Advanced Topics Sorting

Advanced Topics in Sorting

‣ selection
‣ duplicate keys
‣ system sorts
‣ comparators

Reference: http://www.cs.princeton.edu/algs4 Except as otherwise noted, the content of this presentation
is licensed under the Creative Commons Attribution 2.5 License.

Algorithms in Java, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2008 · May 2, 2008 10:45:30 AM

Selection

Goal. Find the kth largest element.
Ex. Min (k = 0), max (k = N-1), median (k = N/2).

Applications.
• Order statistics.
• Find the “top k.”

Use theory as a guide.
• Easy O(N log N) upper bound.
• Easy O(N) upper bound for k = 1, 2, 3.
• Easy Ω(N) lower bound.

Which is true?
• Ω(N log N) lower bound? is selection as hard as sorting?

• O(N) upper bound? is there a linear-time algorithm for all k?

2

Quick-select

Partition array so that:
• Element a[i] is in place.
• No larger element to the left of i.
• No smaller element to the right of i.

Repeat in one subarray, depending on i; finished when i equals k.

public static Comparable select(Comparable[] before int k)
a, v
{
lo hi
StdRandom.shuffle(a);
int lo = 0, hi = a.length - 1;
while (hi > lo) if a[k] is here, if a[k] is here,
{ set hi to i-1 set lo to i+1
int i = partition(a, lo, hi);
if (i < k) lo = i + 1; after v v v
else if (i > k) hi = i - 1;
else return a[k]; lo i hi

}
return a[k];
}

3

Quick-select: mathematical analysis

Proposition. Quick-select takes linear time on average.
Pf sketch.
• Intuitively, each partitioning step roughly splits array in half:
N + N/2 + N/4 + … + 1 ~ 2N compares.
• Formal analysis similar to quicksort analysis yields:

CN = 2 N + k ln ( N / k) + (N - k) ln (N / (N - k))

Ex. (2 + 2 ln 2) N compares to find the median.

Remark. Quick-select might use ~ N2/2 compares, but as with quicksort,
the random shuffle provides a probabilistic guarantee.

4

Theoretical context for selection

Challenge. Design a selection algorithm whose running time is linear in the
worst-case.

Theorem. [Blum, Floyd, Pratt, Rivest, Tarjan, 1973] There exists a compare-
based selection algorithm that takes linear time in the worst case.

Remark. Algorithm is too complicated to be useful in practice.

Use theory as a guide.
• Still worthwhile to seek practical linear-time (worst-case) algorithm.
• Until one is discovered, use quick-select if you don’t need a full sort.

5

Generic methods

In our select() implementation, client needs a cast.

Double[] a = new Double[N];
for (int i = 0; i < N; i++)
a[i] = StdRandom.uniform();
hazardous cast
Double median = (Double) Quick.select(a, N/2);
required

The compiler is also unhappy.

% javac Quick.java
Note: Quick.java uses unchecked or unsafe operations.
Note: Recompile with -Xlint:unchecked for details.

Q. How to fix?
6

Generic methods

Safe version. Compiles cleanly, no cast needed in client.

public class Quick generic type variable
{ (value inferred from argument a[])
public static <Key extends Comparable<Key>> Key select(Key[] a, int k)
{ /* as before */ }

return type matches array type
public static <Key extends Comparable<Key>> void sort(Key[] a)
{ /* as before */ }

private static <Key extends Comparable<Key>> int partition(Key[] a, int lo, int hi)
{ /* as before */ }

private static <Key extends Comparable<Key>> boolean less(Key v, Key w)
{ /* as before */ }

private static <Key extends Comparable<Key>> void exch(Key[] a, int i, int j)
{ Key swap = a[i]; a[i] = a[j]; a[j] = swap; }

} can declare variables of generic type

Remark. Obnoxious code needed in system sort; not in this course (for brevity).
7

‣ selection
‣ duplicate keys
‣ comparators
‣ applications

8

Duplicate keys

Often, purpose of sort is to bring records with duplicate keys together.
• Sort population by age.
• Find collinear points. see Assignment 3

