Shortest Path Tree Research Articles

A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-Off Algorithms

Given a read-only memory for input and a write-only stream for output, an s-workspace algorithm, for a positive integer parameter s, is an algorithm using only O(s) words of workspace in addition to the memory for the input. In this paper, we present an $$O(n^2/s)$$ -time s-workspace algorithm for subdividing a simple n-gon into $$O(\min \{n/s,s\})$$ subpolygons of complexity $$O(\max \{n/s,s\})$$ . As applications of the subdivision, the previously best known time-space trade-offs for the following three geometric problems are improved immediately by adopting the proposed subdivision: (1) computing the shortest path between two points inside a simple n-gon, (2) computing the shortest-path tree from a point inside a simple n-gon, (3) computing a triangulation of a simple n-gon. In addition, we improve the algorithm for problem (2) further by applying different approaches depending on the size of the workspace.

TAPU: Test and pick up-based k-connectivity restoration algorithm for wireless sensor networks

A k-connected wireless sensor network remains connected if any k-1 arbitrary nodes stop working. The aim of movement-assisted k -connectivity restoration is to preserve the k -connectivity of a network by moving the nodes to the necessary positions after possible failures in nodes. This paper proposes an algorithm named TAPU for k-connectivity restoration that guarantees the optimal movement cost. Our algorithm improves the time and space complexities of the previous approach (MCCR) in both best and worst cases. In the proposed algorithm, the nodes are classified into safe and unsafe groups. Failures of safe nodes do not change the k value of the network while failures of unsafe nodes reduce the k value. After an unsafe node’s failure, the shortest path tree of the failed node is generated. Each node moves to its parent location in the tree starting from a safe node with the minimum moving cost to the root. TAPU has been implemented on simulation and testbed environments including Kobuki robots and Iris nodes. The measurements show that TAPU finds the optimum movement up to 79.5% faster with 50% lower memory usage than MCCR and with up to 59% lower cost than the greedy algorithms.

Easy computation of eccentricity approximating trees

A spanning tree T of a graph G=(V,E) is called eccentricityk-approximating if we have eccT(v)≤eccG(v)+k for every v∈V. Let ets(G) be the minimum k such that G admits an eccentricity k-approximating spanning tree. As our main contribution in this paper, we prove that ets(G) can be computed in O(nm)-time along with a corresponding spanning tree. This answers an open question of Dragan et al. (2017). Moreover we also prove that for some classes of graphs such as chordal graphs and hyperbolic graphs, one can compute an eccentricity O(ets(G))-approximating spanning tree in quasi linear time. Our proofs are based on simple relationships between eccentricity approximating trees and shortest-path trees.

Hardness, approximability, and fixed-parameter tractability of the clustered shortest-path tree problem

Given an n-vertex non-negatively real-weighted graph G, whose vertices are partitioned into a set of k clusters, a clustered network design problem on G consists of solving a given network design optimization problem on G, subject to some additional constraints on its clusters. In particular, we focus on the classic problem of designing a single-source shortest-path tree, and we analyse its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the unweighted case, and prove that the problem is $${\textsf {NP}}$$ -hard. However, on the positive side, we show the existence of an approximation algorithm whose quality essentially depends on few parameters, but which remarkably is an O(1)-approximation when the largest out of all the diameters of the clusters is either O(1) or $$\varTheta (n)$$ . Furthermore, we also show that the problem is fixed-parameter tractable with respect to k or to the number of vertices that belong to clusters of size at least 2. Then, we focus on the weighted case, and show that the problem can be approximated within a tight factor of O(n), and that it is fixed-parameter tractable as well. Finally, we analyse the unweighted single-pair shortest path problem, and we show it is hard to approximate within a (tight) factor of $$n^{1-\epsilon }$$ , for any $$\epsilon >0$$ .

Fast approximation of eccentricities and distances in hyperbolic graphs

We show that the eccentricities of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v):=\max\{d_G(u,v): u\in V\}$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree $T$, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$, where $ecc_T(v):=\max\{d_T(u,v): u\in V\}$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating eccentricities and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

Implementing Environmental and Social Responsibility Programs in Supply Networks through Multi-Unit Bilateral Negotiation

