Given a directed weighted graph $G=(V,E)$ undergoing vertex insertions \emph{and} deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of $G$. In two breakthrough results, Italiano and Demetrescu [STOC '03] presented an algorithm that requires $\tilde{O}(n^2)$ \emph{amortized} update time, and Thorup showed in [STOC '05] that \emph{worst-case} update time $\tilde{O}(n^{2+3/4})$ can be achieved. In this article, we make substantial progress on the problem. We present the following new results:
(1) We present the first deterministic data structure that breaks the $\tilde{O}(n^{2+3/4})$ worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to $\tilde{O}(n^{2+5/7}) = \tilde{O}(n^{2.71..})$ and to $\tilde{O}(n^{2+3/5}) = \tilde{O}(n^{2.6})$ for unweighted graphs.
(2) We present a simple deterministic algorithm with $\tilde{O}(n^{2+3/4})$ worst-case update time ($\tilde{O}(n^{2+2/3})$ for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time $\tilde{O}(n^{2+2/3})$ ($\tilde{O}(n^{2 + 1/2})$ for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space $\tilde{O}(n^2)$. These are the first exact dynamic algorithms with truly-subcubic update time \emph{and} space usage. This makes significant progress on an open question posed in multiple articles [COCOON'01, STOC '03, ICALP'04, Encyclopedia of Algorithms '08] and is critical to algorithms in practice [TALG '06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA '17].
In this thesis, we design algorithms for several NP-hard problems in both worst and beyond worst case settings. In the first part of the thesis, we apply the traditional worst case methodology and design approximation algorithms for the Hub Labeling problem; Hub Labeling is a preprocessing technique introduced to speed up shortest path queries. Before this work, Hub Labeling had been extensively studied mainly in the beyond worst case analysis setting, and in particular on graphs with low highway dimension. In this work, we significantly improve our theoretical understanding of the problem and…
We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against …
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…
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…
In this paper I present general outlook on questions relevant to the basic graph algorithms; Finding the Shortest Path with Positive Weights and Minimum Spanning Tree. I will show so far known solution set of basic graph problems and present my own. My solutions to graph problems are characterized by their linear worst-case time complexity. It should be noticed that the algorithms which compute the Shortest Path and Minimum Spanning Tree problems not only analyze the weight of arcs (which is the main and often the only criterion of solution hitherto known algorithms) but also in case of identi…
The minimum-cost flow problem is a classic problem in combinatorial optimization with various applications. Several pseudo-polynomial, polynomial, and strongly polynomial algorithms have been developed in the past decades, and it seems that both the problem and the algorithms are well understood. However, some of the algorithms' running times observed in empirical studies contrast the running times obtained by worst-case analysis not only in the order of magnitude but also in the ranking when compared to each other. For example, the Successive Shortest Path (SSP) algorithm, which has an expone…
In two-stage robust optimization the solution to a problem is built in two stages: In the first stage a partial, not necessarily feasible, solution is exhibited. Then the adversary chooses the "worst" scenario from a predefined set of scenarios. In the second stage, the first-stage solution is extended to become feasible for the chosen scenario. The costs at the second stage are larger than at the first one, and the objective is to minimize the total cost paid in the two stages. We give a 2-approximation algorithm for the robust mincut problem and a (γ+2)-approximation for the robust shortest …
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…
In the Shortest Superstring problem, we are given a set of strings and we are asking for a common superstring, which has the minimum number of characters. The Shortest Superstring problem is NP-hard and several constant-factor approximation algorithms are known for it. Of particular interest is the GREEDY algorithm, which repeatedly merges two strings of maximum overlap until a single string remains. The GREEDY algorithm, being simpler than other well-performing approximation algorithms for this problem, has attracted attention since the 1980s and is commonly used in practical applications. Ta…
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 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 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…
