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 $(1+\epsilon)$-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

NextBestOnce: Achieving Polylog Routing despite Non-greedy Embeddings

Social Overlays suffer from high message delivery delays due to insufficient routing strategies. Limiting connections to device pairs that are owned by individuals with a mutual trust relationship in real life, they form topologies restricted to a subgraph of the social network of their users. While centralized, highly successful social networking services entail a complete privacy loss of their users, Social Overlays at higher performance represent an ideal private and censorship-resistant communication substrate for the same purpose. Routing in such restricted topologies is facilitated by embedding the social graph into a metric space. Decentralized routing algorithms have up to date mainly been analyzed under the assumption of a perfect lattice structure. However, currently deployed embedding algorithms for privacy-preserving Social Overlays cannot achieve a sufficiently accurate embedding and hence conventional routing algorithms fail. Developing Social Overlays with acceptable performance hence requires better models and enhanced algorithms, which guarantee convergence in the presence of local optima with regard to the distance to the target. We suggest a model for Social Overlays that includes inaccurate embeddings and arbitrary degree distributions. We further propose NextBestOnce, a routing algorithm that can achieve polylog routing length despite local optima. We provide analytical bounds on the performance of NextBestOnce assuming a scale-free degree distribution, and furthermore show that its performance can be improved by more than a constant factor when including Neighbor-of-Neighbor information in the routing decisions.Comment: 23 pages, 2 figure

