Improved distributed approximation for Steiner tree in the CONGEST model

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].

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.

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Delta is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Theta(log n) time complexity barrier and the Theta(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?&#13;\n&#13;\nThis paper presents progress towards this question for the distributed ruling set problem in the Congest model. A beta-ruling set is an independent set such that every node in the graph is at most beta hops from a node in the independent set. We present the following results:&#13;\n&#13;\n- Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O(log n/log log n) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon &gt; 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model.&#13;\n&#13;\n- Message Complexity: We show an Omega(n^2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^2 n) messages and runs in O(Delta log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/e)-round algorithm that finds a (1+e)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/e)-round algorithm that finds, with high probability, a (1+e)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/e)-round algorithm that finds, with high probability, a 8(1+e)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+e)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/e)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+e)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from 120^{circ }. This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals.

A super-stabilizing [formula omitted]-approximation algorithm for dynamic Steiner trees

Let \(R\) be a finite set of terminals in a metric space \((M,d)\). We consider finding a minimum size set \(S \subseteq M\) of additional points such that the unit-disc graph \(G[R \cup S]\) of \(R \cup S\) satisfies some connectivity properties. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem \(G[R \cup S]\) should be connected. In the more general Steiner Forest with Minimum Number of Steiner Points (SF-MSP) problem we are given a set \(D \subseteq R \times R\) of demand pairs and \(G[R \cup S]\) should contains a \(uv\)-path for every \(uv \in D\). Let \(\varDelta \) be the maximum number of points in a unit ball such that the distance between any two of them is larger than \(1\). It is known that \(\varDelta =5\) in \(\mathbb {R}^2\). The previous known approximation ratio for ST-MSP was \(\lfloor (\varDelta +1)/2 \rfloor +1+\epsilon \) in an arbitrary normed space [15], and \(2.5+\epsilon \) in the Euclidean space \(\mathbb {R}^2\) [5]. Our approximation ratio for ST-MSP is \(1+\ln (\varDelta -1)+\epsilon \) in an arbitrary normed space, which in \(\mathbb {R}^2\) reduces to \(1+\ln 4+\epsilon < 2.3863 +\epsilon \). For SF-MSP we give a simple \(\varDelta \)-approximation algorithm, improving the folklore ratio \(2(\varDelta -1)\). Finally, we generalize and simplify the \(\varDelta \)-approximation of Calinescu [3] for the \(2\) -Connectivity-MSP problem, where \(G[R \cup S]\) should be \(2\)-connected.KeywordsWireless networkUnit-disc graphSteiner treeSteiner forest \(2\)-connectivityApproximation algorithms

Scan tree architecture can effectively reduce test data volume, test time and test cost for integrated circuits. To reduce the number of leaf nodes and TSVs(through silicon vias）in scan tree for three dimensional integrated circuits, this paper firstly draws the conclusion that the minimum number of leaf nodes is the number of scan cells contained in the maximal compatible group. Then, the necessary and sufficient condition achieving the minimum number of leaf nodes is presented. On the basis above, a heuristic algorithm is proposed, which can minimize the number of leaf nodes and reduce consumed TSVs as many as possible. Experimental results demonstrate the effectiveness of the proposed technique.

The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries.

