001/*
002 * Copyright (C) 2014 The Guava Authors
003 *
004 * Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except
005 * in compliance with the License. You may obtain a copy of the License at
006 *
007 * http://www.apache.org/licenses/LICENSE-2.0
008 *
009 * Unless required by applicable law or agreed to in writing, software distributed under the License
010 * is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express
011 * or implied. See the License for the specific language governing permissions and limitations under
012 * the License.
013 */
014
015package com.google.common.math;
016
017import static com.google.common.base.Preconditions.checkArgument;
018import static java.lang.Double.NEGATIVE_INFINITY;
019import static java.lang.Double.NaN;
020import static java.lang.Double.POSITIVE_INFINITY;
021import static java.util.Arrays.sort;
022import static java.util.Collections.unmodifiableMap;
023
024import com.google.common.annotations.Beta;
025import com.google.common.annotations.GwtIncompatible;
026import com.google.common.primitives.Doubles;
027import com.google.common.primitives.Ints;
028import java.math.RoundingMode;
029import java.util.Collection;
030import java.util.LinkedHashMap;
031import java.util.Map;
032
033/**
034 * Provides a fluent API for calculating <a
035 * href="http://en.wikipedia.org/wiki/Quantile">quantiles</a>.
036 *
037 * <h3>Examples</h3>
038 *
039 * <p>To compute the median:
040 *
041 * <pre>{@code
042 * double myMedian = median().compute(myDataset);
043 * }</pre>
044 *
045 * where {@link #median()} has been statically imported.
046 *
047 * <p>To compute the 99th percentile:
048 *
049 * <pre>{@code
050 * double myPercentile99 = percentiles().index(99).compute(myDataset);
051 * }</pre>
052 *
053 * where {@link #percentiles()} has been statically imported.
054 *
055 * <p>To compute median and the 90th and 99th percentiles:
056 *
057 * <pre>{@code
058 * Map<Integer, Double> myPercentiles =
059 *     percentiles().indexes(50, 90, 99).compute(myDataset);
060 * }</pre>
061 *
062 * where {@link #percentiles()} has been statically imported: {@code myPercentiles} maps the keys
063 * 50, 90, and 99, to their corresponding quantile values.
064 *
065 * <p>To compute quartiles, use {@link #quartiles()} instead of {@link #percentiles()}. To compute
066 * arbitrary q-quantiles, use {@link #scale scale(q)}.
067 *
068 * <p>These examples all take a copy of your dataset. If you have a double array, you are okay with
069 * it being arbitrarily reordered, and you want to avoid that copy, you can use {@code
070 * computeInPlace} instead of {@code compute}.
071 *
072 * <h3>Definition and notes on interpolation</h3>
073 *
074 * <p>The definition of the kth q-quantile of N values is as follows: define x = k * (N - 1) / q; if
075 * x is an integer, the result is the value which would appear at index x in the sorted dataset
076 * (unless there are {@link Double#NaN NaN} values, see below); otherwise, the result is the average
077 * of the values which would appear at the indexes floor(x) and ceil(x) weighted by (1-frac(x)) and
078 * frac(x) respectively. This is the same definition as used by Excel and by S, it is the Type 7
079 * definition in <a
080 * href="http://stat.ethz.ch/R-manual/R-devel/library/stats/html/quantile.html">R</a>, and it is
081 * described by <a
082 * href="http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population">
083 * wikipedia</a> as providing "Linear interpolation of the modes for the order statistics for the
084 * uniform distribution on [0,1]."
085 *
086 * <h3>Handling of non-finite values</h3>
087 *
088 * <p>If any values in the input are {@link Double#NaN NaN} then all values returned are {@link
089 * Double#NaN NaN}. (This is the one occasion when the behaviour is not the same as you'd get from
090 * sorting with {@link java.util.Arrays#sort(double[]) Arrays.sort(double[])} or {@link
091 * java.util.Collections#sort(java.util.List) Collections.sort(List&lt;Double&gt;)} and selecting
092 * the required value(s). Those methods would sort {@link Double#NaN NaN} as if it is greater than
093 * any other value and place them at the end of the dataset, even after {@link
094 * Double#POSITIVE_INFINITY POSITIVE_INFINITY}.)
