Dale Roberts Mergesort Dale Roberts, Lecturer Computer Science, IUPUI E-mail: [email protected] Department of Computer and Information Science, School.

Dale Roberts

MergesortMergesort

Dale Roberts, LecturerComputer Science, IUPUIE-mail: [email protected]

Department of Computer and Information Science,School of Science, IUPUI

Dale Roberts

Why Does It Matter?Why Does It Matter?

1000

Time tosolve a

problemof size

10,000

100,000

million

10 million

1.3 seconds

22 minutes

15 days

41 years

41 millennia

920

3,600

14,000

41,000

1,000

Run time(nanoseconds) 1.3 N3

secondMax sizeproblemsolvedin one

minute

hour

day

10 msec

1 second

1.7 minutes

2.8 hours

1.7 weeks

10,000

77,000

600,000

2.9 million

100

10 N2

0.4 msec

6 msec

78 msec

0.94 seconds

11 seconds

1 million

49 million

2.4 trillion

50 trillion

10+

47 N log2N

0.048 msec

0.48 msec

4.8 msec

48 msec

0.48 seconds

21 million

1.3 billion

76 trillion

1,800 trillion

10

48 N

N multiplied by 10,time multiplied by

Dale Roberts

Orders of MagnitudeOrders of Magnitude

10-10

Meters PerSecond

10-8

10-6

10-4

10-2

1

102

1.2 in / decade

ImperialUnits

1 ft / year

3.4 in / day

1.2 ft / hour

2 ft / minute

2.2 mi / hour

220 mi / hour

Continental drift

Example

Hair growing

Glacier

Gastro-intestinal tract

Ant

Human walk

Propeller airplane

104

106

108

370 mi / min

620 mi / sec

62,000 mi / sec

Space shuttle

Earth in galactic orbit

1/3 speed of light

1

Seconds

102

103

104

105

106

107

108

109

1010

1 second

Equivalent

1.7 minutes

17 minutes

2.8 hours

1.1 days

1.6 weeks

3.8 months

3.1 years

3.1 decades

3.1 centuries

forever

1021 age ofuniverse

210 thousand

220 million

230 billion

. . .

10 10 seconds

Powersof 2

Dale Roberts

Impact of Better AlgorithmsImpact of Better Algorithms

Example 1: N-body-simulation.Example 1: N-body-simulation.Simulate gravitational interactions among N bodies.Simulate gravitational interactions among N bodies.

physicists want N = # atoms in universephysicists want N = # atoms in universe

Brute force method: NBrute force method: N22 steps. steps.Appel (1981). Appel (1981). N log N steps, enables new research. N log N steps, enables new research.

Example 2: Discrete Fourier Transform (DFT).Example 2: Discrete Fourier Transform (DFT).Breaks down waveforms (sound) into periodic components.Breaks down waveforms (sound) into periodic components.

foundation of signal processingfoundation of signal processingCD players, JPEG, analyzing astronomical data, etc.CD players, JPEG, analyzing astronomical data, etc.

Grade school method: NGrade school method: N22 steps. steps.Runge-König (1924), Cooley-Tukey (1965).Runge-König (1924), Cooley-Tukey (1965).FFT algorithm: N log N steps, enables new technology.FFT algorithm: N log N steps, enables new technology.

Dale Roberts

MergesortMergesort

Mergesort (divide-and-conquer)Mergesort (divide-and-conquer)Divide array into two halves.Divide array into two halves.

A L G O R I T H M S

divideA L G O R I T H M S

Dale Roberts

MergesortMergesort


Recursively sort each half.Recursively sort each half.

sort

A L G O R I T H M S


A G L O R H I M S T

Dale Roberts

MergesortMergesort


Recursively sort each half.Recursively sort each half.

Merge two halves to make sorted whole.Merge two halves to make sorted whole.

merge

sort

A L G O R I T H M S


A G L O R H I M S T

A G H I L M O R S T

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

MergingMerging

Merge.Merge.Keep track of smallest element in each sorted half.Keep track of smallest element in each sorted half.