We study the problem of computing shortest paths in so-called dense distance graphs. Every planar graph $G$ on $n$ vertices can be partitioned into a set of $O(n/r)$ edge-disjoint regions (called an $r$-division) with $O(r)$ vertices each, such that each region has $O(\sqrt{r})$ vertices (called boundary vertices) in common with other regions. A dense distance graph of a region is a complete graph containing all-pairs distances between its boundary nodes. A dense distance graph of an $r$-division is the union of the $O(n/r)$ dense distance graphs of the individual pieces. Since the introductio…
We introduce a new problem that combines the well known All Pairs Shortest Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to compute the shortest paths for all pairs of vertices for all possible flow amounts. We call this new problem the All Pairs Shortest Paths for All Flows (APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with unit edge costs and real edge capacities in $\tilde{O}(\sqrt{t}n^{(ω+9)/4}) = \tilde{O}(\sqrt{t}n^{2.843})$ time, where $n$ is the number of vertices, $t$ is the number of distinct edge capacities (flow amounts) and $O(…
Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The distance between two nodes is then the length of a shortest path (with respect to the weights drawn) that connects these nodes. We prove structural properties of the random shortest path metrics generated in this w…
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…
Blomer and Naewe[BN09] modified the randomized sieving algorithm of Ajtai, Kumar and Sivakumar[AKS01] to solve the shortest vector problem (SVP). The algorithm starts with $N = 2^{O(n)}$ randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice. The running time of the sieving procedure is quadratic in $N$. We study this problem for the special but important case of the $\ell_\infty$ norm. We give a new sieving procedure that runs in time linear in $N$, thereby significantly improving the running time of the algorithm for SVP in…
We study the problem of releasing the weights of all-pair shortest paths in a weighted undirected graph with differential privacy (DP). In this setting, the underlying graph is fixed and two graphs are neighbors if their edge weights differ by at most $1$ in the $\ell_1$-distance. We give an $ε$-DP algorithm with additive error $\tilde{O}(n^{2/3} / ε)$ and an $(ε, δ)$-DP algorithm with additive error $\tilde{O}(\sqrt{n} / ε)$ where $n$ denotes the number of vertices. This positively answers a question of Sealfon (PODS'16), who asked whether a $o(n)$-error algorithm exists. We also show that an…
We study the approximation hardness of the Shortest Superstring, the Maximal Compression and the Maximum Asymmetric Traveling Salesperson (MAX-ATSP) problem. We introduce a new reduction method that produces strongly restricted instances of the Shortest Superstring problem, in which the maximal orbit size is eight (with no character appearing more than eight times) and all given strings having length four. Based on this reduction method, we are able to improve the best up to now known approximation lower bound for the Shortest Superstring problem and the Maximal Compression problem by an order…
We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximatio…
We revisit the problem of finding optimal strategies for deterministic Markov Decision Processes (DMDPs), and a closely related problem of testing feasibility of systems of $m$ linear inequalities on $n$ real variables with at most two variables per inequality (2VPI). We give a randomized trade-off algorithm solving both problems and running in $\tilde{O}(nmh+(n/h)^3)$ time using $\tilde{O}(n^2/h+m)$ space for any parameter $h\in [1,n]$. In particular, using subquadratic space we get $\tilde{O}(nm+n^{3/2}m^{3/4})$ running time, which improves by a polynomial factor upon all the known upper bou…
Improved Approximation Ratios for the Shortest Common Superstring Problem with Reverse Complements
Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo（东京大学大学院信息理工学研究科计算机科学系）；Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, The University of Tokyo（东京大学医学科学研究所人类基因组中心医疗数据信息学分部）
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…
Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in…
This paper considers fully dynamic graph algorithms with both faster worst case update time and sublinear space. The fully dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes, process an online sequence of edge insertions, edge deletions, and queries of the form "Is there a path between nodes a and b?" In 2013, the first data structure was presented with worst case time per operation which was polylogarithmic in n. In this paper, we shave off a factor of log n from that time, to O(log^4 n) per update. For sequences which are polynomial in length, our al…
