We present an $O(nm)$ algorithm for all-pairs shortest paths computations in a directed graph with $n$ nodes, $m$ arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup \cite{Thorup1999} for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an $O(m\sqrt{n}\log n)$ preprocessing step. Using these reduced costs, every shortest path query can be solved in $O(m)$ time using an adaptation of Thorup's component hierarchy method. The balancing result can also be applied to the $\ell_\infty$-matrix balancing problem.
This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also des…
Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a~cycle of $G$ with minimum weight. The best previously known algorithm for the former problem, running in $O(n\log^3 n)$ time, can be obtained from the algorithm of \Lacki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the $O(n\log^3 n)$-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in pla…
Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-time algorithms on $n$-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the minimum weight $(2\ell+1)$-Clique problem in edge weighted graphs requires $n^{2\ell+1-o(1)}$ time, we prov…
In this paper, I present an algorithm called Raffica algorithm for Single-Source Shortest Path(SSSP). On random graph, this algorithm has linear time complexity(in expect). More precisely, the random graph uses configuration model, and the weights are distributed mostly positively. It is also linear for random grid graphs. Despite I made an assumption on the weights of the random graph, this algorithm is able to solve SSSP with arbitrary weights; when a negative cycle exists, this algorithm can find it out once traversed. The algorithm has a lot of appliances.
We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in $\widetilde{O}(mn^{4/3}\log{W}/ε)$ total time (where the edge weights are from $[1,W]$) and explicitly maintains a $(1+ε)$-approximate distance matrix. For a fixed $ε>0$, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is $o(n^2)$ regardless of the number of edges. Furthermore, we also show how to imp…
We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^…
A straightforward dynamic programming method for the single-source shortest paths problem (SSSP) in an edge-weighted directed acyclic graph (DAG) processes the vertices in a topologically sorted order. First, we similarly iterate this method alternatively in a breadth-first search sorted order and the reverse order on an input directed graph with both positive and negative real edge weights, $n$ vertices and $m$ edges. For a positive integer $t,$ after $O(t)$ iterations in $O(tm)$ time, we obtain for each vertex $v$ a path distance from the source to $v$ not exceeding that yielded by the short…
Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.
We present an algorithm for the k shortest simple path problem on weighted directed graphs (kSSP) that is based on Eppstein's algorithm for a similar problem in which paths are allowed to contain cycles. In contrast to most other algorithms for kSSP, ours is not based on Yen's algorithm and does not solve replacement path problems. Its worst-case running time is on par with state-of-the-art algorithms for kSSP. Using our algorithm, one may find O(m) simple paths with a single shortest path tree computation and O(n + m) additional time per path in well-behaved cases, where n is the number of no…
The approximate single-source shortest-path problem is as follows: given a graph with nonnegative edge weights and a designated source vertex $s$, return estimates of the distances from~$s$ to each other vertex such that the estimate falls between the true distance and $(1+ε)$ times the distance. This paper provides the first nearly work-efficient parallel algorithm with sublinear span (also called depth) for the approximate shortest-path problem on \emph{directed} graphs. Specifically, for constant $ε$ and polynomially-bounded edge weights, our algorithm has work $\tilde{O}(m)$ and span $n^{1…
Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L_1 and L_2, we show how to approximately count the s-t paths that have length at most L_1 in the first graph and length at most L_2 in the second graph. We bel…
Given a directed graph $G$ with arbitrary real-valued weights, the single source shortest-path problem (SSSP) asks for, given a source $s$ in $G$, finding a shortest path from $s$ to each vertex $v$ in $G$. A classical SSSP algorithm detects a negative cycle of $G$ or constructs a shortest-path tree (SPT) rooted at $s$ in $O(mn)$ time, where $m,n$ are the numbers of edges and vertices in $G$ respectively. In many practical applications, new constraints come from time to time and we need to update the SPT frequently. Given an SPT $T$ of $G$, suppose the weight on a certain edge is modified. We …
Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest non-separating cycle in 2^O(g) n log log n time. Similar algorithms are given to compute a shortest non-contractible or non-null-homologous cycle in 2^O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy clas…
The number balancing (NBP) problem is the following: given real numbers $a_1,\ldots,a_n \in [0,1]$, find two disjoint subsets $I_1,I_2 \subseteq [n]$ so that the difference $|\sum_{i \in I_1}a_i - \sum_{i \in I_2}a_i|$ of their sums is minimized. An application of the pigeonhole principle shows that there is always a solution where the difference is at most $O(\frac{\sqrt{n}}{2^n})$. Finding the minimum, however, is NP-hard. In polynomial time,the differencing algorithm by Karmarkar and Karp from 1982 can produce a solution with difference at most $n^{-Θ(\log n)}$, but no further improvement h…
Let $G = (V,E,w)$ be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental single-source reachability problem (SSR), we are given a fixed source $s$ and the goal is to maintain a data structure that can answer path-queries $s \rightarrowtail v$ for any $v \in V$. In the more general single-source shortest paths (SSSP) problem the goal is to return an approximate shortest path to $v$, and in the SCC problem the goal is to maintain strongly connected components of $G$ and to answer path queries within each component. All of these problems have been very acti…
The betweenness centrality (BC) of a node in a network (or graph) is a measure of its importance in the network. BC is widely used in a large number of environments such as social networks, transport networks, security/mobile networks and more. We present an O(n)-round distributed algorithm for computing BC of every vertex as well as all pairs shortest paths (APSP) in a directed unweighted network, where n is the number of vertices and m is the number of edges. We also present O(n)-round distributed algorithms for computing APSP and BC in a weighted directed acyclic graph (dag). Our algorithms…