• Remove duplicates from mailing list.
• Sort job applicants by college attended.
sorted by time sorted by city (unstable) sorted by cit
Chicago 09:00:00 Chicago 09:25:52 Chicago 09
Typical characteristics of such applications. Phoenix 09:00:03
Houston 09:00:13
Chicago 09:03:13
Chicago 09:21:05
Chicago 09
Chicago 09

• Huge file.
Chicago 09:00:59
Houston 09:01:10
Chicago 09:19:46
Chicago 09:19:32
Chicago 09
Chicago 09

•
Chicago 09:03:13 Chicago 09:00:00 Chicago 09
Small number of key values. Seattle 09:10:11 Chicago 09:35:21 Chicago 09
Seattle 09:10:25 Chicago 09:00:59 Chicago 09
Phoenix 09:14:25 Houston 09:01:10 Houston 09
Chicago 09:19:32 Houston 09:00:13 NOT Houston 09
Chicago 09:19:46 Phoenix 09:37:44 sorted Phoenix 09
Chicago 09:21:05 Phoenix 09:00:03 Phoenix 09
Seattle 09:22:43 Phoenix 09:14:25 Phoenix 09
Seattle 09:22:54 Seattle 09:10:25 Seattle 09
Chicago 09:25:52 Seattle 09:36:14 Seattle 09
Chicago 09:35:21 Seattle 09:22:43 Seattle 09
Seattle 09:36:14 Seattle 09:10:11 Seattle 09
Phoenix 09:37:44 Seattle 09:22:54 Seattle 09

Stability when sorting on a second key
key

9

Duplicate keys

Mergesort with duplicate keys. Always ~ N lg N compares.

Quicksort with duplicate keys.
• Algorithm goes quadratic unless partitioning stops on equal keys!
• 1990s C user found this defect in qsort().

several textbook and system implementations
also have this defect

10

Duplicate keys: the problem

Assume all keys are equal. Recursive code guarantees this case predominates!

Mistake. Put all keys equal to the partitioning element on one side.
Consequence. ~ N2 / 2 compares when all keys equal.

B A A B A B B B C C C A A A A A A A A A A A

Recommended. Stop scans on keys equal to the partitioning element.
Consequence. ~ N lg N compares when all keys equal.

B A A B A B C C B C B A A A A A A A A A A A

Desirable. Put all keys equal to the partitioning element in place.
A A A B B B B B C C C A A A A A A A A A A A
11

3-way partitioning

Goal. Partition array into 3 parts so that:
• Elements between lt and gt equal to partition element v.
• No larger elements to left of lt.
• No smaller elements to right of gt.

before v

lo hi

after <v =v >v

lo lt gt hi

3-way partitioning

Dutch national flag problem. [Edsger Dijkstra]
• Convention wisdom until mid 1990s: not worth doing.
• New approach discovered when fixing mistake in C library qsort().
• Now incorporated into qsort() and Java system sort.
12

3-way partitioning: Dijkstra's solution

3-way partitioning.
• Let v be partitioning element a[lo].
• Scan i from left to right.
- a[i] less than v : exchange a[lt] with a[i] and increment both lt and i
- a[i] greater than v : exchange a[gt] with a[i] and decrement gt
- a[i] equal to v : increment i

before v

All the right properties. lo hi

•
during <v =v >v
In-place.
•
lt i gt
Not much code. >v
after <v =v
• Small overhead if no equal keys. lo lt gt hi

3-way partitioning

13

3-way partitioning: trace

v a[]
lt i gt 0 1 2 3 4 5 6 7 8 9 10 11
0 0 11 R B W W R W B R R W B R
0 1 11 R B W W R W B R R W B R
1 2 11 B R W W R W B R R W B R
1 2 10 B R R W R W B R R W B W
1 3 10 B R R W R W B R R W B W
1 3 9 B R R B R W B R R W W W
2 4 9 B B R R R W B R R W W W
2 5 7 B B R R R R B R W W W W
2 6 7 B B R R R R B R W W W W
3 7 7 B B B R R R R R W W W W
3 8 7 B B B R R R R R W W W W
3-way partitioning trace (array contents after each loop iteration)

14

3-way quicksort: Java implementation

private static void sort(Comparable[] a, int lo, int hi)
{
if (hi <= lo) return;
int lt = lo, gt = hi;
Comparable v = a[lo];
int i = lo;
while (i <= gt)
before v
{
lo hi
int cmp = a[i].compareTo(v);
during <v =v >v
if (cmp < 0) exch(a, lt++, i++);
lt i gt
else if (cmp > 0) exch(a, i, gt--);
after <v =v >v
else i++;
lo lt gt hi
}
3-way partitioning

sort(a, lo, lt - 1);
sort(a, gt + 1, hi);
}

15

3-way quicksort: visual trace

equal to partitioning element

Visual trace of quicksort with 3-way partitioning

16

Duplicate keys: lower bound

Proposition. [Sedgewick-Bentley, 1997] Quicksort with 3-way partitioning
is entropy-optimal.

