Recently we submitted a paper, whose title is A New Fast Unweighted All-pairs Shortest Path Search Algorithm Based on Pruning by Shortest Path Trees, to arXiv. This is related to unweighted graphs. This paper also presents a new fast all-pairs shortest path algorithm for weighted graph based on the same idea. In Dijkstra algorithm which is said to be fast in weighted graphs, the average number of accesses to adjacent vertices (expressed by α) is about equal to the average degree of the graph. On the other hand, our algorithm utilizes the shortest path trees of adjacent vertices of each source vertex in the same manner as the algorithm for unweighted graphs, and reduce α drastically in comparison with Dijkstra algorithm. Roughly speaking α is reduced to the value close to 1, because the average degree of a tree is about 2, and one is used to come in and the other is used to go out, although that does not hold true when the depth of the short path trees is small. In case of weighted graphs, a problem which does not occur in unweighted graphs occurs. It is waiting for the generation of the shortest path tree of an adjacent vertex. Therefore, it is possible that a deadlock occurs. We prove our algorithm is deadlock-free. We compared our algorithm with Dijkstra and Peng algorithms. On Dijkstra algorithm ours outperforms it on speed and α except that Dijkstra algorithm slightly outperforms ours or they are almost the same on CPU time in sparse scale-free graphs. The result on Peng algorithm is as follows: In speed and α, ours outperforms Peng algorithm in hypercube-shaped and dense scale-free graphs, but conversely Peng algorithm outperforms ours in sparse scale-free graphs.
We present a new fast all-pairs shortest path algorithm for unweighted graphs. In breadth-first search which is said to representative and fast in unweighted graphs, the average number of accesses to adjacent vertices (expressed by α) is about equal to the average degree of the graph. On the other hand, our algorithm utilizes the shortest path trees of adjacent vertices of each source vertex, and reduce α drastically. Roughly speaking α is reduced to the value close to 1, because the average degree of a tree is about 2, and one is used to come in and the other is used to go out, although that …
In this paper, we re-evaluate the basic strategies for label correcting algorithms for the multiobjective shortest path (MOSP) problem, i.e., node and label selection. In contrast to common believe, we show that---when carefully implemented---the node-selection strategy usually beats the label-selection strategy. Moreover, we present a new pruning method which is easy to implement and performs very well on real-world road networks. In this study, we test our hypotheses on artificial MOSP instances from the literature with up to 15 objectives and real-world road networks with up to almost 160,0…
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…
Computing shortest paths is one of the most researched topics in algorithm engineering. Currently available algorithms compute shortest paths in mere fractions of a second on continental sized road networks. In the presence of unreliability, however, current algorithms fail to achieve results as impressive as for the static setting. In contrast to speed-up techniques for static route planning, current implementations for the stochastic on-time arrival problem require the computationally expensive step of solving convolution products. Running times can reach hours when considering large scale n…
In most of the shortest path problems like vehicle routing problems and network routing problems, we only need an efficient path between two points source and destination, and it is not necessary to calculate the shortest path from source to all other nodes. This paper concentrates on this very idea and presents an algorithm for calculating shortest path for (i) nonnegative weighted undirected graphs (ii) unweighted undirected graphs. The algorithm completes its execution in O(E) for all graphs except few in which longer path (in terms of number of edges) from source to some node makes it best…
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 distributed network is modeled by a graph having $n$ nodes (processors) and diameter $D$. We study the time complexity of approximating {\em weighted} (undirected) shortest paths on distributed networks with a $O(\log n)$ {\em bandwidth restriction} on edges (the standard synchronous \congest model). The question whether approximation algorithms help speed up the shortest paths (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the are…
A weighted string, also known as a position weight matrix, is a sequence of probability distributions over some alphabet. We revisit the Weighted Shortest Common Supersequence (WSCS) problem, introduced by Amir et al. [SPIRE 2011], that is, the SCS problem on weighted strings. In the WSCS problem, we are given two weighted strings $W_1$ and $W_2$ and a threshold $\mathit{Freq}$ on probability, and we are asked to compute the shortest (standard) string $S$ such that both $W_1$ and $W_2$ match subsequences of $S$ (not necessarily the same) with probability at least $\mathit{Freq}$. Amir et al. s…
Algorithms on grammars/transducers with context-free derivations: hypergraph reachability, shortest path, and inside-outside pruning of 'relatively useless' arcs that are unused by any near-shortest paths.
