使用模擬退火演算法優化 Hash 函式
阿新 • • 發佈：2020-10-04
# 背景
現有個處理股票行情訊息的系統，其架構如下：
symbolAndCount = new TreeMap<>();
for (int i=0; i distribution = findBestDistribution(symbolAndCount);
// 測試效果
int[] buckets = new int[NUM_OF_BUCKETS];
for (Map.Entry entry : symbolAndCount.entrySet()) {
Map.Entry floor = distribution.floorEntry(entry.getKey());
int bucketIndex = floor == null ? 0 : floor.getValue();
buckets[bucketIndex] += entry.getValue();
}
System.out.printf("buckets: %s\n", Arrays.toString(buckets));
}
public static TreeMap findBestDistribution(Map symbolAndCount) {
// 每個桶均勻分佈的情況（最優情況）
int avg = symbolAndCount.values().stream().mapToInt(Integer::intValue).sum() / NUM_OF_BUCKETS;
// 嘗試將 symbol 放入不同的桶
int bucketIdx = 0;
int[] buckets = new int[NUM_OF_BUCKETS];
String[] bulkheads = new String[NUM_OF_BUCKETS-1];
for (Map.Entry entry : symbolAndCount.entrySet()) {
// 如果首個 symbol 資料量過大，則分配給其一個獨立的桶
int count = entry.getValue();
if (count / 2 > avg && bucketIdx == 0 && buckets[0] == 0) {
buckets[bucketIdx] += count;
continue;
}
// 評估將 symbol 放入桶後的效果
// 1. 如果桶中的數量更接近期望，則將其放入當前桶中
// 2. 如果桶中的數量更遠離期望，則將其放入下個桶中
double before = Math.abs(buckets[bucketIdx] - avg);
double after = Math.abs(buckets[bucketIdx] + count - avg);
if (after > before && bucketIdx < buckets.length - 1) {
bulkheads[bucketIdx++] = entry.getKey();
}
buckets[bucketIdx] += count;
}
System.out.printf("expectation: %d\n", avg);
System.out.printf("bulkheads: %s\n", Arrays.toString(bulkheads));
TreeMap distribution = new TreeMap<>();
for (int i=0; i32 the solution of given problem
*/
public abstract class SimulatedAnnealing {
protected final int numberOfIterations; // stopping condition for simulations
protected final double coolingRate; // the percentage by which we reduce the temperature of the system
protected final double initialTemperature; // the starting energy of the system
protected final double minimumTemperature; // optional stopping condition
protected final long simulationTime; // optional stopping condition
protected final int detectionInterval; // optional stopping condition
protected SimulatedAnnealing(int numberOfIterations, double coolingRate) {
this(numberOfIterations, coolingRate, 10000000, 1, 0, 0);
}
protected SimulatedAnnealing(int numberOfIterations, double coolingRate, double initialTemperature, double minimumTemperature, long simulationTime, int detectionInterval) {
this.numberOfIterations = numberOfIterations;
this.coolingRate = coolingRate;
this.initialTemperature = initialTemperature;
this.minimumTemperature = minimumTemperature;
this.simulationTime = simulationTime;
this.detectionInterval = detectionInterval;
}
protected abstract double score(X currentSolution);
protected abstract X neighbourSolution(X currentSolution);
public X simulateAnnealing(X currentSolution) {
final long startTime = System.currentTimeMillis();
// Initialize searching
X bestSolution = currentSolution;
double bestScore = score(bestSolution);
double currentScore = bestScore;
double t = initialTemperature;
for (int i = 0; i < numberOfIterations; i++) {
if (currentScore < bestScore) {
// If the new solution is better, accept it unconditionally
bestScore = currentScore;
bestSolution = currentSolution;
} else {
// If the new solution is worse, calculate an acceptance probability for the worse solution
// At high temperatures, the system is more likely to accept the solutions that are worse
boolean rejectWorse = Math.exp((bestScore - currentScore) / t) < Math.random();
if (rejectWorse || currentScore == bestScore) {
currentSolution = neighbourSolution(currentSolution);
currentScore = score(currentSolution);
}
}
// Stop searching when the temperature is too low
if ((t *= coolingRate) < minimumTemperature) {
break;
}
// Stop searching when simulation time runs out
if (simulationTime > 0 && (i+1) % detectionInterval == 0) {
if (System.currentTimeMillis() - startTime > simulationTime)
break;
}
}
return bestSolution;
}
}
```
```
/**
* Search best hash seed for given key distribution and number of buckets with simulated annealing algorithm
*/
@Data
public class SimulatedAnnealingHashing extends SimulatedAnnealing {
private static final int DISTRIBUTION_BATCH = 100;
static final int SEARCH_BATCH = 200;
private final int[] hashCodes = new int[SEARCH_BATCH];
private final long[][] buckets = new long[SEARCH_BATCH][];
@Data
public class HashingSolution {
