arXiv subjects

Maximilian Probst Gutenberg

Publications and source records attributed to Maximilian Probst Gutenberg.

At least 37 records · Page 2Linked to original sources

arXiv2024

Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality

We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-Räcke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

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arXiv2024

Optimal Electrical Oblivious Routing on Expanders

In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an $m$-edge graph $G = (V, E)$ that is a $Φ$-expander, i.e. where $\lvert \partial S \rvert \geq Φ\cdot \mathrm{vol}(S)$ for every $S \subseteq V, \mathrm{vol}(S) \leq \mathrm{vol}(V)/2$. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for $\ell_{\infty}$ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing. Our main result proves that the electrical routing is an $O(Φ^{-1} \log m)$-competitive oblivious routing in the $\ell_1$- and $\ell_\infty$-norms. We further observe that the oblivious routing is $O(\log^2 m)$-competitive in the $\ell_2$-norm and, in fact, $O(\log m)$-competitive if $\ell_2$-localization is $O(\log m)$ which is widely believed. Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every $p \in [2, \infty]$ and $q$ given by $1/p + 1/q = 1$. Assuming $\ell_2$-localization in $O(\log m)$, we obtain that in $\ell_p$ and $\ell_q$, the electrical oblivious routing is $O(Φ^{-(1-2/p)}\log m)$ competitive. Using the currently known result for $\ell_2$-localization, this ratio deteriorates by at most a sublogarithmic factor for every $p, q \neq 2$. We complement our upper bounds with lower bounds that show that the electrical routing for any such $p$ and $q$ is $Ω(Φ^{-(1-2/p)}\log m)$-competitive. This renders our results in $\ell_1$ and $\ell_{\infty}$ unconditionally tight up to constants, and the result in any $\ell_p$- and $\ell_q$-norm to be tight in case of $\ell_2$-localization in $O(\log m)$.

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arXiv2023

Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, $s$-$t$ Shortest Path, and Minimum-Cost Flow

We give the first almost-linear time algorithms for several problems in incremental graphs including cycle detection, strongly connected component maintenance, $s$-$t$ shortest path, maximum flow, and minimum-cost flow. To solve these problems, we give a deterministic data structure that returns a $m^{o(1)}$-approximate minimum-ratio cycle in fully dynamic graphs in amortized $m^{o(1)}$ time per update. Combining this with the interior point method framework of Brand-Liu-Sidford (STOC 2023) gives the first almost-linear time algorithm for deciding the first update in an incremental graph after which the cost of the minimum-cost flow attains value at most some given threshold $F$. By rather direct reductions to minimum-cost flow, we are then able to solve the problems in incremental graphs mentioned above. At a high level, our algorithm dynamizes the $\ell_1$ oblivious routing of Rozhoň-Grunau-Haeupler-Zuzic-Li (STOC 2022), and develops a method to extract an approximate minimum ratio cycle from the structure of the oblivious routing. To maintain the oblivious routing, we use tools from concurrent work of Kyng-Meierhans-Probst Gutenberg which designed vertex sparsifiers for shortest paths, in order to maintain a sparse neighborhood cover in fully dynamic graphs. To find a cycle, we first show that an approximate minimum ratio cycle can be represented as a fundamental cycle on a small set of trees resulting from the oblivious routing. Then, we find a cycle whose quality is comparable to the best tree cycle. This final cycle query step involves vertex and edge sparsification procedures reminiscent of previous works, but crucially requires a more powerful dynamic spanner which can handle far more edge insertions. We build such a spanner via a construction that hearkens back to the classic greedy spanner algorithm.

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arXiv2023

A Dynamic Shortest Paths Toolbox: Low-Congestion Vertex Sparsifiers and their Applications

We present a general toolbox, based on new vertex sparsifiers, for designing data structures to maintain shortest paths in dynamic graphs. In an $m$-edge graph undergoing edge insertions and deletions, our data structures give the first algorithms for maintaining (a) $m^{o(1)}$-approximate all-pairs shortest paths (APSP) with \emph{worst-case} update time $m^{o(1)}$ and query time $\tilde{O}(1)$, and (b) a tree $T$ that has diameter no larger than a subpolynomial factor times the diameter of the underlying graph, where each update is handled in amortized subpolynomial time. In graphs undergoing only edge deletions, we develop a simpler and more efficient data structure to maintain a $(1+ε)$-approximate single-source shortest paths (SSSP) tree $T$ in a graph undergoing edge deletions in amortized time $m^{o(1)}$ per update. Our data structures are deterministic. The trees we can maintain are not subgraphs of $G$, but embed with small edge congestion into $G$. This is in stark contrast to previous approaches and is useful for algorithms that internally use trees to route flow. To illustrate the power of our new toolbox, we show that our SSSP data structure gives simple deterministic implementations of flow-routing MWU methods in several contexts, where previously only randomized methods had been known. To obtain our toolbox, we give the first algorithm that, given a graph $G$ undergoing edge insertions and deletions and a dynamic terminal set $A$, maintains a vertex sparsifier $H$ that approximately preserves distances between terminals in $A$, consists of at most $|A|m^{o(1)}$ vertices and edges, and can be updated in worst-case time $m^{o(1)}$. Crucially, our vertex sparsifier construction allows us to maintain a low edge-congestion embedding of $H$ into $G$, which is needed for our applications.

