the model to different types of resources. We formally define our problem statement as follows:
Problem statement: Find the placement of network functions among the switches in a network given a set of flows, with each flow taking a precomputed path through the network and requiring a linear partial order of network functions, each of which has an instance cost and a service cost associated with it, such that the overall cost of placing the network functions for all given flows is minimized.
3.2 Proposed solution
In this section, we define the system and its parameters that define the problem and the approach to model them. Then, we describe our divideandconquer approach to solving the NFP problem. Finally, we present a customizable heuristic version of our algorithm which is more agile.
3.2.1 System description
A VNF is characterized by a threetuple containing (i) the name of the function, (ii) the instance cost (I _{j}), and (iii) the service cost per unit data rate of flows (S _{j}). Although the service cost is taken to be linearly increasing with the data rate, this can be replaced in the algorithm by other functions that are nonlinear. A tenant request is defined as a flow taking a given path through the network and having a specific SFC requirement. Thus, a tenant request is described by an ingress node (src _{k}), an egress node (dst _{k}), the path taken by the flow through the network (P _{kl}), the data rate of the flow (BW _{k}), and the sequence of functions that need to service this flow (SFC _{k}). Often, a flow only needs to be serviced by a set of functions or a partial order rather than a specific sequence of them. The algorithm assumes that such requests have been preprocessed (one such heuristic is presented in [12]) and a specific sequence has been provided as input.
Since the path is also predefined for each tenant request, the only information about the network that is required is the resource availability of the network elements. An instance of a network function placed at a specific switch to service a set of one or more tenant requests is called an allocation. Note that this can be a subset of the set of all requests passing through this switch with the given VNF in their network function sequence. This may be caused due to the limited availability of resources on the node. The output of the algorithm is a set of allocations that satisfy as many input tenant requests as can be fully serviced.
An NFPSystem is an instance of the proposed algorithm that solves the NFP problem or a subproblem. By the definition of the NFP problem, an NFPSystem takes as input the list of possible network functions, the allocations that were done a priori, the set of tenant requests that need to be serviced, and the network describing currently available resource capacities of the network elements. It returns with the set of allocations for the given inputs that has the minimum cost among all possible sets of allocations.
3.2.2 Divideandconquer algorithm (DCA)
This section presents the divideandconquer algorithm (DCA) which solves NFP and is shown to be complete and sound. The pseudocode of this algorithm is described in Algorithm 1. We create the first instance of NFPSystem with the complete available set of network functions, an empty set of allocations, the complete set of tenant requests (in the format described earlier), and the network describing the resource capacities of all the network switches. It starts by finding the set of all possible candidate allocations.
Only maximal allocations are considered initially, i.e., when a function is allocated on a node, it services all requests that pass through the node requesting that function. This will be the optimal allocation strategy if there were infinite resources on each switch. However, in reality, the capacity of the switches is constrained.
For each allocation, the algorithm verifies the cost of servicing the given request. Note that we accrue the instance cost of the network function only if an instance of that function is not already allocated on that node. If an allocation is not feasible, then the tenant request that has the maximum dynamic service cost is removed and the process is repeated. The intuition behind this is to minimize the number of instances of the same type of network function (say, firewall) and, in turn, minimize the instance cost.
If the allocation is left with no requests to serve, then the given request cannot be handled in the current NFPSystem, making the NFPSystem infeasible for any allocation of the function on the specified node. Hence, no update of the optimum allocation is done.
On the other hand, if an allocation is feasible, we remove the resource capacity needed for that allocation and the available capacity on the corresponding switch is reduced. Then a divideandconquer method (line number 6 in Algorithm 1) is used to divide the requests that are being serviced by this allocation of the given function into two parts. For example, consider a request with path A…BCD…E and a VNF chaining request F1…F2F3F4…F5 and let the current allocation place F3 at C for this request. As a result, two new requests are created as <A…BC, F1…F2 > and <CD…E, F4…F5 >.
