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