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-problems are uncorrelated, though we can do better/worse than the expected-case under positive/negative correlation (respectively). Whether we observe positive/negative correlation depends on the statistics of the graph in question. In practice, our algorithm is significantly faster than Floyd-Warshall, even for dense graphs.
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…
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…
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…
It is a major open problem whether the $(\min,+)$-product of two $n\times n$ matrices has a truly sub-cubic (i.e. $O(n^{3-ε})$ for $ε>0$) time algorithm, in particular since it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in $n$-vertex graphs. Some restrictions of the $(\min,+)$-product to special types of matrices are known to admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, a…
String similarity, longest common subsequence and shortest edit scripts are the triplets of problem that related to each other. There are different algorithms exist to generate edit script by solving longest common subsequence problem. This paper proposes an algorithm that uses string similarity problem to generate shortest edit script. For this we use the famous Levenshtein distance algorithm, which computes a numerical value that represents similarity between the strings from 0 to n, where n is the length of longest input string, and produce the shortest edit script which contains instructio…
A superstring of a set of strings correspond to a string which contains all the other strings as substrings. The problem of finding the Shortest Linear Superstring is a well-know and well-studied problem in stringology. We present here a variant of this problem, the Shortest Circular Superstring problem where the sought superstring is a circular string. We show a strong link between these two problems and prove that the Shortest Circular Superstring problem is NP-complete. Moreover, we propose a new conjecture on the approximation ratio of the Shortest Circular Superstring problem.
We develop an efficient parallel algorithm for answering shortest-path queries in planar graphs and implement it on a multi-node CPU/GPU clusters. The algorithm uses a divide-and-conquer approach for decomposing the input graph into small and roughly equal subgraphs and constructs a distributed data structure containing shortest distances within each of those subgraphs and between their boundary vertices. For a planar graph with $n$ vertices, that data structure needs $O(n)$ storage per processor and allows queries to be answered in $O(n^{1/4})$ time.
We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set) that intersects each shortest s-t path in a distinct set of vertices. We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe t…
A palindrome is a string that reads the same forward and backward. A palindromic substring $P$ of a string $S$ is called a shortest unique palindromic substring ($\mathit{SUPS}$) for an interval $[x, y]$ in $S$, if $P$ occurs exactly once in $S$, this occurrence of $P$ contains interval $[x, y]$, and every palindromic substring of $S$ which contains interval $[x, y]$ and is shorter than $P$ occurs at least twice in $S$. The $\mathit{SUPS}$ problem is, given a string $S$, to preprocess $S$ so that for any subsequent query interval $[x, y]$ all the $\mathit{SUPS}\mbox{s}$ for interval $[x, y]$ c…
Given a string $T$ of length $n$, a substring $u = T[i..j]$ of $T$ is called a shortest unique substring (SUS) for an interval $[s,t]$ if (a) $u$ occurs exactly once in $T$, (b) $u$ contains the interval $[s,t]$ (i.e. $i \leq s \leq t \leq j$), and (c) every substring $v$ of $T$ with $|v| < |u|$ containing $[s,t]$ occurs at least twice in $T$. Given a query interval $[s, t] \subset [1, n]$, the interval SUS problem is to output all the SUSs for the interval $[s,t]$. In this article, we propose a $4n + o(n)$ bits data structure answering an interval SUS query in output-sensitive $O(\mathit{occ}…
Given a string $T$ of length $n$ over an alphabet $Σ\subset \{1,2,\ldots,n^{O(1)}\}$ of size $σ$, we are to preprocess $T$ so that given a range $[i,j]$, we can return a representation of a shortest string over $Σ$ that is absent in the fragment $T[i]\cdots T[j]$ of $T$. We present an $O(n)$-space data structure that answers such queries in constant time and can be constructed in $O(n\log_σn)$ time.