095 *
096 * <p>Otherwise, {@link Double#NEGATIVE_INFINITY NEGATIVE_INFINITY} and {@link
097 * Double#POSITIVE_INFINITY POSITIVE_INFINITY} sort to the beginning and the end of the dataset, as
098 * you would expect.
099 *
100 * <p>If required to do a weighted average between an infinity and a finite value, or between an
101 * infinite value and itself, the infinite value is returned. If required to do a weighted average
102 * between {@link Double#NEGATIVE_INFINITY NEGATIVE_INFINITY} and {@link Double#POSITIVE_INFINITY
103 * POSITIVE_INFINITY}, {@link Double#NaN NaN} is returned (note that this will only happen if the
104 * dataset contains no finite values).
105 *
106 * <h3>Performance</h3>
107 *
108 * <p>The average time complexity of the computation is O(N) in the size of the dataset. There is a
109 * worst case time complexity of O(N^2). You are extremely unlikely to hit this quadratic case on
110 * randomly ordered data (the probability decreases faster than exponentially in N), but if you are
111 * passing in unsanitized user data then a malicious user could force it. A light shuffle of the
112 * data using an unpredictable seed should normally be enough to thwart this attack.
113 *
114 * <p>The time taken to compute multiple quantiles on the same dataset using {@link Scale#indexes
115 * indexes} is generally less than the total time taken to compute each of them separately, and
116 * sometimes much less. For example, on a large enough dataset, computing the 90th and 99th
117 * percentiles together takes about 55% as long as computing them separately.
118 *
119 * <p>When calling {@link ScaleAndIndex#compute} (in {@linkplain ScaleAndIndexes#compute either
120 * form}), the memory requirement is 8*N bytes for the copy of the dataset plus an overhead which is
121 * independent of N (but depends on the quantiles being computed). When calling {@link
122 * ScaleAndIndex#computeInPlace computeInPlace} (in {@linkplain ScaleAndIndexes#computeInPlace
123 * either form}), only the overhead is required. The number of object allocations is independent of
124 * N in both cases.
125 *
126 * @author Pete Gillin
127 * @since 20.0
128 */
129@Beta
130@GwtIncompatible
131@ElementTypesAreNonnullByDefault
132public final class Quantiles {
133
134  /** Specifies the computation of a median (i.e. the 1st 2-quantile). */
135  public static ScaleAndIndex median() {
136    return scale(2).index(1);
137  }
138
139  /** Specifies the computation of quartiles (i.e. 4-quantiles). */
140  public static Scale quartiles() {
141    return scale(4);
142  }
143
144  /** Specifies the computation of percentiles (i.e. 100-quantiles). */
145  public static Scale percentiles() {
146    return scale(100);
147  }
148
149  /**
150   * Specifies the computation of q-quantiles.
151   *
152   * @param scale the scale for the quantiles to be calculated, i.e. the q of the q-quantiles, which
153   *     must be positive
154   */
155  public static Scale scale(int scale) {
156    return new Scale(scale);
157  }
158
159  /**
160   * Describes the point in a fluent API chain where only the scale (i.e. the q in q-quantiles) has
161   * been specified.
162   *
163   * @since 20.0
164   */
165  public static final class Scale {
166
167    private final int scale;
168
169    private Scale(int scale) {
170      checkArgument(scale > 0, "Quantile scale must be positive");
171      this.scale = scale;
172    }
173
174    /**
175     * Specifies a single quantile index to be calculated, i.e. the k in the kth q-quantile.
176     *
177     * @param index the quantile index, which must be in the inclusive range [0, q] for q-quantiles
178     */
179    public ScaleAndIndex index(int index) {
180      return new ScaleAndIndex(scale, index);
181    }
182
183    /**
184     * Specifies multiple quantile indexes to be calculated, each index being the k in the kth
185     * q-quantile.
