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Bitwise Optimization in Java: Bitfields, Bitboards, and Beyond

by Glen Pepicelli
02/02/2005

Quick quiz: how do you rewrite the statement below, which alternates between two constants, without a conditional?

if (x == a) x= b;
  else x= a;

Answer:

x= a ^ b ^ x;
//where x is equal to either a or b

The ^ character is the logical XOR operator. How does the code work? With XOR, doing the operation twice always gives you back the original result.

Case 1: x equals a Case 2: x equals b
x= a ^ b ^ x x= a ^ b ^ x Use formula
x= a ^ b ^ a x= a ^ b ^ b Substitute
x= b ^ a ^ a x= a ^ b ^ b Reorder variables
x= b x= a The last two variables cancel out

While not a popular technique, delving into the black art of bitwise manipulation can boost performance in Java. The rewrite above benchmarked at 5.2 seconds, versus 5.6 seconds for the conditional statement on my setup. (See Resources below for the sample code.) Avoiding conditionals can often increase performance on modern processors with multiple pipelines.

Related Reading

Better, Faster, Lighter Java
By Justin Gehtland

Of particular interest to Java programmers are certain tricks related to what are called bitsets. In Java, the int and long integer primitives can double as a kind of set of bits with 32 or 64 elements. These simple data structures can be combined to represent larger structures, including arrays. Additionally, certain special operations (called bit-parallel) effectively condense a number of separate operations into one. When using finer grained sized data--bits rather than integers--we will find that operations that do only one thing at a time when applied to integers can sometimes do many things at once when applied to bits. We can think of bit-parallel operations as rediscovered software pipelines in Java systems. (Of course, assembly programmers may not see much new about this!)

Advanced chess-playing programs like Crafty (written in C) use a special chess data structure called a bitboard to represent chess positions faster than arrays could. Java programmers should be even more interested in avoiding arrays than C programmers. While C has a fast implementation of arrays, Java arrays (which have more features like bounds checking and garbage collection) are on the slow side. A simple test I performed on my system (Windows XP, AMD Athlon, HotSpot JVM) comparing integer access to array access shows that array access takes 160 percent longer than integer access. This doesn't even touch on the garbage collection issue. Bitsets, which are implemented as integers, can replace arrays in some situations.

Let's take a look at dusting off some of those old assembly-age tricks with a special emphasis on bitsets. Java actually has good bitwise support for a portable language. Additionally, Tiger adds a number of useful bit manipulation methods to the API.

Considering Bitwise Optimization in the Typical Java Environment

Java programs are pretty far removed from the bit-crunching machines and operating systems on which they run. While modern CPUs often have special instructions to manipulate bitsets, you certainly can't execute these instructions in Java. The JVM supports only signed and unsigned shift, and bitwise XOR, OR, AND, and NOT. Ironically, Java programmers find themselves in the same boat with many assembly programmers who happened to be targeting CPUs, in lacking extra bitwise instructions. The assembly programmers had to emulate these instructions, as quickly as possible, in software. Many of the new Tiger methods do just this in pure Java.

C-language programmers who are micro-tuning may review their actual assembly code and then consult the optimizer's manual for their target CPU, and actually count how many instruction cycles their code will run in. In contrast, a practical problem with doing low-level optimization in Java is determining the results. The Sun JVM spec says nothing about the relative speed at which the given opcodes are executed. You may spoon-feed the JVM the code you think will run fastest, only to be shocked by lack of improvement or even detrimental effects. The JVM may either mangle your optimizations somehow, or it may be optimizing the normal code internally. In any case, any JVM optimizations, even if they are mainstays in compiled languages, should be benchmarked.

The standard tool to look at bytecodes, javap, is included in the JDK. However, users of Eclipse can use the Bytecode Outline plugin from Andrei Loskutov. A reference book about the JVM's design and instruction set is available online on Sun's online Java book page. Note that there are two kinds of memory in every JVM. There is a stack, where local primitives and expressions are stored, and a heap, where objects and arrays are stored. The heap is subject to garbage collection, while the stack is a fixed chunk of memory. While most programs only use a small amount of stack, the JVM specification states that you can expect the stack to be at least 64K in size. Take special note that the JVM is really a 32/64-bit machine, so the byte and short primitives are unlikely to run any faster than the int.

"Two's Complement" Numbers

In Java, all integer primitives are in a signed number format known as "two's complement." To manipulate the primitives, we need to understand them.

There are two types of two's complement numbers: non-negative and negative. The highest bit, the sign bit, usually shown on the left, is set to zero for non-negative numbers. These numbers are "normal" you simply read them left to right and convert to the base you want. However, if the sign bit is on, then the number is negative and the remaining bits represent a negative number.

There are two ways to look at the negative numbers. In the first way, negatives count up starting at the smallest possible number and end at -1. So, for a byte, they start at 10000000 (or -128 decimal), then 10000001 (or -127), all the way up to 11111111 (or -1). The second way to think about them is a little odd. When the sign bit is on, instead of having leading zeros followed one bits, there are leading ones followed by zero bits. However, you must also subtract one from the result. For example, 11111111 is just the sign bit padded by seven ones, which is "negative zero" (-0). We then add (or subtract, depending on how you look at it) one to get -1. 11111110 is -2, 11111101 is -3, and so on.

