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Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction

Shion Fukuzawa, Michael T. Goodrich, Sandy Irani

TL;DR

A quantum algorithm design paradigm called combine and conquer is introduced, which is a quantum version of the"marriage-before-conquest"technique of Kirkpatrick and Seidel, which is better suited for the quantum setting, due to its non-recursive nature.

Abstract

We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in $\tilde{O}(\sqrt{nh})$ time, w.h.p., where $h$ is the size of the output.

Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction

TL;DR

A quantum algorithm design paradigm called combine and conquer is introduced, which is a quantum version of the"marriage-before-conquest"technique of Kirkpatrick and Seidel, which is better suited for the quantum setting, due to its non-recursive nature.

Abstract

We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in time, w.h.p., where is the size of the output.

Paper Structure

This paper contains 9 sections, 9 theorems, 7 equations, 7 figures, 1 table, 10 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Maxima Sets
  4. Quantum Convex Hulls
  5. Discussion
  6. A Review of a Classical Maxima Set Algorithm
  7. A Review of a Classical Convex Hull Algorithm
  8. Quantum Jarvis March Convex Hull Algorithms
  9. A Lower bound for Quantum Convex Hulls

Key Result

Theorem 1

Let $D = [d_0, \ldots, d_{n-1}]$ be a list of $n$ elements represented by $w$ bits, and let $S$ be the time required to prepare the state, and $Q$ the time it takes to query a boolean function $f: D \rightarrow \{0, 1\}$. Let $M$ be the subset of $D$ such that $f(m) = 1$ for all $m \in M$. Also, let $\preceq$ be some ordering of data values in the data register such that comparisons according to

Figures (7)

  • Figure 1: A two-dimensional convex hull.
  • Figure 2: An example instance of maxima set where $n = 32$ and $h = 8$ solved using our quantum combine-and-conquer algorithm. Consecutive blocks of points are colored in alternating purple and green, and the set $T$ of tallest points in each group is circled. The maxima set of $T$ is indicated using solid circles, whereas tallest points that are dominated are circled in dotted lines. To illustrate an example of the conquer step, we focus attention on the group $S_2$ which is highlighted in orange. In this group, the tallest $y$ coordinate is indicated by $T_2$, and $R_2$ is the $y$ coordinate of the tallest point to the right of this group. Thus, the only points that need to be processed are the two points not bound by any blue box. The points that are found in this local check are circled in green.
  • Figure 3: A demonstration of Algorithm \ref{['alg:complete-maxima-set']} where we iteratively search for the maxima point within block $j$ given $T_j$ and $R_j$. The left shows the state of the first iteration, and all points shown are the contents of block $j$ except for the one labeled $R_j$. Points are green if they are not dominated by $T_j$ or $R_j$ and thus return 1 to the Boolean function $f$. Among these points, we search for the lexicographic maximum point $q$ and set this to be the new right boundary for the next iteration. The first $q$ to be discovered in the above instance is marked in blue. This is repeated until there are no remaining green points.
  • Figure 4: An example of our restricted Jarvis March convex hull algorithm. The purple lines are edges that we know are on the convex hull, and the points are the content of some block $S_j$. In a Jarvis March algorithm, we start our search from $p_c$ which is known to be on the convex hull, then search for the point that forms the maximum angle with the incoming edge containing $p_c$. In the above example, the edges we check are in blue, and the solid line denotes the edge that is added to the convex hull. We repeat the search again starting at $p_r$, until we connect to $p_f$.
  • Figure 5: An upper hull with $n = 36$ and $h = 6$. Consecutive blocks of points are colored in alternating purple and green. The bridge edges discovered are shown in purple, and the dotted and dashed lines starting from block $S_2$ indicate how the algorithm handles left turns. The bridge edge between $S_4$ and $S_5$ forms a left turn relative to the previous dotted bridge edge, so both are popped and a new bridge edge is found between $S_3$ and $S_5$. This is repeated until no left turns are formed, giving the solid purple line. Finally, the bridge edges do not close the hull in some blocks, which is where we run the quantum Jarvis march to find the edges of the upper hull shown in red.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1: Quantum Maximum/Minimum Finding durr1999
  • Theorem 2
  • Lemma 3: see, e.g., debergseidelorourke
  • Lemma 4: Sadakane et al. sadakane2002quantum
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Lemma 8: Lanzagorta and Uhlmann lanzagorta2004quantum
  • Theorem 9