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 * <h2>Examples</h2> 038 * 039 * <p>To compute the median: 040 * 041 * {@snippet : 042 * double myMedian = median().compute(myDataset); 043 * } 044 * 045 * where {@link #median()} has been statically imported. 046 * 047 * <p>To compute the 99th percentile: 048 * 049 * {@snippet : 050 * double myPercentile99 = percentiles().index(99).compute(myDataset); 051 * } 052 * 053 * where {@link #percentiles()} has been statically imported. 054 * 055 * <p>To compute median and the 90th and 99th percentiles: 056 * 057 * {@snippet : 058 * Map<Integer, Double> myPercentiles = percentiles().indexes(50, 90, 99).compute(myDataset); 059 * } 060 * 061 * where {@link #percentiles()} has been statically imported: {@code myPercentiles} maps the keys 062 * 50, 90, and 99, to their corresponding quantile values. 063 * 064 * <p>To compute quartiles, use {@link #quartiles()} instead of {@link #percentiles()}. To compute 065 * arbitrary q-quantiles, use {@link #scale scale(q)}. 066 * 067 * <p>These examples all take a copy of your dataset. If you have a double array, you are okay with 068 * it being arbitrarily reordered, and you want to avoid that copy, you can use {@code 069 * computeInPlace} instead of {@code compute}. 070 * 071 * <h2>Definition and notes on interpolation</h2> 072 * 073 * <p>The definition of the kth q-quantile of N values is as follows: define x = k * (N - 1) / q; if 074 * x is an integer, the result is the value which would appear at index x in the sorted dataset 075 * (unless there are {@link Double#NaN NaN} values, see below); otherwise, the result is the average 076 * of the values which would appear at the indexes floor(x) and ceil(x) weighted by (1-frac(x)) and 077 * frac(x) respectively. This is the same definition as used by Excel and by S, it is the Type 7 078 * definition in <a 079 * href="http://stat.ethz.ch/R-manual/R-devel/library/stats/html/quantile.html">R</a>, and it is 080 * described by <a 081 * href="http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population"> 082 * wikipedia</a> as providing "Linear interpolation of the modes for the order statistics for the 083 * uniform distribution on [0,1]." 084 * 085 * <h2>Handling of non-finite values</h2> 086 * 087 * <p>If any values in the input are {@link Double#NaN NaN} then all values returned are {@link 088 * Double#NaN NaN}. (This is the one occasion when the behaviour is not the same as you'd get from 089 * sorting with {@link java.util.Arrays#sort(double[]) Arrays.sort(double[])} or {@link 090 * java.util.Collections#sort(java.util.List) Collections.sort(List<Double>)} and selecting 091 * the required value(s). Those methods would sort {@link Double#NaN NaN} as if it is greater than 092 * any other value and place them at the end of the dataset, even after {@link 093 * Double#POSITIVE_INFINITY POSITIVE_INFINITY}.) 094 * 095 * <p>Otherwise, {@link Double#NEGATIVE_INFINITY NEGATIVE_INFINITY} and {@link 096 * Double#POSITIVE_INFINITY POSITIVE_INFINITY} sort to the beginning and the end of the dataset, as 097 * you would expect. 098 * 099 * <p>If required to do a weighted average between an infinity and a finite value, or between an 100 * infinite value and itself, the infinite value is returned. If required to do a weighted average 101 * between {@link Double#NEGATIVE_INFINITY NEGATIVE_INFINITY} and {@link Double#POSITIVE_INFINITY 102 * POSITIVE_INFINITY}, {@link Double#NaN NaN} is returned (note that this will only happen if the 103 * dataset contains no finite values). 104 * 105 * <h2>Performance</h2> 106 * 107 * <p>The average time complexity of the computation is O(N) in the size of the dataset. There is a 108 * worst case time complexity of O(N^2). You are extremely unlikely to hit this quadratic case on 109 * randomly ordered data (the probability decreases faster than exponentially in N), but if you are 110 * passing in unsanitized user data then a malicious user could force it. A light shuffle of the 111 * data using an unpredictable seed should normally be enough to thwart this attack. 