Divide%20and%20Conquer%20Applications

1
Divide and ConquerApplications

• Sanghyun Park
• Fall 2002
• CSE, POSTECH

2
Merge Sort

• Sort the first half of the array using merge
  sort.
• Sort the second half of the array using merge
  sort.
• Merge the first half of the array with the second
  half.

3
Merge Algorithm

• Merge is an operation that combines two sorted
  arrays.
• Assume the result is to be placed in a separate
  array called result (already allocated).
• The two given arrays are called front and back.
• front and back are in increasing order.
• For the complexity analysis,the size of the
  input, n, is the sum nfront nback.

4
Merge Algorithm

• For each array keep track of the current
  position.
• REPEAT until all the elements of one of the given
  arrays have been copied into result
• Compare the current elements of front and back.
• Copy the smaller into the current position of
  result(break the ties however you like).
• Increment the current position of result and the
  array that was copied from.
• Copy all the remaining elements of the other
  given array into result.

5
Merge Algorithm - Complexity

• Every element in front and back is copied exactly
  once. Each copy is two accesses,so the total
  number of accessing due to copying is 2n.
• The number of comparisons could beas small as
  min(nfront, nback) or as large as n-1.Each
  comparison is two accesses.

6
Merge Algorithm - Complexity

• In the worst casethe total number of accesses
  is2n 2(n-1) O(n).
• In the best casethe total number of accesses
  is2n 2min(nfront,nback) O(n).
• The average case is between the worst and best
  case and is therefore also O(n).

7
Merge Sort Algorithm

• Split anArray into two non-empty parts anyway you
  like.For example,front the first n/2 elements
  in anArrayback the remaining elements in
  anArray
• Sort front and back by recursively calling
  MergeSort.
• Now you have two sorted arrays containing all the
  elements from the original array.Use merge to
  combine them, put the result in anArray.

8
MergeSort Call Graph (n7)

• Each box represents one invocation of MergeSort.
• How many levels are there in generalif the array
  is divided in half each time?

06
02
36
34
56
12
00
33
44
55
66
11
22
9
MergeSort Call Graph (general)

• Suppose n 2k. How many levels?
• How many boxes on level j?
• What values is in each box at level j?

n
n/2
n/2
n/4
n/4
n/4
n/4
1
1
1
1
1
1
1
1
10
MergeSort Complexity Analysis

• Each invocation of mergesort on p array
  positionsdoes the following
• Copies all p positions once (accesses O(p))
• Calls merge (accesses O(p))
• Observe that p is the same for all invocations at
  the same level, therefore total number of
  accesses at a given level j is O((invocations at
  level j) pj).

11
MergeSort Complexity Analysis

• The total number of accesses at level j
  isO((invocations at level j) pj) O(2j-1
  n/2j-1) O(n)
• In other words,the total number of accesses at
  each level is O(n).
• The total number of accesses for the entire
  mergesort is the sum of the accesses for all the
  levels.(levels) O(n) O(log n) O(n) O(n
  log n)

12
MergeSort Complexity Analysis

• Best case O(n log n)
• Worst case O(n log n)
• Average case O(n log n)

13
Quick Sort

• Quicksort can be seen as a variation of
  mergesortin which front and back are defined in
  a different way.

14
Quicksort Algorithm

• Partition anArray into two non-empty parts.
• Pick any value in the array, pivot.
• small the elements in anArray lt pivot
• large the elements in anArray gt pivot
• Place pivot in either part,so as to make sure
  neither part is empty.
• Sort small and large by recursively calling
  QuickSort.
• You could use merge to combine them, but because
  the elements in small are smaller than elements
  in large, simply concatenate small and large, and
  put the result into anArray.