We recursively create parallel NFPSystem instances to identify the optimal allocation by exploiting the independence of one allocation from the other. Once a feasible allocation has been identified, this allocation is added to the current list of incoming set of allocations to be passed to a subsystem to handle the new set of requests.
A parallel NFPSystem is created as a separate thread to solve a subproblem and return the set of optimal allocations satisfying that new NFPSystem. If the subsystem that handles the new requests is infeasible with the current allocation then the tenant request with the highest data rate is removed. We repeat this process until there is one request left in the allocation. If the allocation has only one request, then there are no further possibilities for reducing the allocation. We deem such a NFPSystem infeasible which does not yield a candidate for optimum allocation.
Note that the function chains and the network paths get smaller for the newly created instances of the NFPSystem. Eventually, the new set of requests will contain requests with a trivial path that has only a single node. At this point, a simple feasibility check is used to check if that node can support all the functions and, if so, they are allocated on that node. Otherwise, it is deemed infeasible. Another trivial case occurs when a request has an empty set of functions to be serviced along a path. In other words, there are no functions required to be allocated on a given path. These requests are discarded since they are trivially satisfied.
Given the set of subsystems, each NFPSystem compares the cost of allocations from each subsystem and keeps track of the one with the least cost. Similarly, it compares the cost among all such allocations by waiting for all the threads to join and the overall minimal allocation is returned to the parent system which spawned that system originally. This algorithm, thus, performs exhaustive search and is therefore as complex as the current solutions based on MIQCP. Thus, it has a time complexity of \(\mathcal {O}\left (e^{poly\log {}(poly)}\right)\). We provide the detailed analysis in Section 6. Therefore, a more agile solution that can act as a dynamic NFP solver is required, which is presented in the next section.
3.2.3 Fast heuristic algorithm (DCAH)
The first and most effective way to improve agility is to bound the branching at every NFPSystem instance. This can be enforced by only looking at top T of the allocations possible at every step for the optimal cost solution. Once the branching is curtailed, the algorithm becomes significantly faster. Therefore, to achieve this optimization, only the top T allocations in line 4 of Algorithm 2 are considered. This requires sorting the allocations in a particular order beforehand such that the top T allocations will reasonably represent the overall system. Some of the possible sorting orders are the decreasing order of number of requests each allocation serves, decreasing order of data rate of the flows that each allocation serves, or the decreasing order of cost to the system due to the allocation.
We introduce another heuristic by reducing the backtracking upon discovering the subsystem created by a verified allocation is infeasible to solve. This can be accomplished by making lines 11–17 optional in Algorithm 2. Note that this will have significant performance impact since this polynomially reduces the upper bound on the depth of the NFPSystem visitor tree. The same is true for removing backtracking unverified allocations (allocations that will not meet the resource constraints), which can be done by making lines 22–28 optional in Algorithm 2. This will not drastically improve the time complexity but will significantly reduce the soundness of the algorithm. Thus, it is recommended that this part is left unmodified in DCAH.
To summarize, we proposed a complete and sound algorithm for static NFP in this section and provided a customizable agile heuristic version of the same, which allows tenants to trade off precision for better computation time.
4 Dynamic network function placement
In this next section, we define the problem of dynamic NFP formally and describe a modular approach which harnesses the power of independent optimal solutions of each of the individual components, namely, routing and allocation. While the problem of placement of virtual network functions (VNFs) has been widely studied, most solutions either look at routing of flows and optimal allocation of VNFs on the route individually or are too slow for practical implementation as online VNF solvers.
Note that each VNF has a specific migration delay (Mig _{l}), a start up delay (Stp _{l}), and a tear down time (Trp _{l}). Static values are assumed for migration delay for a function, since usually the migration happens through an overlay network built in the NFV Management and Orchestration (MANO) architecture of the ETSI (European Telecommunications Standards Institute). Also, homogeneity across all commodity switches for static start up and tear down delay is assumed.
