Allocation of Computation-Intensive Graph Jobs over Vehicular Clouds in IoV

TL;DR

This paper proposes a novel framework for allocating computation-intensive graph jobs over vehicular clouds in the Internet of Vehicles, optimizing resource utilization and minimizing job completion time through both optimal and heuristic algorithms.

Contribution

It introduces a new framework for graph job allocation in vehicular clouds, including optimal solutions for low-traffic and a low-complexity randomized approach for rush-hours.

Findings

Optimal solutions for low-traffic scenarios.

A low complexity randomized allocation mechanism for rush-hours.

Simulation results show improved efficiency and reduced costs.

Abstract

Graph jobs represent a wide variety of computation-intensive tasks in which computations are represented by graphs consisting of components (denoting either data sources or data processing) and edges (corresponding to data flows between the components). Recent years have witnessed dramatic growth in smart vehicles and computation-intensive graph jobs, which pose new challenges to the provision of efficient services related to the Internet of Vehicles. Fortunately, vehicular clouds formed by a collection of vehicles, which allows jobs to be offloaded among vehicles, can substantially alleviate heavy on-board workloads and enable on-demand provisioning of computational resources. In this paper, we present a novel framework for vehicular clouds that maps components of graph jobs to service providers via opportunistic vehicle-to-vehicle communication. Then, graph job allocation over vehicular clouds is formulated as a non-linear integer programming with respect to vehicles' contact duration and available resources, aiming to minimize job completion time and data exchange cost. The problem is addressed for two scenarios: low-traffic and rush-hours. For the former, we determine the optimal solutions for the problem. In the latter case, given the intractable computations for deriving feasible allocations, we propose a novel low complexity randomized graph job allocation mechanism by considering hierarchical tree based subgraph isomorphism. We evaluate the performance of our proposed algorithms through extensive simulations.

Equations

$$y_{jj^{\prime}}=\left\{\begin{array}[]{rcl}1,&{\forall e_{ii^{\prime}}^{A}\in{\bm{E}}^{{\bm{A}}},if~{}j\neq j^{\prime}~{}and~{}{x}_{ij}\cdot x_{i^{\prime}j^{\prime}}=1},\\ 0,&{otherwise},\\ \end{array}\right.$$

$$U^t(\mathbf{X})=\max ([{\sum }_{i=1}^n{x}_{ij}{t}_{jm}^{Trans}{]}_{1\le j\le m-1})+t^{Exec},$$

$$U^c(\mathbf{Y}(\mathbf{X}))=\frac{1}{2}\cdot {\sum }_{j=1}^m{\sum }_{j^{\prime }=1}^m{y}_{j{j}^{\prime }}\cdot c_{j{j}^{\prime }}^{Exch},$$

$$\underset{\mathbf{X}}{\min }{\alpha }_1{U}^t\left(\mathbf{X}\right)+{\alpha }_2{U}^c(\mathbf{Y}(\mathbf{X}))$$

$$(a).\sum _{i=1}^n{x}_{ij}\le |{\mathbf{\kappa }}_{\mathbf{j}}|,\forall s_j\in \mathbf{M},$$

$$(b).e^{-({|t}_{jm}^{Trans}-t_{j^{\prime }m}^{Trans}|+{\omega }_{i{i}^{\prime }}^A){\lambda }_{j{j}^{\prime }}}\ge \epsilon ,$$

$$\forall e_{i{i}^{\prime }}^A\in {\mathbf{E}}^{\mathbf{A}},\text{mbox}if{s}_j\ne s_j^{\prime }\text{mbox}and{x}_{ij}\cdot x_{i^{\prime }{j}^{\prime }}=1,$$

$$(c).e^{-{\lambda }_{jm}\sum _{i=1}^n{x}_{ij}{t}_{jm}^{Trans}}\ge \xi ,\forall x_{ij}=1,j\ne m,$$

