Complexity of union-split-find problems

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 45-46).In this thesis, we investigate various interpretations of the Union-Split-Find problem, an extension of the classic Union-Find problem. In the Union-Split Find problem, we maintain disjoint sets of ordered elements subject to the operations of constructing singleton sets, merging two sets together, splitting a set by partitioning it around a specified value, and finding the set that contains a given element. The different interpretations of this problem arise from the different assumptions made regarding when sets can be merged and any special properties the sets may have. We define and analyze the Interval, Cyclic, Ordered, and General Union-Split-Find problems. Previous work implies optimal solutions to the Interval and Ordered Union-Split-Find problems and an (log n/ log log n) lower bound for the Cyclic Union-Split-Find problem in the cell-probe model. We present a new data structure that achieves a matching upper bound of (log n/ log log n) for Cyclic Union-Split Find in the word RAM model. For General Union-Split-Find, no o(n) bound is known. We present a data structure which has an [Omega](log2 n) amortized lower bound in the worst case that we conjecture has polylogarithmic amortized performance. This thesis is the product of joint work with Erik Demaine.by Katherine Jane Lai.M.Eng

Semi-dynamic connectivity in the plane

Motivated by a path planning problem we consider the following procedure. Assume that we have two points $s$ and $t$ in the plane and take $\mathcal{K}=\emptyset$. At each step we add to $\mathcal{K}$ a compact convex set that does not contain $s$ nor $t$. The procedure terminates when the sets in $\mathcal{K}$ separate $s$ and $t$. We show how to add one set to $\mathcal{K}$ in $O(1+k\alpha(n))$ amortized time plus the time needed to find all sets of $\mathcal{K}$ intersecting the newly added set, where $n$ is the cardinality of $\mathcal{K}$, $k$ is the number of sets in $\mathcal{K}$ intersecting the newly added set, and $\alpha(\cdot)$ is the inverse of the Ackermann function

Fine-Grained Complexity Analysis of Two Classic TSP Variants

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in $O(n^2)$ time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in $O(n \log^2 n)$ time and its bottleneck version in $O(n \log^3 n)$ time. Our second set of results concerns the popular $k$-OPT heuristic for TSP in the graph setting. More precisely, we study the $k$-OPT decision problem, which asks whether a given tour can be improved by a $k$-OPT move that replaces $k$ edges in the tour by $k$ new edges. A simple algorithm solves $k$-OPT in $O(n^k)$ time for fixed $k$. For 2-OPT, this is easily seen to be optimal. For $k=3$ we prove that an algorithm with a runtime of the form $\tilde{O}(n^{3-\epsilon})$ exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. The results for $k=2,3$ may suggest that the actual time complexity of $k$-OPT is $\Theta(n^k)$. We show that this is not the case, by presenting an algorithm that finds the best $k$-move in $O(n^{\lfloor 2k/3 \rfloor + 1})$ time for fixed $k \geq 3$. This implies that 4-OPT can be solved in $O(n^3)$ time, matching the best-known algorithm for 3-OPT. Finally, we show how to beat the quadratic barrier for $k=2$ in two important settings, namely for points in the plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016

Fast Algorithm for Partial Covers in Words

A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $\alpha$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $\alpha$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(n\log n)$ time which we apply to compute all shortest $\alpha$-partial covers for a given $\alpha$. We also employ it for an $O(n\log n)$-time algorithm computing a shortest $\alpha$-partial cover for each $\alpha=1,2,\ldots,n$