The First Fully Polynomial Stabilizing Algorithm for BFS Tree Construction

This chapter is devoted to the study of 16 types of greedy algorithms for decision tree construction. The dynamic programming approach is used for construction of optimal decision trees. Optimization is performed relative to minimal values of average depth, depth, number of nodes, number of terminal nodes, and number of nonterminal nodes of decision trees. We compare average depth, depth, number of nodes, number of terminal nodes and number of nonterminal nodes of constructed trees with minimum values of the considered parameters obtained based on a dynamic programming approach. We report experiments performed on data sets from UCI ML Repository and randomly generated binary decision tables. As a result, for depth, average depth, and number of nodes we propose a number of good heuristics.

A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following tree augmentation problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directed edge represents the unique next hop in the routing protocol. Our goal is to direct the undirected edges so that the resulting graph remains acyclic and the number of nodes with outdegree two or more is maximized. These nodes represent those with alternative next hops in their routing paths. We show that tree augmentation is NP-hard in general and present a simple $$\frac{1}{2}$$ -approximation algorithm. We also study 3 special cases. We give exact polynomial-time algorithms for when the input spanning tree consists of exactly 2 directed paths or when the input graph has bounded treewidth. For planar graphs, we present a polynomial-time approximation scheme when the input tree is a breadth-first search tree. To the best of our knowledge, tree augmentation has not been previously studied.

The Steiner tree problem is one of the fundamental optimization problems in distributed graph algorithms. Recently Saikia and Karmakar [27] proposed a deterministic distributed algorithm for the Steiner tree problem that constructs a Steiner tree in \(O(S + \sqrt{n} \log ^* n)\) rounds whose cost is optimal upto a factor of \(2(1 - 1/\ell )\), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and \(\ell \) is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is \(O(\varDelta (n - t) S + n^{3/2})\), which is equivalent to \(O(mS + n^{3/2})\), where \(\varDelta \) is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that \(t < n\)), and m is the number of edges in the given input graph. This algorithm has a better round complexity than the previous best algorithm for Steiner tree construction due to Lenzen and Patt-Shamir [21]. In this paper we present a deterministic distributed algorithm which constructs a Steiner tree in \(\tilde{O}(S + \sqrt{n})\) rounds and \(\tilde{O}(mS)\) messages and still achieves an approximation factor of \(2(1 - 1/\ell )\). Note here that \(\tilde{O}(\cdot )\) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar’s algorithm by dropping the additive term of \(O(n^{3/2})\) at the expense of a logarithmic multiplicative factor in the round complexity. Furthermore, we show that for sufficiently small values of the shortest path diameter \((S=O(\log n))\), a \(2(1 - 1/\ell )\)-approximate Steiner tree can be computed in \(\tilde{O}(\sqrt{n})\) rounds and \(\tilde{O}(m)\) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [9, 12, 23].

The degree distribution of the Internet topology is considered as one of its main properties. However, it is only known through a measurement procedure which gives a biased estimate. This measurement may in first approximation be modeled by a BFS (Breadth-First Search) tree. We explore here our ability to infer the type (Poisson or power-law) of the degree distribution from such a limited knowledge. We design procedures which estimate the degree distribution of a graph from a BFS of it, and show experimentally (on models and real-world data) that this approach succeeds in making the difference between Poisson and power-law graphs.

Tree structures such as breadth-first search (BFS) trees and minimum spanning trees (MST) are among the most fundamental graph structures in distributed network algorithms. However, by definition, these structures are not robust against failures and even a single edge's removal can disrupt their functionality. A well-studied concept which attempts to circumvent this issue is Fault-Tolerant Tree Structures, where the tree gets augmented with additional edges from the network so that the functionality of the structure is maintained even when an edge fails. These structures, or other equivalent formulations, have been studied extensively from a centralized viewpoint. However, despite the fact that the main motivations come from distributed networks, their distributed construction has not been addressed before.

The advances in sensing capabilities of smartphones give rise to a variety of mobile participatory sensing applications that collect users' personal data. Because of the existence of both sensitive, private personal data, and untrusted aggregator, serious privacy concerns on users arise. Currently, existing privacy preserving data collection methods either require bidirectional communications between an untrusted aggregator and mobile users in every aggregation period, or have high computation or communication overhead. To address these problems, we propose an efficient data aggregation approach by which an untrusted aggregator in mobile sensing can collect the statistics over the data contributed by multiple mobile users, while supporting privacy preservation of each user and data integrity verification. In this approach, information hiding and homomorphic encryption are applied to guarantee the data privacy of mobile users. In detail, a breadth-first search tree is first constructed at the initial phase among the mobile users, and then the original datum of each user is perturbed among its neighbors in ciphertext space by using information hiding and homomorphic encryption. The evaluations of our approach show that our protocol requires lower communication and computation overhead and thus more feasible for the computation constrained mobile devices. Copyright © 2016 John Wiley & Sons, Ltd.

