Class Partitioning
The algorithms partition arrays into two or more intervals. They distinguish between synchronously partitioning either one, two or three arrays. They further come in templated versions, either partitioning int[] arrays or double[] arrays.
You may want to start out reading about the simplest case: Partitioning one int[] array into two intervals.
To do so, read partition(int[],int,int,int)
.
Next, building upon that foundation comes a method partitioning int[] arrays into multiple intervals.
See partition(int[],int,int,int[],int,int,int[])
for related documentation.
All other methods are no different than the one's you now already understand, except that they operate on slightly different data types.
Performance
Partitioning into two intervals is O( N ). Partitioning into k intervals is O( N * log(k)). Constants factors are minimized. No temporary memory is allocated; Partitioning is in-place.
- Version:
- 1.0, 03-Jul-99
- See Also:
-
Field Summary
FieldsModifier and TypeFieldDescriptionprivate static final int
private static final int
protected static int
static int
-
Constructor Summary
ConstructorsModifierConstructorDescriptionprotected
Makes this class non instantiable, but still let's others inherit from it. -
Method Summary
Modifier and TypeMethodDescriptionprivate static int
binarySearchFromTo
(int a, int from, int to, IntComparator comp) Finds the given key "a" within some generic data using the binary search algorithm.static int
dualPartition
(double[] list, double[] secondary, int from, int to, double splitter) Same asdualPartition(int[],int[],int,int,int)
except that it synchronously partitions double[] rather than int[] arrays.static void
dualPartition
(double[] list, double[] secondary, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same asdualPartition(int[],int[],int,int,int[],int,int,int[])
except that it synchronously partitions double[] rather than int[] arrays.static int
dualPartition
(int[] list, int[] secondary, int from, int to, int splitter) Same aspartition(int[],int,int,int)
except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array.static void
dualPartition
(int[] list, int[] secondary, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same aspartition(int[],int,int,int[],int,int,int[])
except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array.static void
genericPartition
(int from, int to, int splitFrom, int splitTo, int[] splitIndexes, IntComparator comp, IntComparator comp2, IntComparator comp3, Swapper swapper) Same aspartition(int[],int,int,int[],int,int,int[])
except that it generically partitions arbitrary shaped data (for example matrices or multiple arrays) rather than int[] arrays.private static int
genericPartition
(int from, int to, int splitter, IntComparator comp, Swapper swapper) Same aspartition(int[],int,int,int)
except that it generically partitions arbitrary shaped data (for example matrices or multiple arrays) rather than int[] arrays.private static int
med3
(double[] x, int a, int b, int c) Returns the index of the median of the three indexed elements.private static int
med3
(int[] x, int a, int b, int c) Returns the index of the median of the three indexed elements.private static int
med3
(int a, int b, int c, IntComparator comp) Returns the index of the median of the three indexed chars.private static int
med3
(Object[] x, int a, int b, int c, Comparator comp) Returns the index of the median of the three indexed chars.static int
partition
(double[] list, int from, int to, double splitter) Same aspartition(int[],int,int,int)
except that it partitions double[] rather than int[] arrays.static void
partition
(double[] list, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same aspartition(int[],int,int,int[],int,int,int[])
except that it partitions double[] rather than int[] arrays.static int
partition
(int[] list, int from, int to, int splitter) Partitions (partially sorts) the given list such that all elements falling into the given interval are placed next to each other.static void
partition
(int[] list, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Partitions (partially sorts) the given list such that all elements falling into some intervals are placed next to each other.static void
partition
(DoubleArrayList list, int from, int to, DoubleArrayList splitters, IntArrayList splitIndexes) Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()).static void
partition
(IntArrayList list, int from, int to, IntArrayList splitters, IntArrayList splitIndexes) Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()).static void
partition
(Object[] list, int from, int to, Object[] splitters, int splitFrom, int splitTo, int[] splitIndexes, Comparator comp) Same aspartition(int[],int,int,int[],int,int,int[])
except that it partitions Object[] rather than int[] arrays.static int
partition
(Object[] list, int from, int to, Object splitter, Comparator comp) Same aspartition(int[],int,int,int)
except that it synchronously partitions the objects of the given list by the order of the given comparator.static int
triplePartition
(double[] list, double[] secondary, double[] tertiary, int from, int to, double splitter) Same astriplePartition(int[],int[],int[],int,int,int)
except that it synchronously partitions double[] rather than int[] arrays.static void
triplePartition
(double[] list, double[] secondary, double[] tertiary, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same astriplePartition(int[],int[],int[],int,int,int[],int,int,int[])
except that it synchronously partitions double[] rather than int[] arrays.static int
triplePartition
(int[] list, int[] secondary, int[] tertiary, int from, int to, int splitter) Same aspartition(int[],int,int,int)
except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array.static void
triplePartition
(int[] list, int[] secondary, int[] tertiary, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same aspartition(int[],int,int,int[],int,int,int[])
except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array.