In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter $D$ of the underlying network is constant. We show that for the case where $D = 1$ there is, quite surprisingly, a very simple algorithm that solves the reachability problem in $1$(!) round. In contrast, for networks with $D = 2$, we show that any distributed algorithm (possibly randomized) for this problem requires $Ω(\sqrt{n/ \log{n}}\,)$ rounds. Our results there…
In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph $G=(V,E)$ subject to edge insertions and deletions and a source vertex $s\in V$, and the goal is to maintain the distance $d(s,t)$ for all $t\in V$. Fine-grained complexity has provided strong lower bounds for exact partially dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15]. Thus much focus has been directed towards finding efficient partially dynamic $(1+ε)$-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14, FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20].…
We combine the well known Shortest Paths (SP) problem and the Bottleneck Paths (BP) problem to introduce a new problem called the Shortest Paths for All Flows (SP-AF) problem that has relevance in real life applications. We first solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem on directed graphs with unit edge costs in $O(mn)$ worst case time bound. We then present two algorithms to solve SSSP-AF on directed graphs with integer edge costs bounded by $c$ in $O(m^2 + nc)$ and $O(m^2 + mn\log{(\frac{c}{m})})$ time bounds. Finally we extend our algorithms for the SSSP-AF pro…
Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^ω) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undir…
This doctoral thesis concerns novel distributed algorithms for weight balancing over directed (communication) topologies. A directed topology (digraph) with nonnegative (or positive) weights assigned on each edge is weight-balanced if, for each node, the sum of the weights of in-coming edges equals the sum of the weights of out-going edges. The novel algorithms introduced in this thesis can facilitate the development of strategies for generating weight balanced digraphs, in a distributed manner, and find numerous applications in coordination and control of multi-component systems. In the first…
In this paper we consider shortest path problems in a directed graph where the transitions between nodes are subject to uncertainty. We use a minimax formulation, where the objective is to guarantee that a special destination state is reached with a minimum cost path under the worst possible instance of the uncertainty. Problems of this type arise, among others, in planning and pursuit-evasion contexts, and in model predictive control. Our analysis makes use of the recently developed theory of abstract semicontractive dynamic programming models. We investigate questions of existence and unique…
APSP with small integer weights in undirected graphs [Seidel'95, Galil and Margalit'97] has an $\tilde{O}(n^ω)$ time algorithm, where $ω<2.373$ is the matrix multiplication exponent. APSP in directed graphs with small weights however, has a much slower running time that would be $Ω(n^{2.5})$ even if $ω=2$ [Zwick'02]. To understand this $n^{2.5}$ bottleneck, we build a web of reductions around directed unweighted APSP. We show that it is fine-grained equivalent to computing a rectangular Min-Plus product for matrices with integer entries; the dimensions and entry size of the matrices depend on …
Given an arbitrary, non-negatively weighted, directed graph $G=(V,E)$ we present an algorithm that computes all pairs shortest paths in time $\mathcal{O}(m^* n + m \lg n + nT_ψ(m^*, n))$, where $m^*$ is the number of different edges contained in shortest paths and $T_ψ(m^*, n)$ is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial $n$ times application of $ψ$ that runs in $\mathcal{O}(nT_ψ(m,n))$. In our algorithm we use $ψ$ as a black box and hence any improvement on $ψ$ results also in improvement of our algo…
We present a new pipelined approach to compute all pairs shortest paths (APSP) in a directed graph with nonnegative integer edge weights (including zero weights) in the CONGEST model in the distributed setting. Our deterministic distributed algorithm computes shortest paths of distance at most $Δ$ for all pairs of vertices in at most $2 n \sqrtΔ + 2n$ rounds, and more generally, it computes h-hop shortest paths for k sources in $2\sqrt{nkh} + n + k$ rounds. The algorithm is simple, and it has some novel features and a nontrivial analysis.It uses only the directed edges in the graph for communi…
We introduce stronger notions for approximate single-source shortest-path distances, show how to efficiently compute them from weaker standard notions, and demonstrate the algorithmic power of these new notions and transformations. One application is the first work-efficient parallel algorithm for computing exact single-source shortest paths graphs -- resolving a major open problem in parallel computing. Given a source vertex in a directed graph with polynomially-bounded nonnegative integer lengths, the algorithm computes an exact shortest path tree in $m \log^{O(1)} n$ work and $n^{1/2+o(1)}$…
Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Becker et al. [DISC'17] which deterministically computes $(1+o(1))$-approximate shortest paths in $\tilde O(D+\sqrt n)$ time, where $D$ is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC'04]. The question of exact shortest paths however saw no algorithmic progress for decad…
In the Single Source Replacement Paths (SSRP) problem we are given a graph $G = (V, E)$, and a shortest paths tree $\widehat{K}$ rooted at a node $s$, and the goal is to output for every node $t \in V$ and for every edge $e$ in $\widehat{K}$ the length of the shortest path from $s$ to $t$ avoiding $e$. We present an $\tilde{O}(m\sqrt{n} + n^2)$ time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed. Our new upper …
This paper describes the shortest path problem in weighted graphs and examines the differences in efficiency that occur when using Dijkstra's algorithm with a Fibonacci heap, binary heap, and self-balancing binary tree. Using C++ implementations of these algorithm variants, we find that the fastest method is not always the one that has the lowest asymptotic complexity. Reasons for this are discussed and backed with empirical evidence.
