Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time

In the decremental single-source shortest paths (SSSP) problem we want to maintain the distances between a given source node $s$ and every other node in an $n$-node $m$-edge graph $G$ undergoing edge deletions. While its static counterpart can be solved in near-linear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic $O(mn)$ total update time of Even and Shiloach [JACM 1981] has been the fastest known algorithm for three decades. At the cost of a $(1+\epsilon)$-approximation factor, the running time was recently improved to $n^{2+o(1)}$ by Bernstein and Roditty [SODA 2011]. In this paper, we bring the running time down to near-linear: We give a $(1+\epsilon)$-approximation algorithm with $m^{1+o(1)}$ expected total update time, thus obtaining near-linear time. Moreover, we obtain $m^{1+o(1)} \log W$ time for the weighted case, where the edge weights are integers from $1$ to $W$. The only prior work on weighted graphs in $o(m n)$ time is the $m n^{0.9 + o(1)}$-time algorithm by Henzinger et al. [STOC 2014, ICALP 2015] which works for directed graphs with quasi-polynomial edge weights. The expected running time bound of our algorithm holds against an oblivious adversary. In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a so-called sparse $(h, \epsilon)$-hop set introduced by Cohen [JACM 2000] in the PRAM literature. An $(h, \epsilon)$-hop set of a graph $G=(V, E)$ is a set $F$ of weighted edges such that the distance between any pair of nodes in $G$ can be $(1+\epsilon)$-approximated by their $h$-hop distance (given by a path containing at most $h$ edges) on $G'=(V, E\cup F)$. Our algorithm can maintain an $(n^{o(1)}, \epsilon)$-hop set of near-linear size in near-linear time under edge deletions.Comment: Accepted to Journal of the ACM. A preliminary version of this paper was presented at the 55th IEEE Symposium on Foundations of Computer Science (FOCS 2014). Abstract shortened to respect the arXiv limit of 1920 character

Fast Deterministic Fully Dynamic Distance Approximation

In this paper, we develop deterministic fully dynamic algorithms for computing approximate distances in a graph with worst-case update time guarantees. In particular, we obtain improved dynamic algorithms that, given an unweighted and undirected graph $G=(V,E)$ undergoing edge insertions and deletions, and a parameter $0 < \epsilon \leq 1$, maintain $(1+\epsilon)$-approximations of the $st$-distance between a given pair of nodes $s$ and $t$, the distances from a single source to all nodes ("SSSP"), the distances from multiple sources to all nodes ("MSSP"), or the distances between all nodes ("APSP"). Our main result is a deterministic algorithm for maintaining $(1+\epsilon)$-approximate $st$-distance with worst-case update time $O(n^{1.407})$ (for the current best known bound on the matrix multiplication exponent $\omega$). This even improves upon the fastest known randomized algorithm for this problem. Similar to several other well-studied dynamic problems whose state-of-the-art worst-case update time is $O(n^{1.407})$, this matches a conditional lower bound [BNS, FOCS 2019]. We further give a deterministic algorithm for maintaining $(1+\epsilon)$-approximate single-source distances with worst-case update time $O(n^{1.529})$, which also matches a conditional lower bound. At the core, our approach is to combine algebraic distance maintenance data structures with near-additive emulator constructions. This also leads to novel dynamic algorithms for maintaining $(1+\epsilon, \beta)$-emulators that improve upon the state of the art, which might be of independent interest. Our techniques also lead to improved randomized algorithms for several problems such as exact $st$-distances and diameter approximation.Comment: Changes to the previous version: improved bounds for approximate st distances using new algebraic data structure

A Simple Boosting Framework for Transshipment

Transshipment, also known under the names of earth mover's distance, uncapacitated min-cost flow, or Wasserstein's metric, is an important and well-studied problem that asks to find a flow of minimum cost that routes a general demand vector. Adding to its importance, recent advancements in our understanding of algorithms for transshipment have led to breakthroughs for the fundamental problem of computing shortest paths. Specifically, the recent near-optimal $(1+\varepsilon)$-approximate single-source shortest path algorithms in the parallel and distributed settings crucially solve transshipment as a central step of their approach. The key property that differentiates transshipment from other similar problems like shortest path is the so-called \emph{boosting}: one can boost a (bad) approximate solution to a near-optimal $(1 + \varepsilon)$-approximate solution. This conceptually reduces the problem to finding an approximate solution. However, not all approximations can be boosted -- there have been several proposed approaches that were shown to be susceptible to boosting, and a few others where boosting was left as an open question. The main takeaway of our paper is that any black-box $\alpha$-approximate transshipment solver that computes a \emph{dual} solution can be boosted to an $(1 + \varepsilon)$-approximate solver. Moreover, we significantly simplify and decouple previous approaches to transshipment (in sequential, parallel, and distributed settings) by showing all of them (implicitly) obtain approximate dual solutions. Our analysis is very simple and relies only on the well-known multiplicative weights framework. Furthermore, to keep the paper completely self-contained, we provide a new (and arguably much simpler) analysis of multiplicative weights that leverages well-known optimization tools to bypass the ad-hoc calculations used in the standard analyses