-
Field Details
-
SMALL
private static final int SMALL- See Also:
-
MEDIUM
private static final int MEDIUM- See Also:
-
steps
protected static int steps -
swappedElements
public static int swappedElements
-
-
Constructor Details
-
Partitioning
protected Partitioning()Makes this class non instantiable, but still let's others inherit from it.
-
-
Method Details
-
binarySearchFromTo
Finds the given key "a" within some generic data using the binary search algorithm.- Parameters:
a
- the index of the key to search for.from
- the leftmost search position, inclusive.to
- the rightmost search position, inclusive.comp
- the comparator determining the order of the generic data. Takes as first argument the index a within the generic splitters s. Takes as second argument the index b within the generic data g.- Returns:
- index of the search key, if it is contained in the list; otherwise, (-(insertion point) - 1). The insertion point is defined as the the point at which the value would be inserted into the list: the index of the first element greater than the key, or list.length, if all elements in the list are less than the specified key. Note that this guarantees that the return value will be >= 0 if and only if the key is found.
-
dualPartition
public static void dualPartition(double[] list, double[] secondary, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same asdualPartition(int[],int[],int,int,int[],int,int,int[])
except that it synchronously partitions double[] rather than int[] arrays. -
dualPartition
public static int dualPartition(double[] list, double[] secondary, int from, int to, double splitter) Same asdualPartition(int[],int[],int,int,int)
except that it synchronously partitions double[] rather than int[] arrays. -
dualPartition
public static void dualPartition(int[] list, int[] secondary, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same aspartition(int[],int,int,int[],int,int,int[])
except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array is also moved from index A to B.Use cases:
Image having a large list of 2-dimensional points. If memory consumption and performance matter, it is a good idea to physically lay them out as two 1-dimensional arrays (using something like Point2D objects would be prohibitively expensive, both in terms of time and space). Now imagine wanting to histogram the points. We may want to partially sort the points by x-coordinate into intervals. This method efficiently does the job.
Performance:
Same as for single-partition methods.
-
dualPartition
public static int dualPartition(int[] list, int[] secondary, int from, int to, int splitter) Same aspartition(int[],int,int,int)
except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array is also moved from index A to B.Performance:
Same as for single-partition methods.
-
genericPartition
public static void genericPartition(int from, int to, int splitFrom, int splitTo, int[] splitIndexes, IntComparator comp, IntComparator comp2, IntComparator comp3, Swapper swapper) Same aspartition(int[],int,int,int[],int,int,int[])
except that it generically partitions arbitrary shaped data (for example matrices or multiple arrays) rather than int[] arrays.This method operates on arbitrary shaped data and arbitrary shaped splitters. In fact, it has no idea what kind of data by what kind of splitters it is partitioning. Comparisons and swapping are delegated to user provided objects which know their data and can do the job.
Lets call the generic data g (it may be a matrix, one array, three linked lists or whatever). Lets call the generic splitters s. This class takes a user comparison function operating on two indexes (a,b), namely an
IntComparator
. The comparison function determines whether s[a] is equal, less or greater than g[b]. This method can then decide to swap the data g[b] with the data g[c] (yes, c, not a). It calls a user providedSwapper
object that knows how to swap the data of these two indexes.Again, note the details: Comparisons compare s[a] with g[b]. Swaps swap g[b] with g[c]. Prior to calling this method, the generic splitters s must be sorted ascending and must not contain multiple equal values. These preconditions are not checked; be sure that they are met.