186     *
187     * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for
188     *     q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the
189     *     set will be snapshotted when this method is called
190     * @throws IllegalArgumentException if {@code indexes} is empty
191     */
192    public ScaleAndIndexes indexes(int... indexes) {
193      return new ScaleAndIndexes(scale, indexes.clone());
194    }
195
196    /**
197     * Specifies multiple quantile indexes to be calculated, each index being the k in the kth
198     * q-quantile.
199     *
200     * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for
201     *     q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the
202     *     set will be snapshotted when this method is called
203     * @throws IllegalArgumentException if {@code indexes} is empty
204     */
205    public ScaleAndIndexes indexes(Collection<Integer> indexes) {
206      return new ScaleAndIndexes(scale, Ints.toArray(indexes));
207    }
208  }
209
210  /**
211   * Describes the point in a fluent API chain where the scale and a single quantile index (i.e. the
212   * q and the k in the kth q-quantile) have been specified.
213   *
214   * @since 20.0
215   */
216  public static final class ScaleAndIndex {
217
218    private final int scale;
219    private final int index;
220
221    private ScaleAndIndex(int scale, int index) {
222      checkIndex(index, scale);
223      this.scale = scale;
224      this.index = index;
225    }
226
227    /**
228     * Computes the quantile value of the given dataset.
229     *
230     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
231     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
232     *     this call (it is copied instead)
233     * @return the quantile value
234     */
235    public double compute(Collection<? extends Number> dataset) {
236      return computeInPlace(Doubles.toArray(dataset));
237    }
238
239    /**
240     * Computes the quantile value of the given dataset.
241     *
242     * @param dataset the dataset to do the calculation on, which must be non-empty, which will not
243     *     be mutated by this call (it is copied instead)
244     * @return the quantile value
245     */
246    public double compute(double... dataset) {
247      return computeInPlace(dataset.clone());
248    }
249
250    /**
251     * Computes the quantile value of the given dataset.
252     *
253     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
254     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
255     *     this call (it is copied instead)
256     * @return the quantile value
257     */
258    public double compute(long... dataset) {
259      return computeInPlace(longsToDoubles(dataset));
260    }
261
262    /**
263     * Computes the quantile value of the given dataset.
264     *
265     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
266     *     cast to doubles, and which will not be mutated by this call (it is copied instead)
267     * @return the quantile value
268     */
269    public double compute(int... dataset) {
270      return computeInPlace(intsToDoubles(dataset));
271    }
272
273    /**
274     * Computes the quantile value of the given dataset, performing the computation in-place.
275     *
276     * @param dataset the dataset to do the calculation on, which must be non-empty, and which will
277     *     be arbitrarily reordered by this method call
278     * @return the quantile value
279     */
280    public double computeInPlace(double... dataset) {
281      checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset");
282      if (containsNaN(dataset)) {
283        return NaN;
284      }
285
286      // Calculate the quotient and remainder in the integer division x = k * (N-1) / q, i.e.
287      // index * (dataset.length - 1) / scale. If there is no remainder, we can just find the value
288      // whose index in the sorted dataset equals the quotient; if there is a remainder, we
289      // interpolate between that and the next value.
290
291      // Since index and (dataset.length - 1) are non-negative ints, their product can be expressed
292      // as a long, without risk of overflow:
293      long numerator = (long) index * (dataset.length - 1);
294      // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a
295      // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to get
296      // a rounded ratio and a remainder which can be expressed as ints, without risk of overflow:
297      int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN);
298      int remainder = (int) (numerator - (long) quotient * scale);
299      selectInPlace(quotient, dataset, 0, dataset.length - 1);
300      if (remainder == 0) {
301        return dataset[quotient];
302      } else {
303        selectInPlace(quotient + 1, dataset, quotient + 1, dataset.length - 1);
304        return interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale);
305      }
306    }
307  }
308
309  /**
310   * Describes the point in a fluent API chain where the scale and a multiple quantile indexes (i.e.
311   * the q and a set of values for the k in the kth q-quantile) have been specified.