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arXiv2023

Incremental Approximate Maximum Flow on Undirected Graphs in Subpolynomial Update Time

We provide an algorithm which, with high probability, maintains a $(1-ε)$-approximate maximum flow on an undirected graph undergoing $m$-edge additions in amortized $m^{o(1)} ε^{-3}$ time per update. To obtain this result, we provide a more general algorithm that solves what we call the incremental, thresholded $p$-norm flow problem that asks to determine the first edge-insertion in an undirected graph that causes the minimum $\ell_p$-norm flow to decrease below a given threshold in value. Since we solve this thresholded problem, our data structure succeeds against an adaptive adversary that can only see the data structure's output. Furthermore, since our algorithm holds for $p = 2$, we obtain improved algorithms for dynamically maintaining the effective resistance between a pair of vertices in an undirected graph undergoing edge insertions. Our algorithm builds upon previous dynamic algorithms for approximately solving the minimum-ratio cycle problem that underlie previous advances on the maximum flow problem [Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva, FOCS '22] as well as recent dynamic maximum flow algorithms [v.d.Brand-Liu-Sidford, STOC '23]. Instead of using interior point methods, which were a key component of these recent advances, our algorithm uses an optimization method based on $\ell_p$-norm iterative refinement and the multiplicative weight update method. This ensures a monotonicity property in the minimum-ratio cycle subproblems that allows us to apply known data structures and bypass issues arising from adaptive queries.

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arXiv2023

A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow

We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the first running time improvement for deterministic algorithms that compute maximum-flow in graphs with polynomial bounded capacities since the work of Goldberg-Rao [J.ACM '98]. Our algorithm builds on the framework of Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow by computing a sequence of $m^{1+o(1)}$-approximate undirected minimum-ratio cycles. We develop a deterministic dynamic graph data-structure to compute such a sequence of minimum-ratio cycles in an amortized $m^{o(1)}$ time per edge update. Our key technical contributions are deterministic analogues of the vertex sparsification and edge sparsification components of the data-structure from Chen et al. For the vertex sparsification component, we give a method to avoid the randomness in Chen et al. which involved sampling random trees to recurse on. For the edge sparsification component, we design a deterministic algorithm that maintains an embedding of a dynamic graph into a sparse spanner. We also show how our dynamic spanner can be applied to give a deterministic data structure that maintains a fully dynamic low-stretch spanning tree on graphs with polynomially bounded edge lengths, with subpolynomial average stretch and subpolynomial amortized time per edge update.

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arXiv2023

A Simple Algorithm for Multiple-Source Shortest Paths in Planar Digraphs

Given an $n$-vertex planar embedded digraph $G$ with non-negative edge weights and a face $f$ of $G$, Klein presented a data structure with $O(n\log n)$ space and preprocessing time which can answer any query $(u,v)$ for the shortest path distance in $G$ from $u$ to $v$ or from $v$ to $u$ in $O(\log n)$ time, provided $u$ is on $f$. This data structure is a key tool in a number of state-of-the-art algorithms and data structures for planar graphs. Klein's data structure relies on dynamic trees and the persistence technique as well as a highly non-trivial interaction between primal shortest path trees and their duals. The construction of our data structure follows a completely different and in our opinion very simple divide-and-conquer approach that solely relies on Single-Source Shortest Path computations and contractions in the primal graph. Our space and preprocessing time bound is $O(n\log |f|)$ and query time is $O(\log |f|)$ which is an improvement over Klein's data structure when $f$ has small size.

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arXiv2023

Deterministic Incremental APSP with Polylogarithmic Update Time and Stretch

We provide the first deterministic data structure that given a weighted undirected graph undergoing edge insertions, processes each update with polylogarithmic amortized update time and answers queries for the distance between any pair of vertices in the current graph with a polylogarithmic approximation in $O(\log \log n)$ time. Prior to this work, no data structure was known for partially dynamic graphs, i.e., graphs undergoing either edge insertions or deletions, with less than $n^{o(1)}$ update time except for dense graphs, even when allowing randomization against oblivious adversaries or considering only single-source distances.