We give a Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions. Each update is supported in O(n^{1/2 - c}) expected worst-case time for some constant c > 0 and this worst-case bound holds with probability at least 1 - n^{-d} where d is a constant that can be made arbitrarily large. This is the first data structure achieving an improvement over the O(n^{1/2}) deterministic worst-case update time of Eppstein et al., a bound that has been standing for nearly 25 ye…
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…
We extend the well known bottleneck paths problem in two directions for directed unweighted (unit edge cost) graphs with positive real edge capacities. Firstly we narrow the problem domain and compute the bottleneck of the entire network in $O(n^ω\log{n})$ time, where $O(n^ω)$ is the time taken to multiply two $n$-by-$n$ matrices over ring. Secondly we enlarge the domain and compute the shortest paths for all possible flow amounts. We present a combinatorial algorithm to solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem in $O(mn)$ worst case time, followed by an algorithm …
Given two locations $s$ and $t$ in a road network, a distance query returns the minimum network distance from $s$ to $t$, while a shortest path query computes the actual route that achieves the minimum distance. These two types of queries find important applications in practice, and a plethora of solutions have been proposed in past few decades. The existing solutions, however, are optimized for either practical or asymptotic performance, but not both. In particular, the techniques with enhanced practical efficiency are mostly heuristic-based, and they offer unattractive worst-case guarantees …
The Restricted Shortest Path (RSP) problem, also known as the Delay-Constrained Least-Cost (DCLC) problem, is an NP-hard bicriteria optimization problem on graphs with $n$ vertices and $m$ edges. In a graph where each edge is assigned a cost and a delay, the goal is to find a min-cost path which does not exceed a delay bound. In this paper, we present improved approximation schemes for RSP on several graph classes. For planar graphs, undirected graphs with positive integer resource (= delay) values, and graphs with $m \in Ω(n \log n)$, we obtain $(1 + \varepsilon)$-approximations in time $O(mn…
This paper studies a combinatorial optimization problem which is obtained by combining the flow shop scheduling problem and the shortest path problem. The objective of the obtained problem is to select a subset of jobs that constitutes a feasible solution to the shortest path problem, and to execute the selected jobs on the flow shop machines to minimize the makespan. We argue that this problem is NP-hard even if the number of machines is two, and is NP-hard in the strong sense for the general case. We propose an intuitive approximation algorithm for the case where the number of machines is an…
A synchronizing word of a deterministic finite complete automaton is a word whose action maps every state to a single one. Finding a shortest or a short synchronizing word is a central computational problem in the theory of synchronizing automata and is applied in other areas such as model-based testing and the theory of codes. Because the problem of finding a shortest synchronizing word is computationally hard, among \emph{exact} algorithms only exponential ones are known. We redesign the previously fastest known exact algorithm based on the bidirectional breadth-first search and improve it w…
In this paper, we explore worst-case solutions for the problems of single and multiple matching on strings in the word RAM model with word length w. In the first problem, we have to build a data structure based on a pattern p of length m over an alphabet of size sigma such that we can answer to the following query: given a text T of length n, where each character is encoded using log(sigma) bits return the positions of all the occurrences of p in T (in the following we refer by occ to the number of reported occurrences). For the multi-pattern matching problem we have a set S of d patterns of t…
This paper gives a new deterministic algorithm for the dynamic Minimum Spanning Forest (MSF) problem in the EREW PRAM model, where the goal is to maintain a MSF of a weighted graph with $n$ vertices and $m$ edges while supporting edge insertions and deletions. We show that one can solve the dynamic MSF problem using $O(\sqrt n)$ processors and $O(\log n)$ worst-case update time, for a total of $O(\sqrt n \log n)$ work. This improves on the work of Ferragina [IPPS 1995] which costs $O(\log n)$ worst-case update time and $O(n^{2/3} \log{\frac{m}{n}})$ work.
