This paper introduces a novel algorithm combination designed for fast one-to-many multicriteria shortest path search. A preprocessing algorithm excludes irrelevant vertices by building a smaller cover graph. A modified version of multicriteria label-setting algorithm operates on the cover graph and employs a dimensionality reduction technique for swifter domination checks. While the method itself maintains solution optimality, it is able to additionally incorporate existing heuristics for further speedups. The proposed algorithm has been tested on multiple criteria combinations of varying correlation. The results show the introduced approach provides a speedup of at least 6 times on simple criteria combinations and up to 60 times on hard instances compared to vanilla multicriteria label-setting. Graph preprocessing also decreases memory requirements of queries by up to 13 times.
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 …
We study a scenario for route planning in road networks, where the objective to be optimized may change between every shortest path query. Since this invalidates many of the known speedup techniques for road networks that are based on preprocessing of shortest path structures, we investigate optimizations exploiting solely the topological structure of networks. We experimentally evaluate our technique on a large set of real-world road networks of various data sources. With lightweight preprocessing our technique answers long-distance queries across continental networks significantly faster tha…
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 …
The current paper deals with the subject of shortest path routing in transportation networks (in terms of travelling time), where the speed in several of the network's roads is a function of the time interval. The main contribution of the paper is a procedure that is faster compared to the conventional approaches, that derives the road's traversal time according to the time instant of departure, for the case where the road's speed has a constant value inside each time interval (in general, different value for each time interval). Furthermore, the case where the road's speed is a linear functio…
We study the journey planning problem in public transit networks. Developing efficient preprocessing-based speedup techniques for this problem has been challenging: current approaches either require massive preprocessing effort or provide limited speedups. Leveraging recent advances in Hub Labeling, the fastest algorithm for road networks, we revisit the well-known time-expanded model for public transit. Exploiting domain-specific properties, we provide simple and efficient algorithms for the earliest arrival, profile, and multicriteria problems, with queries that are orders of magnitude faste…
We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph $G$ into regions of at most $r$ vertices each, for some parameter $r < n$. The vertex set of the DDG is the set of $Θ(n/\sqrt r)$ vertices of $G$ that belong to more than one region (boundary vertices). T…
In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n) \log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound on the magnitude of the smallest negative-weight edge. In this work we improve the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an improvement by nearly six log-factors. Some of these log-factors are easy to shave (e.g. replacing the priority queue used in Dijkstra's alg…
We study the problem of computing constrained shortest paths for battery electric vehicles. Since battery capacities are limited, fastest routes are often infeasible. Instead, users are interested in fast routes on which the energy consumption does not exceed the battery capacity. For that, drivers can deliberately reduce speed to save energy. Hence, route planning should provide both path and speed recommendations. To tackle the resulting NP-hard optimization problem, previous work trades correctness or accuracy of the underlying model for practical running times. We present a novel framework…
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…
We present a deterministic O(n log log n) time algorithm for finding shortest cycles and minimum cuts in planar graphs. The algorithm improves the previously known fastest algorithm by Italiano et al. in STOC'11 by a factor of log n. This speedup is obtained through the use of dense distance graphs combined with a divide-and-conquer approach.
We present a new approach called GR (Graph Reduction) algorithm for searching loop-less k-shortest paths (1st to k-th shortest paths) in a graph based on graph reduction. Let a source vertex and a target vertex of k-shortest paths be v_s and v_t respectively. First our approach computes shortest paths to every vertex from v_s and v_t respectively, and reduce a graph to a subgraph that contains all vertices and edges of loop-less k-shortest paths using the already computed shortest paths, and apply an existing k-shortest path search algorithm to the reduced graph. A graph can be reduced quickly…
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.