15
Quicksort Complexity Analysis

• Like mergesort,a single invocation of quicksort
  on an array of size phas complexity O(p)
• p comparisons 2p accesses
• 2p moves (copying) 4p accesses
• Best case every pivot chosen by quicksort
  partitions the array into equal-sized parts. In
  this case quicksort is the same big-O complexity
  as mergesort O(n log n)

16
Quicksort Complexity Analysis

• Worst case the pivot chosen is the largest or
  smallest value in the array. Partition creates
  one part of size 1 (containing only the pivot),
  the other of size p-1.

n
1
n-1
n-2
1
1
1
17
Quicksort Complexity Analysis

• Worst caseThere are n-1 invocations of
  quicksort (not counting base cases) with arrays
  of sizep n, n-1, n-2, , 2Since each of
  these does O(p),the total number of accesses
  isO(n) O(n-1) O(1) O(n2)Ironically
  the worst case occurs when the list is sorted (or
  near sorted)!

18
Quicksort Complexity Analysis

• The average case must be betweenthe best case
  O(n log n) and the worst case is O(n2).
• Analysis yields a complex recurrence relation.
• The average case number of comparisons turns out
  to be approximately 1.386nlog n 2.846n.
• Therefore the average case time complexity isO(n
  log n).

19
Quicksort Complexity Analysis

• Best case O(n log n)
• Worst case O(n2)
• Average case O(n log n)
• Note that the quick sort is inferior to insertion
  sort and merge sort if the list is sorted, nearly
  sorted, or reverse sorted.

20
Closest Pair Of Points

• Given n points in 2D, find the pair that are
  closest.

21
Applications

• We plan to drill holes in a metal sheet.
• If the holes are too close,the sheet will tear
  during drilling.
• Verify that two holes are closer than a threshold
  distance(e.g., holes are at least 1 inch apart.)

22
Air Traffic Control

• 3D --- Locations of airplanes flying in the
  neighborhood of a busy airport are known.
• We want to be sure that no two planes get closer
  than a given threshold distance.

23
Simple Solution

• For each of the n(n-1)/2 pairs of
  points,determine the distance between the points
  in the pair.
• Determine the pair with the minimum distance.
• O(n2) time

24
Divide and Conquer Solution

• When n is small, use simple solution.
• When n is large,
• Divide the point set into two roughly equal parts
  A and B.
• Determine the closest pair of points in A.
• Determine the closest pair of points in B.
• Determine the closest pair of points such that
  one point is in A and the other in B.
• From the three closest pairs computed,select the
  one with least distance.

25
Example

• Divide so thatpoints in A have x coordinate lt
  that of points in B.

26
Example

• Find the closest pair in A.
• Let d1 be the distance between the points in this
  pair.

27
Example

• Find the closest pair in B.
• Let d2 be the distance between the points in this
  pair.

28
Example

• Let d mind1,d2.
• Is there a pair with one point in A and the other
  in B,and their distance lt d?

29
Example

• Candidates lie within d of the dividing line.
• Call these regions RA and RB, respectively.

30
Example

• Let q be a point in RA.
• q need to be paired only with those points in
  RBthat are within d of q.y.

31
Example

• Points that are to be paired with q are in d x 2d
  rectangle of RB (comparing region of q).
• Points in this rectangle are at least d
  apart.(What is the maximum number of points in
  this rectangle?)

32
Example

• So the comparing region of q has at most 6
  points.
• So number of pairs to check is lt 6 RA O(n).

33
Time Complexity

• Create a sorted list of points (by
  x-coordinate)O(n log n) time
• Create a sorted list of points (by
  y-coordinate)O(n log n) time
• Using these two lists, the required pairs of
  points from RA and RB can be constructed in O(n)
  time.
• Let n lt 4 define a small instance.

34
Time Complexity

• Let t(n) be the time to find the closest
  pairexcluding the time to create the two sorted
  lists.
• t(n) c, n lt 4, where c is a constant.
• when n gt 4,t(n) t(ceil(n/2)) t(floor(n/2))
  dn,where d is a constant.
• To solve the recurrence,assume n is a power of 2
  and use repeated substitution.
• t(n) O(n log n)

35
Closest Pair Of Points

• Given n points in 2D, find the pair that are
  closest.
• C routines are given on pp. 695-699 of the
  text.
• The complete program is also available
  athttp//www.mhhe.com/engcs/compsci/sahni/source.
  mhtml.