312   *
313   * @since 20.0
314   */
315  public static final class ScaleAndIndexes {
316
317    private final int scale;
318    private final int[] indexes;
319
320    private ScaleAndIndexes(int scale, int[] indexes) {
321      for (int index : indexes) {
322        checkIndex(index, scale);
323      }
324      checkArgument(indexes.length > 0, "Indexes must be a non empty array");
325      this.scale = scale;
326      this.indexes = indexes;
327    }
328
329    /**
330     * Computes the quantile values of the given dataset.
331     *
332     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
333     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
334     *     this call (it is copied instead)
335     * @return an unmodifiable, ordered map of results: the keys will be the specified quantile
336     *     indexes, and the values the corresponding quantile values. When iterating, entries in the
337     *     map are ordered by quantile index in the same order they were passed to the {@code
338     *     indexes} method.
339     */
340    public Map<Integer, Double> compute(Collection<? extends Number> dataset) {
341      return computeInPlace(Doubles.toArray(dataset));
342    }
343
344    /**
345     * Computes the quantile values of the given dataset.
346     *
347     * @param dataset the dataset to do the calculation on, which must be non-empty, which will not
348     *     be mutated by this call (it is copied instead)
349     * @return an unmodifiable, ordered map of results: the keys will be the specified quantile
350     *     indexes, and the values the corresponding quantile values. When iterating, entries in the
351     *     map are ordered by quantile index in the same order they were passed to the {@code
352     *     indexes} method.
353     */
354    public Map<Integer, Double> compute(double... dataset) {
355      return computeInPlace(dataset.clone());
356    }
357
358    /**
359     * Computes the quantile values of the given dataset.
360     *
361     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
362     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
363     *     this call (it is copied instead)
364     * @return an unmodifiable, ordered map of results: the keys will be the specified quantile
365     *     indexes, and the values the corresponding quantile values. When iterating, entries in the
366     *     map are ordered by quantile index in the same order they were passed to the {@code
367     *     indexes} method.
368     */
369    public Map<Integer, Double> compute(long... dataset) {
370      return computeInPlace(longsToDoubles(dataset));
371    }
372
373    /**
374     * Computes the quantile values of the given dataset.
375     *
376     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
377     *     cast to doubles, and which will not be mutated by this call (it is copied instead)
378     * @return an unmodifiable, ordered map of results: the keys will be the specified quantile
379     *     indexes, and the values the corresponding quantile values. When iterating, entries in the
380     *     map are ordered by quantile index in the same order they were passed to the {@code
381     *     indexes} method.
382     */
383    public Map<Integer, Double> compute(int... dataset) {
384      return computeInPlace(intsToDoubles(dataset));
385    }
386
387    /**
388     * Computes the quantile values of the given dataset, performing the computation in-place.
389     *
390     * @param dataset the dataset to do the calculation on, which must be non-empty, and which will
391     *     be arbitrarily reordered by this method call
392     * @return an unmodifiable, ordered map of results: the keys will be the specified quantile
393     *     indexes, and the values the corresponding quantile values. When iterating, entries in the
394     *     map are ordered by quantile index in the same order that the indexes were passed to the
395     *     {@code indexes} method.
396     */
397    public Map<Integer, Double> computeInPlace(double... dataset) {
398      checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset");
399      if (containsNaN(dataset)) {
400        Map<Integer, Double> nanMap = new LinkedHashMap<>();
401        for (int index : indexes) {
402          nanMap.put(index, NaN);
403        }
404        return unmodifiableMap(nanMap);
405      }
406
407      // Calculate the quotients and remainders in the integer division x = k * (N - 1) / q, i.e.
408      // index * (dataset.length - 1) / scale for each index in indexes. For each, if there is no
409      // remainder, we can just select the value whose index in the sorted dataset equals the
410      // quotient; if there is a remainder, we interpolate between that and the next value.
411
412      int[] quotients = new int[indexes.length];
413      int[] remainders = new int[indexes.length];
414      // The indexes to select. In the worst case, we'll need one each side of each quantile.