Pf. [beyond scope of course]
• Generalize decision tree.
• Tie cost to Shannon entropy.

Ex. Linear-time when only a constant number of distinct keys.

Bottom line. Randomized quicksort with 3-way partitioning reduces running
time from linearithmic to linear in broad class of applications.

17

‣ selection
‣ duplicate keys
‣ comparators
‣ applications

18

Natural order

Comparable interface: sort uses type’s natural order.

public class Date implements Comparable<Date>
{
private final int month, day, year;

public Date(int m, int d, int y)
{
month = m;
day = d;
year = y;
}
…
public int compareTo(Date that)
{
if (this.year < that.year ) return -1;
if (this.year > that.year ) return +1;
if (this.month < that.month) return -1; natural order
if (this.month > that.month) return +1;
if (this.day < that.day ) return -1;
if (this.day > that.day ) return +1;
return 0;
}
}

19

Generalized compare

Comparable interface: sort uses type’s natural order.

Problem 1. May want to use a non-natural order.
Problem 2. Desired data type may not come with a “natural” order.

Ex. Sort strings by:
•
pre-1994 order for digraphs
Natural order. Now is the time ch and ll and rr

• Case insensitive. is Now the time

• Spanish. café cafetero cuarto churro nube ñoño

• British phone book. McKinley Mackintosh

String[] a;
...
Arrays.sort(a);
Arrays.sort(a, String.CASE_INSENSITIVE_ORDER);
Arrays.sort(a, Collator.getInstance(Locale.SPANISH));

import java.text.Collator;
20

Comparators

Solution. Use Java's Comparator interface.

public interface Comparator<Key>
{
public int compare(Key v, Key w);
}

Remark. The compare() method implements a total order like compareTo().

Advantages. Decouples the definition of the data type from the
definition of what it means to compare two objects of that type.
• Can add any number of new orders to a data type.
• Can add an order to a library data type with no natural order.

21

Comparator example

Reverse order. Sort an array of strings in reverse order.

public class ReverseOrder implements Comparator<String>
{
public int compare(String a, String b)
{
return b.compareTo(a);
}
}

comparator implementation

...
Arrays.sort(a, new ReverseOrder());
...

client

22

Sort implementation with comparators

To support comparators in our sort implementations:
• Pass Comparator to sort() and less().
• Use it in less().

type variable
Ex. Insertion sort. (not necessarily Comparable)

public static <Key> void sort(Key[] a, Comparator<Key> comparator)
{
int N = a.length;
for (int i = 0; i < N; i++)
for (int j = i; j > 0; j--)
if (less(comparator, a[j], a[j-1]))
exch(a, j, j-1);
else break;
}

private static <Key> boolean less(Comparator<Key> c, Key v, Key w)
{ return c.compare(v, w) < 0; }

private static <Key> void exch(Key[] a, int i, int j)
{ Key swap = a[i]; a[i] = a[j]; a[j] = swap; }

23

Generalized compare

Comparators enable multiple sorts of a single file (by different keys).

Ex. Sort students by name or by section.

