Elsevier

Ad Hoc Networks

Volume 101, 15 April 2020, 102092
Ad Hoc Networks

An Asynchronous Distributed Algorithm for Minimum s−t Cut Detection in Wireless Multi-hop Networks

https://doi.org/10.1016/j.adhoc.2020.102092Get rights and content

Abstract

Stable communication is a vital and challenging requirement in wireless multi-hop networks because nodes failure may disconnect other active nodes. Minimum s-t cuts can help to measure the reliability of a network regarding the nodes or links failure. A minimum s-t cut is the smallest subset of links that their failure disconnects all communication paths between nodes s and t. Minimum s-t cuts can also reveal the bottlenecks, bridges, and clusters in the network. This paper proposes an asynchronous distributed algorithm for finding minimum s-t cuts in wireless multi-hop networks. The proposed algorithm iteratively finds link-disjoint paths from s to t and changes the direction of the selected edges to ignore them in the next iteration. After finding all paths, the undiscovered nodes in the last search determine the minimum s-t cut. The comprehensive simulation results on different networks with up to 500 nodes show that the proposed algorithm finds minimum s-t cuts in reasonable time with up to 64% and 39% lower energy consumption than the existing central and distributed algorithms, respectively.

Introduction

A Wireless Multi-hop Network (WMhN) is a collection of autonomous nodes that exchange information by sending multi-hop radio messages. One of the common types of WMhNs is Wireless Sensor Network (WSN) in which the nodes collect different information from the environment and send them to a sink node over radio messages. The sink node acts as a bridge to a processing center to transfer the data and commands. WMhNs have growing applications in various fields such internet of things, military, disaster recovery, target tracking, health care, and intelligent structures [1]. In a typical WMhN, each node connects to its neighbors (the nodes that are located in its radio range) and communicates with remote nodes by message passing over multi-hop connections. Hence, besides the main functionality, most of the nodes contribute for preserving the network connectivity. Using ad-hoc connections simplifies the establishment of WMhNs especially in harsh environments. However, failure in intermediate nodes may disconnect the paths between other operative nodes. Hence, an important challenge in creating reliable WMhNs is detecting and reinforcement of the weak points which are critical for preserving the network connectivity.

In graph theory, the minimum s−t cut of a graph is the smallest subset of edges that their removing separates the source node s from the target node t. The cardinality of a minimum s-t cut in a WMhN determines the number of links failure that the network can tolerate before disconnecting node s from t. A minimum s-t cut with cardinality c indicates that the nodes s and t remain connected after losing any c−1 links in the network. A minimum s-t cut with more cardinality indicates a more reliable connection between nodes s and t. Finding minimum s-t cuts may provide significant information about the reliability of connections, weak points in communication paths, bottlenecks, bridges and possible clusters in the network. Also, detecting the minimum cuts can increase the network lifetime by minimizing the usage of links in the minimum cuts [2]. This paper proposes a distributed minimum s-t cut (DMSTC) detection algorithm for WMhNs and has the following contributions;

  • 1.

    The proposed distributed algorithm can find minimum s-t cut between any pair of nodes which helps to find weak points and bottlenecks.

  • 2.

    The proposed algorithm consumes lower energy than the existing central and distributed algorithms.

  • 3.

    To the best of our knowledge, the proposed algorithm is the first asynchronous distributed approach for minimum s-t cut problem.

  • 4.

    The proposed and existing algorithms have been simulated on various topologies to compare their performance.

The remaining sections of the paper are as follows; Section 2 provides a brief survey about existing algorithms. Section 3 contains the problem formulation, network model and motivation. The proposed algorithm and its proof of correctness are presented in Section 4. Section 5 includes the complexity analysis and Section 6 contains the performance evaluation. Finally, the conclusion is drawn in Section 7.

Section snippets

Related Work

Finding minimum cuts is a well-known problem in graph theory which has interesting randomized and deterministic central algorithms [3], [4], [5], [6], [7], [8], [9]. According to the max-flow min-cut theorem [10], the maximum flow passing from node s to node t is equal to the total weight of edges in the minimum cut. Therefore, with a few modifications, a max-flow algorithm can find the minimum s-t cut of graphs. The Ford-Fulkerson algorithm finds the maximum flow of a graph in O(f × m) running

preliminaries

A WMhN can be represented as a graph G(V, E) where V is the set of nodes and E is the set of edges. The nodes located in the radio range of each other may have a link to exchange messages. For example, Fig. 1a shows a sample WMhN where V={0,1,2,...,11} and E={(0−1),(0−2),(0−3),(2−8),...}. We show the undirected edges as (u−v) and directed edges as (u → v). The dashed big circles in Fig. 1a shows the radio transmission range of nodes. Each node usually has a unique Id and we assume that the

The Idea

The main idea is finding the maximum possible link-disjoint paths between the source and target nodes. Two paths are link-disjoint if they pass over different set of edges. The proposed DMSTC algorithm finds a path between nodes s and t, converts the elected undirected edges to directed and selected directed edges to undirected and restarts the process until s have no path to t. The paths are detected using a distributed depth-first search (dDFS) based algorithm.

Fig. 2 shows how the DMSTC

Complexity Analysis

This section presents the bit, time, space and computation complexity analysis of the DMSTC algorithm. We assume that c is the size of minimum s−t cut and sending a single-hop message takes O(1) time units.

Theorem 4

The bit complexity of the DMSTC is O(n × Δ × c × log2n) in the worst case and Ω(D × c × log2n) in the best case.

Proof

The length of every message in the DMSTC is O(log2n). To find a path at most O((n−1)×Δ) Probe and O(n) Confirm or Reject messages are sent. The source node initiates the pathfinding

Performance Evaluation

To evaluate the performance of the proposed algorithm in large networks we implemented the DMSTC, a central (CENT) and a distributed synchronized (SYNC) algorithm in Python using Simpy and NetworkX libraries. In the central algorithm, all nodes send their neighbors list to the source node over an established spanning tree and the source node creates a graph for the network topology and runs a central algorithm [7]. In the implemented synchronized algorithm the source node broadcasts a Round

Conclusion

Reliable communications is a vital requirement in WMhNs and finding minimum cuts provide useful information about the weak points, bridges, bottlenecks and disjoint paths in WMhNs. This study proposed a new asynchronous distributed algorithm that detects minimum s−t cuts by finding link-disjoint paths and changing the direction of selected edges. The comprehensive simulation results showed that the proposed algorithm can find the minimum s−t cut in a reasonable time and respectively up to 64%

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Vahid Khalilpour Akram received the BSc and the MSc degrees in Computer Engineering from Islamic Azad University. He received PhD degree from Ege University, International Computer Institute. He currently is an assistant professor at Bakircay University, Computer Engineering department. His research interests include wireless sensor networks, parallel and distributed algorithms and graph theory.

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    Vahid Khalilpour Akram received the BSc and the MSc degrees in Computer Engineering from Islamic Azad University. He received PhD degree from Ege University, International Computer Institute. He currently is an assistant professor at Bakircay University, Computer Engineering department. His research interests include wireless sensor networks, parallel and distributed algorithms and graph theory.

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