415      int[] requiredSelections = new int[indexes.length * 2];
416      int requiredSelectionsCount = 0;
417      for (int i = 0; i < indexes.length; i++) {
418        // Since index and (dataset.length - 1) are non-negative ints, their product can be
419        // expressed as a long, without risk of overflow:
420        long numerator = (long) indexes[i] * (dataset.length - 1);
421        // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a
422        // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to
423        // get a rounded ratio and a remainder which can be expressed as ints, without risk of
424        // overflow:
425        int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN);
426        int remainder = (int) (numerator - (long) quotient * scale);
427        quotients[i] = quotient;
428        remainders[i] = remainder;
429        requiredSelections[requiredSelectionsCount] = quotient;
430        requiredSelectionsCount++;
431        if (remainder != 0) {
432          requiredSelections[requiredSelectionsCount] = quotient + 1;
433          requiredSelectionsCount++;
434        }
435      }
436      sort(requiredSelections, 0, requiredSelectionsCount);
437      selectAllInPlace(
438          requiredSelections, 0, requiredSelectionsCount - 1, dataset, 0, dataset.length - 1);
439      Map<Integer, Double> ret = new LinkedHashMap<>();
440      for (int i = 0; i < indexes.length; i++) {
441        int quotient = quotients[i];
442        int remainder = remainders[i];
443        if (remainder == 0) {
444          ret.put(indexes[i], dataset[quotient]);
445        } else {
446          ret.put(
447              indexes[i], interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale));
448        }
449      }
450      return unmodifiableMap(ret);
451    }
452  }
453
454  /** Returns whether any of the values in {@code dataset} are {@code NaN}. */
455  private static boolean containsNaN(double... dataset) {
456    for (double value : dataset) {
457      if (Double.isNaN(value)) {
458        return true;
459      }
460    }
461    return false;
462  }
463
464  /**
465   * Returns a value a fraction {@code (remainder / scale)} of the way between {@code lower} and
466   * {@code upper}. Assumes that {@code lower <= upper}. Correctly handles infinities (but not
467   * {@code NaN}).
468   */
469  private static double interpolate(double lower, double upper, double remainder, double scale) {
470    if (lower == NEGATIVE_INFINITY) {
471      if (upper == POSITIVE_INFINITY) {
472        // Return NaN when lower == NEGATIVE_INFINITY and upper == POSITIVE_INFINITY:
473        return NaN;
474      }
475      // Return NEGATIVE_INFINITY when NEGATIVE_INFINITY == lower <= upper < POSITIVE_INFINITY:
476      return NEGATIVE_INFINITY;
477    }
478    if (upper == POSITIVE_INFINITY) {
479      // Return POSITIVE_INFINITY when NEGATIVE_INFINITY < lower <= upper == POSITIVE_INFINITY:
480      return POSITIVE_INFINITY;
481    }
482    return lower + (upper - lower) * remainder / scale;
483  }
484
485  private static void checkIndex(int index, int scale) {
486    if (index < 0 || index > scale) {
487      throw new IllegalArgumentException(
488          "Quantile indexes must be between 0 and the scale, which is " + scale);
489    }
490  }
491
492  private static double[] longsToDoubles(long[] longs) {
493    int len = longs.length;
494    double[] doubles = new double[len];
495    for (int i = 0; i < len; i++) {
496      doubles[i] = longs[i];
497    }
498    return doubles;
499  }
500
501  private static double[] intsToDoubles(int[] ints) {
502    int len = ints.length;
503    double[] doubles = new double[len];
504    for (int i = 0; i < len; i++) {
505      doubles[i] = ints[i];
506    }
507    return doubles;
508  }
509
510  /**
511   * Performs an in-place selection to find the element which would appear at a given index in a
512   * dataset if it were sorted. The following preconditions should hold:
513   *
514   * <ul>
515   *   <li>{@code required}, {@code from}, and {@code to} should all be indexes into {@code array};
516   *   <li>{@code required} should be in the range [{@code from}, {@code to}];
517   *   <li>all the values with indexes in the range [0, {@code from}) should be less than or equal
518   *       to all the values with indexes in the range [{@code from}, {@code to}];
519   *   <li>all the values with indexes in the range ({@code to}, {@code array.length - 1}] should be
520   *       greater than or equal to all the values with indexes in the range [{@code from}, {@code
521   *       to}].