112 * 113 * <p>The time taken to compute multiple quantiles on the same dataset using {@link Scale#indexes 114 * indexes} is generally less than the total time taken to compute each of them separately, and 115 * sometimes much less. For example, on a large enough dataset, computing the 90th and 99th 116 * percentiles together takes about 55% as long as computing them separately. 117 * 118 * <p>When calling {@link ScaleAndIndex#compute} (in {@linkplain ScaleAndIndexes#compute either 119 * form}), the memory requirement is 8*N bytes for the copy of the dataset plus an overhead which is 120 * independent of N (but depends on the quantiles being computed). When calling {@link 121 * ScaleAndIndex#computeInPlace computeInPlace} (in {@linkplain ScaleAndIndexes#computeInPlace 122 * either form}), only the overhead is required. The number of object allocations is independent of 123 * N in both cases. 124 * 125 * @author Pete Gillin 126 * @since 20.0 127 */ 128@J2ktIncompatible 129@GwtIncompatible 130public final class Quantiles { 131 /** 132 * Constructor for a type that is not meant to be instantiated. 133 * 134 * @deprecated Use the static factory methods of the class. There is no reason to create an 135 * instance of {@link Quantiles}. 136 */ 137 @Deprecated 138 public Quantiles() {} 139 140 /** Specifies the computation of a median (i.e. the 1st 2-quantile). */ 141 public static ScaleAndIndex median() { 142 return scale(2).index(1); 143 } 144 145 /** Specifies the computation of quartiles (i.e. 4-quantiles). */ 146 public static Scale quartiles() { 147 return scale(4); 148 } 149 150 /** Specifies the computation of percentiles (i.e. 100-quantiles). */ 151 public static Scale percentiles() { 152 return scale(100); 153 } 154 155 /** 156 * Specifies the computation of q-quantiles. 157 * 158 * @param scale the scale for the quantiles to be calculated, i.e. the q of the q-quantiles, which 159 * must be positive 160 */ 161 public static Scale scale(int scale) { 162 return new Scale(scale); 163 } 164 165 /** 166 * Describes the point in a fluent API chain where only the scale (i.e. the q in q-quantiles) has 167 * been specified. 168 * 169 * @since 20.0 170 */ 171 public static final class Scale { 172 173 private final int scale; 174 175 private Scale(int scale) { 176 checkArgument(scale > 0, "Quantile scale must be positive"); 177 this.scale = scale; 178 } 179 180 /** 181 * Specifies a single quantile index to be calculated, i.e. the k in the kth q-quantile. 182 * 183 * @param index the quantile index, which must be in the inclusive range [0, q] for q-quantiles 184 */ 185 public ScaleAndIndex index(int index) { 186 return new ScaleAndIndex(scale, index); 187 } 188 189 /** 190 * Specifies multiple quantile indexes to be calculated, each index being the k in the kth 191 * q-quantile. 192 * 193 * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for 194 * q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the 195 * set will be snapshotted when this method is called 196 * @throws IllegalArgumentException if {@code indexes} is empty 197 */ 198 public ScaleAndIndexes indexes(int... indexes) { 199 return new ScaleAndIndexes(scale, indexes.clone()); 200 } 201 202 /** 203 * Specifies multiple quantile indexes to be calculated, each index being the k in the kth 204 * q-quantile. 205 * 206 * @param indexes the quantile indexes, each of which must be in the inclusive range [0, q] for 207 * q-quantiles; the order of the indexes is unimportant, duplicates will be ignored, and the 208 * set will be snapshotted when this method is called 209 * @throws IllegalArgumentException if {@code indexes} is empty 210 */ 211 public ScaleAndIndexes indexes(Collection<Integer> indexes) { 212 return new ScaleAndIndexes(scale, Ints.toArray(indexes)); 213 } 214 } 215 216 /** 217 * Describes the point in a fluent API chain where the scale and a single quantile index (i.e. the 218 * q and the k in the kth q-quantile) have been specified. 