- Parameters:
from
- the index of the first element within g to be considered.to
- the index of the last element within g to be considered. The method considers the elements g[from] .. g[to].splitFrom
- the index of the first splitter element to be considered.splitTo
- the index of the last splitter element to be considered. The method considers the splitter elements s[splitFrom] .. s[splitTo].splitIndexes
- a list into which this method fills the indexes of elements delimiting intervals. Upon return splitIndexes[splitFrom..splitTo] will be set accordingly. Therefore, must satisfy splitIndexes.length > splitTo.comp
- the comparator comparing a splitter with an element of the generic data. Takes as first argument the index a within the generic splitters s. Takes as second argument the index b within the generic data g.comp2
- the comparator to determine the order of the generic data. Takes as first argument the index a within the generic data g. Takes as second argument the index b within the generic data g.comp3
- the comparator comparing a splitter with another splitter. Takes as first argument the index a within the generic splitters s. Takes as second argument the index b within the generic splitters g.swapper
- an object that knows how to swap the elements at any two indexes (a,b). Takes as first argument the index b within the generic data g. Takes as second argument the index c within the generic data g.Tip: Normally you will have splitIndexes.length == s.length as well as from==0, to==g.length-1 and splitFrom==0, splitTo==s.length-1.
- See Also:
-
genericPartition
private static int genericPartition(int from, int to, int splitter, IntComparator comp, Swapper swapper) Same aspartition(int[],int,int,int)
except that it generically partitions arbitrary shaped data (for example matrices or multiple arrays) rather than int[] arrays. -
med3
private static int med3(double[] x, int a, int b, int c) Returns the index of the median of the three indexed elements. -
med3
private static int med3(int[] x, int a, int b, int c) Returns the index of the median of the three indexed elements. -
med3
Returns the index of the median of the three indexed chars. -
med3
Returns the index of the median of the three indexed chars. -
partition
public static void partition(double[] list, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same aspartition(int[],int,int,int[],int,int,int[])
except that it partitions double[] rather than int[] arrays. -
partition
public static int partition(double[] list, int from, int to, double splitter) Same aspartition(int[],int,int,int)
except that it partitions double[] rather than int[] arrays. -
partition
public static void partition(int[] list, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Partitions (partially sorts) the given list such that all elements falling into some intervals are placed next to each other. Returns the indexes of elements delimiting intervals.Example:
list = (7, 4, 5, 50, 6, 4, 3, 6), splitters = (5, 10, 30) defines the three intervals [-infinity,5), [5,10), [10,30). Lets define to sort the entire list (from=0, to=7) using all splitters (splitFrom==0, splitTo=2).
The method modifies the list to be list = (4, 4, 3, 6, 7, 5, 6, 50) and returns the splitIndexes = (2, 6, 6). In other words,
- All values list[0..2] fall into [-infinity,5).
- All values list[3..6] fall into [5,10).
- All values list[7..6] fall into [10,30), i.e. no elements, since 7>6.
- All values list[7 .. 7=list.length-1] fall into [30,infinity].
- In general, all values list[splitIndexes[j-1]+1 .. splitIndexes[j]] fall into interval j.
More formally, this method guarantees that upon return for all j = splitFrom .. splitTo there holds:
for all i = splitIndexes[j-1]+1 .. splitIndexes[j]: splitters[j-1] invalid input: '<'= list[i] invalid input: '<' splitters[j].Performance:
Let N=to-from+1 be the number of elements to be partitioned. Let k=splitTo-splitFrom+1 be the number of splitter elements. Then we have the following time complexities
- Worst case: O( N * log(k) ).
- Average case: O( N * log(k) ).
- Best case: O( N ). In general, the more uniform (skewed) the data is spread across intervals, the more performance approaches the worst (best) case. If no elements fall into the given intervals, running time is linear.
Implementation:
The algorithm can be seen as a Bentley/McIlroy quicksort where swapping and insertion sort are omitted. It is designed to detect and take advantage of skew while maintaining good performance in the uniform case.