It may seem strange, but we can do a lot of operations by mixing the bitwise operators together with the arithmetic operators. For example, to change between decimal x and -x, negate and add one: (~x)+1. This can be seen in the following table.

x ~x (~x)+1
or
-x
0111 (7) 1000 (-8) 1001 (-7)
0110 (6) 1001 (-7) 1010 (-6)
0101 (5) 1010 (-6) 1011 (-5)
0100 (4) 1011 (-5) 1100 (-4)
0011 (3) 1100 (-4) 1101 (-3)
0010 (2) 1101 (-3) 1110 (-2)
0001 (1) 1110 (-2) 1111 (-1)
0000 (0) 1111 (-1) 0000 (0)

Boolean Flags and Standard Boolean Bitsets

The bit flag pattern is common knowledge and widely used in the public APIs of GUIs. Perhaps we are writing a Java GUI for a constrained device like a cell phone or a PDA. We have widgets like buttons and drop-down lists that each have a list of Boolean options. With bit flags, we can stuff a large number of options in a single word.

//The constants to use with our GUI widgets:
final int visible      = 1 << 0;  //  1 
final int enabled      = 1 << 1;  //  2 
final int focusable    = 1 << 2;  //  4 
final int borderOn     = 1 << 3;  //  8 
final int moveable     = 1 << 4;  // 16 
final int editable     = 1 << 5;  // 32
final int borderStyleA = 1 << 6;  // 64 
final int borderStyleB = 1 << 7;  //128 
final int borderStyleC = 1 << 8;  //256 
final int borderStyleD = 1 << 9;  //512 
//etc.

myButton.setOptions( 1+2+4 ); 
//set a single widget.

int myDefault= 1+2+4;  //A list of options.
int myExtras = 32+128; //Another list.

myButtonA.setOptions( myDefault );
myButtonB.setOptions( myDefault | myExtras );

Your program can pass around a whole group of Boolean variables in no time with a simple assignment expression. Perhaps the API states that the user can have only one of borderStyleA, borderStyleB, borderStyleC, or borderStyleD at the same time. To check, first select those four bits with a mask and second, check to see that the result has, at most, one bit. The code below uses a little trick we will explain soon.

int illegalOptionCombo=
  2+ 64+ 128+ 512;               // 10 11000010
	
int borderOptionMask= 
  64+ 128+ 256+ 512;             // 11 11000000
	
int temp= illegalOptionCombo &
  borderOptionMask               // 10 11000000
	
int rightmostBit= 
  temp & ( -temp );              // 00 01000000

If temp is not equal to rightMostBit, that means temp must have more than one bit, because rightmostBit will contain zero if temp is zero, otherwise it contains only one bit.

if (temp != rightmostBit)
  throw new IllegalArgumentException();

The example above is a toy example. In the real world, AWT and Swing do use the bit flag pattern, but inconsistently. java.awt.geom.AffineTransform uses it extensively. java.awt.Font uses it, as does java.awt.InputEvent.

Some Common Operations and the New JDK 1.5 Methods

To get very far with bitsets, you need to know the standard "tricks," or operations you can perform. There are new bitset API methods as of the J2SE 5.0 (Tiger) release. If you're using an older release, you can just cut and paste the new methods into your code. A recent book with a lot of material on bitwise algorithms is Hacker's Delight by Henry S. Warren, Jr. (See www.hackersdelight.org or read the book on Safari.)

The following table shows some operations that can be done either with a line of code or one of the API methods:

y= all 0 bits y= 0;
y= all 1 bits y= -1
y= all zeros except for the rightmost or least significant bit y= 1;
y= all zeros except for the leftmost or sign bit y= Integer.MIN_VALUE;
y= the rightmost 1-bit of x y= x & (-x)
y= the leftmost 1-bit of x y= Integer.highestOneBit(x);
y= the rightmost 0-bit of x y= ~x & (x + 1)
y= x with the rightmost 1-bit turned off y= x & (x - 1)
y= x with the rightmost 0-bit turned off y= x | (x + 1)
y= the number of leading zeros in x y= Integer.numberOfLeadingZeros(x);
y= the number of trailing zeros in x y= Integer.numberOfTrailingZeros(x);
y= the number of 1 bits in x y= Integer.bitCount(x);
y= x with the bits reversed y= Integer.reverse(x);
y= x after a rotated shift left by c units y= Integer.rotateLeft(x,c);
y= x after a rotated shift right by c units y= Integer.rotateRight(x,c);
y= x with the bytes reversed y= Integer.reverseBytes(x);

To get an idea of how long these methods take, one can step through the source. Some methods are more cryptic than others. They are all explained in Hacker's Delight. They are either one-liners or a few lines long, like highestOneBit(int) below.

public static int highestOneBit(int i)
{
  i |= (i >> 1);
  i |= (i >> 2);
  i |= (i >> 4);
  i |= (i >> 8);
  i |= (i >> 16);
  return i - (i >>> 1);
}

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