We develop the data structure PReaCH (for Pruned Reachability Contraction Hierarchies) which supports reachability queries in a directed graph, i.e., it supports queries that ask whether two nodes in the graph are connected by a directed path. PReaCH adapts the contraction hierarchy speedup techniques for shortest path queries to the reachability setting. The resulting approach is surprisingly simple and guarantees linear space and near linear preprocessing time. Orthogonally to that, we improve existing pruning techniques for the search by gathering more information from a single DFS-traversa…
We introduce efficient algorithms for finding the $k$ shortest paths of a weighted pushdown automaton (WPDA), a compact representation of a weighted set of strings with potential applications in parsing and machine translation. Both of our algorithms are derived from the same weighted deductive logic description of the execution of a WPDA using different search strategies. Experimental results show our Algorithm 2 adds very little overhead vs. the single shortest path algorithm, even with a large $k$.
In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph $G$ and a source node $s$ the goal is to maintain shortest distances between $s$ and all other nodes in $G$ under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem in unweighted graphs with only $O(mn)$ total update time over all edge deletions. Their classic algorithm was the state of the art for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. A…
We present a deterministic distributed algorithm to compute all-pairs shortest paths(APSP) in an edge-weighted directed or undirected graph. Our algorithm runs in $\tilde{O}(n^{3/2})$ rounds in the Congest model, where $n$ is the number of nodes in the graph. This is the first $o(n^2)$ rounds deterministic distributed algorithm for the weighted APSP problem. Our algorithm is fairly simple and incorporates a deterministic distributed algorithm we develop for computing a `blocker set' \cite{King99}, which has been used earlier in sequential dynamic computation of APSP.
In this paper, we study a set of combinatorial optimization problems on weighted graphs: the shortest path problem with negative weights, the weighted perfect bipartite matching problem, the unit-capacity minimum-cost maximum flow problem and the weighted perfect bipartite $b$-matching problem under the assumption that $\Vert b\Vert_1=O(m)$. We show that each one of these four problems can be solved in $\tilde{O}(m^{10/7}\log W)$ time, where $W$ is the absolute maximum weight of an edge in the graph, which gives the first in over 25 years polynomial improvement in their sparse-graph time compl…
A {\em parametric weighted graph} is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edge-weighted graph. Parametric weighted graph problems are generalizations of weighted graph problems, and arise in various natural scenarios. Parametric weighted graph algorithms consist of two phases. A {\em preprocessing phase} whose input is a parametric weighted graph, and whose output is a data structure, the advice, that is later used by the {\em instantiation phase}, where a specific value for the…
Let $G=(V,E,w)$ be a weighted undirected graph with $n$ vertices and $m$ edges, and fix a set of $s$ sources $S\subseteq V$. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in $S \times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+ε$, for an arbitrarily small constant $ε> 0$ . In this regime existing centralized algorithms require $Ω(\min\{|E|s,n^ω\})$ time, where $ω< 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka tim…
The Floyd-Warshall algorithm is the most popular algorithm for determining the shortest paths between all pairs in a graph. It is very a simple and an elegant algorithm. However, if the graph does not contain any negative weighted edge, using Dijkstra's shortest path algorithm for every vertex as a source vertex to produce all pairs shortest paths of the graph works much better than the Floyd-Warshall algorithm for sparse graphs. Also, for the graphs with negative weighted edges, with no negative cycle, Johnson's algorithm still performs significantly better than the Floyd-Warshall algorithm f…
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 consider the problem of computing all pairs shortest paths (APSP) and shortest paths for k sources in a weighted graph in the distributed CONGEST model. For graphs with non-negative integer edge weights (including zero weights) we build on a recent pipelined algorithm to obtain $\tilde{O}(λ^{1/4}\cdot n^{5/4})$ in graphs with non-negative integer edge-weight at most $λ$, and $\tilde{O}(n \cdot \bigtriangleup^{1/3})$ rounds for shortest path distances at most $\bigtriangleup$. Additionally, we simplify some of the procedures in the earlier APSP algorithms for non-negative edge weights in [HN…
Given an $n$-vertex $m$-edge graph $G$ with non negative edge-weights, the girth of $G$ is the weight of a shortest cycle in $G$. For any graph $G$ with polynomially bounded integer weights, we present a deterministic algorithm that computes, in $\tilde{\cal O}(n^{5/3}+m)$-time, a cycle of weight at most twice the girth of $G$. Our approach combines some new insights on the previous approximation algorithms for this problem (Lingas and Lundell, IPL'09; Roditty and Tov, TALG'13) with Hitting Set based methods that are used for approximate distance oracles and date back from (Thorup and Zwick, J…
For $s$ $>$ 0, we consider an algorithm that computes all $s$-well separated pairs in certain point sets in $\mathbb{R}^{n}$, $n$ $>1$. For an integer $K$ $>1$, we also consider an algorithm that is a permutation of Dijkstra's algorithm, that computes $K$-nearest neighbors using a certain power weighted shortest path metric in $\mathbb{R}^{n}$, $n$ $>$ $1$. We describe each algorithm and their respective dependencies on the input data. We introduce a way to combine both algorithms into a fused algorithm. Several open problems are given for future research.
