Single-source Shortest Paths Tree Research Articles

Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer \(f \ge 1\), we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) \(\sigma \)-approximate single-source shortest-path tree (\(\sigma \)-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched by a factor of at most \(\sigma \). To this respect, we provide an algorithm that efficiently computes an f-EFT \((2|F|+1)\)-ASPT of size O(fn). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of \(3(f+1)\), plus an additive term of \((f+1) \log n\). Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle, that can be built in \(O(f m\, \alpha (m,n)+fn \log ^3 n)\) time, has size \(O(fn \log ^2 n)\), and in \(O(|F|^2 \log ^2 n)\) time is able to report a \((2|F|+1)\)-approximate distance from the source to any node in \(G-F\). Moreover, our oracle can return a corresponding approximate path in the same amount of time plus the path’s size. The oracle is obtained by tackling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G following a batch of k simultaneous modification (i.e., edge insertions, deletions and weight changes). For this problem, we build in \(O(m \log ^3 n)\) time an oracle of size \(O(m \log ^2 n)\), that reports in \(O(k^2 \log ^2 n)\) time the (at most 2k) edges either exiting from or entering into the MSF. Finally, for any integer \(k \ge 1\), we complement all our results with a lower bound of \(\Omega \left( n^{1+\frac{1}{k}}\right) \) to the size of any f-EFT \(\sigma \)-ASPT with \(f \ge \log n\) and \(\sigma < \frac{3k+1}{k+1}\), that holds if the Erdős’ girth conjecture is true.

Hardness of an Asymmetric 2-Player Stackelberg Network Pricing Game

Consider a communication network represented by a directed graph G=(V,E) of n nodes and m edges. Assume that edges in E are partitioned into two sets: a set C of edges with a fixed non-negative real cost, and a set P of edges whose costs are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their use by a follower, whose goal in turn is to select for his communication purposes a subnetwork of Gminimizing a given objective function of the edge costs. In this paper, we study the natural setting in which the follower computes a single-source shortest paths tree of G, and then returns to the leader a payment equal to the sum of the selected priceable edges. Thus, the problem can be modeled as a one-round two-player Stackelberg Network Pricing Game, but with the novelty that the objective functions of the two players are asymmetric, in that the revenue returned to the leader for any of her selected edges is not equal to the cost of such an edge in the follower’s solution. As is shown, for any ϵ&gt;0 and unless P=NP, the leader’s problem of finding an optimal pricing is not approximable within n1/2−ϵ, while, if G is unweighted and the leader can only decide which of her edges enter in the solution, then the problem is not approximable within n1/3−ϵ. On the positive side, we devise a strongly polynomial-time O(n)-approximation algorithm, which favorably compares against the classic approach based on a single-price algorithm. Finally, motivated by practical applications, we consider the special cases in which edges in C are unweighted and happen to form two popular network topologies, namely stars and chains, and we provide a comprehensive characterization of their computational tractability.

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 $$\epsilon >0$$ .

Broadcast tree construction framework in tactile internet via dynamic algorithm

Abstract Extremely low-latency and real-time communications are required in Tactile Internet to transfer physical tactile experiences remotely. In addition, its traffic has stringent requirements on bandwidth and quality of service (QoS). To minimize total costs of establishing the network and satisfy a pre-defined global upper delay-bound on the paths from the server to any other client for message broadcast in Tactile Internet, this paper presents a Rooted Delay-Constrained Minimum Spanning Tree (RDCMST) construction framework based on dynamic algorithm. The network is modeled as a connected weighted and undirected graph. Infeasible and suboptimal edges are discarded first by preprocessing techniques to reduce the processing complexity of the algorithm. Then the edges of the graph are processed based on a dynamic graph algorithm, which can maintain a single-source shortest path tree for the online edge deletions, such that total costs can be minimized while ensuring the delay-constraint and the tree structure. Experimental results show that our proposed approach greatly outperforms existing competing RDCMST formation algorithms, in terms of both average cost and stability of solutions.

