Oblivious Adversary Research Articles

Smoothed Analysis with Adaptive Adversaries

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by \(\tfrac{1}{\sigma }\) times that of the uniform distribution; nature then samples an input from this distribution. Here, σ is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: (1) Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimension d . We bound the regret by \(\tilde{O}(\sqrt {T d\ln (1/\sigma)} + d\ln (T/\sigma))\) and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [ 64 ]. (2) Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ 2 unit ball. We bound the ℓ ∞ norm of the discrepancy vector by \(\tilde{O}(\ln ^2(\frac{nT}{\sigma }))\) . This is as opposed to the worst-case analysis, where the tight discrepancy bound is \(\Theta (\sqrt {T/n})\) . We show such \(\mathrm{polylog}(nT/\sigma)\) discrepancy guarantees are not achievable for non-isotropic σ-smooth distributions. (3) Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where functions with ℓ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is \(({\sigma }/{\sqrt {T\ell }}, \tilde{O}(\sqrt {T\ell }))\) -dispersed. That is, every ball of radius \({\sigma }/{\sqrt {T\ell }}\) is split by \(\tilde{O}(\sqrt {T\ell })\) of the partitions made by these functions. This result matches the dispersion parameters of Balcan et al. [ 13 ] for oblivious smooth adversaries, up to logarithmic factors. On the other hand, worst-case sequences are trivially (0, T )-dispersed. 1

Breaking the 2-competitiveness barrier for two servers in a tree

A randomized on-line algorithm is given for the 2-server problem on a tree, with competitiveness less than 1.94 against the oblivious adversary. This is the first algorithm for this problem with competitive ratio less than 2. The algorithm generalizes earlier work for the line using fractional analysis, and defines a potential in terms of isolation indices from T-theory.

Matching on the Line Admits no \(o(\sqrt {\log n})\) -Competitive Algorithm

We present a simple proof that no randomized online matching algorithm for the line can be \((\sqrt {\log _2(n+1)}/15)\) -competitive against an oblivious adversary for any n = 2 i - 1 : i ∈ ℕ. This is the first super-constant lower bound for the problem, and disproves as a corollary a recent conjecture on the topology-parametrized competitiveness achievable on generic spaces.

Online Markov decision processes with non-oblivious strategic adversary

We study a novel setting in Online Markov Decision Processes (OMDPs) where the loss function is chosen by a non-oblivious strategic adversary who follows a no-external regret algorithm. In this setting, we first demonstrate that MDP-Expert, an existing algorithm that works well with oblivious adversaries can still apply and achieve a policy regret bound of $\mathcal{O}(\sqrt{T \log(L)}+\tau^2\sqrt{ T \log(|A|)})$ where $L$ is the size of adversary's pure strategy set and $|A|$ denotes the size of agent's action space. Considering real-world games where the support size of a NE is small, we further propose a new algorithm: MDP-Online Oracle Expert (MDP-OOE), that achieves a policy regret bound of $\mathcal{O}(\sqrt{T\log(L)}+\tau^2\sqrt{ T k \log(k)})$ where $k$ depends only on the support size of the NE. MDP-OOE leverages the key benefit of Double Oracle in game theory and thus can solve games with prohibitively large action space. Finally, to better understand the learning dynamics of no-regret methods, under the same setting of no-external regret adversary in OMDPs, we introduce an algorithm that achieves last-round convergence result to a NE. To our best knowledge, this is first work leading to the last iteration result in OMDPs.

Online and Approximate Network Construction from Bounded Connectivity Constraints

The Network Construction problem, studied by Angluin et al., Hosoda et al., and others, asks for a minimum-cost network satisfying a set of connectivity constraints which specify subsets of the vertices in the network that have to form connected subgraphs. More formally, given a set [Formula: see text] of vertices, construction costs for all possible edges between pairs of vertices from [Formula: see text], and a sequence [Formula: see text] of connectivity constraints, the objective is to find a set [Formula: see text] of edges such that each [Formula: see text] induces a connected subgraph of the graph [Formula: see text] and the total cost of [Formula: see text] is minimized. First, we study the online version where every constraint must be satisfied immediately after its arrival and edges that have already been added can never be removed. We give an [Formula: see text]-competitive and [Formula: see text]-competitive polynomial-time algorithms, where [Formula: see text] is an upper bound on the size of constraints, while [Formula: see text] denote the number of constraints and the number of vertices, respectively. On the other hand, we observe that an [Formula: see text]-competitive lower bound as well as an [Formula: see text]-competitive lower bound in the cost-uniform case are implied by the known lower bounds for unbounded constraints. For the cost-uniform case with unbounded constraints, we provide an [Formula: see text]-competitive upper bound with high probability. The latter bound is against an oblivious adversary while our other randomized competitive bounds are against an adaptive adversary. Next, we discuss a hybrid approximation method for the (offline) Network Construction problem combining an approximation algorithm of Hosoda et al. with one of Angluin et al. and an application of the hybrid method to bioinformatics. Finally, we consider a natural strengthening of the connectivity requirements in the Network Construction problem, where each constraint has to induce a subgraph (of the constructed graph) of diameter at most [Formula: see text]. Among other things, we provide a polynomial-time [Formula: see text]-approximation algorithm for the Network Construction problem with the [Formula: see text]-diameter requirements, when each constraint has at most [Formula: see text] vertices, and show the APX-completeness of this variant.