$$(d).\sum _{j=1}^m\sum _{i=1}^n{x}_{ij}=n,$$

Full text

I Introduction

The past decade has witnessed great advances in information and communication technologies (ICTs). The proliferation of mobile traffic, smart mobile devices, and numerous related new applications have promoted the rapid development and widespread popularity of wireless communication networks (WCNs), as well as the Internet of Things (IoT). Smart devices such as smartphones and tablets with network access have experienced swift growth in number and variety, among which, smart vehicles are considered as the next frontier in the automotive revolution, in which the number of interconnected vehicles is predicted to reach 250 million by 2020 [1]. As a result, the Internet of Vehicles (IoV) [2] is experiencing the significant transition of merging with wireless networks to create a powerful Intelligent Transportation System (ITS), where autonomous and semi-autonomous driving represents a future trend [3], [4] . This technology promises to substantially improve driving safety through intelligent operations such as obstacle detection, collision avoidance, and experience of on-board entertainment including online games and augmented reality (AR) [5]. Furthermore, technological advances in smart on-board equipment, including computing processors and various sensing devices (e.g., on-board cameras and high-quality sensors), can host perception-related applications with innovative, computation-intensive features such as simultaneous localization and mapping (SLAM), user face/gesture recognition [6], as well as abnormal traffic identification and warning; such applications will greatly influence the way people work and live in the future. Many of such applications can be characterized by graph jobs, wherein a job is represented by a graph, the vertices (components)111A vertex in the graph job can be considered as a component describing a sub-job. We use “component” instead of “vertex” in the rest of this paper so as to accurately represent the physical significance. of which denote either data sources or data processing, with edges corresponding to data flows between vertices [7]-[9]. However, constraints related to computational resources and the capabilities of on-board equipment [7], [10] pose major challenges to IoV, such that the inherent limitation of a single smart vehicle can hinder the fulfillment of job requirements.

To cope with the extensive and ever-changing application demands of smart vehicles, mobile computation offloading (MCO) technology constitutes a new paradigm allowing jobs to be offloaded to cloud servers for execution by integrating communication and computing technologies [11]. Compared with remote cloud servers, deploying multi-access edge computing (MEC) [12] servers located near access points such as road side units (RSUs) can provide services that alleviate severe transmission delays and performance degradation; however, users may still experience signal coverage limitations (e.g., applications such as emergency medical diagnosis in disaster scenes where infrastructures are destroyed and remote regions such as mountains and deserts without RSU coverage) and resource constraints, especially during high-traffic periods. As a result, the vehicular cloud (VC) [13] has been introduced recently as a type of mobile device cloud (MDC), where vehicles with idle resources, named “service providers” (SPs), within a job owner’s (JO’s) communication range can act as mobile computing servers and form a cloud mainly by leveraging vehicle-to-vehicle (V2V) communication technology [14], [15] to process the job in parallel. This technology significantly relieves heavy on-board workloads for JOs and accelerates job completion. The feedback duration of computation results is often ignored given that their data size is much smaller than that of the application data [10], [11]; alternatively, results can be transferred back to the JO via a multi-hop V2V routing path or uploaded to RSUs for future delivery in case the V2V connection is disrupted.

In this paper, we introduce a novel VC-based IoV framework. Without loss of generality, it is assumed that each VC contains one JO with a graph job and a collection of SPs that can interact with the JO via one-hop V2V communications. Moreover, the V2V communication configuration of each VC supports data flows corresponding to graph edges. We utilize slot-based222An idle slot is seen as a resource block that can run one component of a graph job [8], [9]. representation to quantify the available resources at each SP [9], where the number of available slots differs from one vehicle to another based on their heterogeneous resource capabilities. The job completion time and data exchange cost represent key concerns in designing the principal mechanism for allocation of computation-intensive graph jobs over VCs, the latter of which occurs when two connected job components are mapped to slots of different SPs. In our proposed allocation mechanism, each job component is efficiently mapped (offloaded) to an applicable idle slot of a SP under constraints of opportunistic contacts and available resources.

Our main contributions can be summarized as follows:

We first establish an IoV framework containing VCs, which enables computation-intensive graph jobs to be mapped (offloaded) to SPs to overcome limitations in the on-board resources and computational capabilities of JOs by capitalizing on the opportunistic contact duration between vehicles. 2. 2.

We formulate the graph job allocation problem as a nonlinear integer programming (NIP) problem under constraints of opportunistic contacts between SPs and available slots aiming to minimize the job completion time and data exchange cost. 3. 3.

To tackle the aforementioned NIP problem, we first focus on low-traffic IoV scenarios, for which we develop a graph job allocation algorithm to find the optimal solution. This approach relies on addressing the sub-graph isomorphism problem333The subgraph isomorphism problem is a computational task in which two graphs H1 and H2 are given as the input, and one must determine whether H1 contains a subgraph that is isomorphic to H2., which is known to be NP-complete [8, 22]. This makes our first proposed algorithm ineffective upon having a high vehicular density or equivalently large network size. To address this issue, we propose a randomized graph job allocation mechanism by considering hierarchical tree based subgraph isomorphism, which enjoys a low computational complexity, and conforms well to fast-changing environments in IoV. 4. 4.

We conduct a thorough numerical analysis to evaluate the performance of the proposed algorithms under different graph job topologies, traffic densities, vehicular cloud configurations and numbers of idle slots.

