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.HashMap;
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
131public final class Quantiles {
132
133  /** Specifies the computation of a median (i.e. the 1st 2-quantile). */
134  public static ScaleAndIndex median() {
135    return scale(2).index(1);
136  }
137
138  /** Specifies the computation of quartiles (i.e. 4-quantiles). */
139  public static Scale quartiles() {
140    return scale(4);
141  }
142
143  /** Specifies the computation of percentiles (i.e. 100-quantiles). */
144  public static Scale percentiles() {
145    return scale(100);
146  }
147
148  /**
149   * Specifies the computation of q-quantiles.
150   *
151   * @param scale the scale for the quantiles to be calculated, i.e. the q of the q-quantiles, which
152   *     must be positive
153   */
154  public static Scale scale(int scale) {
155    return new Scale(scale);
156  }
157
158  /**
159   * Describes the point in a fluent API chain where only the scale (i.e. the q in q-quantiles) has
160   * been specified.
161   *
162   * @since 20.0
163   */
164  public static final class Scale {
165
166    private final int scale;
167
168    private Scale(int scale) {
169      checkArgument(scale > 0, "Quantile scale must be positive");
170      this.scale = scale;
171    }
172
173    /**
174     * Specifies a single quantile index to be calculated, i.e. the k in the kth q-quantile.
175     *
176     * @param index the quantile index, which must be in the inclusive range [0, q] for q-quantiles
177     */
178    public ScaleAndIndex index(int index) {
179      return new ScaleAndIndex(scale, index);
180    }
181
182    /**
183     * Specifies multiple quantile indexes to be calculated, each index being the k in the kth
184     * q-quantile.
185     *
186     * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for
187     *     q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the
188     *     set will be snapshotted when this method is called
189     */
190    public ScaleAndIndexes indexes(int... indexes) {
191      return new ScaleAndIndexes(scale, indexes.clone());
192    }
193
194    /**
195     * Specifies multiple quantile indexes to be calculated, each index being the k in the kth
196     * q-quantile.
197     *
198     * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for
199     *     q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the
200     *     set will be snapshotted when this method is called
201     */
202    public ScaleAndIndexes indexes(Collection<Integer> indexes) {
203      return new ScaleAndIndexes(scale, Ints.toArray(indexes));
204    }
205  }
206
207  /**
208   * Describes the point in a fluent API chain where the scale and a single quantile index (i.e. the
209   * q and the k in the kth q-quantile) have been specified.
210   *
211   * @since 20.0
212   */
213  public static final class ScaleAndIndex {
214
215    private final int scale;
216    private final int index;
217
218    private ScaleAndIndex(int scale, int index) {
219      checkIndex(index, scale);
220      this.scale = scale;
221      this.index = index;
222    }
223
224    /**
225     * Computes the quantile value of the given dataset.
226     *
227     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
228     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
229     *     this call (it is copied instead)
230     * @return the quantile value
231     */
232    public double compute(Collection<? extends Number> dataset) {
233      return computeInPlace(Doubles.toArray(dataset));
234    }
235
236    /**
237     * Computes the quantile value of the given dataset.
238     *
239     * @param dataset the dataset to do the calculation on, which must be non-empty, which will not
240     *     be mutated by this call (it is copied instead)
241     * @return the quantile value
242     */
243    public double compute(double... dataset) {
244      return computeInPlace(dataset.clone());
245    }
246
247    /**
248     * Computes the quantile value of the given dataset.
249     *
250     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
251     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
252     *     this call (it is copied instead)
253     * @return the quantile value
254     */
255    public double compute(long... dataset) {
256      return computeInPlace(longsToDoubles(dataset));
257    }
258
259    /**
260     * Computes the quantile value of the given dataset.
261     *
262     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
263     *     cast to doubles, and which will not be mutated by this call (it is copied instead)
264     * @return the quantile value
265     */
266    public double compute(int... dataset) {
267      return computeInPlace(intsToDoubles(dataset));
268    }
269
270    /**
271     * Computes the quantile value of the given dataset, performing the computation in-place.
272     *
273     * @param dataset the dataset to do the calculation on, which must be non-empty, and which will
274     *     be arbitrarily reordered by this method call
275     * @return the quantile value
276     */
277    public double computeInPlace(double... dataset) {
278      checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset");
279      if (containsNaN(dataset)) {
280        return NaN;
281      }
282
283      // Calculate the quotient and remainder in the integer division x = k * (N-1) / q, i.e.