Given an undirected, unweighted planar graph $G$ with $n$ vertices, we present a truly subquadratic size distance oracle for reporting exact shortest-path distances between any pair of vertices of $G$ in constant time. For any $\varepsilon > 0$, our distance oracle takes up $O(n^{5/3+\varepsilon})$ space and is capable of answering shortest-path distance queries exactly for any pair of vertices of $G$ in worst-case time $O(\log (1/\varepsilon))$. Previously no truly sub-quadratic size distance oracles with constant query time for answering exact all-pairs shortest paths distance queries existe…
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…
Given an undirected graph $G=(V,E)$, vertices $s,t\in V$, and an integer $k$, Tracking Shortest Paths requires deciding whether there exists a set of $k$ vertices $T\subseteq V$ such that for any two distinct shortest paths between $s$ and $t$, say $P_1$ and $P_2$, we have $T\cap V(P_1)\neq T\cap V(P_2)$. In this paper, we give the first polynomial size kernel for the problem. Specifically we show the existence of a kernel with $\mathcal{O}(k^2)$ vertices and edges in general graphs and a kernel with $\mathcal{O}(k)$ vertices and edges in planar graphs for the Tracking Paths in DAG problem. Th…
The Random-Facet algorithm of Kalai and of Matousek, Sharir and Welzl is an elegant randomized algorithm for solving linear programs and more general LP-type problems. Its expected subexponential time of $2^{\tilde{O}(\sqrt{m})}$, where $m$ is the number of inequalities, makes it the fastest known combinatorial algorithm for solving linear programs. We previously showed that Random-Facet performs an expected number of $2^{\tildeΩ(\sqrt[3]{m})}$ pivoting steps on some LPs with $m$ inequalities that correspond to $m$-action Markov Decision Processes (MDPs). We also showed that Random-Facet-1P, a…
The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in $O(n)$ time, where $n$ is the number of vertices in the input graph $G$. Peleg and Rubinovich (FOCS'99) showed a lower bound of $\tildeΩ(D + \sqrt{n})$ for this problem, where $D$ is the hop-diameter of $G$. Whether or not this problem can be solved in $o(n)$ time when $D$ is relatively small is a major notorious open question. Despite intensive research \cite{LP13,N14,HKN15,EN16,BKKL16} that yielded…
In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all $n\choose 2$ pairs of vertices in an undirected graph can be solved using only $n-1$ calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$, which is $O(n^3)$ when $m = Θ(n^2)$. While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an $\tilde{O}(n^{2})$-time a…
Consider the following distance query for an $n$-node graph $G$ undergoing edge insertions and deletions: given two sets of nodes $I$ and $J$, return the distances between every pair of nodes in $I\times J$. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient $(1+ε)$-approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for Single-Source and All-Pairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Bel…
We provide universally-optimal distributed graph algorithms for $(1+\varepsilon)$-approximate shortest path problems including shortest-path-tree and transshipment. The universal optimality of our algorithms guarantees that, on any $n$-node network $G$, our algorithm completes in $T \cdot n^{o(1)}$ rounds whenever a $T$-round algorithm exists for $G$. This includes $D \cdot n^{o(1)}$-round algorithms for any planar or excluded-minor network. Our algorithms never require more than $(\sqrt{n} + D) \cdot n^{o(1)}$ rounds, resulting in the first sub-linear-round distributed algorithm for transship…
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 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…
Constructing a shortest path between two network nodes is a fundamental task in distributed computing. This work develops schemes for the construction of shortest paths in randomized beeping networks between a predetermined source node and an arbitrary set of destination nodes. Our first scheme constructs a (single) shortest path to an arbitrary destination in $O (D \log\log n + \log^3 n)$ rounds with high probability. Our second scheme constructs multiple shortest paths, one per each destination, in $O (D \log^2 n + \log^3 n)$ rounds with high probability. Our schemes are based on a reduction…
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 …
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 consider differentially private range queries on a graph where query ranges are defined as the set of edges on a shortest path of the graph. Edges in the graph carry sensitive attributes and the goal is to report the sum of these attributes on a shortest path for counting query or the minimum of the attributes in a bottleneck query. We use differential privacy to ensure that the release of these query answers provide protection of the privacy of the sensitive edge attributes. Our goal is to develop mechanisms that minimize the additive error of the reported answers with the given privacy bu…
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…
Suppose that we are given two independent sets I_b and I_r of a graph such that |I_b|=|I_r|, and imagine that a token is placed on each vertex in |I_b|. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I_b into I_r so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. The reconfiguration problems tend to be P…
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…
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…
The shortest path (SP) and shortest paths tree (SPT) problems arise both as direct applications and as subroutines of overlay algorithms solving more complex problems such as the constrained shortest path (CSP) or the constrained minimum Steiner tree (CMST) problems. Often, such algorithms do not use the result of an SP subroutine if its total cost is greater than a given bound. For example, for delay-constrained problems, paths resulting from a least-delay SP run and whose delay is greater than the delay constraint of the original problem are not used by the overlay algorithm to construct its…
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…
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…
给定一个任意的、非负权的有向图$G=(V,E)$，我们提出了一个计算所有对最短路径的算法$\mathcal{O}(m^* n + m \lg n + nT_ψ(m^*, n))$，其中$m^*$是最短路径中包含的不同边的数目，$T_ψ(m^*, n)$是求解单源最短路径问题（SSSP）的算法的运行时间.这是对运行在$\mathcal{O}(nT_ψ(m,n))$中的$ψ$的普通$n$倍应用程序的实质性改进。在我们的算法中，我们使用$ψ$作为一个黑盒，因此任何对$ψ$的改进也会导致我们的算法的改进。此外，结合我们的方法，约翰逊的重新加权技术和拓扑排序的结果在$\mathcal{O}(m^*n + m \lg n)$所有对任意加权有向无环图的最短路径算法。此外，我们还指出了定义在最短路径上的排序问题的复杂性与SSSP之间的联系。
We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely …
Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect $(A+B)$-path packing problem is: given an undirected graph $G$ and two disjoint node subsets $A,B$ with even cardinalities, find a shortest $|A|/2+|B|/2$ disjoint paths whose ends are both in $A$ or both in $B$. Besides its NP…
