Elsevier

Ad Hoc Networks

Volume 59, 1 May 2017, Pages 63-70
Ad Hoc Networks

A maximum flow algorithm based on storage time aggregated graph for delay-tolerant networks

https://doi.org/10.1016/j.adhoc.2017.01.006Get rights and content

Abstract

Delay-tolerant networks (DTNs) (e.g., Internet, satellite networks, sensor networks, ad hoc networks) have attracted considerable attentions in both academia and industry. As a fundamental problem, the maximum flow is of vital importance for routing and service scheduling in networks. For solving the maximum flow problem of the DTN, an appropriate model should be built first. Compared to the conventional snapshot approach to model the DTN topology, the time aggregated graph (TAG) is capable of accurately characterizing the intermittent connectivity and time-varying capacity for each edge, and thus has been acted as a suitable model for modeling DTNs. However, existing TAG-related works only focus on solving the shortest path problem, and neither the correlation between time intervals nor nodes storage of a DTN are described in TAG, resulting in a non-trivial maximum flow problem in TAG. In this paper, we study the maximum flow problem through our proposed storage time aggregated graph (STAG) for DTNs. First, an intermediate quantity named bidirectional storage transfer series is introduced to each node in STAG, and the corresponding transfer rule is also designed for this series to model the correlation between time intervals. Next, on the basis of the storage transfer series, a STAG-based algorithm is proposed and described in detail to maximize the network flow. In addition, we analyze the effectiveness of the proposed algorithm by giving an illustrative example.

Introduction

Recently, Delay-Tolerant Networks (DTNs) [1] have drawn much research attentions due to its wide application in Internet [2], satellite networks [3], [4], ad hoc networks [5], sensor networks [6], Internet of Things (IOT) [7] and many other communication networks [8], [9]. As a fundamental problem, the maximum flow is of vital importance for routing and service scheduling in networks. Especially for DTN networks, there exists no permanent end-to-end path since the topology and links’ characteristics are time-varying. The study on the maximum flow problem could not only adapt to the network dynamics and ensure reliable transmissions, but also provide powerful guarantee for network management (e.g., network planning and optimization), thus playing a significant role in DTN.

In particular, the graph theory has been viewed as an efficient approach to study DTNs in many existing works [10], [11], [12], [13], [14], [15], [16], [17], [18]. In graph theory terminology, snapshots are utilized to model the DTN [11], where each snapshot corresponds to the topology of a DTN at a particular time interval. However, there exists no correlation between snapshots, which can not be utilized to solve the dynamic maximum flow problem. What is more, it would result in a prohibitively large number of snapshots with the time increasing. Time expanded graphs [12], [13], which have been used to model dynamic networks (e.g., DTNs), employ replication of networks across time intervals, also resulting in high storage overhead and computationally complex algorithms. In contrast, time aggregated graphs (TAG), which have been proposed by Betsy George et al. [15], allow the properties of edges to be modeled as a time series. Since the model does not need to replicate the entire graph for each time interval, it uses less memory and the algorithms for common operation are computationally more efficient than those for time expanded graph.

Indeed, time expanded graphs are essentially an expansion of static graphs, and hence many standard flow maximization algorithms (e.g., generic augmenting path algorithm [10]) can be applied to time expanded graphs. In particular, Iosifidis et al. in [17] iteratively updated the minimum cut of the time expanded graph and derived a joint storage capacity management to maximize the amount of data transferred to the destination. Nevertheless, the time expanded graphs connect any two time intervals of the same node via one link and the storage size of this node can be deemed as the link’s capacity, while our proposed TAG approach strives to construct the storage of the node as a time series, namely, bidirectional storage transfer series. Moreover, the augmenting path algorithm only considers single capacity of an edge both in time expanded graphs and traditional static graphs, and thus can not be easily applied to TAG where an edge has a capacity series. Furthermore, existing TAG-related works only focus on solving the shortest path problem [14], [15], and neither the correlation between time intervals nor nodes storage of a DTN are described in TAG, leading to the sub-optimal rather than the optimal solution for the maximum flow problem in a DTN. To the best of our knowledge, the maximum flow problem for a DTN through TAG has not been studied in previous works.

In this paper, we modify the existing TAG as STAG and then exploit it to study the maximum flow problem of a DTN. In order to ameliorate the model, we introduce an intermediate quantity named bidirectional storage transfer series to each node in STAG, and the corresponding transfer rule is also designed for this series to model the correlation between time intervals. As such, it is possible for us to solve the maximum flow problem of a DTN. Furthermore, on the basis of the bidirectional storage transfer series, a STAG-based algorithm is proposed and described in detail to effectively solve the maximum flow problem.

The distinctive features of this paper are summarized as follows:

  • We model the DTN by STAG (a modified TAG) where the time-variant topology (capacity) of a DTN is incorporated. Meanwhile, the bidirectional storage transfer series is introduced to each node in STAG, which could describe the data storage process of each node and the correlation between time intervals of each edge.

  • A transfer rule is formulated to describe bidirectional storage transfer series. It could act as a storage strategy and incorporates two new storage functions, namely forward storage function and reciprocal storage function. Accordingly, some storage transfer series-based definitions are also presented.

  • To solve the DTN’s maximum flow problem with high computational complexity, we propose a Max flow-STAG algorithm on the basis of STAG. The proposed algorithm can obtain the DTN’s maximum flow by giving a feasible routing scheme.

  • The theoretical analysis is presented to validate the proposed algorithm. We analyze the effectiveness of the algorithm by giving an illustrative example.