Arrays.sort(students, Student.BY_NAME);
Arrays.sort(students, Student.BY_SECT);

sort by name then sort by section

Andrews 3 A 664-480-0023 097 Little Fox 1 A 884-232-5341 11 Dickinson

Battle 4 C 874-088-1212 121 Whitman Chen 2 A 991-878-4944 308 Blair

Chen 2 A 991-878-4944 308 Blair Andrews 3 A 664-480-0023 097 Little

Fox 1 A 884-232-5341 11 Dickinson Furia 3 A 766-093-9873 101 Brown

Furia 3 A 766-093-9873 101 Brown Kanaga 3 B 898-122-9643 22 Brown

Gazsi 4 B 665-303-0266 22 Brown Rohde 3 A 232-343-5555 343 Forbes

Kanaga 3 B 898-122-9643 22 Brown Battle 4 C 874-088-1212 121 Whitman

Rohde 3 A 232-343-5555 343 Forbes Gazsi 4 B 665-303-0266 22 Brown

24

Generalized compare

Ex. Enable sorting students by name or by section.

public class Student
{
public static final Comparator<Student> BY_NAME = new ByName();
public static final Comparator<Student> BY_SECT = new BySect();

private final String name;
private final int section;
...
private static class ByName implements Comparator<Student>
{
public int compare(Student a, Student b)
{ return a.name.compareTo(b.name); }
}

private static class BySect implements Comparator<Student>
{
public int compare(Student a, Student b)
{ return a.section - b.section; }
}
} only use this trick if no danger of overflow

25

Generalized compare problem

A typical application. First, sort by name; then sort by section.

Arrays.sort(students, Student.BY_NAME); Arrays.sort(students, Student.BY_SECT);

Andrews 3 A 664-480-0023 097 Little Fox 1 A 884-232-5341 11 Dickinson

Battle 4 C 874-088-1212 121 Whitman Chen 2 A 991-878-4944 308 Blair

Chen 2 A 991-878-4944 308 Blair Kanaga 3 B 898-122-9643 22 Brown

Fox 1 A 884-232-5341 11 Dickinson Andrews 3 A 664-480-0023 097 Little

Furia 3 A 766-093-9873 101 Brown Furia 3 A 766-093-9873 101 Brown

Gazsi 4 B 665-303-0266 22 Brown Rohde 3 A 232-343-5555 343 Forbes

Kanaga 3 B 898-122-9643 22 Brown Battle 4 C 874-088-1212 121 Whitman

Rohde 3 A 232-343-5555 343 Forbes Gazsi 4 B 665-303-0266 22 Brown

@#%&@!!. Students in section 3 no longer in order by name.

A stable sort preserves the relative order of records with equal keys.
26

Stability

Q. Which sorts are stable?
• Selection sort?
• Insertion sort?
• Shellsort?
• Quicksort? sorted by time
Chicago 09:00:00
sorted by city (unstable)
Chicago 09:25:52
sorted by city (stable)
Chicago 09:00:00

•
Phoenix 09:00:03 Chicago 09:03:13 Chicago 09:00:59
Mergesort? Houston 09:00:13 Chicago 09:21:05 Chicago 09:03:13
Chicago 09:00:59 Chicago 09:19:46 Chicago 09:19:32
Houston 09:01:10 Chicago 09:19:32 Chicago 09:19:46
Chicago 09:03:13 Chicago 09:00:00 Chicago 09:21:05
Seattle 09:10:11 Chicago 09:35:21 Chicago 09:25:52
Seattle 09:10:25 Chicago 09:00:59 Chicago 09:35:21
Phoenix 09:14:25 Houston 09:01:10 Houston 09:00:13
Chicago 09:19:32 Houston 09:00:13 NOT Houston 09:01:10 sorted
Chicago 09:19:46 Phoenix 09:37:44 sorted Phoenix 09:00:03
Chicago 09:21:05 Phoenix 09:00:03 Phoenix 09:14:25
Seattle 09:22:43 Phoenix 09:14:25 Phoenix 09:37:44
Seattle 09:22:54 Seattle 09:10:25 Seattle 09:10:11
Chicago 09:25:52 Seattle 09:36:14 Seattle 09:10:25
Chicago 09:35:21 Seattle 09:22:43 Seattle 09:22:43
Seattle 09:36:14 Seattle 09:10:11 Seattle 09:22:54
Phoenix 09:37:44 Seattle 09:22:54 Seattle 09:36:14

Stability when sorting on a second key

Open problem. Stable, inplace, N log N, practical sort??
27

‣ selection
‣ duplicate keys
‣ comparators
‣ system sort

28

Sorting applications

Sorting algorithms are essential in a broad variety of applications:
• Sort a list of names.
• Organize an MP3 library.
• Display Google PageRank results. obvious applications

• List RSS news items in reverse chronological order.