The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e. For unweighted directed graphs on n vertices, the best known algorithm runtime was \tilde{O}(n^{2.5}) by Roditty and Zwick. For graphs with integer weights in {-M,...,M}, Weimann and Yuster recently showed that one can use fast matrix multiplication and solve the problem in O(Mn^{2.584}) time, a runtime which would be O(Mn^{2.33}) if the exponent ωof matrix multiplication is 2. We improve both of these algorit…
Given a directed graph, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs. Methodologically, our algorithm relies on…
We consider dynamic algorithms for maintaining Single-Source Reachability (SSR) and approximate Single-Source Shortest Paths (SSSP) on $n$-node $m$-edge directed graphs under edge deletions (decremental algorithms). The previous fastest algorithm for SSR and SSSP goes back three decades to Even and Shiloach [JACM 1981]; it has $ O(1) $ query time and $ O (mn) $ total update time (i.e., linear amortized update time if all edges are deleted). This algorithm serves as a building block for several other dynamic algorithms. The question whether its total update time can be improved is a major, long…
We study reachability and shortest paths problems in dynamic directed graphs. Whereas algebraic dynamic data structures supporting edge updates and reachability/distance queries have been known for quite a long time, they do not, in general, allow reporting the underlying paths within the same time bounds, especially against an adaptive adversary. In this paper we develop the first known fully dynamic reachability data structures working against an adaptive adversary and supporting edge updates and path queries for two natural variants: (1) point-to-point path reporting, and (2) single-source …
We give an $O(n \log \log n)$ time algorithm for computing the minimum cut (or equivalently, the shortest cycle) of a weighted directed planar graph. This improves the previous fastest $O(n\log^3 n)$ solution. Interestingly, while in undirected planar graphs both min-cut and min $st$-cut have $O(n \log \log n)$ solutions, in directed planar graphs our result makes min-cut faster than min $st$-cut, which currently requires $O(n \log n)$.
Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(gn log n) time with high probability, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [SODA 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tr…
We present two new algorithms for solving the {\em All Pairs Shortest Paths} (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in $\Ot(n^{2+μ})$ time, where $μ$ satisfies the equation $ω(1,μ,1)=1+2μ$ and $ω(1,μ,1)$ is the exponent of the multiplication of an $n\times n^μ$ matrix by an $n^μ\times n$ matrix. Currently, the best available bounds on $ω(1,μ,1)$, obtained by Coppersmith, imply that $μ<0.575$. The…
In this paper, we present a novel and efficient algorithm to find the k longest (shortest) paths between sources and sinks in a directed acyclic graph (DAG). The algorithm does not enumerate paths therefore it is especially useful for very large k values. It is based on the Valued-Sum-of-Product (VSOP) tool, which is an extension of Zero-suppressed Binary Decision Diagrams (ZBDDs). We assessed the performance of this algorithm with a DAG model of a path-intensive combinational circuit, viz. c6288, that has \sim10^{20} paths. We found that it took about 64 minutes to compute all paths in this D…
A shortest-path algorithm finds a path containing the minimal cost between two vertices in a graph. A plethora of shortest-path algorithms is studied in the literature that span across multiple disciplines. This paper presents a survey of shortest-path algorithms based on a taxonomy that is introduced in the paper. One dimension of this taxonomy is the various flavors of the shortest-path problem. There is no one general algorithm that is capable of solving all variants of the shortest-path problem due to the space and time complexities associated with each algorithm. Other important dimension…