Optimal Algorithms for Right-sizing Data Centers

Electricity cost is a dominant and rapidly growing expense in data centers. Unfortunately, much of the consumed energy is wasted, because servers are idle for extended periods of time. We study a capacity management problem that dynamically right-sizes a data center, matching the number of active servers with the varying demand for computing capacity. We resort to a data-center optimization problem introduced by Lin, Wierman, Andrew, and Thereska [ 25 , 27 ] that, over a time horizon, minimizes a combined objective function consisting of operating cost, modeled by a sequence of convex functions, and server switching cost. All prior work addresses a continuous setting in which the number of active servers, at any time, may take a fractional value. In this article, we investigate for the first time the discrete data-center optimization problem where the number of active servers, at any time, must be integer valued. Thereby, we seek truly feasible solutions. First, we show that the offline problem can be solved in polynomial time. Our algorithm relies on a new, yet intuitive graph theoretic model of the optimization problem and performs binary search in a layered graph. Second, we study the online problem and extend the algorithm Lazy Capacity Provisioning (LCP) by Lin et al. [ 25 , 27 ] to the discrete setting. We prove that LCP is 3-competitive. Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than 3. Hence, while LCP does not attain an optimal competitiveness in the continuous setting, it does so in the discrete problem examined here. We prove that the lower bound of 3 also holds in a problem variant with more restricted operating cost functions, introduced by Lin et al. [ 25 ]. In addition, we develop a randomized online algorithm that is 2-competitive against an oblivious adversary. It is based on the algorithm of Bansal et al. [ 7 ] (a deterministic, 2-competitive algorithm for the continuous setting) and uses randomized rounding to obtain an integral solution. Moreover, we prove that 2 is a lower bound for the competitive ratio of randomized online algorithms, so our algorithm is optimal. We prove that the lower bound still holds for the more restricted model. Finally, we address the continuous setting and give a lower bound of 2 on the best competitiveness of online algorithms. This matches an upper bound by Bansal et al. [ 7 ]. A lower bound of 2 was also shown by Antoniadis and Schewior [ 4 ]. We develop an independent proof that extends to the scenario with more restricted operating cost.

Learning Adversarial Markov Decision Processes with Delayed Feedback

Reinforcement learning typically assumes that agents observe feedback for their actions immediately, but in many real-world applications (like recommendation systems) feedback is observed in delay. This paper studies online learning in episodic Markov decision processes (MDPs) with unknown transitions, adversarially changing costs and unrestricted delayed feedback. That is, the costs and trajectory of episode k are revealed to the learner only in the end of episode k+dᵏ, where the delays dᵏ are neither identical nor bounded, and are chosen by an oblivious adversary. We present novel algorithms based on policy optimization that achieve near-optimal high-probability regret of (K+D)¹ᐟ² under full-information feedback, where K is the number of episodes and D=∑ₖ dᵏ is the total delay. Under bandit feedback, we prove similar (K+D)¹ᐟ² regret assuming the costs are stochastic, and (K+D)²ᐟ³ regret in the general case. We are the first to consider regret minimization in the important setting of MDPs with delayed feedback.

Learning-Based Decentralized Offloading Decision Making in an Adversarial Environment

Vehicular fog computing (VFC) pushes the cloud computing capability to the distributed fog nodes at the edge of the Internet, enabling compute-intensive and latency-sensitive computing services for vehicles through task offloading. However, a heterogeneous mobility environment introduces uncertainties in terms of resource supply and demand, which are inevitable bottlenecks for the optimal offloading decision. Also, these uncertainties bring extra challenges to task offloading under the oblivious adversary attack and data privacy risks. In this article, we develop a new adversarial online learning algorithm with bandit feedback based on the adversarial multi-armed bandit theory, to enable scalable and low-complexity offloading decision making. Specifically, we focus on optimizing fog node selection with the aim of minimizing the offloading service costs in terms of delay and energy. The key is to implicitly tune the exploration bonus in the selection process and the assessment rules of the designed algorithm, taking into account volatile resource supply and demand. We theoretically prove that the input-size dependent selection rule allows to choose a suitable fog node without exploring the sub-optimal actions, and also an appropriate score patching rule allows to quickly adapt to evolving circumstances, which reduce variance and bias simultaneously, thereby achieving a better exploitation-exploration balance. Simulation results verify the effectiveness and robustness of the proposed algorithm.