284      // index * (dataset.length - 1) / scale. If there is no remainder, we can just find the value
285      // whose index in the sorted dataset equals the quotient; if there is a remainder, we
286      // interpolate between that and the next value.
287
288      // Since index and (dataset.length - 1) are non-negative ints, their product can be expressed
289      // as a long, without risk of overflow:
290      long numerator = (long) index * (dataset.length - 1);
291      // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a
292      // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to get
293      // a rounded ratio and a remainder which can be expressed as ints, without risk of overflow:
294      int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN);
295      int remainder = (int) (numerator - (long) quotient * scale);
296      selectInPlace(quotient, dataset, 0, dataset.length - 1);
297      if (remainder == 0) {
298        return dataset[quotient];
299      } else {
300        selectInPlace(quotient + 1, dataset, quotient + 1, dataset.length - 1);
301        return interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale);
302      }
303    }
304  }
305
306  /**
307   * Describes the point in a fluent API chain where the scale and a multiple quantile indexes (i.e.
308   * the q and a set of values for the k in the kth q-quantile) have been specified.
309   *
310   * @since 20.0
311   */
312  public static final class ScaleAndIndexes {
313
314    private final int scale;
315    private final int[] indexes;
316
317    private ScaleAndIndexes(int scale, int[] indexes) {
318      for (int index : indexes) {
319        checkIndex(index, scale);
320      }
321      this.scale = scale;
322      this.indexes = indexes;
323    }
324
325    /**
326     * Computes the quantile values of the given dataset.
327     *
328     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
329     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
330     *     this call (it is copied instead)
331     * @return an unmodifiable map of results: the keys will be the specified quantile indexes, and
332     *     the values the corresponding quantile values
333     */
334    public Map<Integer, Double> compute(Collection<? extends Number> dataset) {
335      return computeInPlace(Doubles.toArray(dataset));
336    }
337
338    /**
339     * Computes the quantile values of the given dataset.
340     *
341     * @param dataset the dataset to do the calculation on, which must be non-empty, which will not
342     *     be mutated by this call (it is copied instead)
343     * @return an unmodifiable map of results: the keys will be the specified quantile indexes, and
344     *     the values the corresponding quantile values
345     */
346    public Map<Integer, Double> compute(double... dataset) {
347      return computeInPlace(dataset.clone());
348    }
349
350    /**
351     * Computes the quantile values of the given dataset.
352     *
353     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
354     *     cast to doubles (with any associated lost of precision), and which will not be mutated by
355     *     this call (it is copied instead)
356     * @return an unmodifiable map of results: the keys will be the specified quantile indexes, and
357     *     the values the corresponding quantile values
358     */
359    public Map<Integer, Double> compute(long... dataset) {
360      return computeInPlace(longsToDoubles(dataset));
361    }
362
363    /**
364     * Computes the quantile values of the given dataset.
365     *
366     * @param dataset the dataset to do the calculation on, which must be non-empty, which will be
367     *     cast to doubles, and which will not be mutated by this call (it is copied instead)
368     * @return an unmodifiable map of results: the keys will be the specified quantile indexes, and
369     *     the values the corresponding quantile values
370     */
371    public Map<Integer, Double> compute(int... dataset) {
372      return computeInPlace(intsToDoubles(dataset));
373    }
374
375    /**
376     * Computes the quantile values of the given dataset, performing the computation in-place.
377     *
378     * @param dataset the dataset to do the calculation on, which must be non-empty, and which will
379     *     be arbitrarily reordered by this method call
380     * @return an unmodifiable map of results: the keys will be the specified quantile indexes, and
381     *     the values the corresponding quantile values
382     */
383    public Map<Integer, Double> computeInPlace(double... dataset) {
384      checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset");
385      if (containsNaN(dataset)) {
386        Map<Integer, Double> nanMap = new HashMap<>();
387        for (int index : indexes) {
388          nanMap.put(index, NaN);
389        }
390        return unmodifiableMap(nanMap);
391      }
392
393      // Calculate the quotients and remainders in the integer division x = k * (N - 1) / q, i.e.
