arXiv subjects

Bernhard Haeupler

Publications and source records attributed to Bernhard Haeupler.

At least 19 recordsLinked to original sources

arXiv2026

Heaps and Their Working Sets

We construct a heap with strong beyond-worst-case performance guarantees and explore the analysis of such heaps. First, we unify existing notions of the working-set bound for heaps by proving that essentially all of them are equivalent - with the notable exception of the so-called stack-like bound, which is strictly stronger. This equivalence simplifies the theoretical landscape and extends the range of applications of heaps with working-set bounds. Second, we present the first heap implementation that has the amortized stack-like bound and supports $\mathcal O(1)$-time decrease-key and $o(\log^*n)$-time insert.

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arXiv2026

Splay trees are almost dynamically optimal

Sleator and Tarjan [JACM, 1985] conjectured that splay trees are dynamically optimal -- that on every access sequence, they perform within a constant factor of the optimal offline dynamic binary search tree. Despite four decades of work, no $o(\log n)$ competitive ratio was known. We prove that splay trees are $O(\log\log n \cdot \log^2\log\log n)=\tilde{O}(\log\log n)$-competitive.

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arXiv2026

Dynamic Construction of the Lov\'asz Local Lemma

This paper proves that a wide class of local search algorithms extend as is to the fully dynamic setting with an adaptive adversary, achieving an amortized $\tilde{O}(1)$ number of local-search steps per update. A breakthrough by Moser (2009) introduced the witness-tree and entropy compression techniques for analyzing local resampling processes for the Lov\'asz Local Lemma. These methods have since been generalized and expanded to analyze a wide variety of local search algorithms that can efficiently find solutions to many important local constraint satisfaction problems. These algorithms either extend a partial valid assignment and backtrack by unassigning variables when constraints become violated, or they iteratively fix violated constraints by resampling their variables. These local resampling or backtracking procedures are incredibly flexible, practical, and simple to specify and implement. Yet, they can be shown to be extremely efficient on static instances, typically performing only (sub)-linear number of fixing steps. The main technical challenge lies in proving conditions that guarantee such rapid convergence. This paper extends these convergence results to fully dynamic settings, where an adaptive adversary may add or remove constraints. We prove that applying the same simple local search procedures to fix old or newly introduced violations leads to a total number of resampling steps near-linear in the number of adversarial updates. Our result is very general and yields several immediate corollaries. For example, letting $\Delta$ denote the maximum degree, for a constant $\epsilon$ and $\Delta = \text{poly}(\log n)$, we can maintain a $(1+\epsilon) \Delta$-edge coloring in $\text{poly}(\log n)$ amortized update time against an adaptive adversary. The prior work for this regime has exponential running time in $\sqrt{\log n}$ [Christiansen, SODA '26].

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arXiv2026

Maintaining Random Assignments under Adversarial Dynamics

We study and further develop powerful general-purpose schemes to maintain random assignments under adversarial dynamic changes. The goal is to maintain assignments that are (approximately) distributed similarly as a completely fresh resampling of all assignments after each change, while doing only a few resamples per change. This becomes particularly interesting and challenging when dynamics are controlled by an adaptive adversary. Our work builds on and further develops the proactive resampling technique [Bhattacharya, Saranurak, and Sukprasert ESA'22]. We identify a new ``temporal selection'' attack that adaptive adversaries can use to cause biases, even against proactive resampling. We propose a new ''temporal aggregation'' principle that algorithms should follow to counteract these biases, and present two powerful new resampling schemes based on this principle. We give various applications of our new methods. The main one in maintaining proper coloring of the graph under adaptive adversarial modifications: we maintain $O(\Delta)$ coloring for general graphs with maximum degree $\Delta$ and $O(\frac{\Delta}{\ln \Delta})$ coloring for triangle free graphs, both with sublinear in the number of vertices average work per modification. Other applications include efficiently maintaining random walks in dynamically changing graphs.