Efficient Multiparty Interactive Coding—Part I: Oblivious Insertions, Deletions and Substitutions

In the field of interactive coding, two or more parties wish to carry out a distributed computation over a communication network that may be noisy. The ultimate goal is to develop efficient coding schemes that can tolerate a high level of noise while increasing the communication by only a constant factor (i.e., constant rate). In this work we consider synchronous communication networks over an arbitrary topology, in the powerful adversarial insertion-deletion noise model. Namely, the noisy channel may adversarially alter the content of any transmitted symbol, as well as completely remove a transmitted symbol or inject a new symbol into the channel. We provide an efficient, constant rate scheme that conducts any computation on any arbitrary network, and succeeds with high probability as long as an oblivious adversary corrupts at most \frac εm fraction of the total communication, where m is the number of links in the network and ε is a small constant. In this work (the first part), our scheme assumes that the parties share a random string to which the adversarial noise is oblivious. While previous work considered the insertion-deletion noise model in the two-party setting, to the best of our knowledge, our scheme is the first multiparty scheme that is resilient to insertions and deletions. Furthermore, our scheme is the first computationally efficient scheme in the multiparty setting that is resilient to adversarial noise.

Randomized two-valued bounded delay online buffer management

In the bounded delay buffer management problem unit size packets arrive online to be sent over a network link. The objective is to maximize the total weight of packets sent before their deadline. In this paper we are interested in the two-valued variant of the problem, where every packet has either low (1) or high priority weight (α>1). We show that the optimal randomized competitive ratio against an oblivious adversary is 1+(α−1)∕(α2+α).

Randomized gathering of asynchronous mobile robots

This paper revisits the widely researched gathering problem for two robots without any agreement on the coordinate system in a scenario which allows randomization in the asynchronous scheduling model. The scheduler is considered to be the adversary which determines the activation schedule of the robots. The adversary comes in three flavors, namely, adaptive offline, adaptive online, and oblivious, based on the knowledge of the outcome of random bits. The robots follow wait-look-compute-move cycle. In this paper, we classify the problems based on the capability of the adversary to control the parameters such as wait time, computation delay and the speed of robots and check the feasibility of gathering in terms of adversarial knowledge and capabilities. First, we show the impossibility of gathering under an adaptive offline adversary with non-negative wait time and non-negative computation delay. Gradually relaxing the adversarial capabilities, we present impossibility of gathering for finite random choices and improbability for infinite random choices for an adaptive online adversary. Under an oblivious adversary, we establish the possibility of gathering with zero computation delay. We improve the runtime of the algorithm by restricting the oblivious adversary to choose wait time and computation delay such that the sum of wait time and computation delay is more than a positive constant. Finally, we also extend our algorithm for multiple robots with merging.

Distributed robust time-efficient broadcasting algorithms for multi-channel wireless multi-hop networks with channel disruption

Broadcasting is a fundamental problem for wireless multi-hop networks such as wireless ad hoc and sensor networks. In this paper, we study distributed robust time-efficient global broadcasting for SINR-based (i.e., Signal-to-Interference-plus-Noise-Ratio-based) multi-channel wireless multihop networks subject to channel disruption. In this study, we make the following assumptions: i) for successful packet reception, the SINR at receivers must exceed a certain threshold; ii) an adversary exists in the network, which uniformly and randomly chooses t > 0 channels to disrupt from total F > 1 available channels in every round. To address the broadcasting issue in such networks, we first propose a degree-aware broadcasting algorithm and then propose a degree-independent broadcasting algorithm. The proposed algorithms elaborately integrate distributed channel selection and transmission scheduling at different nodes while considering the following factors: the impact of SINR constraint, availability of channels, sending probability on selected channel, presence of adversary, with and without node degree information. We present the detailed design of the two algorithms. We deduce their worst-case time performance to accomplish the task of global broadcasting with high probability. We further prove that both algorithms can still preserve the same time performance under more powerful oblivious adversary and weakly adaptive adversary. Simulation results show that the proposed algorithms have satisfactory average-case time performance.