Undirected (1+ε)(1+\varepsilon)-Shortest Paths via Minor-Aggregates: Near-Optimal Deterministic Parallel & Distributed Algorithms

This paper presents near-optimal deterministic parallel and distributed algorithms for computing $(1+\varepsilon)$-approximate single-source shortest paths in any undirected weighted graph. On a high level, we deterministically reduce this and other shortest-path problems to $\tilde{O}(1)$ Minor-Aggregations. A Minor-Aggregation computes an aggregate (e.g., max or sum) of node-values for every connected component of some subgraph. Our reduction immediately implies: Optimal deterministic parallel (PRAM) algorithms with $\tilde{O}(1)$ depth and near-linear work. Universally-optimal deterministic distributed (CONGEST) algorithms, whenever deterministic Minor-Aggregate algorithms exist. For example, an optimal $\tilde{O}(HopDiameter(G))$-round deterministic CONGEST algorithm for excluded-minor networks. Several novel tools developed for the above results are interesting in their own right: A local iterative approach for reducing shortest path computations "up to distance $D$" to computing low-diameter decompositions "up to distance $\frac{D}{2}$". Compared to the recursive vertex-reduction approach of [Li20], our approach is simpler, suitable for distributed algorithms, and eliminates many derandomization barriers. A simple graph-based $\tilde{O}(1)$-competitive $\ell_1$-oblivious routing based on low-diameter decompositions that can be evaluated in near-linear work. The previous such routing [ZGY+20] was $n^{o(1)}$-competitive and required $n^{o(1)}$ more work. A deterministic algorithm to round any fractional single-source transshipment flow into an integral tree solution. The first distributed algorithms for computing Eulerian orientations

Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models

We present a method for solving the shortest transshipment problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of (1 + epsilon) in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes epsilon^(-3) polylog(n) iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog(n), where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results: (1) Broadcast CONGEST model: (1 + epsilon)-approximate SSSP using ~O((sqrt(n) + D) epsilon^(-O(1))) rounds, where D is the (hop) diameter of the network. (2) Broadcast congested clique model: (1 + epsilon)-approximate shortest transshipment and SSSP using ~O(epsilon^(-O(1))) rounds. (3) Multipass streaming model: (1 + epsilon)-approximate shortest transshipment and SSSP using ~O(n) space and ~O(epsilon^(-O(1))) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges

Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models

We present a method for solving the shortest transshipment problem-also known as uncapacitated minimum cost flow-up to a multiplicative error of 1 + ϵ in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes ϵ-3 polylog n iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog n, where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results: 1. Broadcast CONGEST model: (1+")-approximate SSSP using Õ((√ n+D) · ϵ-O(1)) rounds, 1 where D is the (hop) diameter of the network. 2. Broadcast congested clique model: (1+ϵ)-approximate shortest transshipment and SSSP using Õ (ϵ-O(1)) rounds. 3. Multipass streaming model: (1+ϵ)-approximate shortest transshipment and SSSP using Õ (n) space and Õ(ϵ-O(1)) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general nonnegative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges

A Unified Framework for Hopsets

Given an undirected graph G = (V,E), an (?,?)-hopset is a graph H = (V,E\u27), so that adding its edges to G guarantees every pair has an ?-approximate shortest path that has at most ? edges (hops), that is, d_G(u,v) ? d_{G?H}^(?)(u,v) ? ?? d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter ?. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^?),O(k^{1-?}))-hopset of size O?(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ? < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (?,?)-hopset of size O(n^{1+1/k}) must have ??? ? ?(k)

Folklore Sampling is Optimal for Exact Hopsets: Confirming the n\sqrt{n} Barrier

For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1/2})$. This improves on the previous lower bound of $\widetilde{\Omega}(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1/4})$. This improves on the previous lower bound of $\Omega(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$