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arXiv2026

DAG Projections: Reducing Distance and Flow Problems to DAGs

We show that every directed graph $G$ with $n$ vertices and $m$ edges admits a directed acyclic graph (DAG) with $m^{1+o(1)}$ edges, called a DAG projection, that can either $(1+1/\text{polylog} (n))$-approximate distances between all pairs of vertices $(s,t)$ in $G$, or $n^{o(1)}$-approximate maximum flow between all pairs of vertex subsets $(S,T)$ in $G$. Previous similar results suffer a $\Omega(\log n)$ approximation factor for distances [Assadi, Hoppenworth, Wein, STOC'25] [Filtser, SODA'26] and, for maximum flow, no prior result of this type is known. Our DAG projections admit $m^{1+o(1)}$-time constructions. Further, they admit almost-optimal parallel constructions, i.e., algorithms with $m^{1+o(1)}$ work and $m^{o(1)}$ depth, assuming the ones for approximate shortest path or maximum flow on DAGs, even when the input $G$ is not a DAG. DAG projections immediately transfer results on DAGs, usually simpler and more efficient, to directed graphs. As examples, we improve the state-of-the-art of $(1+\epsilon)$-approximate distance preservers [Hoppenworth, Xu, Xu, SODA'25] and single-source minimum cut [Cheung, Lau, Leung, SICOMP'13], and obtain simpler construction of $(n^{1/3},\epsilon)$-hop-set [Kogan, Parter, SODA'22] [Bernstein, Wein, SODA'23] and combinatorial max flow algorithms [Bernstein, Blikstad, Saranurak, Tu, FOCS'24] [Bernstein, Blikstad, Li, Saranurak, Tu, FOCS'25]. Finally, via DAG projections, we reduce major open problems on almost-optimal parallel algorithms for exact single-source shortest paths (SSSP) and maximum flow to easier settings: (1) From exact directed SSSP to exact undirected ones, (2) From exact directed SSSP to $(1+1/\text{polylog}(n))$-approximation on DAGs, and (3) From exact directed maximum flow to $n^{o(1)}$-approximation on DAGs.

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arXiv2026

A Constant-Approximation Distance Labeling Scheme under Polynomially Many Edge Failures

A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph $G$ with $n$ vertices so that, for any edge set $F$ of size $|F| \leq f$, one can approximate the distance between $p$ and $q$ in $G \setminus F$ by reading only the labels of $F \cup \{p,q\}$. For any $k$, we present a deterministic polynomial-time scheme with $O(k^{4})$ approximation and $\tilde{O}(f^{4}n^{1/k})$ label size. This is the first scheme to achieve a constant approximation while handling any number of edge faults $f$, resolving the open problem posed by Dory and Parter [DP21]. All previous schemes provided only a linear-in-$f$ approximation [DP21, LPS25]. Our labeling scheme directly improves the state of the art in the simpler setting of distance sensitivity oracles. Even for just $f = \Theta(\log n)$ faults, all previous oracles either have super-linear query time, linear-in-$f$ approximation [CLPR12], or exponentially worse $2^{{\rm poly}(k)}$ approximation dependency in $k$ [HLS24].

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arXiv2026

Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness

All parallel algorithms for directed reachability and shortest paths crucially rely on efficient shortcut constructions. These constructions find directed paths and shortcut them by adding edges, with the goal to reduce the diameter of the graph. A long sequence of works has studied (efficient) shortcut constructions as well as impossibility results on the best diameter and therefore the best parallelism that can be achieved via this approach. This paper introduces a new conceptual tool for this line of research in the form of a simple and natural structural criterion: A shortcut $H$ for a graph $G$ is certified if for any shortcut edge $(u, v) \in H$, there exists a vertex $w$ such that the edges $(u, w)$ and $(w, v)$ are also in $G \cup H$. We show that this criterion captures constructiveness in the following sense: A shortcut $H$ can be constructed in $t$ time by repeatedly spending $\ell$ time on shortcutting a path of length $\ell$, if and only if, there exists a certified shortcut $H' \supseteq H$ of size $\tilde{O}(t)$. Furthermore, all known shortcut constructions with efficient algorithms can be extended to produce certified shortcuts of size $\tilde{O}(m)$. On the other hand, for shortcut constructions for which attempts to find efficient implementations have failed, we can show that this is impossible. We also obtain stronger diameter lower bounds for certified shortcuts and hopsets. For example, no certified shortcut construction with almost-linear size can reduce a graph's diameter below $n^{1/4-o(1)}$. This seems to be the best bound one can hope for with current techniques.