Learning the Truth in Social Networks Using Multi-Armed Bandit

This paper explains how agents in a social network can learn the arbitrary time-varying true state of the network. This is practical in social networks where information is released and updated without any coordination. Most existing literature for learning the true state using the non-Bayesian learning approach, assumes that this true state is fixed, which is impractical. To address this problem, the social network is modeled as a graph network, and the time-varying true state is treated as a multi-armed bandit problem. The few works that have applied multi-armed bandit to a social network did not take into consideration the adversarial effects. Therefore, this paper proposes two non-stochastic multi-armed bandit algorithms that can handle the time-varying true state, even in the presence of an oblivious adversary. Regret bounds on the algorithms are obtained, and the simulation performance shows that all agents can converge to the most stable state. The sublinearity of the proposed algorithms is also compared with two well-known non-stochastic multi-armed bandit algorithms.

Deterministic Dynamic Matching in O(1) Update Time

We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld (in: Proceedings of the ACM symposium on theory of computing (STOC), 2010), this problem has received significant attention in recent years. Very recently, extending the framework of Baswana et al. (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2011) , Solomon (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2016) gave a randomized dynamic algorithm for this problem that has an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm’s past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya et al. (in: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), 2015); it had an approximation ratio of (2+varepsilon ) and an amortized update time of O(log n/varepsilon ^2). Our result can be generalized to give a fully dynamic O(f^3)-approximate algorithm with O(f^2) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices.

Some lower bounds in dynamic networks with oblivious adversaries

This paper considers several closely-related problems in synchronous dynamic networks with oblivious adversaries, and proves novel varOmega (d + text{ poly }(m)) lower bounds on their time complexity (in rounds). Here d is the dynamic diameter of the dynamic network and m is the total number of nodes. Before this work, the only known lower bounds on these problems under oblivious adversaries were the trivial varOmega (d) lower bounds. Our novel lower bounds are hence the first non-trivial lower bounds and also the first lower bounds with a text{ poly }(m) term. Our proof relies on a novel reduction from a certain two-party communication complexity problem. Our central proof technique is unique in the sense that we consider that communication complexity problem with a special leaker. The leaker helps Alice and Bob in the two-party problem, by disclosing to Alice and Bob certain “non-critical” information about the problem instance that they are solving.

Maximum Matching on Trees in the Online Preemptive and the Incremental Graph Models

We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the online preemptive and the incremental graph models. In the Online Preemptive model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms (McGregor, in: Proceedings of the 8th international workshop on approximation, randomization and combinatorial optimization problems, and proceedings of the 9th international conference on randomization and computation: algorithms and techniques, APPROX’05/RANDOM’05, Springer, Berlin, pp. 170–181, 2005; Varadaraja, in: Automata, languages and programming: 38th international colloquium, ICALP 2011, Zurich, Switzerland, proceedings, part I, pp. 379–390, 2011. https://doi.org/10.1007/978-3-642-22006-7_32 ). Epstein et al. (in: 30th international symposium on theoretical aspects of computer science, STACS 2013, Kiel, Germany, pp. 389–399, 2013. https://doi.org/10.4230/LIPIcs.STACS.2013.389 ) gave a 5.356-competitive randomized algorithm for MWM, and also proved a lower bound on the competitive ratio of $$(1+\ln 2) \approx 1.693$$ for MCM. The same lower bound applies for MWM. In the Incremental Graph model, at each step an edge is added to the graph, and the algorithm is supposed to quickly update its current matching. Gupta (in: 34th international conference on foundation of software technology and theoretical computer science, FSTTCS 2014, 15–17 Dec 2014, New Delhi, India, pp. 227–239, 2014. https://doi.org/10.4230/LIPIcs.FSTTCS.2014.227 ) proved that for any $$\epsilon \le 1/2$$ , there exists an algorithm that maintains a $$(1+\epsilon )$$ -approximate MCM for an incremental bipartite graph in an amortized update time of $$O\left( \frac{\log ^2 n}{\epsilon ^4}\right) $$ . No $$(2-\epsilon )$$ -approximation algorithm with a worst case update time of O(1) is known in this model, even for special classes of graphs. In this paper we show that some of the results can be improved for trees, and for some special classes of graphs. In the online preemptive model, we present a 64 / 33-competitive randomized algorithm (which uses only two bits of randomness) for MCM on trees. Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a worst case update time of O(1), in the incremental graph model, which is 3 / 2-approximate (in expectation) on trees, and 1.8-approximate (in expectation) on general graphs with maximum degree 3. Note that this algorithm works only against an oblivious adversary. We derandomize this algorithm, and give a $$(3/2+\epsilon )$$ -approximate deterministic algorithm for MCM on trees, with an amortized update time of $$O(1/\epsilon )$$ . We also present a minor result for MWM in the online preemptive model, a 3-competitive randomized algorithm (that uses only O(1) bits of randomness) on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).