219 * 220 * @since 20.0 221 */ 222 public static final class ScaleAndIndex { 223 224 private final int scale; 225 private final int index; 226 227 private ScaleAndIndex(int scale, int index) { 228 checkIndex(index, scale); 229 this.scale = scale; 230 this.index = index; 231 } 232 233 /** 234 * Computes the quantile value of the given dataset. 235 * 236 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 237 * cast to doubles (with any associated lost of precision), and which will not be mutated by 238 * this call (it is copied instead) 239 * @return the quantile value 240 */ 241 public double compute(Collection<? extends Number> dataset) { 242 return computeInPlace(Doubles.toArray(dataset)); 243 } 244 245 /** 246 * Computes the quantile value of the given dataset. 247 * 248 * @param dataset the dataset to do the calculation on, which must be non-empty, which will not 249 * be mutated by this call (it is copied instead) 250 * @return the quantile value 251 */ 252 public double compute(double... dataset) { 253 return computeInPlace(dataset.clone()); 254 } 255 256 /** 257 * Computes the quantile value of the given dataset. 258 * 259 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 260 * cast to doubles (with any associated lost of precision), and which will not be mutated by 261 * this call (it is copied instead) 262 * @return the quantile value 263 */ 264 public double compute(long... dataset) { 265 return computeInPlace(longsToDoubles(dataset)); 266 } 267 268 /** 269 * Computes the quantile value of the given dataset. 270 * 271 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 272 * cast to doubles, and which will not be mutated by this call (it is copied instead) 273 * @return the quantile value 274 */ 275 public double compute(int... dataset) { 276 return computeInPlace(intsToDoubles(dataset)); 277 } 278 279 /** 280 * Computes the quantile value of the given dataset, performing the computation in-place. 281 * 282 * @param dataset the dataset to do the calculation on, which must be non-empty, and which will 283 * be arbitrarily reordered by this method call 284 * @return the quantile value 285 */ 286 public double computeInPlace(double... dataset) { 287 checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset"); 288 if (containsNaN(dataset)) { 289 return NaN; 290 } 291 292 // Calculate the quotient and remainder in the integer division x = k * (N-1) / q, i.e. 293 // index * (dataset.length - 1) / scale. If there is no remainder, we can just find the value 294 // whose index in the sorted dataset equals the quotient; if there is a remainder, we 295 // interpolate between that and the next value. 296 297 // Since index and (dataset.length - 1) are non-negative ints, their product can be expressed 298 // as a long, without risk of overflow: 299 long numerator = (long) index * (dataset.length - 1); 300 // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a 301 // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to get 302 // a rounded ratio and a remainder which can be expressed as ints, without risk of overflow: 303 int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN); 304 int remainder = (int) (numerator - (long) quotient * scale); 305 selectInPlace(quotient, dataset, 0, dataset.length - 1); 306 if (remainder == 0) { 307 return dataset[quotient]; 308 } else { 309 selectInPlace(quotient + 1, dataset, quotient + 1, dataset.length - 1); 310 return interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale); 311 } 312 } 313 } 314 315 /** 316 * Describes the point in a fluent API chain where the scale and a multiple quantile indexes (i.e. 317 * the q and a set of values for the k in the kth q-quantile) have been specified. 318 * 319 * @since 20.0 320 */ 321 public static final class ScaleAndIndexes { 322 323 private final int scale; 324 private final int[] indexes; 325 326 private ScaleAndIndexes(int scale, int[] indexes) { 327 for (int index : indexes) { 328 checkIndex(index, scale); 329 } 330 checkArgument(indexes.length > 0, "Indexes must be a non empty array"); 331 this.scale = scale; 332 this.indexes = indexes; 333 } 334 335 /** 336 * Computes the quantile values of the given dataset. 