Enumerating consistent global states of a computation is a fundamental problem in parallel computing with applications to debugging, testing and runtime verification of parallel programs. Breadth-first search (BFS) enumeration is especially useful for these applications as it finds an erroneous consistent global state with the least number of events possible. The total number of executed events in a global state is called its rank. BFS also allows enumeration of all global states of a given rank or within a range of ranks. If a computation on n processes has m events per process on average, then the traditional BFS (Cooper-Marzullo and its variants) requires \(\mathcal {O}(\frac{m^{n-1}}{n})\) space in the worst case, whereas our algorithm performs the BFS requires \(\mathcal {O}(m^2n^2)\) space. Thus, we reduce the space complexity for BFS enumeration of consistent global states exponentially, and give the first polynomial space algorithm for this task. In our experimental evaluation of seven benchmarks, traditional BFS fails in many cases by exhausting the 2 GB heap space allowed to the JVM. In contrast, our implementation uses less than 60 MB memory and is also faster in many cases.

Improved distributed approximation for Steiner tree in the CONGEST model

Although any problem in a radio network can be solved using broadcast algorithms, some problems can be solved substantially more efficiently by more specialized algorithms. This paper presents two new approximate algorithms for the local sum problem, in which each node computes a (1±ε)-approximation to the sum of the values held by its incoming neighbors (nodes that have outgoing edges to the node). We propose algorithms both with and without collision detection, as well as for the beeping model, with round complexity \(O({\log^{2} n + \log n\log m \over \epsilon^2})\), where n is the number of nodes and the value held by each node is a real number in {0} ∪ [1,m]. We then show how these algorithms can be used as building blocks to construct applications such as approximate random walk distribution, PageRank, and global sum.Keywordsradio networksalgorithmsmodel with collision detectionmodel without collision detectionbeeping modellocal sumrandom walkPageRank

Breaking the linear-memory barrier in [formula omitted]: Fast [formula omitted] on trees with strongly sublinear memory

The construction of optimum multiway search trees for n keys, n key weights and n+1 gap weights, is investigated. A new general optimality principle is presented, which can be "tuned" for specific applications. Moreover we consider the affects of three additional constraints, namely height, structural and node search restrictions, which lead to a number of new construction algorithms. In particular we concentrate on the construction of optimum t-ary search trees with linear and binary search within their nodes for which we obtain O(n 3 t) and O(n 3log2 t) time algorithms, respectively. Whether these algorithms are or are not optimal remains an important open problem, even in the binary case.

The study of algorithms for decision tree construction was initiated in 1960s. The first algorithms are based on the separation heuristic [13, 31] that at each step tries dividing the set of objects as evenly as possible. Later Garey and Graham [28] showed that such algorithm may construct decision trees whose average depth is arbitrarily far from the minimum. Hyafil and Rivest in [35] proved NP-hardness of DT problem that is constructing a tree with the minimum average depth for a diagnostic problem over 2-valued information system and uniform probability distribution. Cox et al. in [22] showed that for a two-class problem over information system, even finding the root node attribute for an optimal tree is an NP-hard problem.

Recognizing breadth-first search trees in linear time

The wireless sensor network (WSN) has been extensively used to monitor and control the natural ecosystem on a large scale like air quality, natural life, etc. Low battery power,low storage, and limited processing ability are the most critical areas of concern in WSN. To reduce energy utilization, the sensor nodes in WSN work in a cyclic process between active and sleep mode. A certain number of nodes are chosen active and they are responsible for sensing as well as data transmission and rest of the nodes are gone to sleep. In order to lengthen the lifetime of network, in this paper we proposed a level wise periodic tree construction algorithm that uses a specific set of nodes to participate in tree construction, instead of all the nodes, to minimize the energy consumption. In this proposed approach, the main idea is to put the nodes, which are currently active and have already spent a significant amount of energy, to sleep mode, while giving chances to the leaf nodes, which has comparatively spent less energy, to become an active node and maintain connectivity. The performance of the proposed protocol is evaluated using the Castalia simulator. The simulation results show that the proposed level-wise periodic tree construction approach increases the durability of the network in conjunction with the non-level approach.

Internet services provider propose increasingly numerous applications depend on multicast communication patterns, such as database synchronization, newsletter updates, video, and audio group streaming. In spite of, IP multicast is frequently suffered from scaling and instability, which causes internet provider administrators to avoid its utilization. While different multicast protocols in all layers, such as routing and switching, have been created by researchers and providers, through several years, to focus on security, congestion control, and scalability, our work aims to scale multicast regarding the number of supported multicast groups and sources in networks and multicast session. In this paper, we propose an SDN-based IP multicast tree construction module to support and deploy multicast communication-based services. To do so, we take advantage to compute multicast tree, in the centralized way, to achieve a dynamic tree construction algorithm that helps SDN controller to dynamically construct and adjust an optimal multicast tree in all multicast session. The experiments studies on the prototype and simulation system exhibit that our solution gives better performance contrasted with the traditional multicast tree construction algorithms, better also to SDN-based solution proposed in the literature, regarding multicast tree cost, delay, delay variation, and multicast tree construction delay.