Involving suppliers deep in the supply chain is critical for the success of environmental and social responsibility (ESR) initiatives. Administering ESR programs throughout a complex supply network, however, is challenging. In this paper, we apply a multi-unit bilateral bargaining framework to coordinate ESR investments in a general supply network and analyze to what extent an ESR initiator should directly engage the higher-tier suppliers, as opposed to delegating that responsibility to the first-tier suppliers. Our bargaining framework not only generalizes the conventional Shapley value approach by allowing the flexibility of modeling imbalanced power distribution among the firms, but also provides an explicit way of implementing the resulting gain sharing among the firms through negotiated contract terms. We show that the eventual structure of ESR negotiation relationships can be derived by finding a shortest path tree in the supply network with the arc cost defined as the logarithm of the negotiating parties' relative bargaining power. These developments allow us to analyze ESR implementation in generally extended supply networks. We find that the ESR initiator tends to delegate ESR negotiations to a supplier that is strong in negotiations with higher-tier suppliers. When the supply network is complex (i.e., wide and deep), directly engaging all suppliers can lead to a larger gain by the initiator than fully delegating the negotiations with higher-tier suppliers to the first-tier ones. However, as the network gets increasingly complex, the ESR initiator tends to directly engage a reduced percentage of higher-tier suppliers. We further extend our analysis to situations where the ESR relationships are sequentially formed in a decentralized manner, where the benefit of ESR depends on the collective choice of the firms' investment levels, where multiple ESR programs are implemented in the network, and where ESR investments depend on the negotiation relationships.

Design and Implementation of a Wireless Sensor Network for Agricultural Applications

We present the design and implementation of a shortest path tree based, energy efficient data collection wireless sensor network to sense various parameters in an agricultural farm using in-house developed low cost sensors. Nodes follow a synchronized, periodic sleep-wake up schedule to maximize the lifetime of the network. The implemented network consists of 24 sensor nodes in a 3 acre maize farm and its performance is captured by 7 snooper nodes for different data collection intervals: 10 minutes,1 hour and 3 hours. The almost static nature of wireless links in the farm motivated us to use the same tree for a long data collection period (3 days). The imbalance in energy consumption across nodes is observed to be very small and the network architecture uses easy-to-implement protocols to perform different network activities including handling of node failures. We present the results and analysis of extensive tests conducted on our implementation, which provide significant insights.

A Retroactive Approach for Dynamic Shortest Path Problem

Dynamic shortest path algorithms modify the existing shortest path tree or graph, taking into account changes in the underlying graph configuration. In the premise of this paper, the dynamic Dijkstra algorithm is specifically considered which is used for the solution of single source shortest path problem in dynamic graphs. Dynamic Single Source Shortest Path (DSSSP) problem is considered from a completely different perspective. For incorporating the dynamic changes into the solution, the retroactive data structure has been used. Dynamics of retroactive data structures provide a natural order for propagating the desired changes in the underlying graph configuration. DSSSP has been solved in efficient way with worst case complexity O(m log n) and also proved the correctness of our proposed algorithm.

Revisiting the foundations of dominant-strategy mechanisms

An important question in mechanism design is whether there is any theoretical foundation for the use of dominant-strategy mechanisms. This paper studies the maxmin and Bayesian foundations of dominant-strategy mechanisms in general social choice environments with quasi-linear preferences and private values. We propose a condition called the uniform shortest-path tree that, under regularity, ensures the foundations of dominant-strategy mechanisms. This exposes the underlying logic of the existence of such foundations in the single-unit auction setting, and extends the argument to cases where it was hitherto unknown. To prove this result, we adopt the linear programming approach to mechanism design. In settings in which the uniform shortest-path tree condition is violated, maxmin/Bayesian foundations might not exist. We illustrate this by two examples: bilateral trade with ex ante unidentified traders and auction with type-dependent outside option.

Adaptive multicast streaming for videoconferences on software-defined networks

Real-time applications, such as video conferences, have strong Quality of Service requirements for ensuring a decent Quality of Experience. Nowadays, most of these conferences are performed over wireless devices. Thus, an appropriate management of both heterogeneous mobile devices and network dynamics is necessary. Software Defined Networking enables the use of multicasting and stream layering inside the network nodes, two techniques able to enhance the quality of live video streams. In this paper, we propose two algorithms for building and maintaining multicast sessions in a software-defined network. The first algorithm sets up the initial multicast trees for a given call. It optimally places the stream layer adaptation function inside the core network in order to minimize the bandwidth consumption. This algorithm has two versions: the first one, based on shortest path trees is minimizing the latency, while the second one, based on spanning trees is minimizing the bandwidth consumption. The second algorithm adapts the multicast trees according to the network changes occurring during a call. It does not recompute the trees, but only relocates the stream layer adaptation functions. It requires very low computation at the controller, thus making our proposal fast and highly reactive. Extensive simulation results confirm the efficiency of our solution in terms of processing time and bandwidth savings compared to existing solutions such as multiple unicast connections, Multipoint Control Unit solutions and application layer multicast.