The rest of this paper is organized as follows: after describing the motivation, feasibility, and related work in Section II, we introduce system models in Section III. The problem formulation is given in Section IV. Then, in Section V and VI, the optimal allocation mechanism and the hierarchical tree based randomized allocation mechanism are presented, respectively. We evaluate the performance of the proposed mechanisms in Section VII, and provide conclusion, challenges, as well as future works in Section VIII.

II Background and Related work

II-A Background

The increasing demands on autonomous driving coupled with the vast amount of data collected by rapidly developed on-board equipment has bolstered the popularity of computation-intensive applications. However, inherent drawbacks (such as limited computational resources and capability, as well as perceptive sensor range) of a single smart vehicle can often result in unsatisfactory computational performance of such applications [16]. Moreover, constraints on wireless coverage of base stations (BSs) and RSUs, especially in remote regions, render it difficult for smart vehicles to readily enjoy services from cloud computing servers.

VCs can efficiently realize coordination among smart vehicles mainly by utilizing V2V communications, taking full advantage of opportunistic connections, and exploiting idle resources in fast-changing topologies [13, 14].

II-B Related work

The increasing popularity of computation-intensive applications offers great convenience; however, such applications usually place a substantial burden on on-board equipment in IoV, often going beyond the capability of a smart vehicle. This fact has resulted in a growing interest in studying efficient computation offloading strategies. The existing literatures on computation-intensive application offloading can be roughly divided into two categories: 1) those where applications are directly mapped as bit streams and considered as a collection of sub-applications without considering the their dependencies, such as [17]-[19]; 2) those explicitly considering the structure of applications which can be modeled as directed/undirected graphs, such as [6],[7],[9],[10],[20],[21]. A reliability-oriented stochastic optimization model in vehicle-infrastructure systems is proposed in [17] based on the dynamic programming for computation offloading considering the deadline constraint on application execution. A hierarchical IoV framework with backup computing servers is introduced in [18], where a Stackelberg game theoretic approach is adopted to design an optimal multilevel offloading scheme, which maximizes the utilities of both the vehicles and the computing servers. The multiple-vehicle computation offloading in vehicular edge networks is studied in [19] and formulated as a multi-user computation offloading game, where the existence of Nash equilibrium (NE) is proved and a distributed algorithm is proposed to compute the equilibrium. In the aforementioned works, the inner structure of the applications is ignored, which has become more and more important when designing computation offloading strategies.

Over the past years, only a few studies have investigated the allocation of graph-based computation-intensive applications. In the context of cloud computing assuming a static network topology, a low-complexity randomized scheduling algorithm without service preemptions is studied in [9], which stabilizes the system with graph arrivals/departures, and thus facilitates a smooth tradeoff between minimizing the average partitioning cost and average queue length. In the context of mobile computing environments, where service providers are static, a dynamic offloading algorithm for directed-graph-based jobs based on Lyapunov optimization is presented in [6], which satisfies the requirements of energy conservation and application execution time. Applications are modeled by directed taskgraphs in [7] where nodes represent tasks and edges indicate the dependency of tasks. Both sequential and concurrent task offloading algorithms are proposed to minimize the application completion time. It is assumed in [10] that each mobile device has several independent tasks as a set and the task offloading scheduling and transmit power allocation in MEC systems are jointly optimized. Moreover, a low-complexity sub-optimal algorithm is proposed to minimize the weighted sum of the execution delay and device energy consumption based on alternating minimization. The authors in [20] focus on scheduling parallel jobs composed of a set of independent tasks and consider energy consumption while minimizing job completion time. A fast hybrid multi-site computation offloading mechanism is proposed in [21], which finds the offloading solution in a timely manner by considering the application size, where applications are modeled as weighted relation graphs.

The rapid growth of the automobile industry has brought about more and more on-board resources in vehicles, which can be dynamically reallocated to support opportunistic job allocations. However, the aforementioned works paid inadequate attention to on-board resource utilization and optimization and users’ mobilities especially for service providers, which is highly relevant in IoV, where high vehicular mobility leads to difficulties in opportunistic contact between vehicles in fast-changing network topologies. To the best of our knowledge, this paper is among the first to study the allocation of computation-intensive graph jobs over VCs while considering their inherent characteristics, where the graph job allocation problem over vehicular clouds is abstracted as a novel subgraph isomorphism problem under opportunistic communication constraints.

III Problem Overview and System Model

III-A Problem Overview

We propose a framework in which graph jobs can be offloaded to several SPs in the relevant VC via one-hop V2V channels. Different SPs have different numbers of idle slots, and each slot can process one component of a graph job. Note that structural characteristics exist in the offloaded computational jobs and the IoV topology; thus, we model both offloaded jobs and VCs as weighted undirected graphs. Under constraints of limited opportunistic contact duration between vehicles and idle resources, each VC aims to effectively allocate all the components of a graph job to available SPs while minimizing job completion time and data exchange cost.

