Background: Biological systems are often modular: they can be decomposed into nearlyindependent structural units that perform specific functions. The evolutionary origin of modularity is a subject of much current interest. Recent theory suggests that modularity can be enhanced when the environment changes over time. However, this theory has not yet been tested using biological data.

One of the striking features of evolution is the appearance of novel structures in organisms. Recently, Kirschner and Gerhart have integrated discoveries in evolution, genetics, and developmental biology to form a theory of facilitated variation (FV). The key observation is that organisms are designed such that random genetic changes are channeled in phenotypic directions that are potentially useful. An open question is how FV spontaneously emerges during evolution. Here, we address this by means of computer simulations of two well-studied model systems, logic circuits and RNA secondary structure. We find that evolution of FV is enhanced in environments that change from time to time in a systematic way: the varying environments are made of the same set of subgoals but in different combinations. We find that organisms that evolve under such varying goals not only remember their history but also generalize to future environments, exhibiting high adaptability to novel goals. Rapid adaptation is seen to goals composed of the same subgoals in novel combinations, and to goals where one of the subgoals was never seen in the history of the organism. The mechanisms for such enhanced generation of novelty (generalization) are analyzed, as is the way that organisms store information in their genomes about their past environments. Elements of facilitated variation theory, such as weak regulatory linkage, modularity, and reduced pleiotropy of mutations, evolve spontaneously under these conditions. Thus, environments that change in a systematic, modular fashion seem to promote facilitated variation and allow evolution to generalize to novel conditions.

A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. This paper considers breadth-first search (BFS) spanning trees, and addresses the problem of designing a sparse fault-tolerant BFS tree, or FT-BFS tree for short, namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T of T still contains a BFS spanning tree for (the surviving part of) G. For a source node s, a target node t and an edge e ∈ G, the shortest s − t path P s,t,e that does not go through e is known as a replacement path. Thus, our FT-BFS tree contains the collection of all replacement paths P s,t,e for every t ∈ V (G) and every failed edge e ∈ E(G).Our main results are as follows. We present an algorithm that for every n-vertex graph G and source node s constructs a (single edge failure) FT-BFS tree rooted at s with O(n · min{Depth(s), √ n}) edges, where Depth(s) is the depth of the BFS tree rooted at s. This result is complemented by a matching lower bound, showing that there exist n-vertex graphs with a source node s for which any edge (or vertex) FT-BFS tree rooted at s has Ω(n 3/2 ) edges.We then consider fault-tolerant multi-source BFS trees, or FT-MBFS trees for short, aiming to provide (following a failure) a BFS tree rooted at each source s ∈ S for some subset of sources S ⊆ V . Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every n-vertex graph and source set S ⊆ V of size σ constructs a (single failure) FT-MBFS tree T * (S) from each source s i ∈ S, with O( √ σ · n 3/2 ) edges, and on the other hand there exist n-vertex graphs with source sets S ⊆ V of cardinality σ, on which any FT-MBFS tree from S has Ω( √ σ · n 3/2 ) edges. Finally, we propose an O(log n) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no Ω(log n) approximation algorithm for these problems under standard complexity assumptions. In comparison with the randomized FT-BFS construction implicit in [14], our algorithm is deterministic and may improve the number of edges by a factor of up to √ n for some instances. All our algorithms can be extended to deal with one vertex failure as well, with the same performance.

This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions.Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send O(log n)-bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results.Our techniques give a deterministic maximal independent set (MIS) algorithm in the CONGEST model, where the communication graph is identical to the input graph, in O(D log 2 n) rounds, where D is the diameter of the graph. The best known running time in terms of n alone is 2 O( √ log n) , which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based O(∆+log * n)-round algorithm, where ∆ is the maximal degree in the graph. To the best of our knowledge, ours is the first deterministic MIS algorithm for the CONGEST model, which for polylogarithmic values of D is only a polylogarithmic factor off compared with its randomized counterparts.On the way to obtaining the above, we show that in the Congested Clique model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in O(log ∆ log n) rounds.When ∆ = O(n 1/3 ), the bound improves to O(log ∆) and holds also for (∆ + 1)-coloring.In addition, we deterministically construct a (2k − 1)-spanner with O(kn 1+1/k log n) edges in O(k log n) rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a (2k − 1)-spanner with O(kn 1+1/k ) edges runs in O(n 1−1/k ) rounds.A cornerstone family of problems in distributed computing are the so-called local problems. These include finding a maximal independent set (MIS), a (∆ + 1)-coloring where ∆ is the maximal degree in the network graph, finding a maximal matching, constructing multiplicative spanners, and more. Intuitively, as opposed to global problems, local problems admit solutions that do not require communication over the entire network graph.One fundamental characteristic of distributed algorithms for local problems is whether they are deterministic or randomized. Currently, there exists a curious gap between the known complexities of randomized and deterministic solutions for local problems. Interestingly, the main indistinguishabilitybased technique used for obtaining the relatively few lower bounds that are known seems unsuitable for separating these cases. Building upon an important new lower bound technique of Brandt et al. [BFH + 16], a beautiful recent work of Chang et al. [CKP16] sheds some light over this question, by proving that the randomized complexity of any local problem is at least its deterministic complexity on instances of size √ log n. In addition, they show an exponential separation between th...

We present a randomized algorithm that computes a Minimum Spanning Tree (MST) in O(log * n) rounds, with high probability, in the Congested Clique model of distributed computing. In this model, the input is a graph on n nodes, initially each node knows only its incident edges, and per round each two nodes can exchange O(log n) bits.Our key technical novelty is an O(log * n) Graph Connectivity algorithm, the heart of which is a (recursive) forest growth method, based on a combination of two ideas: a sparsity-sensitive sketching aimed at sparse graphs and a random edge sampling aimed at dense graphs.Our result improves significantly over the O(log log log n) algorithm of Hegeman et al. [PODC 2015] and the O(log log n) algorithm of Lotker et al. [SPAA 2003; SICOMP 2005].