Analyzing and Enhancing Routing Protocols for Friend-to-Friend Overlays

The threat of surveillance by governmental and industrial parties is more eminent than ever. As communication moves into the digital domain, the advances in automatic assessment and interpretation of enormous amounts of data enable tracking of millions of people, recording and monitoring their private life with an unprecedented accurateness. The knowledge of such an all-encompassing loss of privacy affects the behavior of individuals, inducing various degrees of (self-)censorship and anxiety. Furthermore, the monopoly of a few large-scale organizations on digital communication enables global censorship and manipulation of public opinion. Thus, the current situation undermines the freedom of speech to a detrimental degree and threatens the foundations of modern society. Anonymous and censorship-resistant communication systems are hence of utmost importance to circumvent constant surveillance. However, existing systems are highly vulnerable to infiltration and sabotage. In particular, Sybil attacks, i.e., powerful parties inserting a large number of fake identities into the system, enable malicious parties to observe and possibly manipulate a large fraction of the communication within the system. Friend-to-friend (F2F) overlays, which restrict direct communication to parties sharing a real-world trust relationship, are a promising countermeasure to Sybil attacks, since the requirement of establishing real-world trust increases the cost of infiltration drastically. Yet, existing F2F overlays suffer from a low performance, are vulnerable to denial-of-service attacks, or fail to provide anonymity. Our first contribution in this thesis is concerned with an in-depth analysis of the concepts underlying the design of state-of-the-art F2F overlays. In the course of this analysis, we first extend the existing evaluation methods considerably, hence providing tools for both our and future research in the area of F2F overlays and distributed systems in general. Based on the novel methodology, we prove that existing approaches are inherently unable to offer acceptable delays without either requiring exhaustive maintenance costs or enabling denial-of-service attacks and de-anonymization. Consequentially, our second contribution lies in the design and evaluation of a novel concept for F2F overlays based on insights of the prior in-depth analysis. Our previous analysis has revealed that greedy embeddings allow highly efficient communication in arbitrary connectivity-restricted overlays by addressing participants through coordinates and adapting these coordinates to the overlay structure. However, greedy embeddings in their original form reveal the identity of the communicating parties and fail to provide the necessary resilience in the presence of dynamic and possibly malicious users. Therefore, we present a privacy-preserving communication protocol for greedy embeddings based on anonymous return addresses rather than identifying node coordinates. Furthermore, we enhance the communication’s robustness and attack-resistance by using multiple parallel embeddings and alternative algorithms for message delivery. We show that our approach achieves a low communication complexity. By replacing the coordinates with anonymous addresses, we furthermore provably achieve anonymity in the form of plausible deniability against an internal local adversary. Complementary, our simulation study on real-world data indicates that our approach is highly efficient and effectively mitigates the impact of failures as well as powerful denial-of-service attacks. Our fundamental results open new possibilities for anonymous and censorship-resistant applications.Die Bedrohung der Überwachung durch staatliche oder kommerzielle Stellen ist ein drängendes Problem der modernen Gesellschaft. Heutzutage findet Kommunikation vermehrt über digitale Kanäle statt. Die so verfügbaren Daten über das Kommunikationsverhalten eines Großteils der Bevölkerung in Kombination mit den Möglichkeiten im Bereich der automatisierten Verarbeitung solcher Daten erlauben das großflächige Tracking von Millionen an Personen, deren Privatleben mit noch nie da gewesener Genauigkeit aufgezeichnet und beobachtet werden kann. Das Wissen über diese allumfassende Überwachung verändert das individuelle Verhalten und führt so zu (Selbst-)zensur sowie Ängsten. Des weiteren ermöglicht die Monopolstellung einiger weniger Internetkonzernen globale Zensur und Manipulation der öffentlichen Meinung. Deshalb stellt die momentane Situation eine drastische Einschränkung der Meinungsfreiheit dar und bedroht die Grundfesten der modernen Gesellschaft. Systeme zur anonymen und zensurresistenten Kommunikation sind daher von ungemeiner Wichtigkeit. Jedoch sind die momentanen System anfällig gegen Sabotage. Insbesondere ermöglichen es Sybil-Angriffe, bei denen ein Angreifer eine große Anzahl an gefälschten Teilnehmern in ein System einschleust und so einen großen Teil der Kommunikation kontrolliert, Kommunikation innerhalb eines solchen Systems zu beobachten und zu manipulieren. F2F Overlays dagegen erlauben nur direkte Kommunikation zwischen Teilnehmern, die eine Vertrauensbeziehung in der realen Welt teilen. Dadurch erschweren F2F Overlays das Eindringen von Angreifern in das System entscheidend und verringern so den Einfluss von Sybil-Angriffen. Allerdings leiden die existierenden F2F Overlays an geringer Leistungsfähigkeit, Anfälligkeit gegen Denial-of-Service Angriffe oder fehlender Anonymität. Der erste Beitrag dieser Arbeit liegt daher in der fokussierten Analyse der Konzepte, die in den momentanen F2F Overlays zum Einsatz kommen. Im Zuge dieser Arbeit erweitern wir zunächst die existierenden Evaluationsmethoden entscheidend und erarbeiten so Methoden, die Grundlagen für unsere sowie zukünftige Forschung in diesem Bereich bilden. Basierend auf diesen neuen Evaluationsmethoden zeigen wir, dass die existierenden Ansätze grundlegend nicht fähig sind, akzeptable Antwortzeiten bereitzustellen ohne im Zuge dessen enorme Instandhaltungskosten oder Anfälligkeiten gegen Angriffe in Kauf zu nehmen. Folglich besteht unser zweiter Beitrag in der Entwicklung und Evaluierung eines neuen Konzeptes für F2F Overlays, basierenden auf den Erkenntnissen der vorangehenden Analyse. Insbesondere ergab sich in der vorangehenden Evaluation, dass Greedy Embeddings hoch-effiziente Kommunikation erlauben indem sie Teilnehmer durch Koordinaten adressieren und diese an die Struktur des Overlays anpassen. Jedoch sind Greedy Embeddings in ihrer ursprünglichen Form nicht auf anonyme Kommunikation mit einer dynamischen Teilnehmermengen und potentiellen Angreifern ausgelegt. Daher präsentieren wir ein Privätssphäre-schützenden Kommunikationsprotokoll für F2F Overlays, in dem die identifizierenden Koordinaten durch anonyme Adressen ersetzt werden. Des weiteren erhöhen wir die Resistenz der Kommunikation durch den Einsatz mehrerer Embeddings und alternativer Algorithmen zum Finden von Routen. Wir beweisen, dass unser Ansatz eine geringe Kommunikationskomplexität im Bezug auf die eigentliche Kommunikation sowie die Instandhaltung des Embeddings aufweist. Ferner zeigt unsere Simulationstudie, dass der Ansatz effiziente Kommunikation mit kurzen Antwortszeiten und geringer Instandhaltungskosten erreicht sowie den Einfluss von Ausfälle und Angriffe erfolgreich abschwächt. Unsere grundlegenden Ergebnisse eröffnen neue Möglichkeiten in der Entwicklung anonymer und zensurresistenter Anwendungen