private final int begin, range; // the begin and range for searching
private int bestSeed; // the best seed found in this search
private long bestScore; // the score corresponding to bestSeed
private long calculateDivergence(long[] bucket) {
long divergence = 0;
for (int i=0; i keyAndCounts, int numOfBuckets) {
super(100000000, .9999);
distributions = buildDistribution(keyAndCounts);
long sum = 0;
for (KeyDistribution[] batch : distributions) {
for (KeyDistribution distribution : batch) {
sum += distribution.getCount();
}
}
this.expectation = sum / numOfBuckets;
this.searchOutset = 0;
for (int i = 0; i< buckets.length; i++) {
buckets[i] = new long[numOfBuckets];
}
}
/**
* SimulatedAnnealingHashing Derivative
* @param prototype prototype simulation
* @param searchOutset the outset for searching
* @param simulationTime the expect time consuming for simulation
*/
private SimulatedAnnealingHashing(SimulatedAnnealingHashing prototype, int searchOutset, long simulationTime) {
super(prototype.numberOfIterations, prototype.coolingRate, prototype.initialTemperature, prototype.minimumTemperature,
simulationTime, 10000);
distributions = prototype.distributions;
expectation = prototype.expectation;
for (int i = 0; i< buckets.length; i++) {
buckets[i] = new long[prototype.buckets[i].length];
}
this.searchOutset = searchOutset;
this.searchMax = searchMin = searchOutset;
}
@Override
public String toString() {
return String.format("expectation: %d, outset:%d, search(min:%d, max:%d)", expectation, searchOutset, searchMin, searchMax);
}
private KeyDistribution[][] buildDistribution(Map symbolCounts) {
int bucketNum = symbolCounts.size() / DISTRIBUTION_BATCH + Integer.signum(symbolCounts.size() % DISTRIBUTION_BATCH);
KeyDistribution[][] distributions = new KeyDistribution[bucketNum][];
int bucketIndex = 0;
List batch = new ArrayList<>(DISTRIBUTION_BATCH);
for (Map.Entry entry : symbolCounts.entrySet()) {
batch.add(new KeyDistribution(entry.getKey().toCharArray(), entry.getValue()));
if (batch.size() == DISTRIBUTION_BATCH) {
distributions[bucketIndex++] = batch.toArray(new KeyDistribution[0]);
batch.clear();
}
}
if (batch.size() > 0) {
distributions[bucketIndex] = batch.toArray(new KeyDistribution[0]);
batch.clear();
}
return distributions;
}
@Override
protected double score(HashingSolution currentSolution) {
return currentSolution.solve().bestScore;
}
@Override
protected HashingSolution neighbourSolution(HashingSolution currentSolution) {
// The default range of neighbourhood is [-100, 100]
int rand = ThreadLocalRandom.current().nextInt(-100, 101);
int next = currentSolution.begin + rand;
searchMin = Math.min(next, searchMin);
searchMax = Math.max(next, searchMax);
return new HashingSolution(next, currentSolution.range);
}
public HashingSolution solve() {
searchMin = searchMax = searchOutset;
HashingSolution initialSolution = new HashingSolution(searchOutset, SEARCH_BATCH);
return simulateAnnealing(initialSolution);
}
public SimulatedAnnealingHashing derive(int searchOutset, long simulationTime) {
return new SimulatedAnnealingHashing(this, searchOutset, simulationTime);
}
}
```
## ForkJoin 框架
為了達到更好的搜尋效果，可以將整個搜尋區域遞迴地劃分為兩兩相鄰的區域，然後在這些區域上執行併發的搜尋，並遞迴地合併相鄰區域的搜尋結果。
使用 JDK 提供的 ForkJoinPool 與 RecursiveTask 能很好地完成以上任務。
```
@Data
@Slf4j
public class HashingSeedCalculator {
/**
* Recursive search task
*/
private class HashingSeedCalculatorSearchTask extends RecursiveTask {
private SimulatedAnnealingHashing simulation;
private final int level;
private final int center, range;
private HashingSeedCalculatorSearchTask() {
this.center = 0;
this.range = Integer.MAX_VALUE / SimulatedAnnealingHashing.SEARCH_BATCH;
this.level = traversalDepth;
this.simulation = hashingSimulation;
}
private HashingSeedCalculatorSearchTask(HashingSeedCalculatorSearchTask parent, int center, int range) {
this.center = center;
this.range = range;
this.level = parent.level - 1;
this.simulation = parent.simulation;
}
@Override
protected HashingSolution compute() {
if (level == 0) {
long actualCenter = center * SimulatedAnnealingHashing.SEARCH_BATCH;
log.info("Searching around center {}", actualCenter);
HashingSolution solution = simulation.derive(center, perShardRunningMills).solve();
log.info("Searching around center {} found {}", actualCenter, solution);
return solution;
} else {
int halfRange = range / 2;
int leftCenter = center - halfRange, rightCenter = center + halfRange;
