Smooth heaps and a dual view of self-adjusting data structures

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures. Roughly speaking, we map an arbitrary heap algorithm within a natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature. Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal. Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. As corollaries of results known for Greedy, we obtain instance-specific upper bounds for the smooth heap, with applications in adaptive sorting. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a "power-of-two-choices" type of heuristic.Comment: Presented at STOC 2018, light revision, additional figure

Maximum Scatter TSP in Doubling Metrics

We study the problem of finding a tour of $n$ points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997), motivated by applications in manufacturing and medical imaging. Arkin et al. gave a $0.5$-approximation for the metric version of the problem and showed that this is the best possible ratio achievable in polynomial time (assuming $P \neq NP$). Arkin et al. raised the question of whether a better approximation ratio can be obtained in the Euclidean plane. We answer this question in the affirmative in a more general setting, by giving a $(1-\epsilon)$-approximation algorithm for $d$-dimensional doubling metrics, with running time $\tilde{O}\big(n^3 + 2^{O(K \log K)}\big)$, where $K \leq \left( \frac{13}{\epsilon} \right)^d$. As a corollary we obtain (i) an efficient polynomial-time approximation scheme (EPTAS) for all constant dimensions $d$, (ii) a polynomial-time approximation scheme (PTAS) for dimension $d = \log\log{n}/c$, for a sufficiently large constant $c$, and (iii) a PTAS for constant $d$ and $\epsilon = \Omega(1/\log\log{n})$. Furthermore, we show the dependence on $d$ in our approximation scheme to be essentially optimal, unless Satisfiability can be solved in subexponential time

On sprays and connections

summary:[For the entire collection see Zbl 0699.00032.] \par A connection structure (M,H) and a path structure (M,S) on the manifold M are called compatible, if $S(v)=H(v,v),\forall v\in TM,$ locally G\sp i(x,y)=y\sp j\Gamma\sp i\sb j(x,y), where G\sp i and \Gamma\sp i\sb j express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: G\sp i(x,y)=\Gamma\sp i\sb{jk}(k)y\sp jy\sp k. On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, y\sp j(\Gamma\sp i\sb j\circ \mu\sb t)=ty\sp j\Gamma\sp i\sb j, whence it follows not that H is linear, i.e. if a connection structure is compatible with a spray, then the connection is not necessarily homogeneous. This fact supplements the investigations of {\it H. B. Levine} [Phys. Fluids 3, 225-245 (1960; Zbl 0106.209)], and {\it M. Crampin} [J. Lond. Math. Soc., II. Ser. 3, 178-182 (1971; Zbl 0215.510)]

Geometriák karakterizálása projektív-metrikus terekben

In the dissertation, we present our research in the fields of projective metric geometries, in the course of which we characterize the hyperbolic and Euclidean geometry among Hilbert, respectively Minkowski geometries by geometric configurations. Most mathematicians know Minkowski geometries as normed vector spaces, and a wide literature counts their attributes and properties. The research of Hilbert geometry falls into line in our days, with further various results. Likewise Minkowski geometry is a straight generalisation of the Euclidean geometry, an immediate generalisation of hyperbolic geometry is Hilbert geometry. That is why investigations of properties of configurations well known from Euclidean and hyperbolic geometries is of prime importance in exploring these geometries. Therefore, the general starting point of our dissertation was the question: what kind of properties do have some configurations - deliberately characterized in other geometries - in Minkowski and Hilbert geometries, furthermore, in case of fulfilling certain conditions, what sort of consequences shall we have take into account? Pursuant to this, a survey of the basic features of geometries under investigation is given. Beyond that of Minkowski geometries, a description of the hyperbolic and Hilbert geometries is given, and afterwards the following questions are investigated: - What properties of the particular configurations continue in the general case, and which properties will stop to characterize the configuration? - What kind of consequences implies for the whole geometry if a certain property of a configuration is required to retain? - Are there any configuration and a particular property of it which block the way of generalisation (i.e. the geometry will be Euclidean, respectively, hyperbolic)? In case of all questions worth to bring up the following problem: if the fulfilment of a property is not a general requirement, but holds only for some specific cases, then will it result in the same result? The dissertation draws a picture of investigations which lead, without exception, to characterisation of classic Euclidean, respectively hyperbolic geometries among Minkowski, respectively Hilbert geometries. We start with investigation of some significant configurations in connection with triangles, analogous theorems about which are well known in hyperbolic geometry, as well. A proof, in the Cayley-Klein model, is shown for the hyperbolic version of the Ceva's and Menelaus' theorems (Hyperbolic Menelaus' theorem and Hyperbolic Ceva's theorem) which automatically come true in Minkowski geometries, as well as for the statement that altitudes, respectively orthogonal bisectors of triangles belong to a bundle (Theorem on hyperbolic orthocentre and Theorem on hyperbolic bisectoral centre). As the Birkhoff-orthogonality is not symmetric, concurrency of perpendicular bisectors and that of altitudes should be treated separately in Minkowski and Hibert geometries in the case of Birkhoff-perpendicularity (left-perpendicularity for later use), and in the case of its inverse relation, called right-perpendicularity, or H-perpendicularity, for the sake of distinction. In the course of our research the inscribed (maximal volume) respectively the circumscribed (minimal volume) ellipsoids (called Loewner, repsectively John ellipsoids) of strictly convex bodies play a prominent role. Some basic statements regarding the tangent points of these ellipsoids to the convex bodies are considered. Besides several statements of technical kind, we prove a result about ellipse characterisation, and a theorem of Ceva type about inscribed triangles of ovals, which are interesting on their own, as well. The ellipse characterisation build upon harmonic division, dual of which is equivalent to - by means of Ceva's and Menelaus' theorem - a theorem of Segre. With respect to Hilbert geometries is proven that hiperbolic geometry is characterized by the property that every trigon possesses - the Ceva property; - the Menelaus property; - a bisectoral centre (i.e., perpendicular bisectors belong to a bundle); - an orthocentre {i.e., altitudes belong to a bundle). As regards latter two statements, we have to mention that they apply the inverse of the Birkhoff-perpendicularity, called H-perpendicularity, as we could not achieve any result in the case of Birkhoff-perpendicularity, and we could not find any reference to a result of that kind in the literature. In Minkowski spaces, it is proven equally for the case of the left- and the right-perpendicularity that the Euclidean geometry is characterised by the property that every trigon possesses - the right-bisectoral centre; - the right-orthocentre; - the left-bisectoral centre; - the left-orthocentre..