Comparison on Routing Processing Performance between Shortest Paths Tree (SPT) and Minimum Spanning Tree (MST)

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n > k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in O(n/k) time (the notation O hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in O(n/Ok) time (for (1 + e)-factor approximation) and in O(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

With technology advancement, Vehicular Ad Hoc Networks (VANETs) have been requested to provide one-to-many group communications (i.e. multicasting) for business and entertainment applications such as video conferences and file sharings. For these group communication applications, the existing broadcasting-based and unicasting-based approaches designed for safety applications however cause high packet generations and redundant data transmissions, respectively. Although the well-known Shortest Path Tree (SPT) and Minimum Spanning Tree (MST) algorithms for multicasting have been proposed, they achieve only a specific goal such as the reduction of delay or tree cost. Thus, for reducing end-to-end delay, delay variation, and transmission number requested for delay-sensitive and cost-efficient group communications and supporting high mobility of vehicles, this paper proposes a multicast protocol which uses the Farthest destination Selection and Shortest path Connection (FSSC) strategy to form a multicast tree. FSSC relies on the road map information and the location information of vehicles and intersections. Simulation results are provided to verify the effectiveness of FSSC.

Compressive sensing (CS) can reduce the number of data transmissions and balance the traffic load throughout networks. However, the total number of data transmissions required in CS method is still large. It is observed that there are many zero elements in the measurement matrix. In each round of data transmission in CS method, the sensor nodes corresponding to the zero elements in the measurement matrix do not have their own data to transmit. To further reduce the number of data transmissions in the network, we aim to compute a data gathering tree by taking advantages of these zero elements in the measurement matrix, such that the total number of data transmissions is minimized. We formulate the problem as linear programming with boolean variables. The problem is NP-hard. We propose heuristic algorithm to compute the Minimum Transmission Tree (MTT) for data gathering in CS methods. The MTT algorithm constructs a spanning tree by iteratively including the edge whose average incremental transmission cost is minimum. The simulation results demonstrate that our algorithm can reduce the number of transmissions significantly, compared with the methods using minimum spanning tree (MST), shortest path tree and the CS method with nonzero measurement coefficient using MST.

We consider a setting where we are given a graph \(\mathcal {G}=(\mathcal {R},E)\), where \(\mathcal {R}=\{R_1,\ldots ,R_n\}\) is a set of polygonal regions in the plane. Placing a point \(p_i\) inside each region \(R_i\) turns \(G\) into an edge-weighted graph \(G_{\varvec{p}}\), \({\varvec{p}}=\{p_1,\ldots ,p_n\}\), where the cost of \((R_i,R_j)\in E\) is the distance between \(p_i\) and \(p_j\). The Shortest Path Problem with Neighborhoods asks, for given \(R_s\) and \(R_t\), to find a placement \(\varvec{p}\) such that the cost of a resulting shortest \(st\)-path in \(\mathcal {G}_{\varvec{p}}\) is minimum among all graphs \(\mathcal {G}_{\varvec{p}}\). The Minimum Spanning Tree Problem with Neighborhoods asks to find a placement \(\varvec{p}\) such that the cost of a resulting minimum spanning tree is minimum among all graphs \(\mathcal {G}_{\varvec{p}}\). We study these problems in the \(L_1\) metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is \(\mathsf {APX}\)-hard, even if the neighborhood regions are segments.

We consider a setting where we are given a graph $$\mathcal {G}=(\mathcal {R},E)$$ , where $$\mathcal {R}=\{R_1,\ldots ,R_n\}$$ is a set of polygonal regions in the plane. Placing a point $$p_i$$ inside each region $$R_i$$ turns $$G$$ into an edge-weighted graph $$G_{\varvec{p}}$$ , $${\varvec{p}}=\{p_1,\ldots ,p_n\}$$ , where the cost of $$(R_i,R_j)\in E$$ is the distance between $$p_i$$ and $$p_j$$ . The Shortest Path Problem with Neighborhoods asks, for given $$R_s$$ and $$R_t$$ , to find a placement $$\varvec{p}$$ such that the cost of a resulting shortest $$st$$ -path in $$\mathcal {G}_{\varvec{p}}$$ is minimum among all graphs $$\mathcal {G}_{\varvec{p}}$$ . The Minimum Spanning Tree Problem with Neighborhoods asks to find a placement $$\varvec{p}$$ such that the cost of a resulting minimum spanning tree is minimum among all graphs $$\mathcal {G}_{\varvec{p}}$$ . We study these problems in the $$L_1$$ metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is $$\mathsf {APX}$$ -hard, even if the neighborhood regions are segments.