A Steiner Minimal Tree (SMT) for a given set P of points is a shortest network interconnecting the points of P whose vertex set may include some additional points in order to get the minimum possible total length in a metric space. When no additional points are allowed the minimum interconnection network is the well-known minimum spanning tree (MST) of P. The Steiner ratio is the greatest lower bound of the ratio of the length of an SMT over that of an MST of P. In this paper we study the Steiner minimal tree problem in which all the edges of SMT have fixed orientations. We call it the SMT problem in the λ-geometry plane, where λ is the number of possible orientations.Here is the summary of our results. 1. We show that the Steiner ratio for ¦P¦≥3 is √3/2 cos(π/2λ), for λ=6m+3 and integer m≥0, and is √3/2, for λ=6k and integer k≥1, disproving a a conjecture of Du et al.[3] that the ratio is √3/2 iff the unit disk in normed planes is an ellipse. 2. We derive the Steiner ratios for ¦P¦ ≤ 4 for all possible λ's and show that for ¦P¦≥3 there exists an SMT whose Steiner points lie in a multi-level Hanan-grid, generalizing a result that holds for rectilinear case, i.e., λ=2. These results show that the Steiner ratio is not a monotonically increasing function of λ, as believed by many researchers. We conjecture that the Steiner ratios obtained above (¦P¦≤4) are actually true for all ¦P¦≥3.KeywordsMinimum Span TreeSteiner TreeRegular PointSteiner PointNormed PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The congested clique model is a message-passing model of distributed computation where k players communicate with each other over a complete network. Here we consider synchronous protocols in which communication happens in rounds (we allow them to be randomized with public coins). In the unicast communication mode, each player i has her own n-bit input \(x_i\) and may send \(k-1\) different b-bit messages through each of her \(k-1\) communication links in each round. On the other end is the broadcast communication mode, where each player can only broadcast a single message over all her links in each round. The goal of this paper is to complete our Brief Announcement at PODC 2015, where we initiated the study of the space that lies between the two extremes. For that purpose, we parametrize the congested clique model by two values: the range r, which is the maximum number of different messages a player is allowed to send in each round, and the bandwidth b, which is the maximum size of these messages. We show that the space between the unicast and broadcast congested clique models is very rich and interesting. For instance, we show that the round complexity of the pairwise set-disjointness function \(\textsc {pwdisj}\) is completely sensitive to the range r. This translates into a \(\varOmega (k)\) gap between the unicast (\(r=k-1\)) and the broadcast (\(r=1\)) modes. Moreover, provided that \(r \ge 2\) and \(rb/\log r = O(k)\), the round complexity of \(\textsc {pwdisj}\) is \(\varTheta (n/ k \log r )\). On the other hand, we also prove that the behavior of \(\textsc {pwdisj}\) is exceptional: almost every boolean function f has maximal round complexity \(\varTheta (n/b)\). Finally, we prove that \(\min \left( \left\lceil \frac{b'}{\lfloor \log r \rfloor } \right\rceil , \left\lceil \frac{r'}{r-1}\right\rceil \left\lceil \frac{b'}{b}\right\rceil \right) \) is an upper bound for the gap between the round complexities with parameters (b, r) and parameters \((b',r')\) of any boolean function.

In a recent paper [5], we addressed the problem of finding a minimum-cost spanning tree T for a given undirected graph G=(V,E) with maximum node-degree at most a given parameter B>1. We developed an algorithm based on Lagrangean relaxation that uses a repeated application of Kruskal’s MST algorithm interleaved with a combinatorial update of approximate Lagrangean node-multipliers maintained by the algorithm.In this paper, we show how to extend this algorithm to the case of Steiner trees where we use a primal-dual approximation algorithm due to Agrawal, Klein, and Ravi [1] in place of Kruskal’s minimum-cost spanning tree algorithm. The algorithm computes a Steiner tree of maximum degree O(B + log n) and total cost that is within a constant factor of that of a minimum-cost Steiner tree whose maximum degree is bounded by B. However, the running time is quasi-polynomial.