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arXiv2025

Clustering with Label Consistency

Designing efficient, effective, and consistent metric clustering algorithms is a significant challenge attracting growing attention. Traditional approaches focus on the stability of cluster centers; unfortunately, this neglects the real-world need for stable point labels, i.e., stable assignments of points to named sets (clusters). In this paper, we address this gap by initiating the study of label-consistent metric clustering. We first introduce a new notion of consistency, measuring the label distance between two consecutive solutions. Then, armed with this new definition, we design new consistent approximation algorithms for the classical $k$-center and $k$-median problems.

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arXiv2025

Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers

We present the first deterministic nearly-linear time algorithm for single-source shortest paths with negative edge weights on directed graphs: given a directed graph $G$ with $n$ vertices, $m$ edges whose weights are integer in $\{-W,\dots,W\}$, our algorithm either computes all distances from a source $s$ or reports a negative cycle in time $\tilde{O}(m)\cdot \log(nW)$ time. All known near-linear time algorithms for this problem have been inherently randomized, as they crucially rely on low-diameter decompositions. To overcome this barrier, we introduce a new structural primitive for directed graphs called the path cover. This plays a role analogous to neighborhood covers in undirected graphs, which have long been central to derandomizing algorithms that use low-diameter decomposition in the undirected setting. We believe that path covers will serve as a fundamental tool for the design of future deterministic algorithms on directed graphs.

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arXiv2025

Reviving Thorup's Shortcut Conjecture

We aim to revive Thorup's conjecture [Thorup, WG'92] on the existence of reachability shortcuts with ideal size-diameter tradeoffs. Thorup originally asked whether, given any graph $G=(V,E)$ with $m$ edges, we can add $m^{1+o(1)}$ ``shortcut'' edges $E_+$ from the transitive closure $E^*$ of $G$ so that $\text{dist}_{G_+}(u,v) \leq m^{o(1)}$ for all $(u,v)\in E^*$, where $G_+=(V,E\cup E_+)$. The conjecture was refuted by Hesse [Hesse, SODA'03], followed by significant efforts in the last few years to optimize the lower bounds. In this paper we observe that although Hesse refuted the letter of Thorup's conjecture, his work~[Hesse, SODA'03] -- and all followup work -- does not refute the spirit of the conjecture, which should allow $G_+$ to contain both new (shortcut) edges and new Steiner vertices. Our results are as follows. (1) On the positive side, we present explicit attacks that break all known shortcut lower bounds when Steiner vertices are allowed. (2) On the negative side, we rule out ideal $m^{1+o(1)}$-size, $m^{o(1)}$-diameter shortcuts whose ``thickness'' is $t=o(\log n/\log \log n)$, meaning no path can contain $t$ consecutive Steiner vertices. (3) We propose a candidate hard instance as the next step toward resolving the revised version of Thorup's conjecture. Finally, we show promising implications. Almost-optimal parallel algorithms for computing a generalization of the shortcut that approximately preserves distances or flows imply almost-optimal parallel algorithms with $m^{o(1)}$ depth for exact shortcut paths and exact maximum flow. The state-of-the-art algorithms have much worse depth of $n^{1/2+o(1)}$ [Rozho\v{n}, Haeupler, Martinsson, STOC'23] and $m^{1+o(1)}$ [Chen, Kyng, Liu, FOCS'22], respectively.

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arXiv2025

Parallel $(1+\epsilon)$-Approximate Multi-Commodity Mincost Flow in Almost Optimal Depth and Work

We present a parallel algorithm for computing $(1+\epsilon)$-approximate mincost flow on an undirected graph with $m$ edges, where capacities and costs are assigned to both edges and vertices. Our algorithm achieves $\hat{O}(m)$ work and $\hat{O}(1)$ depth when $\epsilon > 1/\mathrm{polylog}(m)$, making both the work and depth almost optimal, up to a subpolynomial factor. Previous algorithms with $\hat{O}(m)$ work required $\Omega(m)$ depth, even for special cases of mincost flow with only edge capacities or max flow with vertex capacities. Our result generalizes prior almost-optimal parallel $(1+\epsilon)$-approximation algorithms for these special cases, including shortest paths [Li, STOC'20] [Andoni, Stein, Zhong, STOC'20] [Rozhen, Haeupler, Marinsson, Grunau, Zuzic, STOC'23] and max flow with only edge capacities [Agarwal, Khanna, Li, Patil, Wang, White, Zhong, SODA'24]. Our key technical contribution is the first construction of length-constrained flow shortcuts with $(1+\epsilon)$ length slack, $\hat{O}(1)$ congestion slack, and $\hat{O}(1)$ step bound. This provides a strict generalization of the influential concept of $(\hat{O}(1),\epsilon)$-hopsets [Cohen, JACM'00], allowing for additional control over congestion. Previous length-constrained flow shortcuts [Haeupler, Hershkowitz, Li, Roeyskoe, Saranurak, STOC'24] incur a large constant in the length slack, which would lead to a large approximation factor. To enable our flow algorithms to work under vertex capacities, we also develop a close-to-linear time algorithm for computing length-constrained vertex expander decomposition. Building on Cohen's idea of path-count flows [Cohen, SICOMP'95], we further extend our algorithm to solve $(1+\epsilon)$-approximate $k$-commodity mincost flow problems with almost-optimal $\hat{O}(mk)$ work and $\hat{O}(1)$ depth, independent of the number of commodities $k$.