• Find the median.
• Find the closest pair.
• Binary search in a database. problems become easy once items
• Identify statistical outliers. are in sorted order

• Find duplicates in a mailing list.

• Data compression.
• Computer graphics.
• Computational biology.
non-obvious applications
• Supply chain management.
• Load balancing on a parallel computer.
...
Every system needs (and has) a system sort!
29

Java system sorts

Java uses both mergesort and quicksort.
• Arrays.sort() sorts array of Comparable or any primitive type.

• Uses quicksort for primitive types; mergesort for objects.

import java.util.Arrays;

public class StringSort
{
public static void main(String[] args)
{
String[] a = StdIn.readAll().split("s+");
Arrays.sort(a);
for (int i = 0; i < N; i++)
StdOut.println(a[i]);
}
}

Q. Why use different algorithms, depending on type?

30

Java system sort for primitive types

Engineering a sort function. [Bentley-McIlroy, 1993]
• Original motivation: improve qsort().
• Basic algorithm = 3-way quicksort with cutoff to insertion sort.
• Partition on Tukey's ninther: median of the medians of 3 samples,
each of 3 elements.
approximate median-of-9

nine evenly
spaced elements R L A P M C G A X Z K R B R J J E

groups of 3 R A M G X K B J E

medians M K E

ninther K

Why use Tukey's ninther?
• Better partitioning than sampling.
• Less costly than random.
31

Achilles heel in Bentley-McIlroy implementation (Java system sort)

Based on all this research, Java’s system sort is solid, right?

more disastrous consequences in C
A killer input.
• Blows function call stack in Java and crashes program.
• Would take quadratic time if it didn’t crash first.

% more 250000.txt % java IntegerSort < 250000.txt
0 Exception in thread "main"
218750 java.lang.StackOverflowError
222662 at java.util.Arrays.sort1(Arrays.java:562)
... ...

250,000 integers Java's sorting library crashes, even if
between 0 and 250,000 you give it as much stack space as Windows allows

32

Achilles heel in Bentley-McIlroy implementation (Java system sort)

McIlroy's devious idea. [A Killer Adversary for Quicksort]
• Construct malicious input while running system quicksort,
in response to elements compared.
• If v is partitioning element, commit to (v < a[i]) and (v < a[j]), but don't
commit to (a[i] < a[j]) or (a[j] > a[i]) until a[i] and a[j] are compared.

Consequences.
• Confirms theoretical possibility.
• Algorithmic complexity attack: you enter linear amount of data;
server performs quadratic amount of work.

Remark. Attack is not effective if file is randomly ordered before sort.

Q. Why do you think system sort is deterministic?
33

System sort: Which algorithm to use?

Many sorting algorithms to choose from:

Internal sorts.
• Insertion sort, selection sort, bubblesort, shaker sort.
• Quicksort, mergesort, heapsort, samplesort, shellsort.
• Solitaire sort, red-black sort, splaysort, Dobosiewicz sort, psort, ...

External sorts. Poly-phase mergesort, cascade-merge, oscillating sort.

Radix sorts. Distribution, MSD, LSD, 3-way radix quicksort.

Parallel sorts.
• Bitonic sort, Batcher even-odd sort.
• Smooth sort, cube sort, column sort.
• GPUsort.

34

System sort: Which algorithm to use?

Applications have diverse attributes.
• Stable?
• Multiple keys?
• Deterministic?
• Keys all distinct?
• Multiple key types?
• Linked list or arrays?
• Large or small records?
• Is your file randomly ordered?
• Need guaranteed performance? many more combinations of
attributes than algorithms

Elementary sort may be method of choice for some combination.
Cannot cover all combinations of attributes.

Q. Is the system sort good enough?
A. Usually.
35

Sorting summary

inplace? stable? worst average best remarks

selection x N2/2 N2/2 N2/2 N exchanges

insertion x x N2/2 N2/4 N use for small N or partially ordered

shell x ? ? N tight code, subquadratic

N log N probabilistic guarantee
quick x N2/2 2 N ln N N lg N
fastest in practice
improves quicksort in presence of
3-way quick x N2/2 2 N ln N N
duplicate keys

merge x N lg N N lg N N lg N N log N guarantee, stable

??? x x N lg N N lg N N lg N holy sorting grail

36

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