A Simple Algorithm For Replacement Paths Problem

Let G=(V,E) (|V|=n and |E|=m) be an undirected graph with positive edge weights. Let PG(s,t) be a shortest s−t path in G. Let l be the number of edges in PG(s,t). The Edge Replacement Path problem is to compute a shortest s−t path in G\\{e}, for every edge e in PG(s,t). The Node Replacement Path problem is to compute a shortest s−t path in G\\{v}, for every vertex v in PG(s,t).In this paper we present an O(TSPT(G)+m+l2) time and O(m+l2) space algorithm for both the problems. Where, TSPT(G) is the asymptotic time to compute a single source shortest path tree in G. The proposed algorithm is simple and easy to implement.

Approximations for constructing tree-form structures using specific material with fixed length

Substituting for the ordinary objective to minimize the sum of lengths of all edges in some graph structure of a weighted graph, we propose a new problem of constructing certain tree-form structure in same graph, where all edges needed in such a tree-form structure are supposed to be cut from some pieces of a specific material with fixed length. More precisely, we study a new problem defined as follows: a weighted graph \(G=(V,E; w)\), a tree-form structure \(\mathcal{S}\) and some pieces of specific material with length L, where a length function \(w:E\rightarrow Q^+\), satisfying \(w(u,v) \le L\) for each edge uv in G, we are asked how to construct a required tree-form structure \(\mathcal{S}\) as a subgraph \(G'\) of G such that each edge needed in \(G'\) is constructed by a part of a piece of such a specific material, the new objective is to minimize the number of necessary pieces of such a specific material to construct all edges in \(G'\). For this new objective defined, we obtain three results: (1) We present a \(\frac{3}{2}\)-approximation algorithm to construct a spanning tree of G; (2) We design a \(\frac{3}{2}\)-approximation algorithm to construct a single-source shortest paths tree of G; (3) We provide a 4-approximation algorithms to construct a metric Steiner tree of G.

A Faster Computation of All the Best Swap Edges of a Shortest Paths Tree

We consider a two-edge connected, non-negatively real-weighted graph G with n vertices and m edges, and a single-source shortest paths tree (SPT) of G rooted at an arbitrary vertex. If an edge of the SPT is temporarily removed, a widely recognized approach to reconnect the vertices disconnected from the root consists of joining the two resulting subtrees by means of a single non-tree edge, called a swap edge. This allows to reduce consistently the set-up and computational costs which are incurred if one instead rebuilds a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge, and here we restrict our attention to arguably the two most significant measures: the minimization of either the maximum or the average distance between the root and the disconnected vertices. For the former criteria, we present an $$O(m \log \alpha (m,n))$$O(mlog?(m,n)) time algorithm--where $$\alpha $$? is the inverse of the Ackermann function--to find a best swap edge for every edge of the SPT, thus improving onto the previous $$O(m \log n)$$O(mlogn) time algorithm. Concerning the latter criteria, we provide an $$O(m+n \log n)$$O(m+nlogn) time algorithm for the special but important case where G is unweighted, which compares favourably with the $$O\left( m+n \, \alpha (n,n)\log ^2n\right) $$Om+n?(n,n)log2n time bound that one would get by using the fastest algorithm known for the weighted case--once this is suitably adapted to the unweighted case.

On Dynamic Shortest Paths Problems

We obtain the following results related to dynamic versions of the shortest-paths problem: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of $\tilde {O}(m\sqrt{n})$ (we use $\tilde {O}$ to hide small poly-logarithmic factors) and a worst case query time is O(n3/4). A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.

Exact and Approximate Truthful Mechanisms for the Shortest Paths Tree Problem

Let a communication network be modeled by an undirected graph G=(V,E) of n nodes and m edges, and assume that edges are controlled by selfish agents. In this paper we analyze the problem of designing a truthful mechanism for computing one of the most popular structures in communication networks, i.e., the single-source shortest paths tree. More precisely, we will study several realistic scenarios, in which each agent can own either a single or multiple edges of G. In particular, for the single-edge case, we will show that: (i) in the classic utilitarian case, the problem can be solved efficiently in O(mnlogα(m,n)) time, where ±(m,n) is the inverse of the Ackermanns function; (ii) in a meaningful non-utilitarian case, namely that in which agents valuation functions only depend on the edge lengths, the problem can be solved in O(m+nlogn) time. Conversely, for the multiple-edges case, we will show in the utilitarian case an O(mP+nPlogn) time truthful mechanism, where P=O(n) denotes the number of agents participating in the solution, while in the same non-utilitarian case we will prove a general lower bound to the approximation ratio that can be achieved by any truthful mechanism, by showing that no c-approximate mechanism can exist, for any fixed c < 5+√13 / 3+√13.

Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor

We consider a two-edge connected, undirected graph G = ( V , E ) , with n nodes and m non-negatively real weighted edges, and a single source shortest paths tree (SPT) T of G rooted at an arbitrary node r . If an edge in T is temporarily removed, it makes sense to reconnect the nodes disconnected from the root by adding a single non-tree edge, called a swap edge, instead of rebuilding a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge. In this paper we focus on the most prominent one, that is the minimization of the average distance between the root and the disconnected nodes. To this respect, we present an O ( m log 2 n ) time and O ( m ) space algorithm to find a best swap edge for every edge of T , thus improving for m = o ( n 2 / log 2 n ) the previously known O ( n 2 ) time and space complexity algorithm.

Swapping a Failing Edge of a Single Source Shortest Paths Tree Is Good and Fast

Abstract. Let G=(V,E) be a 2-edge connected, undirected and nonnegatively weighted graph, and let S(r) be a single source shortest paths tree (SPT) of G rooted at r ∈ V . Whenever an edge e in S(r) fails, we are interested in reconnecting the nodes now disconnected from the root by means of a single edge e' crossing the cut created by the removal of e . Such an edge e' is named a swap edge for e . Let S e/e' (r) be the swap tree (no longer an SPT, in general) obtained by swapping e with e' , and let S e be the set of all possible swap trees with respect to e . Let F be a function defined over S e that expresses some feature of a swap tree, such as the average length of a path from the root r to all the nodes below edge e , or the maximum length, or one of many others. A best swap edge for e with respect to F is a swap edge f such that F(S e/f (r)) is minimum. In this paper we present efficient algorithms for the problem of finding a best swap edge, for each edge e of S(r) , with respect to several objectives. Our work is motivated by a scenario in which individual connections in a communication network suffer transient failures. As a consequence of an edge failure, the shortest paths to all the nodes below the failed edge might completely change, and it might be desirable to avoid an expensive switch to a new SPT, because the failure is only temporary. As an aside, what we get is not even far from a new SPT: our analysis shows that the trees obtained from the swapping have features very similar to those of the corresponding SPTs rebuilt from scratch.

Efficient search and hierarchical motion planning by dynamically maintaining single-source shortest paths trees

Hierarchical approximate cell decomposition is a popular approach to the geometric robot motion planning problem. In many cases, the search effort expended at a particular iteration can be greatly reduced by exploiting the work done during previous iterations. In this paper, we describe how this exploitation of past computation can be effected by the use of a dynamically maintained single-source shortest paths tree. We embed a single-source shortest paths tree in the connectivity graph of the approximate representation of the robot configuration space. This shortest paths tree records the most promising path to each vertex in the connectivity graph from the vertex corresponding to the robot's initial configuration. At each iteration, some vertex in the connectivity graph is replaced with a new set of vertices, corresponding to a more detailed representation of the configuration space. Our new, dynamic algorithm is then used to update the single-source shortest paths tree to reflect these changes to the underlying connectivity graph.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A linear algorithm for analysis of minimum spanning and shortest-path trees of planar graphs

We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a single-source shortest-path tree to changes in edge costs, and to analyze the sensitivity of a minimum-cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heap-ordered queue data structure.

A distributed shortest path algorithm for a planar network

An algorithm is presented for finding a single source shortest path tree in a planar undirected distributed network with nonnegative edge costs. The number of messages used by the algorithm is O(n 5 3 ) on an n -node network. Distributed algorithms are also presented for finding a breath-first spanning tree in general network, for finding a shortest path tree in a general network, for finding a separator of a planar network, and for finding a division of a planar network.