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arXiv2025

Simple Length-Constrained Expander Decompositions

Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $\phi$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ while using each edge to an extent at most $1/\phi$. Prior work showed that every $n$-node and $m$-edge graph admits an $(h, s)$-length $\phi$-expander decomposition of size $\log n \cdot s n^{O(1/s)} \cdot \phi m$. In this work, we give a simple proof of the existence of $(h, s)$-length $\phi$-expander decompositions with an improved size of $s n^{O(1/s)}\cdot \phi m$. Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from $\log ^3 n\cdot s^3 n^{O(1/s)}$ to $s\cdot n^{O(1/s)}$.

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arXiv2025

Stronger Directed Low-Diameter Decompositions with Sub-Logarithmic Diameter and Separation

This paper significantly strengthens directed low-diameter decompositions in several ways. We define and give the first results for separated low-diameter decompositions in directed graphs, tighten and generalize probabilistic guarantees, and prove new independence results between (far away) edges. Our results are the first to give meaningful guarantees for decompositions with small diameters $D = \Omega(\log\log n)$ in contrast to the state of the art that only applies to super-logarithmic diameters $D = \omega(\log n)$. These results transfer several important and widely used aspects of undirected low-diameter decompositions to the directed setting. All our results are algorithmic -- small modifications to two existing directed low-diameter decompositions [BFHL25; Li25] can be used to sample decompositions with our new guarantees in near-linear time $\tilde{O}(m)$.

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arXiv2025

Reducing Shortcut and Hopset Constructions to Shallow Graphs

We introduce a blackbox framework that simplifies all known parallel algorithms with near-linear work for single-source reachability and shortest paths in directed graphs. Specifically, existing reachability algorithms rely on constructing shortcuts; our blackbox allows these algorithms that construct shortcuts with hopbound $h$ to assume the input graph $G$ is ``shallow'', meaning if vertex $s$ can reach vertex $t$, it can do so in approximately $h$ hops. This assumption significantly simplifies shortcut construction [Fin18, JLS19], resulting in simpler parallel reachability algorithms. Furthermore, our blackbox extends naturally to simplify parallel algorithms for constructing hopsets and, consequently, for computing shortest paths [CFR20 , CF23 , RHM+23 ].

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arXiv2025

Maintaining Routing Structures under Deletions via Self-Pruning

Expanders are powerful algorithmic structures with two key properties: they are a) routable: for any multi-commodity flow unit demand, there exists a routing with low congestion over short paths, where a demand is unit if the amount of demand sent / received by any vertex is at most the number of edges adjacent to it. b) stable / prunable: for any (sequence of) edge failures, there exists a proportionally small subset of vertices that can be disabled, such that the graph induced on the remaining vertices is an expander. Two natural algorithmic problems correspond to these two existential guarantees: expander routing, i.e. computing a low-congestion routing for a unit multi-commodity demand on an expander, and expander pruning, i.e., maintaining the subset of disabled vertices under a sequence of edge failures. This paper considers the combination of the two problems: maintaining a routing for a unit multi-commodity demand under pruning steps. This is done through the introduction of a family of expander graphs that, like hypercubes, are easy to route in, and are self-pruning: for an online sequence of edge deletions, a simple self-contained algorithm can find a few vertices to prune with each edge deletion, such that the remaining graph always remains an easy-to-route-in expander in the family. Notably, and with considerable technical work, this self-pruning can be made worst-case, i.e., such that every single adversarial deletion only causes a small number of additional deletions. Our results also allow tight constant-factor control over the length of routing paths (with the usual trade-offs in congestion and pruning ratio) and therefore extend to constant-hop and length-constrained expanders in which routing over constant length paths is crucial.