An instance of the NP-hard Quadratic Shortest Path Problem (QSPP) is called linearizable iff it is equivalent to an instance of the classic Shortest Path Problem (SPP) on the same input digraph. The linearization problem for the QSPP (LinQSPP) decides whether a given QSPP instance is linearizable and determines the corresponding SPP instance in the positive case. We provide a novel linear time algorithm for the LinQSPP on acyclic digraphs which runs considerably faster than the previously best algorithm. The algorithm is based on a new insight revealing that the linearizability of the QSPP for…
In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in $\ell_p$ norm ($1\leq p\leq\infty$). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all $\ell_p$ norm ($1\leq p\leq\infty$). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of $2^{2.751n+o(n)}$, which is much less than the $2^{3.849n+o(n)}$ complexity of the previous best a…
For a string $S$, a palindromic substring $S[i..j]$ is said to be a \emph{shortest unique palindromic substring} ($\mathit{SUPS}$) for an interval $[s, t]$ in $S$, if $S[i..j]$ occurs exactly once in $S$, the interval $[i, j]$ contains $[s, t]$, and every palindromic substring containing $[s, t]$ which is shorter than $S[i..j]$ occurs at least twice in $S$. In this paper, we study the problem of answering $\mathit{SUPS}$ queries on run-length encoded strings. We show how to preprocess a given run-length encoded string $\mathit{RLE}_{S}$ of size $m$ in $O(m)$ space and $O(m \log σ_{\mathit{RLE}…
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…
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…
When a network is prone to failures, it is very expensive to compute the shortest paths every time from the scratch. Distance sensitivity oracle provides this privilege to find the new shortest paths faster and with lower cost by once pre-computing an oracle in advance. Although several efficient solutions are proposed in the literature to support the single failure, few efforts are done to devise an efficient method regarding the case of multiple failures. In this paper, we present a novel distance sensitivity oracle based on Markov Tensor Theory \cite{golnari2017markov} to support replacemen…
The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for every possible value of lambda. The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction. The paper presents …
In the decremental $(1+ε)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s \in V$, and we are asked to process edge deletions efficiently and answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(s,v)$ for each $v \in V$, at any stage, such that $\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ ε)\mathbf{dist}_G(s,v)$. In the decremental $(1+ε)$-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance est…
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…
Efficient solution of the single source shortest path (SSSP) problem on road networks is an important requirement for numerous real-world applications. This paper introduces an algorithm for the SSSP problem using compression method. Owning to precomputing and storing all-pairs shortest path (APSP), the process of solving SSSP problem is a simple lookup of a little data from precomputed APSP and decompression. APSP without compression needs at least 1TB memory for a road network with one million vertices. Our algorithm can compress such an APSP into several GB, and ensure a good performance of…
Let $G$ be a directed graph with $n$ vertices, $m$ edges, and non-negative edge costs. Given $G$, a fixed source vertex $s$, and a positive integer $p$, we consider the problem of computing, for each vertex $t\neq s$, $p$ edge-disjoint paths of minimum total cost from $s$ to $t$ in $G$. Suurballe and Tarjan~[Networks, 1984] solved the above problem for $p=2$ by designing a $O(m+n\log n)$ time algorithm which also computes a sparse \emph{single-source $2$-multipath preserver}, i.e., a subgraph containing $2$ edge-disjoint paths of minimum total cost from $s$ to every other vertex of $G$. The ca…
The Shoshan-Zwick algorithm solves the all pairs shortest paths problem in undirected graphs with integer edge costs in the range $\{1, 2, \dots, M\}$. It runs in $\tilde{O}(M\cdot n^ω)$ time, where $n$ is the number of vertices, $M$ is the largest integer edge cost, and $ω< 2.3727$ is the exponent of matrix multiplication. It is the fastest known algorithm for this problem. This paper points out the erroneous behavior of the Shoshan-Zwick algorithm and revises the algorithm to resolve the issues that cause this behavior. Moreover, it discusses implementation aspects of the Shoshan-Zwick algor…
We introduce a new heuristic for the A* algorithm that references a data structure much smaller than the one required by the ALT heuristic. This heuristic's benefits are permitted by a new approach for computing lower bounds using generalized polygon inequalities, leveraging distance information from two landmarks as opposed to the common single landmark paradigm. In this paper, we demonstrate that this heuristic stores a reduced amount of preprocessing information in comparison to previous landmark algorithms while performing faster search queries.