522   * </ul>
523   *
524   * This method will reorder the values with indexes in the range [{@code from}, {@code to}] such
525   * that all the values with indexes in the range [{@code from}, {@code required}) are less than or
526   * equal to the value with index {@code required}, and all the values with indexes in the range
527   * ({@code required}, {@code to}] are greater than or equal to that value. Therefore, the value at
528   * {@code required} is the value which would appear at that index in the sorted dataset.
529   */
530  private static void selectInPlace(int required, double[] array, int from, int to) {
531    // If we are looking for the least element in the range, we can just do a linear search for it.
532    // (We will hit this whenever we are doing quantile interpolation: our first selection finds
533    // the lower value, our second one finds the upper value by looking for the next least element.)
534    if (required == from) {
535      int min = from;
536      for (int index = from + 1; index <= to; index++) {
537        if (array[min] > array[index]) {
538          min = index;
539        }
540      }
541      if (min != from) {
542        swap(array, min, from);
543      }
544      return;
545    }
546
547    // Let's play quickselect! We'll repeatedly partition the range [from, to] containing the
548    // required element, as long as it has more than one element.
549    while (to > from) {
550      int partitionPoint = partition(array, from, to);
551      if (partitionPoint >= required) {
552        to = partitionPoint - 1;
553      }
554      if (partitionPoint <= required) {
555        from = partitionPoint + 1;
556      }
557    }
558  }
559
560  /**
561   * Performs a partition operation on the slice of {@code array} with elements in the range [{@code
562   * from}, {@code to}]. Uses the median of {@code from}, {@code to}, and the midpoint between them
563   * as a pivot. Returns the index which the slice is partitioned around, i.e. if it returns {@code
564   * ret} then we know that the values with indexes in [{@code from}, {@code ret}) are less than or
565   * equal to the value at {@code ret} and the values with indexes in ({@code ret}, {@code to}] are
566   * greater than or equal to that.
567   */
568  private static int partition(double[] array, int from, int to) {
569    // Select a pivot, and move it to the start of the slice i.e. to index from.
570    movePivotToStartOfSlice(array, from, to);
571    double pivot = array[from];
572
573    // Move all elements with indexes in (from, to] which are greater than the pivot to the end of
574    // the array. Keep track of where those elements begin.
575    int partitionPoint = to;
576    for (int i = to; i > from; i--) {
577      if (array[i] > pivot) {
578        swap(array, partitionPoint, i);
579        partitionPoint--;
580      }
581    }
582
583    // We now know that all elements with indexes in (from, partitionPoint] are less than or equal
584    // to the pivot at from, and all elements with indexes in (partitionPoint, to] are greater than
585    // it. We swap the pivot into partitionPoint and we know the array is partitioned around that.
586    swap(array, from, partitionPoint);
587    return partitionPoint;
588  }
589
590  /**
591   * Selects the pivot to use, namely the median of the values at {@code from}, {@code to}, and
592   * halfway between the two (rounded down), from {@code array}, and ensure (by swapping elements if
593   * necessary) that that pivot value appears at the start of the slice i.e. at {@code from}.
594   * Expects that {@code from} is strictly less than {@code to}.
595   */
596  private static void movePivotToStartOfSlice(double[] array, int from, int to) {
597    int mid = (from + to) >>> 1;
598    // We want to make a swap such that either array[to] <= array[from] <= array[mid], or
599    // array[mid] <= array[from] <= array[to]. We know that from < to, so we know mid < to
600    // (although it's possible that mid == from, if to == from + 1). Note that the postcondition
601    // would be impossible to fulfil if mid == to unless we also have array[from] == array[to].
602    boolean toLessThanMid = (array[to] < array[mid]);
603    boolean midLessThanFrom = (array[mid] < array[from]);
604    boolean toLessThanFrom = (array[to] < array[from]);
605    if (toLessThanMid == midLessThanFrom) {
606      // Either array[to] < array[mid] < array[from] or array[from] <= array[mid] <= array[to].