337 * 338 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 339 * cast to doubles (with any associated lost of precision), and which will not be mutated by 340 * this call (it is copied instead) 341 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 342 * indexes, and the values the corresponding quantile values. When iterating, entries in the 343 * map are ordered by quantile index in the same order they were passed to the {@code 344 * indexes} method. 345 */ 346 public Map<Integer, Double> compute(Collection<? extends Number> dataset) { 347 return computeInPlace(Doubles.toArray(dataset)); 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 not 354 * be mutated by this call (it is copied instead) 355 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 356 * indexes, and the values the corresponding quantile values. When iterating, entries in the 357 * map are ordered by quantile index in the same order they were passed to the {@code 358 * indexes} method. 359 */ 360 public Map<Integer, Double> compute(double... dataset) { 361 return computeInPlace(dataset.clone()); 362 } 363 364 /** 365 * Computes the quantile values of the given dataset. 366 * 367 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 368 * cast to doubles (with any associated lost of precision), and which will not be mutated by 369 * this call (it is copied instead) 370 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 371 * indexes, and the values the corresponding quantile values. When iterating, entries in the 372 * map are ordered by quantile index in the same order they were passed to the {@code 373 * indexes} method. 374 */ 375 public Map<Integer, Double> compute(long... dataset) { 376 return computeInPlace(longsToDoubles(dataset)); 377 } 378 379 /** 380 * Computes the quantile values of the given dataset. 381 * 382 * @param dataset the dataset to do the calculation on, which must be non-empty, which will be 383 * cast to doubles, and which will not be mutated by this call (it is copied instead) 384 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 385 * indexes, and the values the corresponding quantile values. When iterating, entries in the 386 * map are ordered by quantile index in the same order they were passed to the {@code 387 * indexes} method. 388 */ 389 public Map<Integer, Double> compute(int... dataset) { 390 return computeInPlace(intsToDoubles(dataset)); 391 } 392 393 /** 394 * Computes the quantile values of the given dataset, performing the computation in-place. 395 * 396 * @param dataset the dataset to do the calculation on, which must be non-empty, and which will 397 * be arbitrarily reordered by this method call 398 * @return an unmodifiable, ordered map of results: the keys will be the specified quantile 399 * indexes, and the values the corresponding quantile values. When iterating, entries in the 400 * map are ordered by quantile index in the same order that the indexes were passed to the 401 * {@code indexes} method. 402 */ 403 public Map<Integer, Double> computeInPlace(double... dataset) { 404 checkArgument(dataset.length > 0, "Cannot calculate quantiles of an empty dataset"); 405 if (containsNaN(dataset)) { 406 Map<Integer, Double> nanMap = new LinkedHashMap<>(); 407 for (int index : indexes) { 408 nanMap.put(index, NaN); 409 } 410 return unmodifiableMap(nanMap); 411 } 412 413 // Calculate the quotients and remainders in the integer division x = k * (N - 1) / q, i.e. 414 // index * (dataset.length - 1) / scale for each index in indexes. For each, if there is no 415 // remainder, we can just select the value whose index in the sorted dataset equals the 416 // quotient; if there is a remainder, we interpolate between that and the next value. 417 418 int[] quotients = new int[indexes.length]; 419 int[] remainders = new int[indexes.length]; 420 // The indexes to select. In the worst case, we'll need one each side of each quantile. 