We study the journey planning problem for multimodal networks consisting of public transit and a non-schedule-based transfer mode (e.g., walking, bicycle, e-scooter). So far, all efficient algorithms for this problem either restrict usage of the transfer mode or Pareto-optimize only two criteria: arrival time and the number of used public transit trips. However, we show that both limitations must be lifted in order to obtain high-quality solutions. In particular, the time spent using the (unrestricted) transfer mode must be optimized as a third criterion. We present McTB, the first algorithm t…
We present the first $m\,\text{polylog}(n)$ work, $\text{polylog}(n)$ time algorithm in the PRAM model that computes $(1+ε)$-approximate single-source shortest paths on weighted, undirected graphs. This improves upon the breakthrough result of Cohen~[JACM'00] that achieves $O(m^{1+ε_0})$ work and $\text{polylog}(n)$ time. While most previous approaches, including Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent developments in \emph{continuous optimization}, studying the shortest path problem from the lens of the closely-related \emph{minimum transshipment} problem…
Faster Goal-Oriented Shortest Path Search for Bulk and Incremental Detailed Routing
IBM Deutschland Research & Development GmbH（IBM德国研究与开发有限公司）；Department of Mathematics, TU Dortmund University（多特蒙德工业大学数学系）；Research Institute for Discrete Math., Hausdorff Center for Mathematics, University of Bonn（波恩大学离散数学研究所、豪斯多夫数学中心）
We present a new fully dynamic algorithm for maintaining betweenness centrality (BC) of vertices in a directed graph $G=(V,E)$ with positive edge weights. BC is a widely used parameter in the analysis of large complex networks. We achieve an amortized $O((ν^*)^2 \log^2 n)$ time per update, where $n = |V| $ and $ν^*$ bounds the number of distinct edges that lie on shortest paths through any single vertex. This result improves on the amortized bound for fully dynamic BC in [Pontecorvi-Ramachandran2015] by a logarithmic factor. Our algorithm uses new data structures and techniques that are extens…
In this paper we present a new fast algorithm finding minimal reset words for finite synchronizing automata. The problem is know to be computationally hard, and our algorithm is exponential. Yet, it is faster than the algorithms used so far and it works well in practice. The main idea is to use a bidirectional BFS and radix (Patricia) tries to store and compare resulted subsets. We give both theoretical and practical arguments showing that the branching factor is reduced efficiently. As a practical test we perform an experimental study of the length of the shortest reset word for random automa…
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…
Finding central nodes is a fundamental problem in network analysis. Betweenness centrality is a well-known measure which quantifies the importance of a node based on the fraction of shortest paths going though it. Due to the dynamic nature of many today's networks, algorithms that quickly update centrality scores have become a necessity. For betweenness, several dynamic algorithms have been proposed over the years, targeting different update types (incremental- and decremental-only, fully-dynamic). In this paper we introduce a new dynamic algorithm for updating betweenness centrality after an …
A long series of recent results and breakthroughs have led to faster and better distributed approximation algorithms for single source shortest paths (SSSP) and related problems in the CONGEST model. The runtime of all these algorithms, however, is $\tildeΩ(\sqrt{n})$, regardless of the network topology, even on nice networks with a (poly)logarithmic network diameter $D$. While this is known to be necessary for some pathological networks, most topologies of interest are arguably not of this type. We give the first distributed approximation algorithms for shortest paths problems that adjust to …
A lattice reduction is an algorithm that transforms the given basis of the lattice to another lattice basis such that problems like finding a shortest vector and closest vector become easier to solve. Some of the famous lattice reduction algorithms are LLL and BKZ reductions. We define a class of bases called \emph{obtuse bases} and show that any lattice basis can be transformed to an obtuse basis in $\mathcal{O}(n^4)$ time. A shortest vector s can be written as $v_1b_1+\cdots+v_nb_n$ where $b_1,\dots,b_n$ are the input basis vectors and $v_1,\dots,v_n$ are integers. When the input basis is ob…