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arXiv2023

Maintaining Expander Decompositions via Sparse Cuts

In this article, we show that the algorithm of maintaining expander decompositions in graphs undergoing edge deletions directly by removing sparse cuts repeatedly can be made efficient. Formally, for an $m$-edge undirected graph $G$, we say a cut $(S, \overline{S})$ is $ϕ$-sparse if $|E_G(S, \overline{S})| < ϕ\cdot \min\{vol_G(S), vol_G(\overline{S})\}$. A $ϕ$-expander decomposition of $G$ is a partition of $V$ into sets $X_1, X_2, \ldots, X_k$ such that each cluster $G[X_i]$ contains no $ϕ$-sparse cut (meaning it is a $ϕ$-expander) with $\tilde{O}(ϕm)$ edges crossing between clusters. A natural way to compute a $ϕ$-expander decomposition is to decompose clusters by $ϕ$-sparse cuts until no such cut is contained in any cluster. We show that even in graphs undergoing edge deletions, a slight relaxation of this meta-algorithm can be implemented efficiently with amortized update time $m^{o(1)}/ϕ^2$. Our approach naturally extends to maintaining directed $ϕ$-expander decompositions and $ϕ$-expander hierarchies and thus gives a unifying framework while having simpler proofs than previous state-of-the-art work. In all settings, our algorithm matches the run-times of previous algorithms up to subpolynomial factors. Moreover, our algorithm provides stronger guarantees for $ϕ$-expander decompositions. For example, for graphs undergoing edge deletions, our approach is the first to maintain a dynamic expander decomposition where each updated decomposition is a refinement of the previous decomposition, and our approach is the first to guarantee a sublinear $ϕm^{1+o(1)}$ bound on the total number of edges that cross between clusters across the entire sequence of dynamic updates.

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arXiv2022

A Simple Framework for Finding Balanced Sparse Cuts via APSP

We present a very simple and intuitive algorithm to find balanced sparse cuts in a graph via shortest-paths. Our algorithm combines a new multiplicative-weights framework for solving unit-weight multi-commodity flows with standard ball growing arguments. Using Dijkstra's algorithm for computing the shortest paths afresh every time gives a very simple algorithm that runs in time $\widetilde{O}(m^2/ϕ)$ and finds an $\widetilde{O}(ϕ)$-sparse balanced cut, when the given graph has a $ϕ$-sparse balanced cut. Combining our algorithm with known deterministic data-structures for answering approximate All Pairs Shortest Paths (APSP) queries under increasing edge weights (decremental setting), we obtain a simple deterministic algorithm that finds $m^{o(1)}ϕ$-sparse balanced cuts in $m^{1+o(1)}/ϕ$ time. Our deterministic almost-linear time algorithm matches the state-of-the-art in randomized and deterministic settings up to subpolynomial factors, while being significantly simpler to understand and analyze, especially compared to the only almost-linear time deterministic algorithm, a recent breakthrough by Chuzhoy-Gao-Li-Nanongkai-Peng-Saranurak (FOCS 2020).

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arXiv2022

Derandomizing Directed Random Walks in Almost-Linear Time

In this article, we present the first deterministic directed Laplacian L systems solver that runs in time almost-linear in the number of non-zero entries of L. Previous reductions imply the first deterministic almost-linear time algorithms for computing various fundamental quantities on directed graphs including stationary distributions, personalized PageRank, hitting times and escape probabilities. We obtain these results by introducing partial symmetrization, a new technique that makes the Laplacian of an Eulerian directed graph ``less directed'' in a useful sense, which may be of independent interest. The usefulness of this technique comes from two key observations: Firstly, the partially symmetrized Laplacian preconditions the original Eulerian Laplacian well in Richardson iteration, enabling us to construct a solver for the original matrix from a solver for the partially symmetrized one. Secondly, the undirected structure in the partially symmetrized Laplacian makes it possible to sparsify the matrix very crudely, i.e. with large spectral error, and still show that Richardson iterations convergence when using the sparsified matrix as a preconditioner. This allows us to develop deterministic sparsification tools for the partially symmetrized Laplacian. Together with previous reductions from directed Laplacians to Eulerian Laplacians, our technique results in the first deterministic almost-linear time algorithm for solving linear equations in directed Laplacians. To emphasize the generality of our new technique, we show that two prominent existing (randomized) frameworks for solving linear equations in Eulerian Laplacians can be derandomized in this way: the squaring-based framework of Cohen, Kelner, Peebles, Peng, Rao, Sidford and Vladu (STOC 2017) and the sparsified Cholesky-based framework of Peng and Song (STOC 2022).