394      // index * (dataset.length - 1) / scale for each index in indexes. For each, if there is no
395      // remainder, we can just select the value whose index in the sorted dataset equals the
396      // quotient; if there is a remainder, we interpolate between that and the next value.
397
398      int[] quotients = new int[indexes.length];
399      int[] remainders = new int[indexes.length];
400      // The indexes to select. In the worst case, we'll need one each side of each quantile.
401      int[] requiredSelections = new int[indexes.length * 2];
402      int requiredSelectionsCount = 0;
403      for (int i = 0; i < indexes.length; i++) {
404        // Since index and (dataset.length - 1) are non-negative ints, their product can be
405        // expressed as a long, without risk of overflow:
406        long numerator = (long) indexes[i] * (dataset.length - 1);
407        // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a
408        // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to
409        // get a rounded ratio and a remainder which can be expressed as ints, without risk of
410        // overflow:
411        int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN);
412        int remainder = (int) (numerator - (long) quotient * scale);
413        quotients[i] = quotient;
414        remainders[i] = remainder;
415        requiredSelections[requiredSelectionsCount] = quotient;
416        requiredSelectionsCount++;
417        if (remainder != 0) {
418          requiredSelections[requiredSelectionsCount] = quotient + 1;
419          requiredSelectionsCount++;
420        }
421      }
422      sort(requiredSelections, 0, requiredSelectionsCount);
423      selectAllInPlace(
424          requiredSelections, 0, requiredSelectionsCount - 1, dataset, 0, dataset.length - 1);
425      Map<Integer, Double> ret = new HashMap<>();
426      for (int i = 0; i < indexes.length; i++) {
427        int quotient = quotients[i];
428        int remainder = remainders[i];
429        if (remainder == 0) {
430          ret.put(indexes[i], dataset[quotient]);
431        } else {
432          ret.put(
433              indexes[i], interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale));
434        }
435      }
436      return unmodifiableMap(ret);
437    }
438  }
439
440  /** Returns whether any of the values in {@code dataset} are {@code NaN}. */
441  private static boolean containsNaN(double... dataset) {
442    for (double value : dataset) {
443      if (Double.isNaN(value)) {
444        return true;
445      }
446    }
447    return false;
448  }
449
450  /**
451   * Returns a value a fraction {@code (remainder / scale)} of the way between {@code lower} and
452   * {@code upper}. Assumes that {@code lower <= upper}. Correctly handles infinities (but not
453   * {@code NaN}).
454   */
455  private static double interpolate(double lower, double upper, double remainder, double scale) {
456    if (lower == NEGATIVE_INFINITY) {
457      if (upper == POSITIVE_INFINITY) {
458        // Return NaN when lower == NEGATIVE_INFINITY and upper == POSITIVE_INFINITY:
459        return NaN;
460      }
461      // Return NEGATIVE_INFINITY when NEGATIVE_INFINITY == lower <= upper < POSITIVE_INFINITY:
462      return NEGATIVE_INFINITY;
463    }
464    if (upper == POSITIVE_INFINITY) {
465      // Return POSITIVE_INFINITY when NEGATIVE_INFINITY < lower <= upper == POSITIVE_INFINITY:
466      return POSITIVE_INFINITY;
467    }
468    return lower + (upper - lower) * remainder / scale;
469  }
470
471  private static void checkIndex(int index, int scale) {
472    if (index < 0 || index > scale) {
473      throw new IllegalArgumentException(
474          "Quantile indexes must be between 0 and the scale, which is " + scale);
475    }
476  }
477
478  private static double[] longsToDoubles(long[] longs) {
479    int len = longs.length;
480    double[] doubles = new double[len];
481    for (int i = 0; i < len; i++) {
482      doubles[i] = longs[i];
483    }
484    return doubles;
485  }
486
487  private static double[] intsToDoubles(int[] ints) {
488    int len = ints.length;
489    double[] doubles = new double[len];
490    for (int i = 0; i < len; i++) {
491      doubles[i] = ints[i];
492    }
493    return doubles;
494  }
495
496  /**
497   * Performs an in-place selection to find the element which would appear at a given index in a
498   * dataset if it were sorted. The following preconditions should hold:
499   *
500   * <ul>
501   *   <li>{@code required}, {@code from}, and {@code to} should all be indexes into {@code array};
502   *   <li>{@code required} should be in the range [{@code from}, {@code to}];
503   *   <li>all the values with indexes in the range [0, {@code from}) should be less than or equal
504   *       to all the values with indexes in the range [{@code from}, {@code to}];
505   *   <li>all the values with indexes in the range ({@code to}, {@code array.length - 1}] should be
506   *       greater than or equal to all the values with indexes in the range [{@code from}, {@code
507   *       to}].
508   * </ul>
509   *
510   * This method will reorder the values with indexes in the range [{@code from}, {@code to}] such
511   * that all the values with indexes in the range [{@code from}, {@code required}) are less than or
512   * equal to the value with index {@code required}, and all the values with indexes in the range
513   * ({@code required}, {@code to}] are greater than or equal to that value. Therefore, the value at
514   * {@code required} is the value which would appear at that index in the sorted dataset.
515   */
516  private static void selectInPlace(int required, double[] array, int from, int to) {
517    // If we are looking for the least element in the range, we can just do a linear search for it.
518    // (We will hit this whenever we are doing quantile interpolation: our first selection finds
519    // the lower value, our second one finds the upper value by looking for the next least element.)
520    if (required == from) {
521      int min = from;
522      for (int index = from + 1; index <= to; index++) {
523        if (array[min] > array[index]) {
524          min = index;
525        }
526      }
527      if (min != from) {
528        swap(array, min, from);
529      }
530      return;
531    }
532
533    // Let's play quickselect! We'll repeatedly partition the range [from, to] containing the
534    // required element, as long as it has more than one element.
535    while (to > from) {
536      int partitionPoint = partition(array, from, to);
537      if (partitionPoint >= required) {
538        to = partitionPoint - 1;
539      }
540      if (partitionPoint <= required) {
541        from = partitionPoint + 1;
542      }
543    }
544  }
545
546  /**
547   * Performs a partition operation on the slice of {@code array} with elements in the range [{@code
548   * from}, {@code to}]. Uses the median of {@code from}, {@code to}, and the midpoint between them
549   * as a pivot. Returns the index which the slice is partitioned around, i.e. if it returns {@code
550   * ret} then we know that the values with indexes in [{@code from}, {@code ret}) are less than or
551   * equal to the value at {@code ret} and the values with indexes in ({@code ret}, {@code to}] are
552   * greater than or equal to that.
553   */
554  private static int partition(double[] array, int from, int to) {
555    // Select a pivot, and move it to the start of the slice i.e. to index from.
556    movePivotToStartOfSlice(array, from, to);
557    double pivot = array[from];
558
559    // Move all elements with indexes in (from, to] which are greater than the pivot to the end of
560    // the array. Keep track of where those elements begin.
561    int partitionPoint = to;
562    for (int i = to; i > from; i--) {
563      if (array[i] > pivot) {
564        swap(array, partitionPoint, i);
565        partitionPoint--;
566      }
567    }
568
569    // We now know that all elements with indexes in (from, partitionPoint] are less than or equal
570    // to the pivot at from, and all elements with indexes in (partitionPoint, to] are greater than
571    // it. We swap the pivot into partitionPoint and we know the array is partitioned around that.
572    swap(array, from, partitionPoint);
573    return partitionPoint;
574  }
575
576  /**
577   * Selects the pivot to use, namely the median of the values at {@code from}, {@code to}, and
578   * halfway between the two (rounded down), from {@code array}, and ensure (by swapping elements if
579   * necessary) that that pivot value appears at the start of the slice i.e. at {@code from}.
580   * Expects that {@code from} is strictly less than {@code to}.
581   */
582  private static void movePivotToStartOfSlice(double[] array, int from, int to) {
583    int mid = (from + to) >>> 1;
584    // We want to make a swap such that either array[to] <= array[from] <= array[mid], or
585    // array[mid] <= array[from] <= array[to]. We know that from < to, so we know mid < to
586    // (although it's possible that mid == from, if to == from + 1). Note that the postcondition
587    // would be impossible to fulfil if mid == to unless we also have array[from] == array[to].