III-B Vehicular Contact Model

An opportunistic contact event between vehicle $j$ and $j^{\prime}$ occurs during $\tau\mathrm{\in(}\tau_{\mathrm{1}},\tau_{\mathrm{2}}\mathrm{)}$ if the following conditions are satisfied: $\|L_{j}(\tau_{\mathrm{1}})-L_{j^{\prime}}\mathrm{(}\tau_{\mathrm{1}}\mathrm{)}\|>R$, $\|L_{j}(\tau)-L_{j^{\prime}}\mathrm{(}\tau\mathrm{)}\|\leq R$ and $\|L_{j}(\tau_{\mathrm{2}})-L_{j^{\prime}}\mathrm{(}\tau_{\mathrm{2}}\mathrm{)}\|>R$, where $L_{j}\mathrm{(}\tau\mathrm{)}$ and $L_{j^{\prime}}\mathrm{(}\tau\mathrm{)}$ denote vehicular locations at time $\tau$, $\|\cdot\|$ represents the Euclidean distance, and $R$ indicates the communication radius of vehicles. It is assumed that the contact duration between two vehicles obeys the exponential distribution [23] with parameter ${\mathrm{\lambda}}_{jj^{\prime}}$; hence, the probability that the contact duration $\mathrm{\Delta}\tau_{jj^{\prime}}$ between vehicle $j\mathrm{}$ and $j^{\prime}$ is larger than $T$, $T>0$ can be calculated as $P(\Delta\tau_{jj^{\prime}}>T)=e^{-{T\lambda}_{jj^{\prime}}}$. Correspondingly, the larger the value of this probability is, the more ensurance can be achieved to protect successful data transmission between moving vehicles.

III-C Model of Graph Jobs

Consider a collection of graph jobs ${\bm{J}}$ with different types, each of which can be described as a graph** ${\bm{G}}_{{\bm{a}}}^{{\bm{A}}}=({\bm{V}}_{{\bm{a}}}^{{\bm{A}}},{\bm{E}}_{{\bm{a}}}^{{\bm{A}}},{\bm{W}}_{{\bm{a}}}^{{\bm{A}}})$**, containing a set of components (computations) ${\bm{V}}_{{\bm{a}}}^{{\bm{A}}}=\{\upsilon_{i}^{A_{a}}|i\in{\bm{N}}_{{\bm{a}}}\}$,

where each component requires a slot, along with a set of edges** ${\bm{E}}_{{\bm{a}}}^{{\bm{A}}}=\{e_{ii^{\prime}}^{A_{a}}|i\in{\bm{N}}_{{\bm{a}}},\mathbf{}i^{\prime}\in{\bm{N}}_{{\bm{a}}}\mathbf{},i\neq i^{\prime}\}$** with associated weight** ${\bm{W}}_{{\bm{a}}}^{{\bm{A}}}=\{\omega_{ii^{\prime}}^{A_{a}}|i\in{\bm{N}}_{{\bm{a}}},i^{\prime}\in{\bm{N}}_{{\bm{a}}}\mathbf{},i\neq i^{\prime}\}$**, and $\left|{\bm{N}}_{{\bm{a}}}\right|$ denotes the number of components of job type $a\in\bm{J}$. Each graph ${\bm{G}}_{{\bm{a}}}^{{\bm{A}}}$ represents how the computation is supposed to be splitted among components in ${\bm{V}}_{{\bm{a}}}^{{\bm{A}}}$. In the above notations, the superscript $\bm{A}$ denotes the application. Edges represent data flows between these components; the weight $\omega_{ii^{\prime}}^{A_{a}}$ of edge $e_{ii^{\prime}}^{A_{a}}$ indicates the requested connecting duration between components $\upsilon_{i}^{A_{a}}$ and $\upsilon_{i^{\prime}}^{A_{a}}$. In other words, the contact duration of SPs that are handling components $\upsilon_{i}^{A_{a}}$ and $\upsilon_{i^{\prime}}^{A_{a}}$ should be equal to or larger than $\omega_{ii^{\prime}}^{A_{a}}$. An example of a graph job with five components is illustrated in Fig. 1(a).