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arXiv2022

Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities in $m^{1+o(1)}$ time. Our algorithm builds the flow through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized $m^{o(1)}$ time using a new dynamic graph data structure. Our framework extends to algorithms running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, $p$-norm flows, and $p$-norm isotonic regression on arbitrary directed acyclic graphs.

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arXiv2022

Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets

We study linear equations in combinatorial Laplacians of $k$-dimensional simplicial complexes ($k$-complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial role in homology and are a central tool in topology. Beyond this, they have various applications in data analysis and physical modeling problems. It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes. This paper shows that linear equations in combinatorial Laplacians of 2-complexes are as hard to solve as general linear equations. More precisely, for any constant $c \geq 1$, if we can solve linear equations in combinatorial Laplacians of 2-complexes up to high accuracy in time $\tilde{O}((\# \text{ of nonzero coefficients})^c)$, then we can solve general linear equations with polynomially bounded integer coefficients and condition numbers up to high accuracy in time $\tilde{O}((\# \text{ of nonzero coefficients})^c)$. We prove this by a nearly-linear time reduction from general linear equations to combinatorial Laplacians of 2-complexes. Our reduction preserves the sparsity of the problem instances up to poly-logarithmic factors.

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arXiv2021

Incremental SSSP for Sparse Digraphs Beyond the Hopset Barrier

Given a directed, weighted graph $G=(V,E)$ undergoing edge insertions, the incremental single-source shortest paths (SSSP) problem asks for the maintenance of approximate distances from a dedicated source $s$ while optimizing the total time required to process the insertion sequence of $m$ edges. Recently, Gutenberg, Williams and Wein [STOC'20] introduced a deterministic $\tilde{O}(n^2)$ algorithm for this problem, achieving near linear time for very dense graphs. For sparse graphs, Chechik and Zhang [SODA'21] recently presented a deterministic $\tilde{O}(m^{5/3})$ algorithm, and an adaptive randomized algorithm with run-time $\tilde{O}(m\sqrt{n} + m^{7/5})$. This algorithm is remarkable for two reasons: 1) in very spare graphs it reaches the directed hopset barrier of $\tildeΩ(n^{3/2})$ that applied to all previous approaches for partially-dynamic SSSP [STOC'14, SODA'20, FOCS'20] \emph{and} 2) it does not resort to a directed hopset technique itself. In this article we introduce \emph{propagation synchronization}, a new technique for controlling the error build-up on paths throughout batches of insertions. This leads us to a significant improvement of the approach in [SODA'21] yielding a \emph{deterministic} $\tilde{O}(m^{3/2})$ algorithm for the problem. By a very careful combination of our new technique with the sampling approach from [SODA'21], we further obtain an adaptive randomized algorithm with total update time $\tilde{O}(m^{4/3})$. This is the first partially-dynamic SSSP algorithm in sparse graphs to bypass the notorious directed hopset barrier which is often seen as the fundamental challenge towards achieving truly near-linear time algorithms.

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arXiv2021

New Techniques and Fine-Grained Hardness for Dynamic Near-Additive Spanners

Maintaining and updating shortest paths information in a graph is a fundamental problem with many applications. As computations on dense graphs can be prohibitively expensive, and it is preferable to perform the computations on a sparse skeleton of the given graph that roughly preserves the shortest paths information. Spanners and emulators serve this purpose. This paper develops fast dynamic algorithms for sparse spanner and emulator maintenance and provides evidence from fine-grained complexity that these algorithms are tight. Under the popular OMv conjecture, we show that there can be no decremental or incremental algorithm that maintains an $n^{1+o(1)}$ edge (purely additive) $+n^δ$-emulator for any $δ<1/2$ with arbitrary polynomial preprocessing time and total update time $m^{1+o(1)}$. Also, under the Combinatorial $k$-Clique hypothesis, any fully dynamic combinatorial algorithm that maintains an $n^{1+o(1)}$ edge $(1+ε,n^{o(1)})$-spanner or emulator must either have preprocessing time $mn^{1-o(1)}$ or amortized update time $m^{1-o(1)}$. Both of our conditional lower bounds are tight. As the above fully dynamic lower bound only applies to combinatorial algorithms, we also develop an algebraic spanner algorithm that improves over the $m^{1-o(1)}$ update time for dense graphs. For any constant $ε\in (0,1]$, there is a fully dynamic algorithm with worst-case update time $O(n^{1.529})$ that whp maintains an $n^{1+o(1)}$ edge $(1+ε,n^{o(1)})$-spanner. Our new algebraic techniques and spanner algorithms allow us to also obtain (1) a new fully dynamic algorithm for All-Pairs Shortest Paths (APSP) with update and path query time $O(n^{1.9})$; (2) a fully dynamic $(1+ε)$-approximate APSP algorithm with update time $O(n^{1.529})$; (3) a fully dynamic algorithm for near-$2$-approximate Steiner tree maintenance.