Data Structures for Weighted Matching and Extensions to b -matching and f -factors

This article shows the weighted matching problem on general graphs can be solved in time O ( n ( m + n log n )) for n and m the number of vertices and edges, respectively. This was previously known only for bipartite graphs. The crux is a data structure for blossom creation. It uses a dynamic nearest-common-ancestor algorithm to simplify blossom steps, so they involve only back edges rather than arbitrary nontree edges. The rest of the article presents direct extensions of Edmonds’ blossom algorithm to weighted b -matching and f -factors. Again, the time bound is the one previously known for bipartite graphs: for b -matching the time is O (min { b ( V ), n log n }( m + n log n )), and for f -factors the time is O (min { f ( V ), m log n }( m + n log n )), where b ( V ) and f ( V ) both denote the sum of all degree constraints. Several immediate applications of the f -factor algorithm are given: The generalized shortest path structure of Reference [19], i.e., the analog of the shortest-paths tree for conservative undirected graphs, is shown to be a version of the blossom structure for f -factors. This structure is found in time O (| N |( m + n log n )) for N , the set of negative edges (0 < | N | < n ). A shortest T -join is found in time O ( n ( m + n log n )) or O (| T |( m + n log n )) when all costs are nonnegative. These bounds are all slight improvements of previously known ones, and are simply achieved by proper initialization of the f -factor algorithm.

Localized topology control and on-demand power-efficient routing for wireless ad hoc and sensor networks

Power use is a crucial design concern in wireless ad hoc and sensor networks since it corresponds directly to the network operational time. In this paper, we study the issue of power-efficient use in the following two aspects: Selecting power-efficient routes and performing efficient localized topology control to assign reduced transmit powers at nodes while preserving the global optimal connectivity. We proposed a localized topology control algorithm using two-hop neighborhood knowledge, which works to build local shortest path tree at each node independently in order to generate a reduced topology while preserving the global optimal connectivity. We derive the energy stretch ratio and maximum degree performance of our proposed algorithm as well as several existing algorithms in this aspect. We then devise three power-efficient on-demand routing protocols on top of various localized topology control algorithms, which are to acquire minimum-power paths while minimizing the associated protocol overhead for route discovery by utilizing local network state information and also packet receipt status at neighbor nodes. We further derive the asymptotical performance of the routing strategy in our protocols in terms of energy stretch ratio and route acquisition latency when network nodes operate at limited discrete power levels. Simulation results are provided to demonstrate the high performance of our topology control algorithm and also the devised routing protocols.

A Time-Space Trade-off for the Shortest Path Tree in a Simple Polygon

A [Formula: see text]-workspace algorithm may use a workspace of [Formula: see text] words which can be read and written, in addition to their input, which is provided as a read-only array of [Formula: see text] items. We present a [Formula: see text]-workspace algorithm for constructing the shortest path tree of a simple polygon [Formula: see text] with [Formula: see text] vertices, with respect to a given point inside the polygon in [Formula: see text] time, where [Formula: see text] is the time for computing the visibility polygon of a given point inside a simple polygon with [Formula: see text] additional words of space.

A MILP-based VND for the min-max regret Shortest Path Tree Problem with interval costs

The min-max regret Shortest Path Tree problem (RSPT) is a NP-Hard Robust Optimization counterpart of the Shortest Path Tree problem, where arcs costs are modeled as intervals of possible values. This problem arises from the uncertainty in link quality the routing protocols for IPv6 Low Wireless Personal Area Networks have to handle. In this paper, we propose a Variable Neighborhood Descent (VND) heuristic based on a Mixed Integer Linear Programming formulation. An exact algorithm based on the same formulation is used to assess the quality of this heuristic. Computational experiments show that VND has an average optimality gap of 0.91%, being smaller that with the best heuristic in literature for RSPT.

An Efficient Multipath Routing Algorithm

This paper studies how to employ the incremental shortest path first algorithm to reduce the computational overhead of the DC (Downstream Criterion) implementation, and proposes an efficient Intra-domain routing protection algorithm based on i-SPF (ERPISPF). Theoretical analysis indicates that the time complexity of ERPISPF is less than that of constructing a shortest path tree. The experiment results show that ERPISPF reduce more than 93% computation overhead compared to the DC, and can provide the same protection ratio with DC.

Dynamic Merging of Frontiers for Accelerating the Evaluation of Betweenness Centrality

Betweenness Centrality (BC) is a widely used metric of the relevance of a node in a network. The fastest-known algorithm for the evaluation of BC on unweighted graphs builds a tree representing information about the shortest paths for each vertex to calculate its contribution to the BC score. Actually, for specific vertices, the shortest-path trees of neighboring nodes could be leveraged to reduce the computational burden, but existing BC algorithms do not exploit that information and carry out redundant computations. We propose a new algorithm, called dynamic merging of frontiers, which makes use of such information to derive the BC score of degree-2 vertices by re-using the results of the sub-trees of the neighbors. We implemented our idea in parallel fashion exploiting Graphics Processing Units. Compared to state-of-the-art implementations, our approach achieves a linear improvement in the number of degree-2 vertices and an average improvement of × over a variety of real-world graphs.