Dynamics of spectral algorithms for distributed routing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 109-117).In the past few decades distributed systems have evolved from man-made machines to organically changing social, economic and protein networks. This transition has been overwhelming in many ways at once. Dynamic, heterogeneous, irregular topologies have taken the place of static, homogeneous, regular ones. Asynchronous, ad hoc peer-to-peer networks have replaced carefully engineered super-computers, governed by globally synchronized clocks. Modern network scales have demanded distributed data structures in place of traditionally centralized ones. While the core problems of routing remain mostly unchanged, the sweeping changes of the computing environment invoke an altogether new science of algorithmic and analytic techniques. It is these techniques that are the focus of the present work. We address the re-design of routing algorithms in three classical domains: multi-commodity routing, broadcast routing and all-pairs route representation. Beyond their practical value, our results make pleasing contributions to Mathematics and Theoretical Computer Science. We exploit surprising connections to NP-hard approximation, and we introduce new techniques in metric embeddings and spectral graph theory. The distributed computability of "oblivious routes", a core combinatorial property of every graph and a key ingredient in route engineering, opens interesting questions in the natural and experimental sciences as well. Oblivious routes are "universal" communication pathways in networks which are essentially unique. They are magically robust as their quality degrades smoothly and gracefully with changes in topology or blemishes in the computational processes. While we have only recently learned how to find them algorithmically, their power begs the question whether naturally occurring networks from Biology to Sociology to Economics have their own mechanisms of finding and utilizing these pathways. Our discoveries constitute a significant progress towards the design of a self-organizing Internet, whose infrastructure is fueled entirely by its participants on an equal citizen basis. This grand engineering challenge is believed to be a potential technological solution to a long line of pressing social and human rights issues in the digital age. Some prominent examples include non-censorship, fair bandwidth allocation, privacy and ownership of social data, the right to copy information, non-discrimination based on identity, and many others.by Petar Maymounkov.Ph.D

Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, ss-tt 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 Rozho\v{n}-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

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

Bootstrapping Dynamic Distance Oracles

Designing approximate all-pairs distance oracles in the fully dynamic setting is one of the central problems in dynamic graph algorithms. Despite extensive research on this topic, the first result breaking the O(√n) barrier on the update time for any non-trivial approximation was introduced only recently by Forster, Goranci and Henzinger [SODA'21] who achieved m^{1/ρ+o(1)} amortized update time with a O(log n)^{3ρ-2} factor in the approximation ratio, for any parameter ρ ≥ 1. In this paper, we give the first constant-stretch fully dynamic distance oracle with small polynomial update and query time. Prior work required either at least a poly-logarithmic approximation or much larger update time. Our result gives a more fine-grained trade-off between stretch and update time, for instance we can achieve constant stretch of O(1/(ρ²))^{4/ρ} in amortized update time Õ(n^{ρ}), and query time Õ(n^{ρ/8}) for any constant parameter 0 < ρ < 1. Our algorithm is randomized and assumes an oblivious adversary. A core technical idea underlying our construction is to design a black-box reduction from decremental approximate hub-labeling schemes to fully dynamic distance oracles, which may be of independent interest. We then apply this reduction repeatedly to an existing decremental algorithm to bootstrap our fully dynamic solution