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arXiv2025

Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric

A function $\varphi:\{0,1\}^n \to \{0,1\}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x,y\in\{0,1\}^n$, the Hamming distance between $x$ and $y$ is equal to the edit distance between $\varphi(x)$ and $\varphi(y)$. The rate of such an embedding is defined as the ratio $n/N$. It is well known in the literature how to construct isometric embeddings with rate $\Omega(1/\log n)$. However, achieving even near-isometric embeddings with positive constant rate has remained elusive until now. In this paper, we present an isometric embedding with rate $1/8$ by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called misaligners. As an immediate consequence of our constant-rate isometric embedding, we improve known conditional lower bounds for various optimization problems in the edit metric, now with optimal dependence on the dimension. We complement our results by showing that no isometric embedding $\varphi:\{0,1\}^n \to \{0,1\}^N$ can have rate greater than $15/32$ for all positive integers $n$. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets. Finally, we consider embeddings $\varphi:\Sigma_{\mathrm{in}}^n \to \Sigma_{\mathrm{out}}^N$ between different input and output alphabets, where the rate is given by $\frac{n\log|\Sigma_{\mathrm{in}}|}{N\log|\Sigma_{\mathrm{out}}|}$. In this setting, we show that the rate can be made arbitrarily close to $1$.

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arXiv2025

Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts

We show the existence of length-constrained expander decomposition in directed graphs and undirected vertex-capacitated graphs. Previously, its existence was shown only in undirected edge-capacitated graphs [Haeupler-R\"acke-Ghaffari, STOC 2022; Haeupler-Hershkowitz-Tan, FOCS 2024]. Along the way, we prove the multi-commodity maxflow-mincut theorems for length-constrained expansion in both directed and undirected vertex-capacitated graphs. Based on our decomposition, we build a length-constrained flow shortcut for undirected vertex-capacitated graphs, which roughly speaking is a set of edges and vertices added to the graph so that every multi-commodity flow demand can be routed with approximately the same vertex-congestion and length, but all flow paths only contain few edges. This generalizes the shortcut for undirected edge-capacitated graphs from [Haeupler-Hershkowitz-Li-Roeyskoe-Saranurak, STOC 2024]. Length-constrained expander decomposition and flow shortcuts have been crucial in the recent algorithms in undirected edge-capacitated graphs [Haeupler-Hershkowitz-Li-Roeyskoe-Saranurak, STOC 2024; Haeupler-Long-Saranurak, FOCS 2024]. Our work thus serves as a foundation to generalize these concepts to directed and vertex-capacitated graphs.

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arXiv2025

Adversarially-Robust Gossip Algorithms for Approximate Quantile and Mean Computations

This paper presents gossip algorithms for aggregation tasks that demonstrate both robustness to adversarial corruptions of any order of magnitude and optimality across a substantial range of these corruption levels. Gossip algorithms distribute information in a scalable and efficient way by having random pairs of nodes exchange small messages. Value aggregation problems are of particular interest in this setting, as they occur frequently in practice, and many elegant algorithms have been proposed for computing aggregates and statistics such as averages and quantiles. An important and well-studied advantage of gossip algorithms is their robustness to message delays, network churn, and unreliable message transmissions. However, these crucial robustness guarantees only hold if all nodes follow the protocol and no messages are corrupted. In this paper, we remedy this by providing a framework to model both adversarial participants and message corruptions in gossip-style communications by allowing an adversary to control a small fraction of the nodes or corrupt messages arbitrarily. Despite this very powerful and general corruption model, we show that robust gossip algorithms can be designed for many important aggregation problems. Our algorithms guarantee that almost all nodes converge to an approximately correct answer with optimal efficiency and essentially as fast as without corruptions. The design of adversarially-robust gossip algorithms poses completely new challenges. Despite this, our algorithms remain very simple variations of known non-robust algorithms with often only subtle changes to avoid non-compliant nodes gaining too much influence over outcomes. While our algorithms remain simple, their analysis is much more complex and often requires a completely different approach than the non-adversarial setting.

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