421 int[] requiredSelections = new int[indexes.length * 2]; 422 int requiredSelectionsCount = 0; 423 for (int i = 0; i < indexes.length; i++) { 424 // Since index and (dataset.length - 1) are non-negative ints, their product can be 425 // expressed as a long, without risk of overflow: 426 long numerator = (long) indexes[i] * (dataset.length - 1); 427 // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a 428 // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to 429 // get a rounded ratio and a remainder which can be expressed as ints, without risk of 430 // overflow: 431 int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN); 432 int remainder = (int) (numerator - (long) quotient * scale); 433 quotients[i] = quotient; 434 remainders[i] = remainder; 435 requiredSelections[requiredSelectionsCount] = quotient; 436 requiredSelectionsCount++; 437 if (remainder != 0) { 438 requiredSelections[requiredSelectionsCount] = quotient + 1; 439 requiredSelectionsCount++; 440 } 441 } 442 sort(requiredSelections, 0, requiredSelectionsCount); 443 selectAllInPlace( 444 requiredSelections, 0, requiredSelectionsCount - 1, dataset, 0, dataset.length - 1); 445 Map<Integer, Double> ret = new LinkedHashMap<>(); 446 for (int i = 0; i < indexes.length; i++) { 447 int quotient = quotients[i]; 448 int remainder = remainders[i]; 449 if (remainder == 0) { 450 ret.put(indexes[i], dataset[quotient]); 451 } else { 452 ret.put( 453 indexes[i], interpolate(dataset[quotient], dataset[quotient + 1], remainder, scale)); 454 } 455 } 456 return unmodifiableMap(ret); 457 } 458 } 459 460 /** Returns whether any of the values in {@code dataset} are {@code NaN}. */ 461 private static boolean containsNaN(double... dataset) { 462 for (double value : dataset) { 463 if (Double.isNaN(value)) { 464 return true; 465 } 466 } 467 return false; 468 } 469 470 /** 471 * Returns a value a fraction {@code (remainder / scale)} of the way between {@code lower} and 472 * {@code upper}. Assumes that {@code lower <= upper}. Correctly handles infinities (but not 473 * {@code NaN}). 474 */ 475 private static double interpolate(double lower, double upper, double remainder, double scale) { 476 if (lower == NEGATIVE_INFINITY) { 477 if (upper == POSITIVE_INFINITY) { 478 // Return NaN when lower == NEGATIVE_INFINITY and upper == POSITIVE_INFINITY: 479 return NaN; 480 } 481 // Return NEGATIVE_INFINITY when NEGATIVE_INFINITY == lower <= upper < POSITIVE_INFINITY: 482 return NEGATIVE_INFINITY; 483 } 484 if (upper == POSITIVE_INFINITY) { 485 // Return POSITIVE_INFINITY when NEGATIVE_INFINITY < lower <= upper == POSITIVE_INFINITY: 486 return POSITIVE_INFINITY; 487 } 488 return lower + (upper - lower) * remainder / scale; 489 } 490 491 private static void checkIndex(int index, int scale) { 492 checkArgument( 493 index >= 0 && index <= scale, 494 "Quantile indexes must be between 0 and the scale, which is %s", 495 scale); 496 } 497 498 private static double[] longsToDoubles(long[] longs) { 499 int len = longs.length; 500 double[] doubles = new double[len]; 501 for (int i = 0; i < len; i++) { 502 doubles[i] = longs[i]; 503 } 504 return doubles; 505 } 506 507 private static double[] intsToDoubles(int[] ints) { 508 int len = ints.length; 509 double[] doubles = new double[len]; 510 for (int i = 0; i < len; i++) { 511 doubles[i] = ints[i]; 512 } 513 return doubles; 514 } 515 516 /** 517 * Performs an in-place selection to find the element which would appear at a given index in a 518 * dataset if it were sorted. The following preconditions should hold: 519 * 520 * <ul> 521 * <li>{@code required}, {@code from}, and {@code to} should all be indexes into {@code array}; 522 * <li>{@code required} should be in the range [{@code from}, {@code to}]; 523 * <li>all the values with indexes in the range [0, {@code from}) should be less than or equal 524 * to all the values with indexes in the range [{@code from}, {@code to}]; 525 * <li>all the values with indexes in the range ({@code to}, {@code array.length - 1}] should be 526 * greater than or equal to all the values with indexes in the range [{@code from}, {@code 527 * to}]. 