The remainder of this paper is organized as follows. In Section 2, the system model is presented. Following it, Section 3 provides some basic definitions in STAG. In Section 4, we propose and describe the STAG-based maximum flow algorithm in detail. In Section 5, we provide the validness of the proposed algorithm. Finally, we conclude this work and discuss the future works in Section 6.

Section snippets

System model

In this section, the system model for a DTN with some predictable characteristics is presented. For the sake of presentation and without loss of generality, as in [19], we neglect both the transmission delay and the propagation delay on the edge.

Consider a DTN with predictable characteristics [20], e.g., the motion period, the topology structure and the capacity of each edge. Assume a large time period T=[t0,th), V and E are the set of nodes and edges, respectively. Edge eE is a directed

Basic definitions in STAG

Let s be the source of the network, and d be the sink. The feasible flow in STAG is also a series fT=(fτ1,,fτq,,fτh), satisfying the following two properties:

  • 1)

    Capacity constraint: 0fτqu,vcτqu,v1qh,(u,v)E.

  • 2)

    Flow conservation: vVq=1hfτqu,vvVq=1hfτqv,u={q=1hfτq,u=s,0,us,d,q=1hfτq,u=d.

Intuitively, given a flow network STAG and a flow fT, the residual network STAG (rSTAG) consists of new added edges with capacities (for amending the flow), and nodes with the bidirectional storage

Maximum flow algorithm for storage time aggregated graph

The maximum flow problem can be stated as that we wish to send as much flow as possible between two special nodes (e.g., a source node s and a sink node d), under the capacity constraints in STAG. That is, fTM=maxlifTli=maxliq=1hfτqli,where li is the ith augmenting path in the residual network rSTAG, and fTli=(fτ1li,,fτqli,,fτhli) is the maximum feasible flow of li (which also satisfies the aforementioned two properties of the feasible flow). Moreover, fTM represents the maximum flow of

Algorithm complexity analysis

Theorem 1

The time complexity of the Max flow-STAG algorithm isO((m+nh)|fTM|), where h is the number of time intervals (for a given time period T, we partition T into a set of h small time intervals),fTM denotes the value of the maximum flow in a DTN, n is the number of nodes and m represents the number of edges in STAG.

Proof

We combine the depth-first search method with the earliest start time of nodes to find an augmenting path in rSTAG, and the time complexity for this step reaches O(m+n). Besides, both

Conclusion and future work

In this paper, we described a model named as STAG to represent the DTN. First, the bidirectional storage transfer series was introduced to each node in STAG. Second, a bidirectional storage transfer series-based maximum flow algorithm was proposed and described in detail to maximize the network flow. Finally, the effectiveness of the proposed algorithm was also illustrated.

As an innovative approach, the proposed STAG-based maximum flow algorithm (i.e., Max-flow STAG algorithm) has broken

Acknowledgments

This work is supported by the National Science Foundation (91338115, 61231008), National S&T Major Project (2015ZX03002006), the Fundamental Research Funds for the Central Universities (WRYB142208, JB140117), Program for Changjiang Scholars and Innovative Research Team in University (IRT0852), the 111 Project (B08038), SAST (201454). Furthermore, we thank the reviewers for their detailed reviews and constructive comments, which have helped to improve the quality of this paper.

Hongyan Li (M’08) received the M.S. degree in control engineering from Xi’an Jiaotong University, Xi’an, China, in 1991 and the Ph.D. degree in signal and information processing from Xidian University, Xi’an, in 2000. She is currently a Professor with the State Key Laboratory of Integrated Service Networks, Xidian University. Her research interests include wireless networking, cognitive networks, integration of heterogeneous network, and mobile ad hoc networks.

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    Hongyan Li (M’08) received the M.S. degree in control engineering from Xi’an Jiaotong University, Xi’an, China, in 1991 and the Ph.D. degree in signal and information processing from Xidian University, Xi’an, in 2000. She is currently a Professor with the State Key Laboratory of Integrated Service Networks, Xidian University. Her research interests include wireless networking, cognitive networks, integration of heterogeneous network, and mobile ad hoc networks.

    Tao Zhang received the B.S. degree in telecommunication engineering from HeFei University of Technology, Hefei, China, in 2015. He is currently pursuing the Ph.D. degree with the State Key Laboratory of Integrated Service Networks, Institute of Information and Science, Xidian University, Xi’an, China. His current research interests include wireless networking, cognitive networks, integration of heterogeneous network, and mobile ad hoc networks.

    Yangkun Zhang is currently an undergraduate student in the Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, China. His current research interests is network optimization.

    Kan Wang received the B.S. degree in broadcasting and television engineering from Zhejiang University of Media and Communications, Hangzhou, China, in 2009. He is currently working toward the Ph.D. degree in military communications with the State Key Lab of ISN, Xidian University, Xi’an, China. From Oct. 2014 to Oct. 2015, he was also with Carleton University, Ottawa, ON, Canada, as a visiting scholar funded by China Scholarship Council (CSC). His current research interests include 5G cellular networks, resource management, and interference alignment.

    Jiandong Li received the B.E., M.S., and Ph.D. degrees in communications engineering from Xidian University, Xi’an, China, in 1982, 1985, and 1991, respectively. He has been a Faculty Member of the School of Telecommunications Engineering at Xidian University since 1985, where he is currently a Professor and Vice Director of the Academic Committee of State Key Laboratory of Integrated Service Networks. He was a Visiting Professor to the Department of Electrical and Computer Engineering at Cornell University from 2002 to 2003. He served as the General Vice Chair for ChinaCom 2009 and TPC Chair of IEEE ICCC 2013. He was awarded as Distinguished Young Researcher from NSFC and Changjiang Scholar from Ministry of Education, China, respectively. His major research interests include wireless communication theory, cognitive radio, and signal processing.

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