A Self-Stabilizing Algorithm for Maximal Matching in Anonymous Networks

This full paper presents an innovative approach in teaching anonymity algorithms. Teaching end-to-end anonymity algorithms, such as onion routing, has been a challenging task. This is primarily because of the inaccessibility of all the components that make up the end-to-end anonymity network. For example, an onion is the data structure formed by "wrapping" a message with successive layers of encryption to be decrypted by as many intermediary computers as there are layers before arriving at its destination. The intent of the onion routing algorithms is to protect the privacy of its users as well as their freedom and ability to conduct confidential communication by keeping their Internet activities from being monitored. In recent years, community research, specifically at the university level, into onion routing has ground to a halt and the limited resources for establishing private testing networks have fallen into a derelict state. The proposed solution is an elaborate walk-through, detailing the construction of an isolated onion routing network in which onion routing can be observed and verified in a controlled environment. Achieving this will require the configuration of several Directory Authorities, multiple nodes configured as Entry, Middle, and Exit relays, and the generation of a consensus. Once the consensus has been generated, the network will be able to build circuits across the nodes and fulfill the client's requests for information. When a packet capturing tool is applied to the various entities, the life of the packet as it traverses the network will be observed. The product of this endeavor was the creation of a complete virtual environment and detailed documentation of the steps necessary to construct a private onion routing network that can be readily utilized by educators and students to teach and learn the innerworkings of the OR algorithms. Constructing an isolated OR network, where all the network traffic is observable, is an essential first step in peeling back the next layer of onion routing.

For a combined input and output queued (CIOQ) switch, maximal matching (MM) algorithm with a speedup of 2 has been known to deliver 100% throughput under any admissible traffic, where in each time slot, two independent schedulings and two cell transferrings are made respectively. This paper proposes a new kind of extended maximal matching (EMM) algorithm, which just utilizes a little information of VOQ length. We show that the implementation complexity of CIOQ switches is reduced by using the EMM(2) algorithm, where only one scheduling and two cell transferrings (data speedup of 2) are necessary in each time slot to achieve 100% throughput when input traffic is admissible. The EMM(2) algorithm can be easily realized by modifying the existing MM algorithms. In this paper, we provide a practical instance of EMM(2) algorithm, named EiSLIP(1,2), based on the well-known iSLIP algorithm. Simulations show that EiSLIP(1,2) algorithm with a data speedup of 2 achieves almost the same delay performance as output queued policy under uniform and non-uniform traffic with Bernoulli arrival, as well as the bursty traffic.

We consider networks of small, autonomous devices that communicate with each other wirelessly. Minimizing energy usage is an important consideration in designing algorithms for such networks, as battery life is a crucial and limited resource. Working in a model where both sending and listening for messages deplete energy, we consider the problem of finding a maximal matching of the nodes in a radio network of arbitrary and unknown topology. We present a distributed randomized algorithm that produces, with high probability, a maximal matching. The maximum energy cost per node is O(log2 n), and the time complexity is O(Δ log(n)). Here n is any upper bound on the number of nodes, and Δ is any upper bound on the maximum degree; n and Δ are parameters of our algorithm that we assume are known a priori to all the processors. We note that there exist families of graphs for which our bounds on energy cost and time complexity are simultaneously optimal up to polylog factors, so any significant improvement would need additional assumptions about the network topology. We also consider the related problem of assigning, for each node in the network, a neighbor to back up its data in case of eventual node failure. Here, a key goal is to minimize the maximum load, defined as the number of nodes assigned to a single node. We present an efficient decentralized low-energy algorithm that finds a neighbor assignment whose maximum load is at most a polylog(n) factor bigger that the optimum.

The delay and throughput characteristics of a packet switch depend mainly on the queueing scheme and the scheduling algorithm deployed at the switch. Early research on scheduling algorithms has mainly focused on maximum weight matching scheduling algorithms. It is known that maximum weight matching algorithms guarantee the stability of input-queued switches, but are impractical due to their high computational complexity. Later research showed that the less complex maximal matching algorithms can stabilize input-queued switches when they are deployed with a speed-up of two. For practical purposes, neither a high computational complexity nor a speed-up of two is desirable.In this paper, we investigate the application of matching algorithms that approximate maximum weight matching algorithms to scheduling problems. We show that while having a low computational complexity, they guarantee the stability of input queued switches when they are deployed with a moderate speed-up.In particular, we show that the improve_matching algorithm stabilizes input-queued switches when it is deployed with a speed-up of 3<over>2+ε.In a second step, we further improve on these results by proposing a class of maximal weight matching algorithms that stabilize an input-queued switch without any speed-up.Whereas initial research has only focused on scheduling algorithms that guarantee the stability of a single switch, recent work has shown how scheduling algorithms for single switches can be modified in order to design distributed scheduling algorithms that stabilize networks of input-queued switches. Using those results, we show that the switching algorithms proposed in this paper do not only stabilize a single switch, but also networks of input-queued switches.

