4641897: Faster string conversion of large integers

Summary: Accelerate conversion to string by means of Schoenhage recursive base conversion.
Reviewed-by: bpb, alanb
Contributed-by: NAlan Eliasen <eliasen@mindspring.com>

src/share/classes/java/math/BigInteger.java

@@ -33,6 +33,7 @@ import java.io.IOException;
 import java.io.ObjectInputStream;
 import java.io.ObjectOutputStream;
 import java.io.ObjectStreamField;
+import java.util.ArrayList;
 import java.util.Arrays;
 import java.util.Random;
 import sun.misc.DoubleConsts;
@@ -213,6 +214,16 @@ public class BigInteger extends Number implements Comparable<BigInteger> {
      */
     private static final int TOOM_COOK_SQUARE_THRESHOLD = 140;
 
+    /**
+     * The threshold value for using Schoenhage recursive base conversion. If
+     * the number of ints in the number are larger than this value,
+     * the Schoenhage algorithm will be used.  In practice, it appears that the
+     * Schoenhage routine is faster for any threshold down to 2, and is
+     * relatively flat for thresholds between 2-25, so this choice may be
+     * varied within this range for very small effect.
+     */
+    private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 8;
+
     //Constructors
 
     /**
@@ -1026,6 +1037,19 @@ public class BigInteger extends Number implements Comparable<BigInteger> {
     private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
     private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
 
+    /**
+     * The cache of powers of each radix.  This allows us to not have to
+     * recalculate powers of radix^(2^n) more than once.  This speeds
+     * Schoenhage recursive base conversion significantly.
+     */
+    private static ArrayList<BigInteger>[] powerCache;
+
+    /** The cache of logarithms of radices for base conversion. */
+    private static final double[] logCache;
+
+    /** The natural log of 2.  This is used in computing cache indices. */
+    private static final double LOG_TWO = Math.log(2.0);
+
     static {
         for (int i = 1; i <= MAX_CONSTANT; i++) {
             int[] magnitude = new int[1];
@@ -1033,6 +1057,22 @@ public class BigInteger extends Number implements Comparable<BigInteger> {
             posConst[i] = new BigInteger(magnitude,  1);
             negConst[i] = new BigInteger(magnitude, -1);
         }
+
+        /*
+         * Initialize the cache of radix^(2^x) values used for base conversion
+         * with just the very first value.  Additional values will be created
+         * on demand.
+         */
+        powerCache = (ArrayList<BigInteger>[])
+            new ArrayList[Character.MAX_RADIX+1];
+        logCache = new double[Character.MAX_RADIX+1];
+
+        for (int i=Character.MIN_RADIX; i<=Character.MAX_RADIX; i++)
+        {
+            powerCache[i] = new ArrayList<BigInteger>(1);
+            powerCache[i].add(BigInteger.valueOf(i));
+            logCache[i] = Math.log(i);
+        }
     }
 
     /**
@@ -1357,7 +1397,7 @@ public class BigInteger extends Number implements Comparable<BigInteger> {
             if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD))
                 return multiplyKaratsuba(this, val);
             else
-               return multiplyToomCook3(this, val);
+                return multiplyToomCook3(this, val);
     }
 
     private static BigInteger multiplyByInt(int[] x, int y, int sign) {
@@ -3299,6 +3339,28 @@ public class BigInteger extends Number implements Comparable<BigInteger> {
         if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
             radix = 10;
 
+        // If it's small enough, use smallToString.
+        if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
+           return smallToString(radix);
+
+        // Otherwise use recursive toString, which requires positive arguments.
+        // The results will be concatenated into this StringBuilder
+        StringBuilder sb = new StringBuilder();
+        if (signum < 0) {
+            toString(this.negate(), sb, radix, 0);
+            sb.insert(0, '-');
+        }
+        else
+            toString(this, sb, radix, 0);
+
+        return sb.toString();
+    }
+
+    /** This method is used to perform toString when arguments are small. */
+    private String smallToString(int radix) {
+        if (signum == 0)
+            return "0";
+
         // Compute upper bound on number of digit groups and allocate space
         int maxNumDigitGroups = (4*mag.length + 6)/7;
         String digitGroup[] = new String[maxNumDigitGroups];
@@ -3337,6 +3399,78 @@ public class BigInteger extends Number implements Comparable<BigInteger> {
         return buf.toString();
     }
 