III-D Model of Vehicular Clouds

Consider a scenario with $|{\bm{O}}|+|{\bm{S}}|$ vehicles moving on north-south and east-west roads, where $o_{i}\in{\bm{O}}$ are JOs and $s_{j}\in{\bm{S}}$ are SPs, each of which has a different number of idle slots. In one snapshot, vehicles are partitioned into $|{\bm{O}}|$ VCs, ${\bm{Q}}=\{{VC}_{1},{VC}_{2},...,{VC}_{|{\bm{O}}|}\}$. Each ${VC}_{i}\in{\bm{Q}}$ contains one JO $o_{i}$ and several SPs, all of which can communicate with $o_{i}$ via a one-hop V2V channel. Notably, because $o_{i}$ may have idle slots, each JO is considered as a virtual SP; as such, each VC can be regarded as an undirected connected graph. In this article, ${VC}_{i}$ is considered to provide computational service to process the graph job owned by $o_{i}$ in parallel. Suppose that there is no interference among VCs, we take one VC in ${\bm{Q}}$ as an example in analyzing graph job allocation mechanisms in different traffic scenarios.

Consider a VC containing a collection of SPs ${\bm{M}}$, where each SP $s_{j}\in{\bm{M}}$** has a set of idle slots $\mathbf{\bm{\kappa}}_{{j}}$ **that are fully connected with each other and can be used to run up to $\mathrm{\mathbf{|}}{\bm{\kappa}}_{{j}}\mathbf{|}$ processes (components) in parallel. For notational simplicity, $m=|{\bm{M}}|$ represents the number of SPs (including JO) in this VC and $s_{m}$ denotes the virtual SP corresponding to the JO. The VC can be represented as a graph ${\bm{G}}^{{\bm{S}}}=({\bm{V}}^{{\bm{S}}},{\bm{E}}^{{\bm{S}}},{\bm{W}}^{{\bm{S}}})$, consisting of a SP set ${\bm{V}}^{{\bm{S}}}=\{s_{j}|s_{j}\in{\bm{M}}\}$; an edge set ${\bm{E}}^{{\bm{S}}}=\{e_{jj^{\prime}}^{S}|s_{j}\in{\bm{M}},s_{j^{\prime}}\in{\bm{M}},j\neq j^{\prime}\}$, where each $e_{jj^{\prime}}^{S}$ takes a binary value indicating that $s_{j}$ can communicate with $s_{j^{\prime}}$ via a one-hop V2V channel ($e_{jj^{\prime}}^{S}=1$) or not ($e_{jj^{\prime}}^{S}=0)$; and the corresponding weight set ${\bm{W}}^{{\bm{S}}}=\{\lambda_{jj^{\prime}}|s_{j}\in{\bm{M}},s_{j^{\prime}}\in{\bm{M}},j\neq j^{\prime}\}$ where $\lambda_{jj^{\prime}}$ denotes the parameter of the corresponding exponential distribution on contact duration between $s_{j}$ and $s_{j^{\prime}}$. The superscript $\bm{S}$ in the notations denotes the service. Please note that there are two graph models considered in this study, one for the structure of the graph jobs discussed in III.C and the other for the topology of VC discussed here. Each idle slot can run one component of a graph job and the execution time of which is denoted as $t^{Exec}$, which is assumed to be the same for all components of the graph job. The proposed model for VCs is depicted in Fig. 1(b).

III-E Candidate

An important construct in this article is the concept of a candidate. For any job type, there are several ways (an exponentially large number) in which a job can be distributed over the VC as depicted in Fig. 1(a). A candidate corresponds to a sub-graph with the same structure as the job in a VC, which satisfies the requested connecting durations for all pairs of connected components.

III-F Data Exchange Cost

We assume the cost of data exchange between two slots on the same service provider to be zero; otherwise, $c_{jj^{\prime}}^{Exch}$ is defined as the cost if two connected components are allocated to slots of different SPs, $s_{j}$ and $s_{j^{\prime}}$. This captures the cost incurred from the total traffic exchange among different SPs in one VC. Several examples of the data exchange cost are presented in Fig. 1(a).

IV Problem Formulation of Computation-Intensive Graph Job Allocation Over Vehicular Clouds