528 * </ul> 529 * 530 * This method will reorder the values with indexes in the range [{@code from}, {@code to}] such 531 * that all the values with indexes in the range [{@code from}, {@code required}) are less than or 532 * equal to the value with index {@code required}, and all the values with indexes in the range 533 * ({@code required}, {@code to}] are greater than or equal to that value. Therefore, the value at 534 * {@code required} is the value which would appear at that index in the sorted dataset. 535 */ 536 private static void selectInPlace(int required, double[] array, int from, int to) { 537 // If we are looking for the least element in the range, we can just do a linear search for it. 538 // (We will hit this whenever we are doing quantile interpolation: our first selection finds 539 // the lower value, our second one finds the upper value by looking for the next least element.) 540 if (required == from) { 541 int min = from; 542 for (int index = from + 1; index <= to; index++) { 543 if (array[min] > array[index]) { 544 min = index; 545 } 546 } 547 if (min != from) { 548 swap(array, min, from); 549 } 550 return; 551 } 552 553 // Let's play quickselect! We'll repeatedly partition the range [from, to] containing the 554 // required element, as long as it has more than one element. 555 while (to > from) { 556 int partitionPoint = partition(array, from, to); 557 if (partitionPoint >= required) { 558 to = partitionPoint - 1; 559 } 560 if (partitionPoint <= required) { 561 from = partitionPoint + 1; 562 } 563 } 564 } 565 566 /** 567 * Performs a partition operation on the slice of {@code array} with elements in the range [{@code 568 * from}, {@code to}]. Uses the median of {@code from}, {@code to}, and the midpoint between them 569 * as a pivot. Returns the index which the slice is partitioned around, i.e. if it returns {@code 570 * ret} then we know that the values with indexes in [{@code from}, {@code ret}) are less than or 571 * equal to the value at {@code ret} and the values with indexes in ({@code ret}, {@code to}] are 572 * greater than or equal to that. 573 */ 574 private static int partition(double[] array, int from, int to) { 575 // Select a pivot, and move it to the start of the slice i.e. to index from. 576 movePivotToStartOfSlice(array, from, to); 577 double pivot = array[from]; 578 579 // Move all elements with indexes in (from, to] which are greater than the pivot to the end of 580 // the array. Keep track of where those elements begin. 581 int partitionPoint = to; 582 for (int i = to; i > from; i--) { 583 if (array[i] > pivot) { 584 swap(array, partitionPoint, i); 585 partitionPoint--; 586 } 587 } 588 589 // We now know that all elements with indexes in (from, partitionPoint] are less than or equal 590 // to the pivot at from, and all elements with indexes in (partitionPoint, to] are greater than 591 // it. We swap the pivot into partitionPoint and we know the array is partitioned around that. 592 swap(array, from, partitionPoint); 593 return partitionPoint; 594 } 595 596 /** 597 * Selects the pivot to use, namely the median of the values at {@code from}, {@code to}, and 598 * halfway between the two (rounded down), from {@code array}, and ensure (by swapping elements if 599 * necessary) that that pivot value appears at the start of the slice i.e. at {@code from}. 600 * Expects that {@code from} is strictly less than {@code to}. 601 */ 602 private static void movePivotToStartOfSlice(double[] array, int from, int to) { 603 int mid = (from + to) >>> 1; 604 // We want to make a swap such that either array[to] <= array[from] <= array[mid], or 605 // array[mid] <= array[from] <= array[to]. We know that from < to, so we know mid < to 606 // (although it's possible that mid == from, if to == from + 1). Note that the postcondition 607 // would be impossible to fulfil if mid == to unless we also have array[from] == array[to]. 608 boolean toLessThanMid = (array[to] < array[mid]); 609 boolean midLessThanFrom = (array[mid] < array[from]); 610 boolean toLessThanFrom = (array[to] < array[from]); 611 if (toLessThanMid == midLessThanFrom) { 612 // Either array[to] < array[mid] < array[from] or array[from] <= array[mid] <= array[to]. 613 swap(array, mid, from); 614 } else if (toLessThanMid != toLessThanFrom) { 615 // Either array[from] <= array[to] < array[mid] or array[mid] <= array[to] < array[from]. 