+    /**
+     * Converts the specified BigInteger to a string and appends to
+     * <code>sb</code>.  This implements the recursive Schoenhage algorithm
+     * for base conversions.
+     * <p/>
+     * See Knuth, Donald,  _The Art of Computer Programming_, Vol. 2,
+     * Answers to Exercises (4.4) Question 14.
+     *
+     * @param u      The number to convert to a string.
+     * @param sb     The StringBuilder that will be appended to in place.
+     * @param radix  The base to convert to.
+     * @param digits The minimum number of digits to pad to.
+     */
+    private static void toString(BigInteger u, StringBuilder sb, int radix,
+                                 int digits) {
+        /* If we're smaller than a certain threshold, use the smallToString
+           method, padding with leading zeroes when necessary. */
+        if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
+            String s = u.smallToString(radix);
+
+            // Pad with internal zeros if necessary.
+            // Don't pad if we're at the beginning of the string.
+            if ((s.length() < digits) && (sb.length() > 0))
+                for (int i=s.length(); i<digits; i++) // May be a faster way to
+                    sb.append('0');                    // do this?
+
+            sb.append(s);
+            return;
+        }
+
+        int b, n;
+        b = u.bitLength();
+
+        // Calculate a value for n in the equation radix^(2^n) = u
+        // and subtract 1 from that value.  This is used to find the
+        // cache index that contains the best value to divide u.
+        n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
+        BigInteger v = getRadixConversionCache(radix, n);
+        BigInteger[] results;
+        results = u.divideAndRemainder(v);
+
+        int expectedDigits = 1 << n;
+
+        // Now recursively build the two halves of each number.
+        toString(results[0], sb, radix, digits-expectedDigits);
+        toString(results[1], sb, radix, expectedDigits);
+    }
+
+    /**
+     * Returns the value radix^(2^exponent) from the cache.
+     * If this value doesn't already exist in the cache, it is added.
+     * <p/>
+     * This could be changed to a more complicated caching method using
+     * <code>Future</code>.
+     */
+    private static synchronized BigInteger getRadixConversionCache(int radix,
+                                                                   int exponent) {
+        BigInteger retVal = null;
+        ArrayList<BigInteger> cacheLine = powerCache[radix];
+        int oldSize = cacheLine.size();
+        if (exponent >= oldSize) {
+            cacheLine.ensureCapacity(exponent+1);
+            for (int i=oldSize; i<=exponent; i++) {
+                retVal = cacheLine.get(i-1).square();
+                cacheLine.add(i, retVal);
+            }
+        }
+        else
+            retVal = cacheLine.get(exponent);
+
+        return retVal;
+    }
 
     /* zero[i] is a string of i consecutive zeros. */
     private static String zeros[] = new String[64];

test/java/math/BigInteger/BigIntegerTest.java

@@ -61,10 +61,13 @@ public class BigIntegerTest {
     // KARATSUBA_SQUARE_THRESHOLD = 90  ints = 2880 bits
     // TOOM_COOK_SQUARE_THRESHOLD = 140 ints = 4480 bits
     //
+    // SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 8 ints = 256 bits
+    //
     static final int BITS_KARATSUBA = 1600;
     static final int BITS_TOOM_COOK = 2400;
     static final int BITS_KARATSUBA_SQUARE = 2880;
     static final int BITS_TOOM_COOK_SQUARE = 4480;
+    static final int BITS_SCHOENHAGE_BASE = 256;
 
     static final int ORDER_SMALL = 60;
     static final int ORDER_MEDIUM = 100;
@@ -467,12 +470,13 @@ public class BigIntegerTest {
     public static void stringConv() {
         int failCount = 0;
 
+        // Generic string conversion.
         for (int i=0; i<100; i++) {
             byte xBytes[] = new byte[Math.abs(rnd.nextInt())%100+1];
             rnd.nextBytes(xBytes);
             BigInteger x = new BigInteger(xBytes);
 
-            for (int radix=2; radix < 37; radix++) {
+            for (int radix=Character.MIN_RADIX; radix < Character.MAX_RADIX; radix++) {
                 String result = x.toString(radix);
                 BigInteger test = new BigInteger(result, radix);
                 if (!test.equals(x)) {
@@ -483,6 +487,32 @@ public class BigIntegerTest {
                 }
             }
         }
+
+        // String conversion straddling the Schoenhage algorithm crossover
+        // threshold, and at twice and four times the threshold.
+        for (int k = 0; k <= 2; k++) {
+            int factor = 1 << k;
+            int upper = factor * BITS_SCHOENHAGE_BASE + 33;
+            int lower = upper - 35;
+
+            for (int bits = upper; bits >= lower; bits--) {
+                for (int i = 0; i < 50; i++) {
+                    BigInteger x = BigInteger.ONE.shiftLeft(bits - 1).or(new BigInteger(bits - 2, rnd));
+
+                    for (int radix = Character.MIN_RADIX; radix < Character.MAX_RADIX; radix++) {
+                        String result = x.toString(radix);
+                        BigInteger test = new BigInteger(result, radix);
+                        if (!test.equals(x)) {
+                            failCount++;
+                            System.err.println("BigInteger toString: " + x);
+                            System.err.println("Test: " + test);
+                            System.err.println(radix);
+                        }
+                    }
+                }
+            }
+        }
+
         report("String Conversion", failCount);
     }