To better analyze the framework of graph job allocation, we mainly study the problem in one VC since the proposed mechanism is universal across all job types and VCs. Hence, we use simpler notations in which a graph job is represented as: ${\bm{G}}^{{\bm{A}}}=({\bm{V}}^{{\bm{A}}}$, ${\bm{E}}^{{\bm{A}}}$, ${\bm{W}}^{{\bm{A}}})$ that contains components set ${\bm{V}}^{{\bm{A}}}=\{\upsilon_{i}^{A}|i\in\textbf{{N}}\}$; edge set ${\bm{E}}^{{\bm{A}}}=\{e_{ii^{\prime}}^{A}|i\in{\bm{N}}$, $i^{\prime}\in{\bm{N}},\mathbf{}i\neq i^{\prime}\}$ and the associated weight set ${\bm{W}}^{{\bm{A}}}=\{\omega_{ii^{\prime}}^{A}|i\in{\bm{N}},i^{\prime}\in{\bm{N}},i\neq i^{\prime}\}$, where $n=\left|{\bm{N}}\right|$ denotes the total number of components in the graph job. Moreover, a VC is defined as graph ${\bm{G}}^{{\bm{S}}}=({\bm{V}}^{{\bm{S}}},{\bm{E}}^{{\bm{S}}},{\bm{W}}^{{\bm{S}}})$ with SP set ${\bm{V}}^{{\bm{S}}}$, edge set ${\bm{E}}^{{\bm{S}}}$, and a weight set corresponding to edges ${\bm{W}}^{{\bm{S}}}$, as mentioned in Section III.D. We denote the data transmission duration of one component from $s_{m}$ (the JO) to $s_{j}$ as $t_{jm}^{Trans}$ ($1\leq j\leq m$), which is related to channel conditions, packet loss and retransmissions as well as transmission power. Consider the binary indicator $x_{ij}$ for which $x_{ij}=1$ if component $\upsilon_{i}^{A}$ is assigned to $s_{j}$, and $x_{ij}=0$, otherwise. As for the cost model of job partitioning among different SPs, let the binary indicator $y_{jj^{\prime}}=1$ denote the existence of data exchange between $s_{j}$ and $s_{j^{\prime}}$; otherwise, $y_{jj^{\prime}}=0$.

Based on the above notations, the indicator $y_{jj^{\prime}}$ can be represented as a piecewise function of $x_{ij}$ and $x_{i^{\prime}j^{\prime}}$ shown as equation (1).

[TABLE]

The job completion time can be calculated as:

[TABLE]

where ${\bm{X}}=[x_{ij}]_{{1\leq i\leq n,1\leq j\leq m}}$ defines the matrix of indicator** $x_{ij}$**, the first term describes the data transmission duration given that each JO can offload different numbers of components to different SPs in parallel and the second term represents the job execution time, which is introduced in Section III.D. We can then formulate the total cost of job allocation as the following function::

[TABLE]

where ${\bm{Y}(\bm{X})}\mathrm{\mathbf{=}}\left[y_{jj^{\prime}}\right]_{{1\leq j\leq m,1\leq}{j}^{\mathbf{{}^{\prime}}}{\bm{\leq}m}}$ denotes the matrix of indicator ${y}_{jj^{\prime}}$, which is a function of $\bm{X}$ according to (1).

Correspondingly, we formulate the graph job allocation problem in a VC, aiming to minimize the sum of job completion time and data exchange cost under opportunistic contact and idle resource constraints, as follows:

[TABLE]

where $\alpha_{1}$ and $\alpha_{2}$ denote non-negative weight parameters which describe the preference between job completion time and data exchange cost, respectively, with $\alpha_{1}+\alpha_{2}=1$. In this formulation, constraint $(a)$ imposes restrictions on idle resources where $|\bm{\kappa}_{\bm{j}}|$ denotes the number of idle slots on each SP. Since the contact duration between vehicles follows exponential distribution as mentioned in the aforementioned section, constraint $(b)$ and $(c)$ are probabilistic constraints, where $(b)$ ensures that if two connected components $\upsilon_{i}^{A}$ and $\upsilon_{i^{\prime}}^{A}$ with edge $e_{ii^{\prime}}^{A}$ and associated weight $\omega_{ii^{\prime}}^{A}$ are allocated to different SPs $s_{j}$ and $s_{j^{\prime}}$, respectively, the probability of the contact duration between $s_{j}$ and $s_{j}$ being larger than $\omega_{ii^{\prime}}^{A}$ should be greater than threshold $\varepsilon$. Notably, $|{t}_{jm}^{Trans}-t_{j^{\prime}m}^{Trans}|+\omega_{ii^{\prime}}^{A}$ is the total time that adjacent vehicles need to maintain their contact, which consists of the effective transmission time and processing time. Similarly, constraint $(c)$ confirms successful data transmission of each component from JO to SPs, where $\xi$ denotes a positive constant which is less than or equal to 1. Notably, the larger the values of $\varepsilon$ and $\xi$ are, the higher guarantees can be met for the graph job allocation requirements. Also, constraint $(d)$ ensures the assignment of all the components to available slots in the related VC. In this study, it is assumed that the system parameters such as $t_{j^{\prime}m}^{Trans}$, $\lambda_{jj^{\prime}}$ and $|\bm{\kappa}_{\bm{j}}|$ involved in (4) are average statistics and will stay unchanged during graph job allocation [23][24].

Given that the objective function in (4) represents an NIP problem that is NP-hard, a JO can rarely identify a solution to reconfigure the IoV extemporaneously, as the running time needed to solve large and real-life network cases increases sharply in line with vehicular density. To solve the graph job allocation problem, we first propose an optimal method for low-traffic conditions (e.g., fewer than five SPs in a VC). Afterward, in rush-hour scenarios, we design a randomized graph job allocation mechanism via hierarchical tree based subgraph isomorphism, through which near-optimal solutions can be obtained with low computational complexity.

