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Consider the task of performing a sequence of searches in a binary search tree. After each search, an algorithm is allowed to arbitrarily restructure the tree, at a cost proportional to the amount of restructuring performed. The cost of an execution is the sum of the time spent searching and the time spent optimizing those searches with restructuring operations. This notion was introduced by Sleator and Tarjan in (JACM, 1985), along with an algorithm and a conjecture.The algorithm, Splay, is an elegant procedure for performing adjustments while moving searched items to the top of the tree. The conjecture, called dynamic optimality, is that the cost of splaying is always within a constant factor of the optimal algorithm for performing searches. The conjecture stands to this day. In this work, we attempt to lay the foundations for a proof of the dynamic optimality conjecture.Central to our methods are simulation embeddings and approximate monotonicity. A simulation embedding maps each execution to a list of keys that induces a target algorithm to simulate the execution. Approximately monotone algorithms are those whose cost does not increase by more than a constant factor when keys are removed from the list. As we shall see, approximately monotone algorithms with simulation embeddings are dynamically optimal. Building on these ideas:• We construct a simulation embedding for Splay by inducing Splay to perform arbitrary subtree transformations. Thus, if Splay is approximately monotone then it is dynamically optimal. We also show that approximate monotonicity is a necessary condition for dynamic optimality. (Section 2)• We show that if Splay is dynamically optimal, then with respect to optimal cost, its additive overhead is at most linear in the sum of initial tree size and the number of requests. (Section 3)• We prove that a known lower bound on optimal execution cost by Wilber [58] is approximately monotone. (Section 4 and Appendix C)• We speculate about how one might establish dynamic optimality by adapting the proof of approximate monotonicity from the lower bound to Splay. (Section 5)• We demonstrate that two related conjectures, traversal [48] and deque [55], also follow if Splay is approximately monotone, and that many results in this paper extend to a broad class of "path-based" algorithms. (Section 6) Appendix A generalizes the tree transformations used to build simulation embeddings, and Appendix B includes proofs of selected pieces of "folklore" that have appeared throughout the literature.

Any astrophysical object can, in principle, serve as a probe of the interaction between Dark Matter and regular, baryonic matter. This method is based on the potential observable consequences annihilations of captured Dark Matter has on the surface temperature of the object itself. In a series of previous papers we developed and validated simple analytic approximations for the total capture rates of Dark Matter (DM) valid in four distinct regions of the DM-nucleon scattering cross section (σ) vs. DM particle mass (m X ) parameter space. In this work we summarize those previous results and extend them significantly, by deriving a completely general, closed form solution for the total capture rate of Dark Matter in the multiscatter regime. Moreover, we demonstrate the existence of a region in the σ vs. m X parameter space where the constraining power of any astrophysical object heated by annihilations of captured DM is lost. This corresponds to a maximal temperature (T crit ) any astrophysical object can have, such that it can still serve as a DM probe. Any object with observed temperature T obs > T crit loses its DM constraining power. We provide analytic formulae that can be used to estimate T crit for any object.

Let T be a binary search tree of n nodes with root r, left subtree L = left(r), and right subtree R = right(r). The preorder and postorder of T are defined as follows: the preorder and postorder of the empty tree is the empty sequence, andwhere ⊕ denotes sequence concatenation. 1 We prove the following results about the behavior of splaying [21] preorders and postorders: 1. Inserting the nodes of preorder(T ) into an empty tree via splaying costs O(n). (Theorem 2.) 2. Inserting the nodes of postorder(T ) into an empty tree via splaying costs O(n). (Theorem 3.) 3. If T has the same keys as T and T is weight-balanced [18] then splaying either preorder(T ) or postorder(T ) starting from T costs O(n). (Theorem 4.) * Research at Princeton University partially supported by an innovation research grant from Princeton and a gift from Microsoft.