616 swap(array, from, to); 617 } 618 // The postcondition now holds. So the median, our chosen pivot, is at from. 619 } 620 621 /** 622 * Performs an in-place selection, like {@link #selectInPlace}, to select all the indexes {@code 623 * allRequired[i]} for {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. These 624 * indexes must be sorted in the array and must all be in the range [{@code from}, {@code to}]. 625 */ 626 private static void selectAllInPlace( 627 int[] allRequired, int requiredFrom, int requiredTo, double[] array, int from, int to) { 628 // Choose the first selection to do... 629 int requiredChosen = chooseNextSelection(allRequired, requiredFrom, requiredTo, from, to); 630 int required = allRequired[requiredChosen]; 631 632 // ...do the first selection... 633 selectInPlace(required, array, from, to); 634 635 // ...then recursively perform the selections in the range below... 636 int requiredBelow = requiredChosen - 1; 637 while (requiredBelow >= requiredFrom && allRequired[requiredBelow] == required) { 638 requiredBelow--; // skip duplicates of required in the range below 639 } 640 if (requiredBelow >= requiredFrom) { 641 selectAllInPlace(allRequired, requiredFrom, requiredBelow, array, from, required - 1); 642 } 643 644 // ...and then recursively perform the selections in the range above. 645 int requiredAbove = requiredChosen + 1; 646 while (requiredAbove <= requiredTo && allRequired[requiredAbove] == required) { 647 requiredAbove++; // skip duplicates of required in the range above 648 } 649 if (requiredAbove <= requiredTo) { 650 selectAllInPlace(allRequired, requiredAbove, requiredTo, array, required + 1, to); 651 } 652 } 653 654 /** 655 * Chooses the next selection to do from the required selections. It is required that the array 656 * {@code allRequired} is sorted and that {@code allRequired[i]} are in the range [{@code from}, 657 * {@code to}] for all {@code i} in the range [{@code requiredFrom}, {@code requiredTo}]. The 658 * value returned by this method is the {@code i} in that range such that {@code allRequired[i]} 659 * is as close as possible to the center of the range [{@code from}, {@code to}]. Choosing the 660 * value closest to the center of the range first is the most efficient strategy because it 661 * minimizes the size of the subranges from which the remaining selections must be done. 662 */ 663 private static int chooseNextSelection( 664 int[] allRequired, int requiredFrom, int requiredTo, int from, int to) { 665 if (requiredFrom == requiredTo) { 666 return requiredFrom; // only one thing to choose, so choose it 667 } 668 669 // Find the center and round down. The true center is either centerFloor or halfway between 670 // centerFloor and centerFloor + 1. 671 int centerFloor = (from + to) >>> 1; 672 673 // Do a binary search until we're down to the range of two which encloses centerFloor (unless 674 // all values are lower or higher than centerFloor, in which case we find the two highest or 675 // lowest respectively). If centerFloor is in allRequired, we will definitely find it. If not, 676 // but centerFloor + 1 is, we'll definitely find that. The closest value to the true (unrounded) 677 // center will be at either low or high. 678 int low = requiredFrom; 679 int high = requiredTo; 680 while (high > low + 1) { 681 int mid = (low + high) >>> 1; 682 if (allRequired[mid] > centerFloor) { 683 high = mid; 684 } else if (allRequired[mid] < centerFloor) { 685 low = mid; 686 } else { 687 return mid; // allRequired[mid] = centerFloor, so we can't get closer than that 688 } 689 } 690 691 // Now pick the closest of the two candidates. Note that there is no rounding here. 692 if (from + to - allRequired[low] - allRequired[high] > 0) { 693 return high; 694 } else { 695 return low; 696 } 697 } 698 699 /** Swaps the values at {@code i} and {@code j} in {@code array}. */ 700 private static void swap(double[] array, int i, int j) { 701 double temp = array[i]; 702 array[i] = array[j]; 703 array[j] = temp; 704 } 705}