V Optimal Graph Job Allocation Mechanism

In this section, we propose an optimal method for low-traffic scenarios, the pseudo code of which is given in Algorithm 1. Our method is composed of two stages: in stage 1, we find all candidates from VC graph ${\bm{G}}^{{\bm{S}}}$ by analyzing the adjacency matrices of graph jobs and related VCs shown in step 2 and step 3; in stage 2, we select one candidate that can minimize the value of the objective function given in (4). In this method, going through all possible candidates ensures identification of the optimal solution for the graph job allocation; however, the computational complexity will be $O({n!}C(K,n))$, where $n$ represents the number of components in the graph job, $K=\sum_{s_{j}}{|{\bm{\kappa}_{j}}|}$ indicates the number of available slots in the related VC, and $C(K,n)$ stands for the K-choose-n operation. As can be seen, this computational complexity is prohibitive as the vehicular density is high.

VI Randomized Graph Job Allocation Mechanism via Hierarchical Tree based Subgraph Isomorphism

Motivated by the intensive required computations of our previous method, we aim to develop a low-complexity graph job allocation method, to better solve larger and real-life cases. To this end, we develop a randomized graph job allocation algorithm by proposing a hierarchical tree based subgraph isomorphism algorithm. In our algorithm, for each graph job, we randomly choose one component and put it on an idle slot first. Then, the graph job is constructed as a hierarchical tree with the first chosen component as the root. Afterward, we randomly match each component in the hierarchical tree with one of the idle slots on the basis of different layers while satisfying all constraints shown in (4). Our randomized subgraph isomorphism can greatly improve the efficiency with computational-complexity of $O(n)$; moreover, the hierarchical tree based scheme ensures that components in a graph job can be matched to idle slots while satisfying the topologies of both the graph job and the vehicular cloud. The pseudo code of our algorithm is given in Algorithm 2.

In this algorithm, steps 1-4 describe the initialization procedure, and steps 6–11 handle the cases in which the graph job cannot be distributed due to insufficient available slots in the related VC. In Steps 12–34, each job is seen as a hierarchical tree with layers by regarding the first randomly chosen component as the root, examples of which are depicted in Fig. 2(b). Concretely, we choose components that have one-hop connections with the root as the second layer, and components in the third layer will have one-hop connections with a component in the second layer; the remaining layers can be formed in a same manner. To satisfy edge relations between components in the hierarchical tree corresponding to a graph job, all components in the same layer should be mapped randomly into idle slots in each allocation procedure while meeting constraints (c-1), (c-2), and (c-3) shown in steps 20-30, where (c-1) ensures the data transmission between JO and SPs, (c-2) guarantees the connection between two slots on different SPs if they are assigned to handle connected components, and (c-3) ensures the contacts between SPs that deal with components in adjacent layers. Among the candidates obtained in each iteration, the best one will be reserved through comparing the value of the objective function given in (4), through steps 34-40.

VII Numerical Results and Performance Evaluation

This section presents numerical results illustrating the performance of the proposed mechanisms. For the convenience of analysis, the optimal solution and the randomized solution via hierarchical tree based subgraph isomorphism are abbreviated as “Opt“ and “R$\_$HTSI“, respectively. The graph job topologies considered in simulations are depicted in Fig. 2(a). We assume $\varepsilon=\xi=0.9$, $t_{jm}^{Trans}\mathrm{\in[0.2,0.6]}$, $c_{jj^{\prime}}^{Exch}\mathrm{\in[0.2,0.6]}$, $\omega_{ii^{\prime}}^{A}\mathrm{\in[0.1,0.4]}$, and $\lambda_{jj^{\prime}}\in[0.01,0.06]$, where each parameter is randomly chosen in its respective interval.