For a graph G=(V,E), finding a set of disjoint edges that do not share any vertices is called a matching problem, and finding the maximum matching is a fundamental problem in the theory of distributed graph algorithms. Although local algorithms for the approximate maximum matching problem have been widely studied, exact algorithms has not been much studied. In fact, no exact maximum matching algorithm that is faster than the trivial upper bound of O(n^2) rounds is known for the general instance. In this paper, we propose a randomized O(s_{max}^{3/2}+log n)-round algorithm in the CONGEST model, where s_{\max} is the size of maximum matching. This is the first exact maximum matching algorithm in o(n^2) rounds for general instances in the CONGEST model. The key technical ingredient of our result is a distributed algorithms of finding an augmenting path in O(s_{\max}) rounds, which is based on a novel technique of constructing a sparse certificate of augmenting paths, which is a subgraph of the input graph preserving at least one augmenting path. To establish a highly parallel construction of sparse certificates, we also propose a new characterization of sparse certificates, which might also be of independent interest.

A matrix-algebraic formulation of distributed-memory maximal cardinality matching algorithms in bipartite graphs

Obtaining a matching in a graph satisfying a certain objective is an important class of graph problems. Matching algorithms have received attention for several decades. However, while there are efficient algorithms to obtain a maximum weight matching, not much is known about the maximum weight maximum cardinality, and maximum cardinality maximum weight matching problems for general graphs. Our contribution in this work is to show that for bounded weight input graphs one can obtain an algorithm for both maximum weight maximum cardinality (for real weights), and maximum cardinality maximum weight matching (for integer weights) by modifying the input and running the existing maximum weight matching algorithm. Also, given the current state of the art in maximum weight matching algorithms, we show that, for bounded weight input graphs, both maximum weight maximum cardinality, and maximum cardinality maximum weight matching have algorithms of similar complexities to that of maximum weight matching. Subsequently, we also obtain approximation algorithms for maximum weight maximum cardinality, and maximum cardinality maximum weight matching.

In this paper, we propose a depth-first search (DFS) algorithm for finding maximum matching in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating-paths in the super-vertices shrinking from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending the time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with m edges and n vertices in 𝑂(𝑚𝑛) time with space complexity 𝑂(𝑛).

A fault-containing self-stabilizing [formula omitted]-approximation algorithm for vertex cover in anonymous networks

Devising methods to publish social network data in a form that affords utility without compromising privacy remains a longstanding challenge, while many existing methods based on k-anonymity algorithms on social networks may result in nontrivial utility loss without analyzing the social network topological structure and without considering the attributes of sparse distribution. Toward this objective, we explore the impact of the attributes of sparse distribution on data utility. Firstly, we propose a new utility metric that emphasizes network structure distortion and attribute value loss. Furthermore, we design and implement a differentiated k-anonymity l-diversity social network anonymity algorithm, which seeks to protect users’ privacy in social networks and increase the usability of the published anonymized data. Its key idea is that it divides a node into two child nodes and only anonymizes sensitive values to satisfy anonymity requirements. The evaluation results show that our method can effectively improve the data utility as compared to generalized anonymizing algorithms.

We present a derandomization process which leads to efficient parallel algorithms for the maximal independent set and maximal matching problems. The derandomization of the general pairs PROFIT/COST problem depends on the derandomization of the bit pairs PROFIT/COST problem which follows Luby’s approach of using an O(n) sized sample space. This simplifies the approach presented in [16].

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and $$(\Delta +1)$$ -coloring algorithms by Barenboim and Elkin (Distrib Comput 22(5–6):363–379, 2010), by Kuhn (2009), and by Panconesi and Srinivasan (J Algorithms 20(2):356–374, 1996), as well as the $$O\mathopen {}(\Delta ^2)$$ -coloring algorithm by Linial (J Comput 21:193, 1992). Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree $$\Delta $$ or the number of nodes $$n$$ . This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem (In particular, it applies to all aforementioned algorithms). To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.

Port partitioning and dynamic queueing for IP forwarding

Motivated by an application in controllability we consider maximum matchings in random bipartite graphs G = (A, B). First we analyse Karp–Sipser's algorithm to determine the asymptotic size of maximum matchings in random bipartite graphs with a fixed degree distribution. We then allow an adversary to delete one edge adjacent to every vertex in A in the more restricted model where each vertex in A chooses d neighbours uniformly at random from B.

The maximum matching algorithm for bipartite graphs can be used to provide optimal scheduling for crossbar based interconnection networks. Unfortunately, maximum matching requires O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) time for an N times N communication system, which has limited its application to real-time network scheduling. In this paper, we show how maximum matching can be reformulated in terms of Boolean operations rather than the more traditional formulations. By taking advantage of the inherent parallelism available in custom hardware design, we introduce three maximum matching implementations in hardware and show how we can trade design complexity for performance. Specifically, we examine a pure logic scheduler with three dimensions of parallelism, a matrix scheduler with two dimensions of parallelism and a vector scheduler with one dimension of parallelism. These designs reduce the algorithmic time complexity down to O(1), O(K), and O(KN), respectively, where K is the number of optimization steps. While an optimal scheduling algorithm requires K=2N-1 steps, our simulation results show that the scheduler can achieve 99% of the optimal schedule when K=9. We examine hardware and time complexity of these architectures for crossbar sizes of up to N=1024. Using FPGA synthesis results, we show that a greedy schedule for various sized crossbars, ranging from 8 times 8 to 256 times 256, can be optimized in less than 20 ns per optimization step. For crossbars reaching 1024 times 1024 the scheduling can be completed in approximately 10 s with current technology and could reach under 90 ns with future technologies