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arXiv2021

Deterministic Decremental SSSP and Approximate Min-Cost Flow in Almost-Linear Time

In the decremental single-source shortest paths problem, the goal is to maintain distances from a fixed source $s$ to every vertex $v$ in an $m$-edge graph undergoing edge deletions. In this paper, we conclude a long line of research on this problem by showing a near-optimal deterministic data structure that maintains $(1+ε)$-approximate distance estimates and runs in $m^{1+o(1)}$ total update time. Our result, in particular, removes the oblivious adversary assumption required by the previous breakthrough result by Henzinger et al. [FOCS'14], which leads to our second result: the first almost-linear time algorithm for $(1-ε)$-approximate min-cost flow in undirected graphs where capacities and costs can be taken over edges and vertices. Previously, algorithms for max flow with vertex capacities, or min-cost flow with any capacities required super-linear time. Our result essentially completes the picture for approximate flow in undirected graphs. The key technique of the first result is a novel framework that allows us to treat low-diameter graphs like expanders. This allows us to harness expander properties while bypassing shortcomings of expander decomposition, which almost all previous expander-based algorithms needed to deal with. For the second result, we break the notorious flow-decomposition barrier from the multiplicative-weight-update framework using randomization.

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arXiv2021

Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing

Let $G = (V,E,w)$ be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental single-source reachability problem (SSR), we are given a fixed source $s$ and the goal is to maintain a data structure that can answer path-queries $s \rightarrowtail v$ for any $v \in V$. In the more general single-source shortest paths (SSSP) problem the goal is to return an approximate shortest path to $v$, and in the SCC problem the goal is to maintain strongly connected components of $G$ and to answer path queries within each component. All of these problems have been very actively studied over the past two decades, but all the fast algorithms are randomized and, more significantly, they can only answer path queries if they assume a weaker model: they assume an oblivious adversary which is not adaptive and must fix the update sequence in advance. This assumption significantly limits the use of these data structures, most notably preventing them from being used as subroutines in static algorithms. All the above problems are notoriously difficult in the adaptive setting. In fact, the state-of-the-art is still the Even and Shiloach tree, which dates back all the way to 1981 and achieves total update time $O(mn)$. We present the first algorithms to break through this barrier: 1) deterministic decremental SSR/SCC with total update time $mn^{2/3 + o(1)}$ 2) deterministic decremental SSSP with total update time $n^{2+2/3+o(1)}$. To achieve these results, we develop two general techniques of broader interest for working with dynamic graphs: 1) a generalization of expander-based tools to dynamic directed graphs, and 2) a technique that we call congestion balancing and which provides a new method for maintaining flow under adversarial deletions. Using the second technique, we provide the first near-optimal algorithm for decremental bipartite matching.

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arXiv2020

Near-Optimal Algorithms for Reachability, Strongly-Connected Components and Shortest Paths in Partially Dynamic Digraphs

In this thesis, we present new techniques to deal with fundamental algorithmic graph problems where graphs are directed and partially dynamic, i.e. undergo either a sequence of edge insertions or deletions: - Single-Source Reachability (SSR), - Strongly-Connected Components (SCCs), and - Single-Source Shortest Paths (SSSP). These problems have recently received an extraordinary amount of attention due to their role as subproblems in various more complex and notoriously hard graph problems, especially to compute flows, bipartite matchings and cuts. Our techniques lead to the first near-optimal data structures for these problems in various different settings. Letting $n$ denote the number of vertices in the graph and by $m$ the maximum number of edges in any version of the graph, we obtain - the first randomized data structure to maintain SSR and SCCs in near-optimal total update time $\tilde{O}(m)$ in a graph undergoing edge deletions. - the first randomized data structure to maintain SSSP in partially dynamic graphs in total update time $\tilde{O}(n^2)$ which is near-optimal in dense graphs. - the first deterministic data structures for SSR and SCC for graphs undergoing edge deletions, and for SSSP in partially dynamic graphs that improve upon the $O(mn)$ total update time by Even and Shiloach from 1981 that is often considered to be a fundamental barrier.

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