Insert smallest of two elements into auxiliary array.Insert smallest of two elements into auxiliary array.

Repeat until done.Repeat until done.

A

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A

MergingMerging




G

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A G

MergingMerging




H

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A G H

MergingMerging




I

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I

MergingMerging




L

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L

MergingMerging




M

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L M

MergingMerging




O

Dale Roberts

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L M O

MergingMerging




R

Dale Roberts

auxiliary array

first halfexhausted smallest

A G L O R H I M S T

A G H I L M O R

MergingMerging




S

Dale Roberts

auxiliary array

first halfexhausted smallest

A G L O R H I M S T

A G H I L M O R S

MergingMerging




T

Dale Roberts

Implementing MergesortImplementing Mergesort

Item aux[MAXN];

void mergesort(Item a[], int left, int right) { int mid = (right + left) / 2; if (right <= left) return; mergesort(a, left, mid); mergesort(a, mid + 1, right); merge(a, left, mid, right);}

mergesort (see Sedgewick Program 8.3)

uses scratch array

Dale Roberts

Implementing Merge (Idea 0)Implementing Merge (Idea 0)mergeAB(Item c[], Item a[], int N, Item b[], int M )mergeAB(Item c[], Item a[], int N, Item b[], int M ) { int i, j, k;{ int i, j, k; for (i = 0, j = 0, k = 0; k < N+M; k++)for (i = 0, j = 0, k = 0; k < N+M; k++) {{ if (i == N) { c[k] = b[j++]; continue; }if (i == N) { c[k] = b[j++]; continue; } if (j == M) { c[k] = a[i++]; continue; }if (j == M) { c[k] = a[i++]; continue; } c[k] = (less(a[i], b[j])) ? a[i++] : b[j++];c[k] = (less(a[i], b[j])) ? a[i++] : b[j++]; }} }}

Dale Roberts

Implementing MergesortImplementing Mergesort

void merge(Item a[], int left, int mid, int right) { int i, j, k; for (i = mid+1; i > left; i--) aux[i-1] = a[i-1]; for (j = mid; j < right; j++) aux[right+mid-j] = a[j+1];

for (k = left; k <= right; k++) if (ITEMless(aux[i], aux[j])) a[k] = aux[i++]; else a[k] = aux[j--];}

merge (see Sedgewick Program 8.2)

copy to temporary array

merge two sorted sequences

Dale Roberts

Mergesort DemoMergesort Demo

Mergesort Mergesort The auxilliary array used in the merging The auxilliary array used in the merging operation is shown to the right of the array a[], going operation is shown to the right of the array a[], going from (N+1, 1) to (2N, 2N). from (N+1, 1) to (2N, 2N).

The demo is a dynamic representation of the algorithm The demo is a dynamic representation of the algorithm in action, sorting an array a containing a permutation of in action, sorting an array a containing a permutation of the integers 1 through N. For each i, the array element the integers 1 through N. For each i, the array element a[i] is depicted as a black dot plotted at position (i, a[i]). a[i] is depicted as a black dot plotted at position (i, a[i]). Thus, the end result of each sort is a diagonal of black Thus, the end result of each sort is a diagonal of black dots going from (1, 1) at the bottom left to (N, N) at the dots going from (1, 1) at the bottom left to (N, N) at the top right. Each time an element is moved, a green dot is top right. Each time an element is moved, a green dot is left at its old position. Thus the moving black dots give a left at its old position. Thus the moving black dots give a dynamic representation of the progress of the sort and dynamic representation of the progress of the sort and the green dots give a history of the data-movement cost. the green dots give a history of the data-movement cost.

Dale Roberts Mergesort Dale Roberts, Lecturer Computer Science, IUPUI E-mail: [email protected] Department of Computer and Information Science, School.

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