ForkJoinTask leftTask = new HashingSeedCalculatorSearchTask(this, leftCenter, halfRange).fork();
ForkJoinTask rightTask = new HashingSeedCalculatorSearchTask(this, rightCenter, halfRange).fork();
HashingSolution left = leftTask.join();
HashingSolution right = rightTask.join();
return left.getBestScore() < right.getBestScore() ? left : right;
}
}
}
private final int poolParallelism;
private final int traversalDepth;
private final long perShardRunningMills;
private final SimulatedAnnealingHashing hashingSimulation;
/**
* HashingSeedCalculator
* @param numberOfShards the shard of the whole search range [Integer.MIN_VALUE, Integer.MAX_VALUE]
* @param totalRunningHours the expect total time consuming for searching
* @param symbolCounts the key and it`s distribution
* @param numOfBuckets the number of buckets
*/
public HashingSeedCalculator(int numberOfShards, int totalRunningHours, Map symbolCounts, int numOfBuckets) {
int n = (int) (Math.log(numberOfShards) / Math.log(2));
if (Math.pow(2, n) != numberOfShards) {
throw new IllegalArgumentException();
}
this.traversalDepth = n;
this.poolParallelism = Math.max(ForkJoinPool.getCommonPoolParallelism() / 3 * 2, 1); // conservative estimation for parallelism
this.perShardRunningMills = TimeUnit.HOURS.toMillis(totalRunningHours * poolParallelism) / numberOfShards;
this.hashingSimulation = new SimulatedAnnealingHashing(symbolCounts, numOfBuckets);
}
@Override
public String toString() {
int numberOfShards = (int) Math.pow(2, traversalDepth);
int totalRunningHours = (int) TimeUnit.MILLISECONDS.toHours(perShardRunningMills * numberOfShards) / poolParallelism;
return "HashingSeedCalculator(" +
"numberOfShards: " + numberOfShards +
", perShardRunningMinutes: " + TimeUnit.MILLISECONDS.toMinutes(perShardRunningMills) +
", totalRunningHours: " + totalRunningHours +
", poolParallelism: " + poolParallelism +
", traversalDepth: " + traversalDepth + ")";
}
public synchronized HashingSolution searchBestSeed() {
long now = System.currentTimeMillis();
log.info("SearchBestSeed start");
ForkJoinTask root = new HashingSeedCalculatorSearchTask().fork();
HashingSolution initSolution = hashingSimulation.derive(0, perShardRunningMills).solve();
HashingSolution bestSolution = root.join();
log.info("Found init solution {}", initSolution);
log.info("Found best solution {}", bestSolution);
if (initSolution.getBestScore() < bestSolution.getBestScore()) {
bestSolution = initSolution;
}
long cost = System.currentTimeMillis() - now;
log.info("SearchBestSeed finish (cost:{}ms)", cost);
return bestSolution;
}
}
```
# 效果
將改造後的程式碼部署到測試環境後，某日訓練日誌：
> 12:49:15.227 85172866 INFO hash.HashingSeedCalculator - Found init solution (seed:15231, score:930685828341164)
12:49:15.227 85172866 INFO hash.HashingSeedCalculator - Found best solution (seed:362333, score:793386389726926)
12:49:15.227 85172866 INFO hash.HashingSeedCalculator - SearchBestSeed finish (cost:10154898ms)
12:49:15.227 85172866 INFO hash.TrainingService -
Training result: (seed:362333, score:793386389726926)
Buckets: 15
Expectation: 44045697
Result of Hashing.HashCode(seed=362333): 21327108 [42512742, 40479608, 43915771, 47211553, 45354264, 43209190, 43196570, 44725786, 41999747, 46450288, 46079231, 45116615, 44004021, 43896194, 42533877]
Result of Hashing.HashCode(seed=31): 66929172 [39723630, 48721463, 43365391, 46301448, 43931616, 44678194, 39064877, 45922454, 43171141, 40715060, 33964547, 49709090, 58869949, 34964729, 47581868]
當晚使用 `BKDRHash(seed=31)` 對新的交易日資料的進行分片：
> 04:00:59.001 partition messages per minute [45171, 68641, 62001, 80016, 55977, 61916, 55102, 49322, 55982, 57081, 51100, 70437, 135992, 37823, 58552] , messages total [39654953, 48666261, 43310578, 46146841, 43834832, 44577454, 38990331, 45871075, 43106710, 40600708, 33781629, 49752592, 58584246, 34928991, 47545369]
當晚使用 `BKDRHash(seed=362333)` 對新的交易日資料的進行分片：
> 04:00:59.001 partition messages per minute [62424, 82048, 64184, 47000, 57206, 69439, 64430, 60096, 46986, 58182, 54557, 41523, 64310, 72402, 100326] , messages total [44985772, 48329212, 39995385, 43675702, 45216341, 45524616, 41335804, 44917938, 44605376, 44054821, 43371892, 42068637, 44000817, 42617562, 44652695]
對比日誌發現 hash 經過優化後，分割槽的均勻程度有了顯著的上升，並且熱點分片也被消除了，基本達到當初設想的優化
- 根據所有 symbol 的訊息總數計算一個期望的分佈均值
(expectation)。 - 將每個 symbol 的訊息數按照 symbol 的順序進行排列，最後將這組陣列劃分為 15 個區間，並且儘可能使得每個區間元素之和與 expection 接近。
- 使用一個有序查詢表記錄每個區間的首個 symbol，後續就可以按照這個表對資料進行劃分。