607      swap(array, mid, from);
608    } else if (toLessThanMid != toLessThanFrom) {
609      // Either array[from] <= array[to] < array[mid] or array[mid] <= array[to] < array[from].
610      swap(array, from, to);
611    }
612    // The postcondition now holds. So the median, our chosen pivot, is at from.
613  }
614
615  /**
616   * Performs an in-place selection, like {@link #selectInPlace}, to select all the indexes {@code
617   * allRequired[i]} for {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. These
618   * indexes must be sorted in the array and must all be in the range [{@code from}, {@code to}].
619   */
620  private static void selectAllInPlace(
621      int[] allRequired, int requiredFrom, int requiredTo, double[] array, int from, int to) {
622    // Choose the first selection to do...
623    int requiredChosen = chooseNextSelection(allRequired, requiredFrom, requiredTo, from, to);
624    int required = allRequired[requiredChosen];
625
626    // ...do the first selection...
627    selectInPlace(required, array, from, to);
628
629    // ...then recursively perform the selections in the range below...
630    int requiredBelow = requiredChosen - 1;
631    while (requiredBelow >= requiredFrom && allRequired[requiredBelow] == required) {
632      requiredBelow--; // skip duplicates of required in the range below
633    }
634    if (requiredBelow >= requiredFrom) {
635      selectAllInPlace(allRequired, requiredFrom, requiredBelow, array, from, required - 1);
636    }
637
638    // ...and then recursively perform the selections in the range above.
639    int requiredAbove = requiredChosen + 1;
640    while (requiredAbove <= requiredTo && allRequired[requiredAbove] == required) {
641      requiredAbove++; // skip duplicates of required in the range above
642    }
643    if (requiredAbove <= requiredTo) {
644      selectAllInPlace(allRequired, requiredAbove, requiredTo, array, required + 1, to);
645    }
646  }
647
648  /**
649   * Chooses the next selection to do from the required selections. It is required that the array
650   * {@code allRequired} is sorted and that {@code allRequired[i]} are in the range [{@code from},
651   * {@code to}] for all {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. The
652   * value returned by this method is the {@code i} in that range such that {@code allRequired[i]}
653   * is as close as possible to the center of the range [{@code from}, {@code to}]. Choosing the
654   * value closest to the center of the range first is the most efficient strategy because it
655   * minimizes the size of the subranges from which the remaining selections must be done.
656   */
657  private static int chooseNextSelection(
658      int[] allRequired, int requiredFrom, int requiredTo, int from, int to) {
659    if (requiredFrom == requiredTo) {
660      return requiredFrom; // only one thing to choose, so choose it
661    }
662
663    // Find the center and round down. The true center is either centerFloor or halfway between
664    // centerFloor and centerFloor + 1.
665    int centerFloor = (from + to) >>> 1;
666
667    // Do a binary search until we're down to the range of two which encloses centerFloor (unless
668    // all values are lower or higher than centerFloor, in which case we find the two highest or
669    // lowest respectively). If centerFloor is in allRequired, we will definitely find it. If not,
670    // but centerFloor + 1 is, we'll definitely find that. The closest value to the true (unrounded)
671    // center will be at either low or high.
672    int low = requiredFrom;
673    int high = requiredTo;
674    while (high > low + 1) {
675      int mid = (low + high) >>> 1;
676      if (allRequired[mid] > centerFloor) {
677        high = mid;
678      } else if (allRequired[mid] < centerFloor) {
679        low = mid;
680      } else {
681        return mid; // allRequired[mid] = centerFloor, so we can't get closer than that
682      }
683    }
684
685    // Now pick the closest of the two candidates. Note that there is no rounding here.
686    if (from + to - allRequired[low] - allRequired[high] > 0) {
687      return high;
688    } else {
689      return low;
690    }
691  }
692
693  /** Swaps the values at {@code i} and {@code j} in {@code array}. */
694  private static void swap(double[] array, int i, int j) {
695    double temp = array[i];
696    array[i] = array[j];
697    array[j] = temp;
698  }
699}