The allocation component of the problem deals with the least cost allocation of network functions given a static set of requests and the corresponding routes that they will take. Note that the allocation optimization problem remains the same as in dynamic NFP as static NFP. The routing part of the problem tries to look for the best possible route for a VNF chaining request where a set of VNFs are statically allocated and routes may loop to satisfy function chains. Our approach ignores the possibility of dynamic VNF migration and may end up using inefficient routes to meet the SFC requirements. However, even the fastest VNF migration [20] is too slow for migration during a flow, hence justifying our assumption.
The combination of the routing and allocation aspects of the NFP problem makes it intractable and most of the complete and sound solutions of the above problems are too slow for deployment in real networks. However, there exist fast heuristics which solve both of the above problems that significantly improve the time complexity. In this section, we propose a consolidated solution which combines such heuristic approaches. However, since the solution is heuristic, neither the soundness nor the completeness of such a solution is guaranteed. To summarize, the proposed solution is a consolidated approach for VNF placement, which combines the objectives of optimal routing as well as leastcost placement VNFs, to provide a realworld online solution for NFP.
4.1 Hybrid Dijkstra’s algorithm
In this section, we discuss an approach for the routing aspect of the dynamic NFP problem that also considers the importance of the reutilizing previously allocated functions which are servicing existing flow requests already. Traditionally, link state routing protocols such as the Open Shortest Path First (OSPF) protocol are used for (i) discovering and maintaining the link state of the network and (ii) using algorithm such as Dijkstra’s to compute shortest paths based on administrative weight or path attributes such as latency. We use the overall latency of allocation as the optimization criterion [21].
However, such path computation is completely oblivious to the already allocated network functions in the network and conceivably, a longer path with network functions which can be reused for the request being serviced, can provide lower latency by avoiding the migration and setup latency of the network functions. Algorithm 3 is a hybrid Dijkstra’s algorithm that addresses this problem by considering the presence of network functions on the commodity switches to compute a path from source to destination.
The algorithm proceeds as follows. The shortest path between the source and the destination using the traditional Dijkstra’s algorithm (line 3) is first identified. Then, to provide alternatives to the shortest path, kshortest paths are computed between the source and destination. These paths are retrieved by excluding individual links of the shortest path. This approach is similar to the ones taken by Yen [16, 17] to find loopless kshortest paths. Thus, the path computed upon removal of each edge is computed and added to the set of paths to be checked (lines 4–6).
where matchingAllocs refers to the number of allocations on the path, where the VNF of the allocation occurs in the function sequence of the request and totalFuncs denotes the total number of functions in the SFC of the request. Note that in the above formula, the coefficient of \(\frac{\text{matchingAllocs}}{\text{totalFuncs}}\) (α_{r}) represents the ratio of importance given to previous allocations versus the actual latency/cost of the path. We use α_{r} = 0.25 for our results but it can be replaced with any other number depending on the ratio of importance you want to give the existing allocated network function. Using the above cost heuristic, the cost of the shortest path (the one with the lowest latency) is found (line 8). Then for all candidate paths in allPaths, we calculate that cost and if it is less than the cost of the currently known lowest cost path, then it is used to replace the shortest path (lines 915). Finally, we return the path with the lowest cost (line 16).
4.2 Fast Centralized NFP Solution (FDCAH)
In this section, a heuristic approach is discussed for finding the optimal allocations of functions to resolve the SFC requirements of a tenant request using the path computed using the abovementioned algorithm. When a request for finding new allocations is handled, there may be allocations already done on the given path. Thus, either existing allocated functions can be used to serve this flow or due to resource constraints, new virtual network functions must be spawned. Using existing allocated functions has the advantage of not waiting for the migration and setup of the VNF instances to be complete, while instantiating new functions will allow load balancing, which may facilitate better service of future NFV requests.
This fast heuristic approach is described to solve the allocation problem based on the paths computed by Algorithm 3 builds upon the solution described in Algorithm 1. Note that, in this approach, the problem always reduces at every computation depth and thus guarantees completion. We will show in Section 6.1.1 that this algorithm is neither complete nor sound, but the deficit in performance is compensated by the immense decrease in time complexity.