Constructing multicast routing tree for inter-cloud data transmission: an approximation algorithmic perspective

Networking plays a crucial role in cloud computing especially in an inter-cloud environment, where data communications among data centers located at different geographical sites form the foundation of inter-cloud federation. Data transmissions required for inter-cloud federation in the complex inter-cloud networking system are often point-to-multi points, which calls for a more effective and efficient multicast routing algorithm in complex networking systems. In this paper, we investigate the multicast routing problem in the inter-cloud context with K constraints where K U+2265 2. Unlike most of existing algorithms that are too complex to be applied in practical scenarios, a novel and fast algorithm for establishing multicast routing tree for interclouds is proposed. The proposed algorithm leverages an entropybased process to aggregate all weights into a comprehensive metric, and then uses it to search a multicast tree U+0028 MT U+0029 on the basis of the shortest path tree U+0028 SPT U+0029.We conduct complexity analysis and extensive simulations for the proposed algorithm from the approximation perspective. Both analytical and experimental results demonstrate that the algorithm is more efficient than a representative multi-constrained multicast routing algorithm in terms of both speed and accuracy, and thus we believe that the proposed algorithm is applicable to the inter-cloud environment.

A Dynamic traffic-aware energy-efficient algorithm based on sleep-scheduling for autonomous systems

Over recent years, energy conservation of Internet infrastructure has attracted a great deal of attention to reduce the economical cost and environmental pollution produced by networking sectors. Due to over-provisioning of networking resources for peak traffic periods, an important proportion of these resources may remain under-utilized in low traffic periods. In this paper, we propose an online green algorithm which switches off some extra nodes and links in order to conserve energy while guaranteeing the network performance. To achieve this, nodes with less occurrence in the shortest path tree and their connected links are selected to be switched-off if their utilization is not more than the average network utilization. Due to the dynamic nature of network traffic, our solution is equipped with a dynamic engine which re-switches on some nodes and links in higher traffic loads. The proposed approach does not require any centralized controller for switching off devices and uses a link-state protocol like OSPF to share required information with minimal overhead. Simulation results obtained using a real network topology show that our proposal switches off up to 28% of network nodes and up to 74% of the maximum number of switchable links with low effect in network performance.

Two optimization problems for unit disks

We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in O(nlog⁡n) worst-case time, where n is the number of disks.In the minimum-separation problem, we are given n unit disks and two points s and t, not contained in any of the disks, and we want to compute the minimum number of disks one needs to retain so that any curve connecting s to t intersects some of the retained disks. We present a new algorithm solving this problem in O(n2log3⁡n) worst-case time and its implementation.

Проблема отката в ориентированной распределенной системе

For a distributed system based on a directed graph without multiple edges and loops, the backtracing problem is considered: how to transfer a message from the final vertex of the arc to its initial vertex. The task is to create a structure on the graph that allows the message to be transmitted from the final vertex of the arc to its initial vertex in the minimum time, i.e. on the shortest path. Such a structure at each vertex a is given by mapping the number of vertex b to the number of the outgoing arc through which the shortest path passes from a to b . In particular, such a mapping makes it possible to simulate, in directed distributed systems, algorithms for solving problems on a graph, developed for unditected distributed systems. This increases the running time of such algorithms by not more than k times, where k does not exceed the diameter of the graph, k < n , where n is the number of vertices of the graph. Section 2 describes the asynchronous model of the distributed system used. Section 3 contains the basic definitions and notation, and Section 4 - the statement of the problem. Section 5 describes two auxiliary algorithms for subtree correction, the application of which makes it possible to construct spanning trees of shortest paths: a out-tree and an in-tree. Section 6 contains a description of the various methods for transmitting messages over the graph. In Section 7, two algorithms are proposed for constructing in the memory of the graph root automaton the descriptions of spanning out- and in- shortest path trees, and in Section 8, the algorithms for constructing the required mapping based on them: a "fast" algorithm with T = O ( n ) and N = O ( n ) and an "economical" algorithm with T = O ( n 2 ) and N = O (1), where T is the running time of the algorithm, N is the number of messages simultaneously transmitted along the arc. In Section 9 it is proved that these estimates of time are not improved. In Section 10, the "fast" algorithm is modified for a synchronous model with N = 1. The conclusion sums up and outlines directions for further research.