The edit distance between two ordered trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worstcase O(n 3 )-time algorithm for this problem, improving the previous best O(n 3 log n)time algorithm [6]. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems (which is interesting in its own right), together with a deeper understanding of the previous algorithms for the problem. We also prove the optimality of our algorithm among the family of decomposition strategy algorithms-which also includes the previous fastest algorithms-by tightening the known lower bound of Ω(n 2 log 2 n) [4] to Ω(n 3 ), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of Θ(nm 2 (1 + log n m )) when the two trees have different sizes m and n, where m < n.

Given a planar graph G on n vertices and an integer parameter r < n, an r-division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O( √ r). We provide a linear-time algorithm for computing r-divisions with few holes. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r-divisions for essentially all values of r. In particular, given an increasing sequence r = (r1, r2, ...), our algorithm can produce a recursive r-division with few holes in linear time.r-divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st-cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs).

The utilization of a d-level partially entangled state, shared by two parties wishing to communicate classical information without errors over a noiseless quantum channel, is discussed. We analytically construct deterministic dense coding schemes for certain classes of non-maximally entangled states, and numerically obtain schemes in the general case. We study the dependency of the information capacity of such schemes on the partially entangled state shared by the two parties. Surprisingly, for d > 2 it is possible to have deterministic dense coding with less than one ebit. In this case the number of alphabet letters that can be communicated by a single particle, is between d and 2d. In general we show that the alphabet size grows in "steps" with the possible values d, d + 1, . . . , d 2 − 2. We also find that states with less entanglement can have greater communication capacity than other more entangled states.

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We present an O(n 1.5 )-space distance oracle for directed planar graphs that answers distance queries in O(log n) time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses O(n 5/3 )-space and answers queries in O(log n) time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs.We further show a smooth tradeoff between space and query-time. For any S ∈ [n, n 2 ], we show an oracle of size S that answers queries inÕ(max{1, n 1.5 /S}) time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of S and improves by polynomial factors over all previously known tradeoffs for the range S ∈ [n, n 5/3 ].