Energy conservation is a critical issue in ad hoc wireless networks for node and network life, as the nodes are powered by batteries only. One major approach for energy conservation is to route a communication session along the route which requires the lowest total energy consumption. This optimization problem is referred to as Minimum-Energy Routing. While the minimum-energy unicast routing problem can be solved in polynomial time by shortest-path algorithms, it remains open whether the minimum-energy broadcast routing problem can be solved in polynomial time, despite the NP-hardness of its general graph version. Recently three greedy heuristics were proposed in [11]: MST (minimum spanning tree), SPT (shortest-path tree), and BIP (broadcasting incremental power). They have been evaluated through simulations in [11], but little is known about their analytical performances. The main contribution of this paper is a quantitative characterization of their performances in terms of approximation ratios. By exploring geometric structures of Euclidean MSTs, we have been able to prove that the approximation ratio of MST is between 6 and 12, and the approximation ratio of BIP is between 13/3 and 12. On the other hand, we show that the approximation ratio of SPT is at least n/2, where n is the number of receiving nodes. To the best of our knowledge, these are the first analytical results for the minimum-energy broadcasting problem.

The minimum spanning tree problem with non-terminal set

We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log2n) work per modification. No previous dynamic geometric MST algorithm was known.

On the K shortest path trees problem

The wind farm layout optimisation problem is similar to the classic mathematical problem of finding the minimum spanning tree (MST) of a weighted undirected graph. Due to the cable current‐carrying capacity limitation, the cable sectional area should be carefully selected to meet the system operational requirement and this constraint should be considered during the MST formulation process. Hence, traditional MST algorithm cannot ensure a minimal cable investment layout. In this study, a new method to optimise the offshore wind farm cable connection layout is presented. The algorithm is formulated based on the concept of MST and further improved by adaptive particle swarm optimisation algorithm. Since the location of the offshore substation (OS) has a significant impact on both the layout formulation and total cost of cables, the optimised location of OS is expected to be found together with the optimised cable connection layout. The proposed method is compared with the MST and dynamic MST methods and simulation results show the effectiveness of the proposed method.

Fully Dynamic Algorithms for Maintaining Shortest Paths Trees

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].

In order to solve the path planning problem of time-dependent road network(TDRN), an dynamic A* landmarks triangle algorithm(ALT) is proposed based on landmark-oriented technique and short-path tree(SPT). There are three main contributions: (1) constructing the shortest path tree in the preprocessing stage and calculating the distance between the landmark and other nodes; (2) using the dynamic shortest path tree to optimize the query in the point-to-point heuristic path planning process; (3) When the edge weight of the network changes, the shortest path tree is dynamically updated, and the structural characteristics of the tree are used to reduce the redundancy calculation. Experimental results indicate that the DALT algorithm not only outperforms the ALT implementation in point-to-point shortest path problem as the average query time is reduced by up to 51.71%, but also computes economically for updating shortest path tree compared with previous dynamic update algorithm as the average update times for increments are reduced by up to 9.90% with less modifications.

We propose a simple mechanism for supporting routing control in unstructured P2P systems. In the proposal, each node keeps the information on the hop-limited shortest path trees whose roots are located around the neighbor. Here, the hop-limited tree means the one that is made of nodes within a certain number of hops (say n hops) from the root. When a node receives a unicast message, it transmits the message to the upstream neighbor on the shortest-path tree rooted at the message destination only if the destination is located within the n-hop radius. Otherwise, it generates duplicates of the original message and forwards them to downstream neighbors on the shortest-path tree rooted at the message source. The information of upstream and downstream neighbors on the shortest path trees is stored in the message forwarding or the routing tables, which are constructed and updated in a fully distributed fashion. Numerical examples show that our proposal can largely reduce the number of queries generated in a file search.

The Minimum Spanning Tree problem aims to create a subset of a graph where all the vertices are connected with the minimum edge weights and with no cycle. In this field, an innovative method entitled Dhouib-Matrix-MSTP (DM-MSTP) is designed in this research work with a time complexity independently of the number of edges O(n*log(n)) where n is the number of vertices. DM-MSTP is a constructive algorithm based on a matrix navigation with two new lists (Min-Columns and MST-Path) in order to organize the steering flow. DM-MSTP is composed of four simple steps where the first and the fourth steps are repeated only once, whereas the second and third are reiterated (n-1). For more clarification, a step-by-step application of the proposed method is presented in details. Besides, the performance of DM-MSTP is proved through different examples from the literature including a graph with negative weighted edges and complete graphs from TSP-LIB. Moreover, with a simple modification (Min by Max) the DM-MSTP is tested on the Maximum (Largest) Spanning Tree Problem. Also, DM-MSTP is applied on eight case studies and compared to six methods developed in the literature. All these experimental results in the above different environments show that DM-MSTP can rapidly plan the shortest spanning tree with a stable performance and convivial representation of the optimal tree. Hence, DM-MSTP is developed under Python programming language using Matplotlib and Numpy standard libraries.