- Parameters:
list
- the list to be partially sorted.from
- the index of the first element within list to be considered.to
- the index of the last element within list to be considered. The method considers the elements list[from] .. list[to].splitters
- the values at which the list shall be split into intervals. Must be sorted ascending and must not contain multiple identical values. These preconditions are not checked; be sure that they are met.splitFrom
- the index of the first splitter element to be considered.splitTo
- the index of the last splitter element to be considered. The method considers the splitter elements splitters[splitFrom] .. splitters[splitTo].splitIndexes
- a list into which this method fills the indexes of elements delimiting intervals. Upon return splitIndexes[splitFrom..splitTo] will be set accordingly. Therefore, must satisfy splitIndexes.length > splitTo.Tip: Normally you will have splitIndexes.length == splitters.length as well as from==0, to==list.length-1 and splitFrom==0, splitTo==splitters.length-1.
- See Also:
-
partition
public static int partition(int[] list, int from, int to, int splitter) Partitions (partially sorts) the given list such that all elements falling into the given interval are placed next to each other. Returns the index of the element delimiting the interval.Example:
list = (7, 4, 5, 50, 6, 4, 3, 6), splitter = 5 defines the two intervals [-infinity,5), [5,+infinity].
The method modifies the list to be list = (4, 4, 3, 50, 6, 7, 5, 6) and returns the split index 2. In other words,
- All values list[0..2] fall into [-infinity,5).
- All values list[3=2+1 .. 7=list.length-1] fall into [5,+infinity].
More formally, this method guarantees that upon return there holds:
- for all i = from .. returnValue: list[i] invalid input: '<' splitter and
- for all i = returnValue+1 .. list.length-1: !(list[i] invalid input: '<' splitter).
Performance:
Let N=to-from+1 be the number of elements to be partially sorted. Then the time complexity is O( N ). No temporary memory is allocated; the sort is in-place.
- Parameters:
list
- the list to be partially sorted.from
- the index of the first element within list to be considered.to
- the index of the last element within list to be considered. The method considers the elements list[from] .. list[to].splitter
- the value at which the list shall be split.- Returns:
- the index of the largest element falling into the interval [-infinity,splitter), as seen after partitioning.
-
partition
public static void partition(Object[] list, int from, int to, Object[] splitters, int splitFrom, int splitTo, int[] splitIndexes, Comparator comp) Same aspartition(int[],int,int,int[],int,int,int[])
except that it partitions Object[] rather than int[] arrays. -
partition
Same aspartition(int[],int,int,int)
except that it synchronously partitions the objects of the given list by the order of the given comparator. -
partition
public static void partition(DoubleArrayList list, int from, int to, DoubleArrayList splitters, IntArrayList splitIndexes) Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()). -
partition
public static void partition(IntArrayList list, int from, int to, IntArrayList splitters, IntArrayList splitIndexes) Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()). -
triplePartition
public static void triplePartition(double[] list, double[] secondary, double[] tertiary, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same astriplePartition(int[],int[],int[],int,int,int[],int,int,int[])
except that it synchronously partitions double[] rather than int[] arrays. -
triplePartition
public static int triplePartition(double[] list, double[] secondary, double[] tertiary, int from, int to, double splitter) Same astriplePartition(int[],int[],int[],int,int,int)
except that it synchronously partitions double[] rather than int[] arrays. -
triplePartition
public static void triplePartition(int[] list, int[] secondary, int[] tertiary, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same aspartition(int[],int,int,int[],int,int,int[])
except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array as well as the corresponding element within the tertiary array are also moved from index A to B.Use cases:
Image having a large list of 3-dimensional points. If memory consumption and performance matter, it is a good idea to physically lay them out as three 1-dimensional arrays (using something like Point3D objects would be prohibitively expensive, both in terms of time and space). Now imagine wanting to histogram the points. We may want to partially sort the points by x-coordinate into intervals. This method efficiently does the job.
Performance:
Same as for single-partition methods.
-
triplePartition
public static int triplePartition(int[] list, int[] secondary, int[] tertiary, int from, int to, int splitter) Same aspartition(int[],int,int,int)
except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array as well as the corresponding element within the tertiary array are also moved from index A to B.Performance:
Same as for single-partition methods.
-