We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverse-Ackermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algo-rithms are based on an efficient split-findmin data structure, which main-tains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our split-findmin algorithm is the first with superlinear but sub-inverse-Ackermann complexity. We also give a reduction from MST sensitivity to the MST problem it-self. Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensi-tivity algorithm.

In the path minima problem on trees each tree edge is assigned a weight and a query asks for the edge with minimum weight on a path between two nodes. For the dynamic version of the problem on a tree, where the edge-weights can be updated, we give comparison-based and RAM data structures that achieve optimal query time. These structures support inserting a node on an edge, inserting a leaf, and contracting edges. When only insertion and deletion of leaves in a tree are needed, we give two data structures that achieve optimal and significantly lower query times than when updating the edge-weights is allowed. One is a semigroup structure for which the edge-weights are from an arbitrary semigroup and queries ask for the semigroup-sum of the edge-weights on a given path. For the other structure the edge-weights are given in the word RAM. We complement these upper bounds with lower bounds for different variants of the problem.

One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of Davenport-Schinzel sequences, where an order-s DS sequence is defined to be one over an n-letter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · with length s + 2. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since become an indispensable tool in computational geometry and the analysis of discrete geometric structures. Let λs(n) be the extremal function for such sequences. What is λs asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch) when s is even or s ≤ 3. However, since the work of Agarwal, Sharir, and Shor in the 1980s there has been a persistent gap in our understanding of the odd orders, a gap that is just as much qualitative as quantitative. In this paper we establish the following bounds on λs(n) for every order s. n s = 1 2n − 1 s = 2 ⎪ ⎨ 2nα(n) + O(n) s = 3 λs(n) = Θ(n2 α(n) ) s = 4 Θ(nα(n)2 α(n) ) s = 5 n2 (1+o(1))αt (n)/t! s ≥ 6, t = ⌊ s−2 2 ⌋ These results refute a conjecture of Alon, Kaplan, Nivasch, Sharir, and Smorodinsky and run counter to common sense. When s is odd, λs behaves essentially like λs−1.

In the path minimum query problem, we preprocess a tree on n weighted nodes, such that given an arbitrary path, we can locate the node with the smallest weight along this path. We design novel succinct indices for this problem; one of our index structures supports queries in O((m;n)) time, and occupies O(m) bits of space in addition to the space required for the input tree, where m is an integer greater than or equal to n and (m;n) is the inverse-Ackermann function. These indices give us the first succinct data structures for the path minimum problem, and allow us to obtain new data structures for path reporting queries, which report the nodes along a query path whose weights are within a query range. We achieve three different time/space tradeoffs for path reporting by designing (a) an O(n)-word structure with O(lgϵ n+ occ lgϵ n) query time, where occ is the number of nodes reported; (b) an O(n lg lgn)-word structure with O(lg lg n+ occ lg lgn) query time; and (c) an O(n lgϵ n)-word structure with O(lg lg n+occ) query time. These tradeoffs match the state of the art of two-dimensional orthogonal range reporting queries [8] which can be treated as a special case of path reporting queries. When the number of distinct weights is much smaller than n, we further improve both the query time and the space cost of these three results.

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(σ, n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to σ. It has been proved that for every σ, Ex(σ, n) is either linear or very close to linear; in particular it is O(n2 α(n)O(1)), where α is the inverse-Ackermann function and O(1) depends on σ. However, very little is known about the properties of σ that induce superlinearity of Ex(σ, n). In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 prototypical superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences.