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.GwtIncompatible; 025import com.google.common.annotations.J2ktIncompatible; 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<Double>)} 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@J2ktIncompatible 130@GwtIncompatible 131public final class Quantiles { 132 /** 133 * Constructor for a type that is not meant to be instantiated. 134 * 135 * @deprecated Use the static factory methods of the class. There is no reason to create an 136 * instance of {@link Quantiles}. 137 */ 138 @Deprecated 139 public Quantiles() {} 140 141 /** Specifies the computation of a median (i.e. the 1st 2-quantile). */ 142 public static ScaleAndIndex median() { 143 return scale(2).index(1); 144 } 145 146 /** Specifies the computation of quartiles (i.e. 4-quantiles). */ 147 public static Scale quartiles() { 148 return scale(4); 149 } 150 151 /** Specifies the computation of percentiles (i.e. 100-quantiles). */ 152 public static Scale percentiles() { 153 return scale(100); 154 } 155 156 /** 157 * Specifies the computation of q-quantiles. 158 * 159 * @param scale the scale for the quantiles to be calculated, i.e. the q of the q-quantiles, which 160 * must be positive 161 */ 162 public static Scale scale(int scale) { 163 return new Scale(scale); 164 } 165 166 /** 167 * Describes the point in a fluent API chain where only the scale (i.e. the q in q-quantiles) has 168 * been specified. 169 * 170 * @since 20.0 171 */ 172 public static final class Scale { 173 174 private final int scale; 175 176 private Scale(int scale) { 177 checkArgument(scale > 0, "Quantile scale must be positive"); 178 this.scale = scale; 179 } 180 181 /** 182 * Specifies a single quantile index to be calculated, i.e. the k in the kth q-quantile. 183 * 184 * @param index the quantile index, which must be in the inclusive range [0, q] for q-quantiles 185 */ 186 public ScaleAndIndex index(int index) { 187 return new ScaleAndIndex(scale, index); 188 } 189 190 /** 191 * Specifies multiple quantile indexes to be calculated, each index being the k in the kth 192 * q-quantile. 193 * 194 * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for 195 * q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the 196 * set will be snapshotted when this method is called 197 * @throws IllegalArgumentException if {@code indexes} is empty 198 */ 199 public ScaleAndIndexes indexes(int... indexes) { 200 return new ScaleAndIndexes(scale, indexes.clone()); 201 } 202 203 /** 204 * Specifies multiple quantile indexes to be calculated, each index being the k in the kth 205 * q-quantile. 206 * 207 * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for 208 * q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the 209 * set will be snapshotted when this method is called 210 * @throws IllegalArgumentException if {@code indexes} is empty 211 */ 212 public ScaleAndIndexes indexes(Collection<Integer> indexes) { 213 return new ScaleAndIndexes(scale, Ints.toArray(indexes)); 214 } 215 } 216 217 /** 218 * Describes the point in a fluent API chain where the scale and a single quantile index (i.e. the 219 * q and the k in the kth q-quantile) have been specified. 220 * 221 * @since 20.0 222 */ 223 public static final class ScaleAndIndex { 224 225 private final int scale; 226 private final int index; 227 228 private ScaleAndIndex(int scale, int index) { 229 checkIndex(index, scale); 230 this.scale = scale; 231 this.index = index; 232 } 233 234 /** 235 * Computes the quantile value of the given dataset. 236 * 237 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 238 * cast to doubles (with any associated lost of precision), and which will not be mutated by 239 * this call (it is copied instead) 240 * @return the quantile value 241 */ 242 public double compute(Collection<? extends Number> dataset) { 243 return computeInPlace(Doubles.toArray(dataset)); 244 } 245 246 /** 247 * Computes the quantile value of the given dataset. 248 * 249 * @param dataset the dataset to do the calculation on, which must be non-empty, which will not 250 * be mutated by this call (it is copied instead) 251 * @return the quantile value 252 */ 253 public double compute(double... dataset) { 254 return computeInPlace(dataset.clone()); 255 } 256 257 /** 258 * Computes the quantile value of the given dataset. 259 * 260 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 261 * cast to doubles (with any associated lost of precision), and which will not be mutated by 262 * this call (it is copied instead) 263 * @return the quantile value 264 */ 265 public double compute(long... dataset) { 266 return computeInPlace(longsToDoubles(dataset)); 267 } 268 269 /** 270 * Computes the quantile value of the given dataset. 271 * 272 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 273 * cast to doubles, and which will not be mutated by this call (it is copied instead) 274 * @return the quantile value 275 */ 276 public double compute(int... dataset) { 277 return computeInPlace(intsToDoubles(dataset)); 278 } 279 280 /** 281 * Computes the quantile value of the given dataset, performing the computation in-place. 282 * 283 * @param dataset the dataset to do the calculation on, which must be non-empty, and which will 284 * be arbitrarily reordered by this method call 285 * @return the quantile value 286 */ 287 public double computeInPlace(double... dataset) { 288 checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset"); 289 if (containsNaN(dataset)) { 290 return NaN; 291 } 292 293 // Calculate the quotient and remainder in the integer division x = k * (N-1) / q, i.e. 294 // index * (dataset.length - 1) / scale. If there is no remainder, we can just find the value 295 // whose index in the sorted dataset equals the quotient; if there is a remainder, we 296 // interpolate between that and the next value. 297 298 // Since index and (dataset.length - 1) are non-negative ints, their product can be expressed 299 // as a long, without risk of overflow: 300 long numerator = (long) index * (dataset.length - 1); 301 // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a 302 // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to get 303 // a rounded ratio and a remainder which can be expressed as ints, without risk of overflow: 304 int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN); 305 int remainder = (int) (numerator - (long) quotient * scale); 306 selectInPlace(quotient, dataset, 0, dataset.length - 1); 307 if (remainder == 0) { 308 return dataset[quotient]; 309 } else { 310 selectInPlace(quotient + 1, dataset, quotient + 1, dataset.length - 1); 311 return interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale); 312 } 313 } 314 } 315 316 /** 317 * Describes the point in a fluent API chain where the scale and a multiple quantile indexes (i.e. 318 * the q and a set of values for the k in the kth q-quantile) have been specified. 319 * 320 * @since 20.0 321 */ 322 public static final class ScaleAndIndexes { 323 324 private final int scale; 325 private final int[] indexes; 326 327 private ScaleAndIndexes(int scale, int[] indexes) { 328 for (int index : indexes) { 329 checkIndex(index, scale); 330 } 331 checkArgument(indexes.length > 0, "Indexes must be a non empty array"); 332 this.scale = scale; 333 this.indexes = indexes; 334 } 335 336 /** 337 * Computes the quantile values of the given dataset. 338 * 339 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 340 * cast to doubles (with any associated lost of precision), and which will not be mutated by 341 * this call (it is copied instead) 342 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 343 * indexes, and the values the corresponding quantile values. When iterating, entries in the 344 * map are ordered by quantile index in the same order they were passed to the {@code 345 * indexes} method. 346 */ 347 public Map<Integer, Double> compute(Collection<? extends Number> dataset) { 348 return computeInPlace(Doubles.toArray(dataset)); 349 } 350 351 /** 352 * Computes the quantile values of the given dataset. 353 * 354 * @param dataset the dataset to do the calculation on, which must be non-empty, which will not 355 * be mutated by this call (it is copied instead) 356 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 357 * indexes, and the values the corresponding quantile values. When iterating, entries in the 358 * map are ordered by quantile index in the same order they were passed to the {@code 359 * indexes} method. 360 */ 361 public Map<Integer, Double> compute(double... dataset) { 362 return computeInPlace(dataset.clone()); 363 } 364 365 /** 366 * Computes the quantile values of the given dataset. 367 * 368 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 369 * cast to doubles (with any associated lost of precision), and which will not be mutated by 370 * this call (it is copied instead) 371 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 372 * indexes, and the values the corresponding quantile values. When iterating, entries in the 373 * map are ordered by quantile index in the same order they were passed to the {@code 374 * indexes} method. 375 */ 376 public Map<Integer, Double> compute(long... dataset) { 377 return computeInPlace(longsToDoubles(dataset)); 378 } 379 380 /** 381 * Computes the quantile values of the given dataset. 382 * 383 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 384 * cast to doubles, and which will not be mutated by this call (it is copied instead) 385 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 386 * indexes, and the values the corresponding quantile values. When iterating, entries in the 387 * map are ordered by quantile index in the same order they were passed to the {@code 388 * indexes} method. 389 */ 390 public Map<Integer, Double> compute(int... dataset) { 391 return computeInPlace(intsToDoubles(dataset)); 392 } 393 394 /** 395 * Computes the quantile values of the given dataset, performing the computation in-place. 396 * 397 * @param dataset the dataset to do the calculation on, which must be non-empty, and which will 398 * be arbitrarily reordered by this method call 399 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 400 * indexes, and the values the corresponding quantile values. When iterating, entries in the 401 * map are ordered by quantile index in the same order that the indexes were passed to the 402 * {@code indexes} method. 403 */ 404 public Map<Integer, Double> computeInPlace(double... dataset) { 405 checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset"); 406 if (containsNaN(dataset)) { 407 Map<Integer, Double> nanMap = new LinkedHashMap<>(); 408 for (int index : indexes) { 409 nanMap.put(index, NaN); 410 } 411 return unmodifiableMap(nanMap); 412 } 413 414 // Calculate the quotients and remainders in the integer division x = k * (N - 1) / q, i.e. 415 // index * (dataset.length - 1) / scale for each index in indexes. For each, if there is no 416 // remainder, we can just select the value whose index in the sorted dataset equals the 417 // quotient; if there is a remainder, we interpolate between that and the next value. 418 419 int[] quotients = new int[indexes.length]; 420 int[] remainders = new int[indexes.length]; 421 // The indexes to select. In the worst case, we'll need one each side of each quantile. 422 int[] requiredSelections = new int[indexes.length * 2]; 423 int requiredSelectionsCount = 0; 424 for (int i = 0; i < indexes.length; i++) { 425 // Since index and (dataset.length - 1) are non-negative ints, their product can be 426 // expressed as a long, without risk of overflow: 427 long numerator = (long) indexes[i] * (dataset.length - 1); 428 // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a 429 // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to 430 // get a rounded ratio and a remainder which can be expressed as ints, without risk of 431 // overflow: 432 int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN); 433 int remainder = (int) (numerator - (long) quotient * scale); 434 quotients[i] = quotient; 435 remainders[i] = remainder; 436 requiredSelections[requiredSelectionsCount] = quotient; 437 requiredSelectionsCount++; 438 if (remainder != 0) { 439 requiredSelections[requiredSelectionsCount] = quotient + 1; 440 requiredSelectionsCount++; 441 } 442 } 443 sort(requiredSelections, 0, requiredSelectionsCount); 444 selectAllInPlace( 445 requiredSelections, 0, requiredSelectionsCount - 1, dataset, 0, dataset.length - 1); 446 Map<Integer, Double> ret = new LinkedHashMap<>(); 447 for (int i = 0; i < indexes.length; i++) { 448 int quotient = quotients[i]; 449 int remainder = remainders[i]; 450 if (remainder == 0) { 451 ret.put(indexes[i], dataset[quotient]); 452 } else { 453 ret.put( 454 indexes[i], interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale)); 455 } 456 } 457 return unmodifiableMap(ret); 458 } 459 } 460 461 /** Returns whether any of the values in {@code dataset} are {@code NaN}. */ 462 private static boolean containsNaN(double... dataset) { 463 for (double value : dataset) { 464 if (Double.isNaN(value)) { 465 return true; 466 } 467 } 468 return false; 469 } 470 471 /** 472 * Returns a value a fraction {@code (remainder / scale)} of the way between {@code lower} and 473 * {@code upper}. Assumes that {@code lower <= upper}. Correctly handles infinities (but not 474 * {@code NaN}). 475 */ 476 private static double interpolate(double lower, double upper, double remainder, double scale) { 477 if (lower == NEGATIVE_INFINITY) { 478 if (upper == POSITIVE_INFINITY) { 479 // Return NaN when lower == NEGATIVE_INFINITY and upper == POSITIVE_INFINITY: 480 return NaN; 481 } 482 // Return NEGATIVE_INFINITY when NEGATIVE_INFINITY == lower <= upper < POSITIVE_INFINITY: 483 return NEGATIVE_INFINITY; 484 } 485 if (upper == POSITIVE_INFINITY) { 486 // Return POSITIVE_INFINITY when NEGATIVE_INFINITY < lower <= upper == POSITIVE_INFINITY: 487 return POSITIVE_INFINITY; 488 } 489 return lower + (upper - lower) * remainder / scale; 490 } 491 492 private static void checkIndex(int index, int scale) { 493 if (index < 0 || index > scale) { 494 throw new IllegalArgumentException( 495 "Quantile indexes must be between 0 and the scale, which is " + scale); 496 } 497 } 498 499 private static double[] longsToDoubles(long[] longs) { 500 int len = longs.length; 501 double[] doubles = new double[len]; 502 for (int i = 0; i < len; i++) { 503 doubles[i] = longs[i]; 504 } 505 return doubles; 506 } 507 508 private static double[] intsToDoubles(int[] ints) { 509 int len = ints.length; 510 double[] doubles = new double[len]; 511 for (int i = 0; i < len; i++) { 512 doubles[i] = ints[i]; 513 } 514 return doubles; 515 } 516 517 /** 518 * Performs an in-place selection to find the element which would appear at a given index in a 519 * dataset if it were sorted. The following preconditions should hold: 520 * 521 * <ul> 522 * <li>{@code required}, {@code from}, and {@code to} should all be indexes into {@code array}; 523 * <li>{@code required} should be in the range [{@code from}, {@code to}]; 524 * <li>all the values with indexes in the range [0, {@code from}) should be less than or equal 525 * to all the values with indexes in the range [{@code from}, {@code to}]; 526 * <li>all the values with indexes in the range ({@code to}, {@code array.length - 1}] should be 527 * greater than or equal to all the values with indexes in the range [{@code from}, {@code 528 * to}]. 529 * </ul> 530 * 531 * This method will reorder the values with indexes in the range [{@code from}, {@code to}] such 532 * that all the values with indexes in the range [{@code from}, {@code required}) are less than or 533 * equal to the value with index {@code required}, and all the values with indexes in the range 534 * ({@code required}, {@code to}] are greater than or equal to that value. Therefore, the value at 535 * {@code required} is the value which would appear at that index in the sorted dataset. 536 */ 537 private static void selectInPlace(int required, double[] array, int from, int to) { 538 // If we are looking for the least element in the range, we can just do a linear search for it. 539 // (We will hit this whenever we are doing quantile interpolation: our first selection finds 540 // the lower value, our second one finds the upper value by looking for the next least element.) 541 if (required == from) { 542 int min = from; 543 for (int index = from + 1; index <= to; index++) { 544 if (array[min] > array[index]) { 545 min = index; 546 } 547 } 548 if (min != from) { 549 swap(array, min, from); 550 } 551 return; 552 } 553 554 // Let's play quickselect! We'll repeatedly partition the range [from, to] containing the 555 // required element, as long as it has more than one element. 556 while (to > from) { 557 int partitionPoint = partition(array, from, to); 558 if (partitionPoint >= required) { 559 to = partitionPoint - 1; 560 } 561 if (partitionPoint <= required) { 562 from = partitionPoint + 1; 563 } 564 } 565 } 566 567 /** 568 * Performs a partition operation on the slice of {@code array} with elements in the range [{@code 569 * from}, {@code to}]. Uses the median of {@code from}, {@code to}, and the midpoint between them 570 * as a pivot. Returns the index which the slice is partitioned around, i.e. if it returns {@code 571 * ret} then we know that the values with indexes in [{@code from}, {@code ret}) are less than or 572 * equal to the value at {@code ret} and the values with indexes in ({@code ret}, {@code to}] are 573 * greater than or equal to that. 574 */ 575 private static int partition(double[] array, int from, int to) { 576 // Select a pivot, and move it to the start of the slice i.e. to index from. 577 movePivotToStartOfSlice(array, from, to); 578 double pivot = array[from]; 579 580 // Move all elements with indexes in (from, to] which are greater than the pivot to the end of 581 // the array. Keep track of where those elements begin. 582 int partitionPoint = to; 583 for (int i = to; i > from; i--) { 584 if (array[i] > pivot) { 585 swap(array, partitionPoint, i); 586 partitionPoint--; 587 } 588 } 589 590 // We now know that all elements with indexes in (from, partitionPoint] are less than or equal 591 // to the pivot at from, and all elements with indexes in (partitionPoint, to] are greater than 592 // it. We swap the pivot into partitionPoint and we know the array is partitioned around that. 593 swap(array, from, partitionPoint); 594 return partitionPoint; 595 } 596 597 /** 598 * Selects the pivot to use, namely the median of the values at {@code from}, {@code to}, and 599 * halfway between the two (rounded down), from {@code array}, and ensure (by swapping elements if 600 * necessary) that that pivot value appears at the start of the slice i.e. at {@code from}. 601 * Expects that {@code from} is strictly less than {@code to}. 602 */ 603 private static void movePivotToStartOfSlice(double[] array, int from, int to) { 604 int mid = (from + to) >>> 1; 605 // We want to make a swap such that either array[to] <= array[from] <= array[mid], or 606 // array[mid] <= array[from] <= array[to]. We know that from < to, so we know mid < to 607 // (although it's possible that mid == from, if to == from + 1). Note that the postcondition 608 // would be impossible to fulfil if mid == to unless we also have array[from] == array[to]. 609 boolean toLessThanMid = (array[to] < array[mid]); 610 boolean midLessThanFrom = (array[mid] < array[from]); 611 boolean toLessThanFrom = (array[to] < array[from]); 612 if (toLessThanMid == midLessThanFrom) { 613 // Either array[to] < array[mid] < array[from] or array[from] <= array[mid] <= array[to]. 614 swap(array, mid, from); 615 } else if (toLessThanMid != toLessThanFrom) { 616 // Either array[from] <= array[to] < array[mid] or array[mid] <= array[to] < array[from]. 617 swap(array, from, to); 618 } 619 // The postcondition now holds. So the median, our chosen pivot, is at from. 620 } 621 622 /** 623 * Performs an in-place selection, like {@link #selectInPlace}, to select all the indexes {@code 624 * allRequired[i]} for {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. These 625 * indexes must be sorted in the array and must all be in the range [{@code from}, {@code to}]. 626 */ 627 private static void selectAllInPlace( 628 int[] allRequired, int requiredFrom, int requiredTo, double[] array, int from, int to) { 629 // Choose the first selection to do... 630 int requiredChosen = chooseNextSelection(allRequired, requiredFrom, requiredTo, from, to); 631 int required = allRequired[requiredChosen]; 632 633 // ...do the first selection... 634 selectInPlace(required, array, from, to); 635 636 // ...then recursively perform the selections in the range below... 637 int requiredBelow = requiredChosen - 1; 638 while (requiredBelow >= requiredFrom && allRequired[requiredBelow] == required) { 639 requiredBelow--; // skip duplicates of required in the range below 640 } 641 if (requiredBelow >= requiredFrom) { 642 selectAllInPlace(allRequired, requiredFrom, requiredBelow, array, from, required - 1); 643 } 644 645 // ...and then recursively perform the selections in the range above. 646 int requiredAbove = requiredChosen + 1; 647 while (requiredAbove <= requiredTo && allRequired[requiredAbove] == required) { 648 requiredAbove++; // skip duplicates of required in the range above 649 } 650 if (requiredAbove <= requiredTo) { 651 selectAllInPlace(allRequired, requiredAbove, requiredTo, array, required + 1, to); 652 } 653 } 654 655 /** 656 * Chooses the next selection to do from the required selections. It is required that the array 657 * {@code allRequired} is sorted and that {@code allRequired[i]} are in the range [{@code from}, 658 * {@code to}] for all {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. The 659 * value returned by this method is the {@code i} in that range such that {@code allRequired[i]} 660 * is as close as possible to the center of the range [{@code from}, {@code to}]. Choosing the 661 * value closest to the center of the range first is the most efficient strategy because it 662 * minimizes the size of the subranges from which the remaining selections must be done. 663 */ 664 private static int chooseNextSelection( 665 int[] allRequired, int requiredFrom, int requiredTo, int from, int to) { 666 if (requiredFrom == requiredTo) { 667 return requiredFrom; // only one thing to choose, so choose it 668 } 669 670 // Find the center and round down. The true center is either centerFloor or halfway between 671 // centerFloor and centerFloor + 1. 672 int centerFloor = (from + to) >>> 1; 673 674 // Do a binary search until we're down to the range of two which encloses centerFloor (unless 675 // all values are lower or higher than centerFloor, in which case we find the two highest or 676 // lowest respectively). If centerFloor is in allRequired, we will definitely find it. If not, 677 // but centerFloor + 1 is, we'll definitely find that. The closest value to the true (unrounded) 678 // center will be at either low or high. 679 int low = requiredFrom; 680 int high = requiredTo; 681 while (high > low + 1) { 682 int mid = (low + high) >>> 1; 683 if (allRequired[mid] > centerFloor) { 684 high = mid; 685 } else if (allRequired[mid] < centerFloor) { 686 low = mid; 687 } else { 688 return mid; // allRequired[mid] = centerFloor, so we can't get closer than that 689 } 690 } 691 692 // Now pick the closest of the two candidates. Note that there is no rounding here. 693 if (from + to - allRequired[low] - allRequired[high] > 0) { 694 return high; 695 } else { 696 return low; 697 } 698 } 699 700 /** Swaps the values at {@code i} and {@code j} in {@code array}. */ 701 private static void swap(double[] array, int i, int j) { 702 double temp = array[i]; 703 array[i] = array[j]; 704 array[j] = temp; 705 } 706}