5 SDNenabled framework for online NFP
As in all SDNenabled technologies, only the centralized controller will have the full view of the NFT and be able to query the remaining bandwidth or resources of its elements. Also the controller will be a reactive agent, which will respond based on the packets or requests that it receives. We also assume that the MANO layer of the NFV architecture can communicate with the SDN controller, as is prevalent in stateoftheart SDNNFV combined approaches. Whenever a new flow enters the NFT, the first switch receives a description packet, which is forwarded to the SDN controller. This description packet contains the description of the VNF sequence that the incoming flow requires, the destination node, the bandwidth and the time for which the flow of packets will occur. While the controller is finding the optimal route and allocation for the flow, the packets are routed through the controller to the destination via a high latency path. The controller is assumed to have sufficient resources to run all possible VNFs always to service the chaining request of this flow while the dynamic NFP solution is being calculated [22].
The controller first looks for a set of candidate paths that could possibly be optimal routes for this flow. The main constraining resource for routing is the bandwidth of the links in the NFT. Using appropriate cost functions, the controller finds the best possible route of the incoming request. Once the appropriate route is found, this route along with the sequence of VNFs that need to be serviced for this flow is supplied to the online NFP solver. The NFP solver uses the divideandconquer approach to divide the request into smaller requests and find the optimal NFP solution.
It may so happen that there is no possible route or that for the chosen route there is no feasible network function placement found. In such cases, the solution for the request is delayed by a certain amount described as the backoff delay of the system. The solution is reattempted periodically until a threshold upper bound value denoted as the denial cutoff above which the request is denied and the high latency path through the controller is used for the duration of the flow. However, if the solution is found, then the SDN controller removes the bandwidth and resource capacities of the links and nodes respectively according to the allocation in its internal view of the NFT. It then invokes the MANO layer of the NFV architecture to migrate the required VNF containers to the required nodes and to scale each of the VNFs to the updated bandwidth. Thus, the migration delay for a request is defined as the maximum migration delay out of all the migrations of VNFs required for the request. We also define startup delay as the maximum sum of migration and startup delays of the VNFs required to meet the needs of the specified allocations.
Next, the flow rules for the request are pushed by the controller in the respective switches. At every switch, the rule will be as follows:
The rules are added in sequence starting from the destination switch and the last rule is added at the source switch in the path. After this, the incoming flow packets are switched across the data plane. Once the time defined in the description packet is elapsed, the flow is said to have completed. Once the flow is completed, the request ID is removed from all allocations stored by the SDN controller. At this point, the bandwidth capacity is freed in the links and dynamic cost for the given request for all relevant allocations in the SDN controller. This may also lead to some allocations serving no active requests in the NFP system. The controller triggers a teardown for all such allocations—i.e., it tells the MANO layer that the VNF container is no longer required and can be vacated to free the resources. After a corresponding teardown delay, for each such null allocation, the instantiation cost for the VNF container is freed at the switch. This SDNenabled framework harnesses the power of individual heuristics for each of the components and allows a consolidated feedbackbased approach for optimal online service of VNF chaining requests.
6 Experimentation and results
In this section, details of the experimentation and theoretical and empirical analysis of various approaches are presented and it is shown that these approaches are significantly better than prior solutions to the NFP problem.
6.1 Analysis of static NFP
In this section, Algorithm 1 (DCA) is shown to be complete and sound. Then, the time complexity of the DCAH algorithm is discussed to justify that it is polynomial when all the subsystems of a given system are run in parallel. Further, the solution is empirically studied with various parameters and finally shown to scale to even large stateoftheart DCN topologies.
6.1.1 Completeness and soundness of DCA
Completeness of an NFP algorithm is its ability to consider all possible inputs and generate a solution if one exists. This is automatically taken care of for each subproblem in our case by the definition of the inputs. This is guaranteed for a given linear function chain and a given path for every request and given resource capacity of a network node. Thus, DCA is proven to be complete.