Considering the graph job types shown in Fig. 2(a), Fig. 3 depicts the running time performance comparison between the optimal and the proposed R$\_$HTSI mechanisms for different sizes and configurations of the VC network. Since the gap between the running time of the R$\_$HTSI and the optimal mechanism becomes too large as the graph job structure becomes more complicated (i.e., contains more nodes and edges), 10-based logarithm of the running time is presented in the left sub-plot of Fig. 3(b), Fig. 3(c), Fig. 3(d) and Fig. 3(e). Compared with the R$\_$HTSI, the left sub-plot in each sub-figure indicates that as the traffic density and the number of available slots of SPs in a VC grow, the running time of the optimal mechanism may rise sharply, which makes it unsuitable for fast-changing and large IoV networks. Notably, different topological complexity of VC configurations (e.g., existence of more service providers and connections between vehicles in VC graph) may lead to a dramatic change in the running time performance. For example, the left sub-plot of Fig. 3(c) shows that the running time of the optimal mechanism increases approximately 10 times when the number of SPs increases from 4 to 5 with the average number of slots per SP equal to 4. Also, in the left sub-plot of Fig. 3(d), considering the number of SPs to be 5, as the average number of slots per SP grows from 3 to 5, the running time of the optimal mechanism experiences about a hundredfold increase, while that of the proposed R$\_$HTSI remains almost unchanged. Note that a small gap in the running time of the optimal mechanism upon increasing the size of the network in some scenarios, e.g., in the left sub-plot of Fig. 3(a) upon increasing the number of SPs from 7 to 8 when the average number of slots per SP equals to 6, is due to the possibility of the existene of more edges between the vehicles in smaller VCs. The right sub-plot of each sub-figure depicts the running time of the R$\_$HTSI considering different number of iterations. Comparing the values depicted in the right sub-plot with the running time of the optimal mechanism in the left subplot, using the proposed R$\_$HTSI mechanism results in significant decrease in the running time of graph job allocation. For instance, the utilizing of the R$\_$HTSI in Fig. 3(d) can achieve about 4500 times running time reduction when the number of SPs and slots per SP are both equal to 5, and offers approximately 3 thousand fold decrease in the running time on average, as compared to the optimal mechanism. Moreover, Fig. 3(e) also shows about 3 thousand times running time reduction on average upon utilizing the R$\_$HTSI mechanism. The right sub-plot in each sub-figure reveals that the running time of the proposed R$\_$HTSI mechanism remains low as the number of iterations increases, allowing this mechanism to be implemented efficiently in rapidly changing and large-scale IoV networks, particularly during rush hours.

By utilizing Monte-Carlo iterations, the comparisons of the objective function values (given in (4)) between the optimal and the R$\_$HTSI mechanisms for different graph job types and vehicular cloud sizes as well as configurations are shown in Fig. 4 with $\alpha_{1}=\alpha_{2}=0.5$. For each type of graph job, low-traffic and rush-hour cases reveal that as the number of iterations increases, the performance of the R$\_$HTSI mechanism can approach the optimal one with a much lower computational complexity, which is more applicable in large and real-life IoV scenarios.

Fig. 5 and Fig. 6 depict the job completion time and data exchange cost upon using different values of $\alpha_{1}$ and $\alpha_{2}$, respectively, under low-traffic and rush hour scenarios. As can be seen from Fig. 5, for each graph job type, the larger the value of $\alpha_{1}$ is, the more sensitive the JO will be to minimize the job completion time rather than the data exchange cost. On the contrary, larger $\alpha_{2}$ leads to a lower data exchange cost as illustrated in Fig. 6 under various traffic scenarios.

VIII Conclusion, Challenges, and Future Work

This paper studies novel allocation mechanisms for computation-intensive graph jobs over VCs via opportunistic V2V communications in IoV frameworks, which is modeled as a nonlinear integer programming problem with constraints. For low-traffic scenarios, an optimal approach is introduced. For rush-hour scenarios, we propose a randomized graph job allocation mechanism via hierarchical tree based subgraph isomorphism with low complexity, which efficiently addresses the graph job allocation problem in large IoV networks. Based on comprehensive simulations, the effectiveness of the proposed methods in low-traffic scenarios is revealed. In high-traffic scenarios, we demonstrate a significant running time gap between the optimal and proposed sub-optimal R$\_$HTSI mechanisms; moreover, sub-optimal solutions can closely approach the optimal ones as the number of iterations increases.

Several challenges remain in the context of graph job allocation in IoV environments, such as considering load balancing issues so as to improve on-board resource utilizations, and designing more efficient subgraph isomorphism algorithms to accommodate fast changing topologies in the IoV environment. Furthermore, increasingly complex traffic conditions and job types lead to growing challenges for low-complexity mechanisms in larger networks. One main goal of our future work is to explore more efficient models for designing graph job allocation mechanisms in IoV, while ensuring load balancing and low computation complexity.

Acknowledgment

This work is supported in part by the 2015 National Natural Science Foundation of China (grant no. 61401381), the Major Research Plan of the National Natural Science Foundation of China (grant no. 91638204), the State Key Program of the National Natural Science Foundation of China (grant no. 61731012), 2018 National Natural Science Foundation of China (grant no. 61871339), Digital Fujian Province Key Laboratory of IoT Communication, Architecture and Safety Technology (grant no. 2010499) and the US National Science Foundation under Grants ECCS-1444009 and CNS-1824518.