588    boolean toLessThanMid = (array[to] < array[mid]);
589    boolean midLessThanFrom = (array[mid] < array[from]);
590    boolean toLessThanFrom = (array[to] < array[from]);
591    if (toLessThanMid == midLessThanFrom) {
592      // Either array[to] < array[mid] < array[from] or array[from] <= array[mid] <= array[to].
593      swap(array, mid, from);
594    } else if (toLessThanMid != toLessThanFrom) {
595      // Either array[from] <= array[to] < array[mid] or array[mid] <= array[to] < array[from].
596      swap(array, from, to);
597    }
598    // The postcondition now holds. So the median, our chosen pivot, is at from.
599  }
600
601  /**
602   * Performs an in-place selection, like {@link #selectInPlace}, to select all the indexes {@code
603   * allRequired[i]} for {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. These
604   * indexes must be sorted in the array and must all be in the range [{@code from}, {@code to}].
605   */
606  private static void selectAllInPlace(
607      int[] allRequired, int requiredFrom, int requiredTo, double[] array, int from, int to) {
608    // Choose the first selection to do...
609    int requiredChosen = chooseNextSelection(allRequired, requiredFrom, requiredTo, from, to);
610    int required = allRequired[requiredChosen];
611
612    // ...do the first selection...
613    selectInPlace(required, array, from, to);
614
615    // ...then recursively perform the selections in the range below...
616    int requiredBelow = requiredChosen - 1;
617    while (requiredBelow >= requiredFrom && allRequired[requiredBelow] == required) {
618      requiredBelow--; // skip duplicates of required in the range below
619    }
620    if (requiredBelow >= requiredFrom) {
621      selectAllInPlace(allRequired, requiredFrom, requiredBelow, array, from, required - 1);
622    }
623
624    // ...and then recursively perform the selections in the range above.
625    int requiredAbove = requiredChosen + 1;
626    while (requiredAbove <= requiredTo && allRequired[requiredAbove] == required) {
627      requiredAbove++; // skip duplicates of required in the range above
628    }
629    if (requiredAbove <= requiredTo) {
630      selectAllInPlace(allRequired, requiredAbove, requiredTo, array, required + 1, to);
631    }
632  }
633
634  /**
635   * Chooses the next selection to do from the required selections. It is required that the array
636   * {@code allRequired} is sorted and that {@code allRequired[i]} are in the range [{@code from},
637   * {@code to}] for all {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. The
638   * value returned by this method is the {@code i} in that range such that {@code allRequired[i]}
639   * is as close as possible to the center of the range [{@code from}, {@code to}]. Choosing the
640   * value closest to the center of the range first is the most efficient strategy because it
641   * minimizes the size of the subranges from which the remaining selections must be done.
642   */
643  private static int chooseNextSelection(
644      int[] allRequired, int requiredFrom, int requiredTo, int from, int to) {
645    if (requiredFrom == requiredTo) {
646      return requiredFrom; // only one thing to choose, so choose it
647    }
648
649    // Find the center and round down. The true center is either centerFloor or halfway between
650    // centerFloor and centerFloor + 1.
651    int centerFloor = (from + to) >>> 1;
652
653    // Do a binary search until we're down to the range of two which encloses centerFloor (unless
654    // all values are lower or higher than centerFloor, in which case we find the two highest or
655    // lowest respectively). If centerFloor is in allRequired, we will definitely find it. If not,
656    // but centerFloor + 1 is, we'll definitely find that. The closest value to the true (unrounded)
657    // center will be at either low or high.
658    int low = requiredFrom;
659    int high = requiredTo;
660    while (high > low + 1) {
661      int mid = (low + high) >>> 1;
662      if (allRequired[mid] > centerFloor) {
663        high = mid;
664      } else if (allRequired[mid] < centerFloor) {
665        low = mid;
666      } else {
667        return mid; // allRequired[mid] = centerFloor, so we can't get closer than that
668      }
669    }
670
671    // Now pick the closest of the two candidates. Note that there is no rounding here.
672    if (from + to - allRequired[low] - allRequired[high] > 0) {
673      return high;
674    } else {
675      return low;
676    }
677  }
678
679  /** Swaps the values at {@code i} and {@code j} in {@code array}. */
680  private static void swap(double[] array, int i, int j) {
681    double temp = array[i];
682    array[i] = array[j];
683    array[j] = temp;
684  }
685}