Soundness of an NFP algorithm is the property of the output solution being of optimum cost and if the algorithm does not find a solution, then there does not exist a solution. If the algorithm produces a solution, it is clear that at each stage it has found the minimum cost subsystem after each allocation. Also, at each stage, we find and use the allocation which gives us the minimum cost. After each system and its subsystems return the possible allocations, the minimum cost among all possibilities is returned to the parent system. This guarantees that given a feasible NFPSystem, the algorithm finds the optimal solution for all subsystems that are visited. Using this argument recursively, DCA is proven to be sound.
6.1.2 Theoretical complexity analysis
For this analysis, N denotes the number of network elements, M the number of network functions, and K the number of tenant requests.
DCA complexity: A given NFPSystem is started by finding all possible allocations. The total number of such allocations is \(\mathcal {O}(NM)\). Then for each allocation, the allocation is verified and the network is updated, both of which takes \(\mathcal {O}(1)\). The divideandconquer algorithm is then executed to find a new set of requests, which is \(\mathcal {O}(K)\). In the worst case, the number of new requests is twice the number of current requests. Thus the complexity of each run of NFPSystem is \(\mathcal {O}(NM+K)\). Then, the depth that the search tree can grow to is analyzed, which has an upper bound of the number of possible allocations \(\mathcal {O}(NM)\) multiplied by the maximum number of times a function can be allocated on the node, which is \(\mathcal {O}(K)\). Thus, the upper bound of the depth is \(\mathcal {O}(NMK)\). The branching at each step is determined by the possible number of allocations, which is \(\mathcal {O}(NM)\). Thus, the worst case time complexity is \(\mathcal {O}\left ((NM+K)\cdot \left (NM^{(NMK)}\right)\right)\) which is \(\mathcal {O}\left (e^{poly\log {}(poly)}\right)\).
Heuristic complexity: In the heuristic version, the allocations are sorted, which is done in \(\mathcal {O}(NM\log {}(NM))\) time. Thus, the worsecase complexity with the sorting, but without any heuristic bounding becomes \(\mathcal {O}\left ((NM\log {}(NM)+K)\cdot \left (NM^{(NMK)}\right)\right)\). Looking at only the top T allocations will bound the number of branches at each step by T (a constant). This will change the multiplicative factor to \(\mathcal {O}\left (\ldots \cdot \left (T^{(NMK)}\right)\right)\). This leads to the overall complexity of \(\mathcal {O}\left (e^{poly}\right)\), where the exponent is actually quite small in most of the typical scenarios. For T=1, the algorithm becomes polynomial time with complexity \(\mathcal {O}(NM\log {}(NM)+K)\). The next optimization step is to reduce the backtracking that happens when we find an infeasible subsystem which reduces the first term of the complexity to \(\mathcal {O}((NM\log {}(NM))\cdot \ldots)\). Moreover, for T=1, the complexity reduces to \(\mathcal {O}(NM\log {}(NM))\). Note that this complexity is independent of K and is only a function of the network topology and set of possible network functions that can be allocated. This is very important to give performance guarantees in a network system.
6.1.3 Empirical time complexity analysis
Base case parameters: Given a set of nodes N, we generate \(\sqrt {N}\) requests. Each node has a resource capacity of N^{0.8} and each request has a random number of nodes in its path size, chosen uniformly from the range \(\sqrt [\leftroot {2}\uproot {2}3]{N}\) to \(\sqrt {N}\). Each request has a data rate of one unit and the nodes on the path are randomly chosen from 1 to N. The number of functions requested for each path are chosen randomly from 1 to \(\sqrt [\leftroot {2}\uproot {2}4]{N}\). Each of those functions are chosen from a predecided set of 10 functions. The value of T chosen for the base case is 1.