We revisit the problem of privately releasing the all-pairs shortest path distances of a weighted undirected graph up to low additive error, which was first studied by Sealfon [Sea16]. In this paper, we improve significantly on Sealfon's results, both for arbitrary weighted graphs and for bounded-weight graphs on $n$ nodes. Specifically, we provide an approximate-DP algorithm that outputs all-pairs shortest path distances up to maximum additive error $\tilde{O}(\sqrt{n})$, and a pure-DP algorithm that outputs all pairs shortest path distances up to maximum additive error $\tilde{O}(n^{2/3})$ (…
The single-source shortest path problem is a classical problem in the research field of graph algorithm. In this paper, a new single-source shortest path algorithm for nonnegative weight graph is proposed. The algorithm can compress multi-round Fibonacci heap operations to one round to save running time relative to Dijkstra's algorithm using Fibonacci heap. The time complexity of the algorithm is also O(m+nlogn) in the worst case, where m is the number of edges and n is the number of nodes. However, the bound can be linear in some case, for example, when edge weights of a graph are all the sam…
The technique of Schroeppel and Shamir (SICOMP, 1981) has long been the most efficient way to trade space against time for the SUBSET SUM problem. In the random-instance setting, however, improved tradeoffs exist. In particular, the recently discovered dissection method of Dinur et al. (CRYPTO 2012) yields a significantly improved space--time tradeoff curve for instances with strong randomness properties. Our main result is that these strong randomness assumptions can be removed, obtaining the same space--time tradeoffs in the worst case. We also show that for small space usage the dissection …
Finding the shortest path between two points in a graph is a fundamental problem that has been well-studied over the past several decades. Shortest path algorithms are commonly applied to modern navigation systems, so our study aims to improve the efficiency of an existing algorithm on large-scale Euclidean networks. The current literature lacks a deep understanding of certain algorithms' performance on these types of networks. Therefore, we incorporate a new heuristic function, called the $k$-step look-ahead, into the A* search algorithm and conduct a computational experiment to evaluate and …
In this paper, we study the single-source shortest-path (SSSP) problem with positive edge weights, which is a notoriously hard problem in the parallel context. In practice, the $Δ$-stepping algorithm proposed by Meyer and Sanders has been widely adopted. However, $Δ$-stepping has no known worst-case bounds for general graphs. The performance of $Δ$-stepping also highly relies on the parameter $Δ$. There have also been lots of algorithms with theoretical bounds, such as Radius-stepping, but they either have no implementations available or are much slower than $Δ$-stepping in practice. We propos…
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…
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…
Various classes of induced subgraphs are involved in the deepest results of graph theory and graph algorithms. A prominent example concerns the {\em perfection} of $G$ that the chromatic number of each induced subgraph $H$ of $G$ equals the clique number of $H$. The seminal Strong Perfect Graph Theorem confirms that the perfection of $G$ can be determined by detecting odd holes in $G$ and its complement. Chudnovsky et al. show in 2005 an $O(n^9)$ algorithm for recognizing perfect graphs, which can be implemented to run in $O(n^{6+ω})$ time for the exponent $ω<2.373$ of square-matrix multiplica…
The GC problem is to identify a pre-determined number of center vertices such that the distances or costs from (or to) the centers to (or from) other vertices is minimized. The bottleneck of a path is the minimum capacity of edges on the path. The Bottleneck Paths (BP) problem is to compute the paths that give us the maximum bottleneck values between pairs of vertices. The Graph Bottleneck (GB) problem is to find the minimum bottleneck value out of bottleneck paths for all possible pairs of vertices. We give two similar algorithms that are based on binary search to solve the 1-center GC proble…
A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior…
In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal $\min(O(\log n), f)$ approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters denoting the maximum number of sets, number of elements, frequency, and the cost range.) In the high-frequency range, when $f=Ω(\log n)$, this was achieved by a deterministic $O(\log n)$-approximation algorithm with $O(f \log n)$ amortized update time [Gupta et al. STOC'17]. In the low-frequency range, the line of work by Gupta et al. [STOC'17], Abboud et al. [STOC'19], and Bhat…