Efficient randomized test-and-set implementations

We study randomized test-and-set (TAS) implementations from registers in the asynchronous shared memory model with n processes. We introduce the problem of group election, a natural variant of leader election, and propose a framework for the implementation of TAS objects from group election objects. We then present two group election algorithms, each yielding an efficient TAS implementation. The first implementation has expected max-step complexity $$O(\log ^*k)$$ in the location-oblivious adversary model, and the second has expected max-step complexity $$O(\log \log k)$$ against any read/write-oblivious adversary, where $$k\le n$$ is the contention. These algorithms improve the previous upper bound by Alistarh and Aspnes (in: Proceedings of the 25th International Symposium on Distributed Computing, 2011) of $$O(\log \log n)$$ expected max-step complexity in the oblivious adversary model. We also propose a modification to a TAS algorithm devised by Alistarh, Attiya, Gilbert, Giurgiu, and Guerraoui (in: Proceedings of the 24th International Symposium on Distributed Computing, DISC 2010) for the strong adaptive adversary, which improves its space complexity from super-linear to linear, while maintaining its $$O(\log n)$$ expected max-step complexity. We then describe how this algorithm can be combined with any randomized TAS algorithm that has expected max-step complexity T(n) in a weaker adversary model, so that the resulting algorithm has $$O(\log n)$$ expected max-step complexity against any strong adaptive adversary and O(T(n)) in the weaker adversary model. Finally, we prove that for any randomized 2-process TAS algorithm, there exists a schedule determined by an oblivious adversary such that with probability at least $$1/4^t$$ one of the processes needs at least t steps to finish its TAS operation. This complements a lower bound by Attiya and Censor-Hillel (SIAM J Comput 39(8):3885–3904, 2010) on a similar problem for $$n\ge 3$$ processes.

A Hybrid RF-VLC System for Energy Efficient Wireless Access

In this paper, we propose a new paradigm in designing and realizing energy efficient wireless indoor access networks, namely, a hybrid system enabled by traditional RF access, such as WiFi, as well as the emerging visible light communication (VLC). VLC facilitates the great advantage of being able to jointly perform illumination and communications, and little extra power beyond illumination is required to empower communications, thus rendering wireless access with almost zero power consumption. On the other hand, when illumination is not required from the light source, the energy consumed by VLC could be more than that consumed by the RF. By capitalizing on the above properties, the proposed hybrid RF-VLC system is more energy efficient and more adaptive to the illumination conditions than the individual VLC or RF systems. To demonstrate the viability of the proposed system, we first formulate the problem of minimizing the power consumption of the hybrid RF-VLC system while satisfying the users requests and maintaining acceptable level of illumination, which is NP-complete. Therefore, we divide the problem into two subproblems. In the first subproblems, we determine the set of VLC access points (AP) that needs to be turned on to satisfy the illumination requirements. Given this set, we turn our attention to satisfying the users’ requests for real-time communications, and we propose a randomized online algorithm that, against an oblivious adversary, achieves a competitive ratio of ${\log (N)\log (M)}$ with probability of success $ {(1-({1}/{N}))}$ , where ${N}$ is the number of users and ${M}$ is the number of VLC and RF APs. We also show that the best online algorithm to solve this problem can achieve a competitive ratio of $ {\log } {M}$ . Simulation results further demonstrate the advantages of the hybrid system.

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 [16] has been the fastest known algorithm for three decades. At the cost of a (1+ϵ)-approximation factor, the running time was recently improved to n 2+ o (1) by Bernstein and Roditty [9]. In this article, we bring the running time down to near-linear: We give a (1+ϵ)-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 ( mn ) time is the mn 0.9 + o (1) -time algorithm by Henzinger et al. [18, 19], 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 , ϵ )- hop set introduced by Cohen [12] in the PRAM literature. An ( h , ϵ)-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+ϵ)-approximated by their h -hop distance (given by a path containing at most h edges) on G ′ =( V , E ∪ F ). Our algorithm can maintain an ( n o (1) , ϵ)-hop set of near-linear size in near-linear time under edge deletions. It is the first of its kind to the best of our knowledge. To maintain approximate distances using this hop set, we extend the monotone Even-Shiloach tree of Henzinger et al. [20] and combine it with the bounded-hop SSSP technique of Bernstein [4, 5] and Mądry [27]. These two new tools might be of independent interest.

Cover time and mixing time of random walks on dynamic graphs

AbstractThe application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.