Next, the base case is compared with a system having N requests instead of \(\sqrt {N}\). Increasing the number of requests increases computation time exponentially, as seen in Fig. 7b. This is because increase in the number of requests increases the number of possible allocations exponentially. This reaffirms the results of our complexity analysis. Then, T is varied from 1 to 4 to find the complexity of increasing the number of top T allocations considered. As expected, increasing T results in an exponential increase in the time taken for running the system. This is shown in Fig. 7c, where time is plotted on a log scale.
Another important dimension to analyze the solution is to find at what values of N does it become impractical to be run for different values of T. For this analysis, the base case is checked with values of T varying from 2 to 10 and values of N increasing in jumps of 25 each. The cutoff time for the simulation is taken as 30 min beyond which the solution becomes impractical. The same experiment is repeated for DCA as well with N increasing in jumps of 5 each. The results are shown in Fig. 7d. It is seen that to make this algorithm useful for practical purpose for values of T beyond 2, significant additional algorithm optimization is required. Also, with increase in values of T, the maximum value of N, for which it is reasonable, decreases exponentially. It should also be noted that the DCA did not finish in 48+ h for N=45.
6.1.4 Precision time tradeoff analysis
Precision Time TradeOff
Resource units  T=1  T=3  T=5  T= ∞ 

N=25  7.2  7.2  7.2  7.2 
N=50  14  14  14  14 
N=100  17.6  17.6  17.6  17.6 
6.1.5 Scalability in DCNs
Since NFV is well suited for data center networks (DCN), a DCN scenario is also considered. Here, the flows are primarily between leaf nodes or between core nodes and leaf nodes. The topology used is the fattree topology described in [23].
The capacity of each leaf node was kept at \(\frac {N}{2}\), each edge and aggregate node at \(\frac {N^{2}}{4}\), and each core node at \(\frac {N^{3}}{8}\) in a Npod FatTree topology. The data rate of each flow was kept constant at 1.0 unit. This was done so as to not affect the symmetry of flow requests from each node. The number of requests generated for an Npod FatTree was \(\sqrt {\mathcal {N}}\), where \(\mathcal {N}\) denotes is the number of nodes is an Npod FatTree. The number of functions in the function chain were specified to be randomly chosen between 1 to 3 for coretoend paths and 3 to 5 for endtoend paths proportional to the worst case number of nodes in the paths. N was varied from 2 to 48 with jumps of 2 each.
6.2 Analysis of dynamic NFP
In this section, the execution time (in ms) of the proposed algorithm is studied with varying parameters. The execution times for randomly generated connected topologies are studied first, followed by DCN topologies of Kpod Fat Trees.
6.2.1 Nonsymmetric topologies

Heavy:N requests arriving in N ms.

Medium:N\(^{\frac {4}{5}}\) requests arriving in N ms.

Sparse:\(\sqrt {N}\) requests in arriving in N ms.
A backoff delay of 0.1N ms and a denial delay as 2N ms are used. Each request is generated within these parameters with random ingress and egress nodes. The number of functions in the functionSequence (chosen randomly) are Min(\(\sqrt {N}\),Random(5,...,15)). Each NFVenabled node is provided with a capacity support approximately Min(\(\sqrt {N}\),10) flows on it. Each link is given enough capacity to support an average of five requests (chosen randomly). The whole experiment is repeated 10 times to discard results due to anomalous behavior.

Path computation time/delay: This is the time taken by the hybrid Dijkstra’s algorithm at the SDN controller to find the candidate paths and choose one out of them for a given request.

NFP computation time/delay: This is the time taken by the heuristic NFP algorithm at the SDN controller to find the optimal allocation for the given request on the chosen path.

Feasibility delay: This is the delay of backoff and denial caused due to lack of feasible solutions at a given time for the incoming request.

Migration time/delay: This is the time taken by the VNF containers to be migrated to the appropriate NFVenabled switches.