Important problems across multiple disciplines involve computations on the semiring $(\times, \max)$ (or its equivalents, the negated version $(\times, \min)$), the log-transformed version $(+, \max)$, or the negated log-transformed version $(+, \min)$): max-convolution, all-pairs shortest paths in a weighted graph, and finding the largest $k$ values in $x_i+y_j$ for two lists $x$ and $y$. However, fast algorithms such as those enabling FFT convolution, sub-cubic matrix multiplication, \emph{etc.}, require inverse operations, and thus cannot be computed on semirings. This manuscript generalize…
A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $α$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $α$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(n\log n)$ time which we apply to compute all shortest $α$-partial covers for a given $α$. We also e…
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.…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Cyber Science and Technology, University of Science and Technology of China（中国科学技术大学网络空间安全学院）
We give an $\tilde{O}(n^{7/5} \log (nC))$-time algorithm to compute a minimum-cost maximum cardinality matching (optimal matching) in $K_h$-minor free graphs with $h=O(1)$ and integer edge weights having magnitude at most $C$. This improves upon the $\tilde{O}(n^{10/7}\log{C})$ algorithm of Cohen et al. [SODA 2017] and the $O(n^{3/2}\log (nC))$ algorithm of Gabow and Tarjan [SIAM J. Comput. 1989]. For a graph with $m$ edges and $n$ vertices, the well-known Hungarian Algorithm computes a shortest augmenting path in each phase in $O(m)$ time, yielding an optimal matching in $O(mn)$ time. The Hop…
We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in term…
We consider the question of speeding up classic graph algorithms with machine-learned predictions. In this model, algorithms are furnished with extra advice learned from past or similar instances. Given the additional information, we aim to improve upon the traditional worst-case run-time guarantees. Our contributions are the following: (i) We give a faster algorithm for minimum-weight bipartite matching via learned duals, improving the recent result by Dinitz, Im, Lavastida, Moseley and Vassilvitskii (NeurIPS, 2021); (ii) We extend the learned dual approach to the single-source shortest path …
It has been shown by Alon et al. that the so-called 'all-pairs shortest-path' problem can be solved in O((MV)^2.688 * log^3(V)) for graphs with V vertices, with integer distances bounded by M. We solve the more general problem for graphs in R (assuming no negative cycles), with expected-case running time O(V^2.5 * log(V)). While our result appears to violate the Omega(V^3) requirement of "Funny Matrix Multiplication" (due to Kerr), we find that it has a sub-cubic expected time solution subject to reasonable conditions on the data distribution. The expected time solution arises when certain sub…
We consider the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph $G=(V,E)$ with a positive real weight on each edge. We present two fully dynamic algorithms for this problem in which an update supports either weight increases or weight decreases on a subset of edges incident to a vertex. Our first algorithm runs in amortized $O({ν^*}^2 \cdot \log^3 n)$ time per update, where $n = |V|$, and $ν^*$ bounds the number of edges that lie on shortest paths through any single vertex. Our APASP algorithm leads to t…
Given an undirected graph and two disjoint vertex pairs $s_1,t_1$ and $s_2,t_2$, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting $s_1$ with $t_1$, and $s_2$ with $t_2$, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs…
We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain $poly(\log\log n)$-round algorithms for the following problems in unweighted undirected $n$-vertex graphs: -- $(1+ε)$-approximation of multi-source shortest paths (MSSP) from $O(\sqrt{n})$ sources. -- $(2+ε)$-approximation of all pairs shortest paths (APSP). -- $(1+ε,β)$-approximation of APSP where $β=O(\frac{\log\log n}ε)^{\log\log n}$. These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., P…
