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A temulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant temulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior. In particular, our main result is that, for (2k1)emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any nnode input graph, we construct a 5emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that Ω(n^{4/3}) edges are necessary even if the 5emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k.more » « less

The best known solutions for kmessage broadcast in dynamic networks of size n require Ω(nk) rounds. In this paper, we see if these bounds can be improved by smoothed analysis. To do so, we study perhaps the most natural randomized algorithm for disseminating tokens in this setting: at every time step, choose a token to broadcast randomly from the set of tokens you know. We show that with even a small amount of smoothing (i.e., one random edge added per round), this natural strategy solves kmessage broadcast in Õ(n+k³) rounds, with high probability, beating the best known bounds for k = o(√n) and matching the Ω(n+k) lower bound for static networks for k = O(n^{1/3}) (ignoring logarithmic factors). In fact, the main result we show is even stronger and more general: given 𝓁smoothing (i.e., 𝓁 random edges added per round), this simple strategy terminates in O(kn^{2/3}log^{1/3}(n)𝓁^{1/3}) rounds. We then prove this analysis close to tight with an almostmatching lower bound. To better understand the impact of smoothing on information spreading, we next turn our attention to static networks, proving a tight bound of Õ(k√n) rounds to solve kmessage broadcast, which is better than what our strategy can achieve in the dynamic setting. This confirms the intuition that although smoothed analysis reduces the difficulties induced by changing graph structures, it does not eliminate them altogether. Finally, we apply tools developed to support our smoothed analysis to prove an optimal result for kmessage broadcast in socalled wellmixed networks in the absence of smoothing. By comparing this result to an existing lower bound for wellmixed networks, we establish a formal separation between oblivious and strongly adaptive adversaries with respect to wellmixed token spreading, partially resolving an open question on the impact of adversary strength on the kmessage broadcast problem.more » « less

The research area of algorithms with predictions has seen recent success showing how to incorporate machine learning into algorithm design to improve performance when the predictions are correct, while retaining worstcase guarantees when they are not. Most previous work has assumed that the algorithm has access to a single predictor. However, in practice, there are many machine learning methods available, often with incomparable generalization guarantees, making it hard to pick a best method a priori. In this work we consider scenarios where multiple predictors are available to the algorithm and the question is how to best utilize them. Ideally, we would like the algorithm's performance to depend on the quality of the {\em best} predictor. However, utilizing more predictions comes with a cost, since we now have to identify which prediction is best. We study the use of multiple predictors for a number of fundamental problems, including matching, load balancing, and nonclairvoyant scheduling, which have been wellstudied in the single predictor setting. For each of these problems we introduce new algorithms that take advantage of multiple predictors, and prove bounds on the resulting performance.more » « less

null (Ed.)Recent work has pinned down the existentially optimal size bounds for vertex faulttolerant spanners: for any positive integer k, every nnode graph has a (2k – 1)spanner on O(f^{1–1/k} n^{1+1/k}) edges resilient to f vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by analyzing the output spanner of a certain exponentialtime greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized algorithm which takes Õ(f^{1–1/k} n^{2+1/k} + mf2) time. We also derandomize our algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential improvement in runtime over [BodwinPatel PODC '19], the only previously known algorithm for constructing optimal vertex faulttolerant spanners.more » « less

A recent line of research investigates how algorithms can be augmented with machinelearned predictions to overcome worst case lower bounds. This area has revealed interesting algorithmic insights into problems, with particular success in the design of competitive online algorithms. However, the question of improving algorithm running times with predictions has largely been unexplored. We take a first step in this direction by combining the idea of machinelearned predictions with the idea of "warmstarting" primaldual algorithms. We consider one of the most important primitives in combinatorial optimization: weighted bipartite matching and its generalization to bmatching. We identify three key challenges when using learned dual variables in a primaldual algorithm. First, predicted duals may be infeasible, so we give an algorithm that efficiently maps predicted infeasible duals to nearby feasible solutions. Second, once the duals are feasible, they may not be optimal, so we show that they can be used to quickly find an optimal solution. Finally, such predictions are useful only if they can be learned, so we show that the problem of learning duals for matching has low sample complexity. We validate our theoretical findings through experiments on both real and synthetic data. As a result we give a rigorous, practical, and empirically effective method to compute bipartite matchings.more » « less