The NFP computation time (Fig. 9b) for 1000 nodes and heavy requests is 920 ms at an average and is only a function of the size of the path and the function sequence. Since the path size is constrained to 1000 in the worst case and the function sequence to 15, this result is pretty reasonable to characterize realworld situations. The feasibility delay (Fig. 9c) of 2600 ms for 1000 nodes and heavy requests is actually more than the NFP computation delay which suggests that, either better NFP solutions can be considered until the feasibility delay becomes less than the NFP computation delay, or the backoff delay and the limit for denial delay can be reduced. The migration delay (Fig. 9d) for 1000 nodes and heavy requests is 1463 ms though realworld values of migration delays will give a better view on this particular factor.
6.2.2 DCN topologies: Kpod Fat Trees

Endtoend flows: These are primarily between leaf nodes (such as for MapReduce tasks). The path computation just has to look at all cores and once a core is fixed, the path is automatically defined (direct path).

Coretoend flows: These are primarily between core nodes and leaf nodes (such as for content delivery).
The topology to be used for experimentation is the fattree topology described in [23]. For a Kpod fat tree, K flows are initiated every K^{3} ms and K^{2} flows every K^{3} ms. The backoff delay is taken as K ms and the denial delay as K^{3} ms. Each request is generated as defined above between random core and leaf node or random leaf and leaf node. The number of functions in the function sequence (chosen randomly) Random (1,...,3) for Core to Leaf Node flows and Random(1,...,5) nodes for Leaf to Leaf Node flows. The whole experiment is repeated 10 times to discard data from anomalous behavior.
7 Conclusions
The network function placement (NFP) problem that requires meeting the service function chaining (SFC) requirements of data flows in softwaredefined networks is a critical problem in the field of network function virtualization. This paper proposes a solution based on the divideandconquer strategy for the NFP problem and shows that it is complete and sound. The solution is applied to both static NFP requests and dynamic NFP requests. It is shown that this model can be customized to become a heuristic algorithm which trades off precision for time complexity using various parameters. It is theoretically shown that there exists a set of parameters for which our algorithm has a complexity that is just a function of the topology and not dependent on the number of requests received. The algorithm was studied for various parameter values to show their influence on the time complexity of the heuristic algorithm. Based on the results, the algorithm scales to large stateoftheart DCN topologies. In the case of dynamic NFP, existing solutions to optimize individual components of this complex problem are too slow for being used as online NFP solvers. This paper combines the divideandconquer approach to NFP with a modified Dijkstra’s algorithm to propose an online NFP solver. Moreover, a framework based on SDN controllers and the ETSI MANO architecture is presented for deploying flows with corresponding VNF chains. The proposed protocol is simulated on nonsymmetric as well as DCN topologies and it is shown that even for DCNs beyond the scale of current use, the algorithm takes time in the order of seconds. With further work, these heuristic components can mature into fast and more optimal algorithms meeting the constraint of solving the problem in real time. There is also a possibility of adding fault tolerance to the proposed protocol to handle failures of links, nodes, or VNFs.
Notes
Acknowledgements
The authors would like to acknowledge their fellow researchers at the DonLab Research Group of the Department of Computer Science and Engineering, IIT Madras for their valuable input and feedback.
Funding
There are no external sources of funding for this research work.
Authors’ contributions
AG is the main researcher for this work and the first author of this paper. The simulation framework was programmed by AG and used to generate the results. A major portion of the work was completed while AG was a B.TechM.Tech Dual Degree student at Indian Institute of Technology Madras. He is currently a Ph.D. scholar at Carnegie Mellon University.
AA is a coresearcher in this work. He has been working on network function placementrelated problems, analyzing various aspects of both static and dynamic versions of the problem as well as comparison of centralized and distributed approaches. AA is a Ph.D. scholar at the Indian Institute of Technology Madras. He is responsible for final editing of the manuscript.
KMS provided key input to defining the problem and various approaches to solution as well as analysis and verification of the results. He is currently the head of the Department of Computer Science and Engineering at Indian Institute of Technology Madras. He also leads the DonLab Research Group, which, among other things, is researching network function virtualization. He is an IEEE Fellow, ACM Distinguished Scientist and INAE Fellow.
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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