Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random k-colourings on the ∆-ary tree for large k. Bhatnagar et. al. [2] showed non-reconstruction when ∆ ≤ 1 2k log k − o(k log k). We tighten this result and show non-reconstruction when ∆ ≤ k[log k+loglog k+1−ln2− o(1)] which is very close to the known upper bound on the number of colours needed for reconstruction, ∆ ≥ k[log k + log log k + 1 + o(1)]. 1

We introduce a new phylogenetic reconstruction algorithm which, unlike most previous rigorous inference techniques, does not rely on assumptions regarding the branch lengths or the depth of the tree. The algorithm returns a forest which is guaranteed to contain all edges that are: 1) sufficiently long and 2) sufficiently close to the leaves. How much of the true tree is recovered depends on the sequence length provided. The algorithm is distance-based and runs in polynomial time. 1

Phylogenetic reconstruction is the problem of reconstructing an evolutionary tree from sequences corresponding to leaves of that tree. A central goal in phylogenetic reconstruction is to be able to reconstruct the tree as accurately as possible from as short as possible input sequences. The sequence length required for correct topological reconstruction depends on certain properties of the tree, such as its depth and minimal edge-weight. Fast converging reconstruction algorithms are considered state-of the-art in this sense, as they require asymptotically minimal sequence length in order to guarantee (with high probability) correct topological reconstruction of the entire tree. However, when the original phylogenetic tree contains very short edges, this minimal sequence-length is still too long for practical purposes. Short

Consider k-colorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze ” the entire tree when k ≤ ∆ + 1. For k ≥ ∆ + 2, Jonasson proved the root is “unbiased ” for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a non-vanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Non-reconstruction is equivalent to extremality of the Gibbs measure. When k < ∆ / ln ∆, it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for C> 2 and k = C∆ / ln ∆, non-reconstruction holds, i.e., the Gibbs measure is extremal. We prove a strong form of extremality: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. These are the first results coming close to the threshold for extremality for colorings. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.

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Abstract: Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.

Abstract. The reconstruction problem on the tree has been studied in numerous contexts including statistical physics, information theory and computational biology. However, rigorous reconstruction thresholds have only been established in a small number of models. We prove the first exact reconstruction threshold in a non-binary model establishing the Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten-Stigum bound is not tight for the q-state Potts model when q ≥ 5. Moreover, we determine asymptotics for the reconstruction thresholds. 1.

One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than p ∗ = ( √ 2 − 1)/2 3/2, then the tree can be recovered from sequences of length O(log n). The value p ∗ is exactly the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel’s conjecture was proven by the second author in the special case where the tree is “balanced”. The second author also proved that if all edges have mutation probability larger than p ∗ then the length needed is n Ω(1). Here we complete the proof of Steel’s conjecture and give a reconstruction algorithm that requires optimal (up to a multiplicative constant) sequence length. Our results further extend to obtain an optimal reconstruction algorithm for the Jukes-Cantor model with short edges. All reconstruction algorithms run in polynomial time. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.

Abstract. Markov random fields are often used to model high dimensional distributions in a number of applied areas. A number of recent papers have studied the problem of reconstructing a dependency graph of bounded degree from independent samples from the Markov random field. These results require observing samples of the distribution at all nodes of the graph. It was heuristically recognized that the problem of reconstructing the model where there are hidden variables (some of the variables are not observed) is much harder. Here we prove that the problem of reconstructing bounded-degree models with hidden nodes is hard. Specifically, we show that unless NP = RP, – It is impossible to decide in randomized polynomial time if two models generate distributions whose statistical distance is at most 1/3 or at least 2/3. – Given two generating models whose statistical distance is promised to be at least 1/3, and oracle access to independent samples from one of the models, it is impossible to decide in randomized polynomial time which of the two samples is consistent with the model. The second problem remains hard even if the samples are generated efficiently, albeit under a stronger assumption. 1

Abstract. Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following question: is the amount of sequence data required to reconstruct a tree accurately significantly more than the amount required to test whether or not a candidate tree was the ‘true ’ tree? By ‘significantly’, we mean that the two quantities behave the same way as a function of the number of species being considered. We prove that, for a certain model, the amount of information required is not significantly different; while for another model, the information required to test a tree is independent of the number of leaves, while that required to reconstruct it grows with this number. Our results combine probabilistic and combinatorial arguments.