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We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces.Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t log 2 n), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t log O(1) n-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t.The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting. * An extended abstract of this work appeared in ICALP 2016.Proof. The proof is similar to the proof of Lemma 3.Proof. The fact that (u, v) ∈ R implies that at the beginning of Step 4 there must have been a path P = {a 1 = u, a 2 , . . . , a m = v} from u to v such that (a i , a i+1 ) ∈ R for all i ∈ {1, . . . , m − 1}. Since M is a tree quasimetric space, the shortest path is the single unique path from u to v for any u, v ∈ V (T ). From Lemma 2 we have that one of the following three cases is true:Case 1: u is in the shortest path from t to v. We have d M (u, v) ≤ r/2 from Lemma 3. Case 2: v is in the shortest path from u to t. We have d M (u, v) ≤ r/2 from Lemma 4. Case 3: There exists a j that lies on the shortest path from u to t and on the shortest path from t to v. From Lemmas 3 and 4 we have that d M (u, a j ) ≤ r/2 and d M (a j , v) ≤ r/2. By the triangle inequality we get d M (u, v) ≤ r. Lemma 6. Any (u, v) ∈ E(T ) is removed with probability at most 2d M (u, v)/r in Step 4 of the algorithm.Proof. Since M is a tree quasimetric space there are exactly two cases:Let i be the largest integer such that i · r/2 ≤ d M (v, t) . (u, v) is removed from R if z is chosen between d M (v, t) − i · r/2 and d M (u, t) − i · r/2. The probability of that event is bounded by d M (u,t)−i·r/2 d M (v,t)−i·r/2 p(z)dz = 2 r (d M (u, t) − d M (v, t)) ≤ 2d M (u, v)/r by the triangle inequality. Lemma 7. Pr[(u, v) ∈ R] ≤ 2d M (u, v)/r for...

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a b s t r a c tWe show that the Density of States (DoS) for lattice Self Avoiding Walks can be estimated by using an inverse algorithm, called flatIGW, whose step-growth rules are dynamically adjusted by requiring the energy histogram to be locally flat. Here, the (attractive) energy associated with a configuration is taken to be proportional to the number of non-bonded nearest neighbor pairs (contacts). The energy histogram is able to explicitly direct the growth of a walk because the step-growth rule of the Interacting Growth Walk [5]) samples the available nearest neighbor sites according to the number of contacts they would make. We have obtained the complex Fisher zeros corresponding to the DoS, estimated for square lattice walks of various lengths, and located the θ temperature by extrapolating the finite size values of the real zeros to their asymptotic value, ∼1.49 (reasonably close to the known value, ∼1.50 (Barkema et al. (1998) [14]).

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