Here the All-pairs shortest path problem on weighted undirected sparse graphs is being considered. For the problem considered, we propose ``disassembly and assembly of a graph'' algorithm which uses a solution of the problem on a small-dimensional graph to obtain the solution for the given graph. The proposed algorithm has been compared to one of the fastest classic algorithms on data from an open public source.
We study computing {\em all-pairs shortest paths} (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits $(1+o(1))$-approximation $\tilde O(n)$-time algorithms ~\cite{LenzenP-podc15,Nanongkai-STOC14}, which are matched with $\tilde Ω(n)$-time lower bounds~\cite{Nanongkai-STOC14,LenzenP_stoc13,FrischknechtHW12}\footnote{$\tilde Θ$, $\tilde O$ and $\tilde Ω$ hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the we…
We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slig…
Bulgarian Academy of Sciences（保加利亚科学院）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
We present the first polynomial time approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain $\D$, consisting of $n$ tetrahedra with positive weights, and a real number $\eps\in(0,1)$, our algorithm constructs paths in $\D$ from a fixed source vertex to all vertices of $\D$, whose costs are at most $1+\eps$ times the costs of (weighted) shortest paths, in $O(\C(\D)\frac{n}{\eps^{2.5}}\log\frac{n}{\eps}\log^3\frac{1}{\eps})$ time, where $\C(\D)$ is a geometric parameter related to the aspect ratios of tetrahedra. The efficiency of th…
Nagoya University（名古屋大学）；Kyoto University（京都大学）；National Institute of Informatics（国立信息学研究所）
Finding diverse solutions in combinatorial problems recently has received considerable attention (Baste et al. 2020; Fomin et al. 2020; Hanaka et al. 2021). In this paper we study the following type of problems: given an integer $k$, the problem asks for $k$ solutions such that the sum of pairwise (weighted) Hamming distances between these solutions is maximized. Such solutions are called diverse solutions. We present a polynomial-time algorithm for finding diverse shortest $st$-paths in weighted directed graphs. Moreover, we study the diverse version of other classical combinatorial problems …
In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has $\tilde{O}(nm+(n/d)^3)$ work and $\tilde{O}(d)$ depth for any depth parameter $d\in [1,n]$. To the best of our knowledge, such a trade-off has only been previously described for the real-weighted single-source shortest paths problem using randomization [Bringmann et al., ICALP'17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has $\tilde{O}(nm+n^3/d^2)$ work and …
A solution of the $k$ shortest paths problem may output paths that are identical up to a single edge. On the other hand, a solution of the $k$ independent shortest paths problem consists of paths that share neither an edge nor an intermediate node. We investigate the case in which the number of edges that are not shared among any two paths in the output $k$-set is a parameter. We study two main directions: exploring \emph{near-shortest} paths and exploring \emph{exactly shortest paths}. We assume that the weighted graph $G=(V,E,w)$ has no parallel edges and that the edge lengths (weights) are …
We design fast deterministic algorithms for distance computation in the congested clique model. Our key contributions include: -- A $(2+ε)$-approximation for all-pairs shortest paths in $O(\log^2{n} / ε)$ rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model. -- A $(1+ε)$-approximation for multi-source shortest paths from $O(\sqrt{n})$ sources in $O(\log^2{n} / ε)$ rounds on weighted undirected graphs. This is the first sub-polynomial algorithm…
Graph theory is increasingly commonly utilised in genetics, proteomics and neuroimaging. In such fields, the data of interest generally constitute weighted graphs. Analysis of such weighted graphs often require the integration of topological metrics with respect to the density of the graph. Here, density refers to the proportion of the number of edges present in that graph. When topological metrics based on shortest paths are of interest, such density-integration usually necessitates the iterative application of Dijkstra's algorithm in order to compute the shortest path matrix at each density …
Python implementation of selected weighted graph algorithms is presented. The minimal graph interface is defined together with several classes implementing this interface. Graph nodes can be any hashable Python objects. Directed edges are instances of the Edge class. Graphs are instances of the Graph class. It is based on the adjacency-list representation, but with fast lookup of nodes and neighbors (dict-of-dict structure). Other implementations of this class are also possible. In this work, many algorithms are implemented using a unified approach. There are separate classes and modules devot…
Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the $\mathcal{O}(n^3)$ time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time $\mathcal{O}(n^{2.842})$, but no combinatorial, truly subcubic algorithm is known. Motivated by the recent framework of efficient parameterized algorithms (or "FPT i…
When deciding where to place access points in a wireless network, it is useful to model the signal propagation loss between a proposed antenna location and the areas it may cover. The indoor dominant path (IDP) model, introduced by Wölfle et al., is shown in the literature to have good validation and generalization error, is faster to compute than competing methods, and is used in commercial software such as WinProp, iBwave Design, and CellTrace. Previously, the algorithms known for computing it involved a worst-case exponential-time tree search, with pruning heuristics to speed it up. We prov…
The \textit{Multi-Constraint Shortest Path (MCSP)} problem aims to find the shortest path between two nodes in a network subject to a given constraint set. It is typically processed as a \textit{skyline path} problem. However, the number of intermediate skyline paths becomes larger as the network size increases and the constraint number grows, which brings about the dramatical growth of computational cost and further makes the existing index-based methods hardly capable of obtaining the complete exact results. In this paper, we propose a novel high-dimensional skyline path concatenation method…
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…
Several papers have achieved time $O(\sqrt n m)$ for cardinality matching, starting from first principles. This results in a long derivation. We simplify the task by employing well-known concepts for maximum weight matching. We use Edmonds' algorithm to derive the structure of shortest augmenting paths. We extend this to a complete algorithm for maximum cardinality matching in time $O(\sqrt n m)$.
We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an $n$-vertex $m$-edge graph $G$ with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is to support approximate shortest-path queries: given a pair $x,y$ of vertices of $G$, return a path $P$ connecting $x$ to $y$, whose length is within factor $α$ of the length of the shortest $x$-$y$ path, in time $\tilde O(|E(P)|)$, where $α$ is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic…
We revisit a classical graph-theoretic problem, the \textit{single-source shortest-path} (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in $O(n \log^2 n)$ time using linear space, where $n$ is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA'15] which uses $O(n^{1+δ})$ time and $O(n^{1+δ})$ space (for any small constant $δ>0$) and the previous randomized algorithm by Kaplan et al. [SODA'17] which uses $O(n \log^{12+o(1)} n)$ expect…
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…
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…
Let G=(V,E) be a graph with f:V\to Z_+ a function assigning degree bounds to vertices. We present the first efficient algebraic algorithm to find an f-factor. The time is \tilde{O}(f(V)^ω). More generally for graphs with integral edge weights of maximum absolute value W we find a maximum weight f-factor in time \tilde{O}(Wf(V)^ω). (The algorithms are randomized, correct with high probability and Las Vegas; the time bound is worst-case.) We also present three specializations of these algorithms: For maximum weight perfect f-matching the algorithm is considerably simpler (and almost identical to…
We propose a new exact method for shortest-path distance queries on large-scale networks. Our method precomputes distance labels for vertices by performing a breadth-first search from every vertex. Seemingly too obvious and too inefficient at first glance, the key ingredient introduced here is pruning during breadth-first searches. While we can still answer the correct distance for any pair of vertices from the labels, it surprisingly reduces the search space and sizes of labels. Moreover, we show that we can perform 32 or 64 breadth-first searches simultaneously exploiting bitwise operations.…
Due to the computational complexity of finding almost shortest simple paths, we propose that identifying a larger collection of (nonbacktracking) paths is more efficient than finding almost shortest simple paths on positively weighted real-world networks. First, we present an easy to implement $O(m\log m+kL)$ solution for finding all (nonbacktracking) paths with bounded length $D$ between two arbitrary nodes on a positively weighted graph, where $L$ is an upperbound for the number of nodes in any of the $k$ outputted paths. Subsequently, we illustrate that for undirected Chung-Lu random graphs…
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…