Given an edge-weighted graph and a set of known seed vertices of interest, a network scientist often desires to understand the graph relationships to explain connections between the seed vertices. If the size of the seed set is 2, shortest path calculations are an attractive computational kernel to explore the connections between the two vertices. When the seed set is 3 or larger (say up to 1,000s) Steiner minimal tree – min-weight acyclic connected subgraph (of the input graph) that contains all the seed vertices – is an attractive generalization of shortest weighted paths. In general, computing a Steiner minimal tree is NP-hard, but decades ago several polynomial-time algorithms were designed and proven to yield Steiner trees whose total weight is bounded within 2 times the minimal Steiner tree. Despite its rich theoretical literature, works related to parallel Steiner minimal tree computation and their scalable implementations are rather scarce. In this paper, we present a parallel 2-approximation Steiner minimal tree algorithm (with theoretical guarantees) and its MPI-based distributed implementation. In place of distance computation between all pairs of seed vertices, an expensive phase in many approximation algorithms, the solution we employ, exploits Voronoi cell computation. Also, this approach has higher parallel efficiency than others that involve minimum spanning tree computation on the entire graph. Furthermore, our distributed design exploits asynchronous processing and a message prioritization scheme to accelerate convergence of distance computation, employs techniques to avoid inefficient distributed spanning tree computation on the entire graph, and harnesses a combination of vertex and edge centric processing to offer fast time-to-solution. We demonstrate scalability and performance of our solution using real-world graphs with up to 128 billion edges and 512 compute nodes (8K processes), show the ability to find Steiner trees with up to 10K seed vertices in under one minute, and present in-depth analyses that highlight the benefits of our design choices. Using four real-world graphs and three seed sets for each, we compare our solution with the state-of-the-art exact Steiner minimal tree solver, SCIP-Jack, and two sequential algorithms with the same approximation bound as our algorithm. Our distributed solution comfortably outperforms these related works on graphs with 10s million edges and offers decent strong scaling – up to 90% efficient. We empirically show that, on average, the total distance (sum of edge weights) of the Steiner tree identified by our solution is 1.0527 times greater than the Steiner minimal tree (i.e., the optimal solution) – well within the theoretical bound of less than equal to 2.

We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the congested clique model. The input graph in the congested clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange O(log n) bits. As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches, we do not aim at simulating the standard Boruvka's algorithm, at least at initial stages of the CC algorithm. Instead, we develop a new technique which combines connected components of sample sparse subgraphs of the input graph in order to accelerate the process of uncovering connected components of the original input graph. More specifically, we develop a sparsification technique which reduces an initial CC problem in O(1) rounds to its two restricted instances. The former instance has a graph with maximal degree O(log log n) as the input - here our sample-combining technique helps. In the latter instance, a partition of the input graph into O(n/ log log n) connected components is known. This gives an opportunity to apply previous algorithms to determine connected components in O(1) rounds. Our result addresses a problem proposed by Lotker et al. [SPAA 2003; SICOMP 2005] and improves over previous O(log* n) algorithm of Ghaffari et al. [PODC 2016], and O(log log log n) algorithm of Hegeman et al. [PODC 2015]. It also determines Θ(1) round complexity in the congested clique for MST, as well as other graph problems, including bipartiteness, cut verification, s-t connectivity, and cycle containment.

스타이너 트리의 생성은 NP-Complete 영역에 속하므로, 이것을 위한 휴리스틱들은, 다수의 입력 노드에 대해서 많은 시간과 계산을 요구한다. 본 논문에서는 많은 입력노드에 대해, 들로네 삼각망과 Prim의 최소신장트리를 결합한 효과적인 유클리드 스타이너 최소트리 구성방법을 제안한다. 이 방법은 Prim의 최소신장트리와 최소신장트리기반 스타이너 트리와 각각 비교 분석되었다. 제안된 방법은 30,000개의 입력노드에 대해 최소신장트리에 비해 연결 길이는 2.1% 감소, 실행시간은 138.2% 증가하였고, 최소신장트리기반 스타이너최소트리에 비해 실행시간 18.9% 감소, 연결 길이 0.013% 감소의 실험결과를 보였다. 따라서 본 연구의 제안방법은 실행시간이 주요 요인이 되지 않는 환경에서 연결 길이를 단축해야 할 응용에 잘 적용될 수 있을 것이다. As Steiner minimum tree building belongs to NP-Complete problem domain, heuristics for the problem ask for immense amount execution time and computations in numerous inputs. In this paper, we propose an efficient mechanism of euclidean Steiner minimum tree construction for numerous inputs using combination of Delaunay triangulation and Prim's minimum spanning tree algorithm. Trees built by proposed mechanism are compared respectively with the Prim's minimum spanning tree and minimums spanning tree based Steiner minimum tree. For 30,000 input nodes, Steiner minimum tree by proposed mechanism shows about 2.1% tree length less and 138.2% execution time more than minimum spanning tree, and does about 0.013% tree length less and 18.9% execution time less than minimum spanning tree based Steiner minimum tree in experimental results. Therefore the proposed mechanism can work moderately well to many useful applications where execution time is not critical but reduction of tree length is a key factor.