CSCE 310: Data Structures

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CSCE 310 Data Structures Algorithms

• Divide and Conquer (III)
• Dr. Ying Lu
• ylu_at_cse.unl.edu

slides referred to http//www.aw-bc.com/info/lev
itin
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Divide and conquer examples

• Closest-pair problem
• Convex-hull problem

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Closest pair problem

• Problem find closest among n points in
  k-dimensional space
• Algorithm Compute distance between each pair of
  points
• The basic operation of the algorithm
• Efficiency

?(n2)
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Closest-pair problem divide and conquer
approach

• Presentation Slides by Tsvika Klein

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Quickhull algorithm

• Presentation slides by João Comba

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QuickHull Algorithm

• Inspired by Quicksort, it compute Convex Hull
• Assume points are sorted by x-coordinate values
• Identify extreme points P1 and P2 (part of hull)
• Compute upper hull (Lower hull computation is
  similar)
• find point Pmax that is farthest away from line
  P1P2
• compute the hull of the points to the left of
  line P1Pmax
• compute the hull of the points to the left of
  line PmaxP2

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Efficiency of QuickHull algorithm

• Finding point farthest away from line P1P2 can be
  done in linear time
• This gives same efficiency as quicksort
• Worst case T( n2)
• Average case T( n log n)
• If points are not initially sorted by
  x-coordinate value, this can be accomplished in
  T( n log n) no increase in asymptotic
  efficiency class
• Other algorithms for convex hull
• Grahams scan
• DCHull
• also in T( n log n)

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In-Class Exercise 1

• A k-way merge operation. Suppose you have k
  sorted arrays, each with n elements, and you want
  to combine them into a single sorted array of kn
  elements.
• A) Heres one strategy Using the merge procedure
  from Section 4.1, merge the first two arrays,
  then merge in the third, then merge in the
  fourth, and so on. What is the time complexity of
  this algorithm, in terms of k and n?
• B) Give a more efficient solution to this
  problem, using divide-and-conquer.

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In-Class Exercise 2

• An array A1 n is said to have a majority
  element if more than half of its entries are the
  same. Given an array, the task is to design an
  efficient algorithm to tell whether the array has
  a majority element, and, if so, to find that
  element. The elements of the array are not
  necessarily from some ordered domain like the
  integers, and so there can be no comparisons of
  the form is Ai gt Aj?. (Think of the array
  elements as GIF files, say.) However you can
  answer questions of the form is Ai Aj?
  in constant time.

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Reading List of the Week

• Chapter 5.1 5.2, 5.3 5.4