Lecture 13: Masked Vectors

COSC 273: Parallel and Distributed Computing

Spring 2023

Announcements

1. Homework 02: Now Due Next Friday (03/10)
2. Lab 03 will be due after spring break

Outline

1. Vectors and Masking
2. Benchmarking Notes
3. The Mandelbrot Set

Vector Operations in Pictures

For each i, set c[i] = a[i] + b[i]

Vector Operations in Code

	int step = SPECIES.length();
	int bound = SPECIES.loopBound(a.length);
	int i = 0;
	for (; i < bound; i += step) {
	    var va = FloatVector.fromArray(SPECIES, a, i);
	    var vb = FloatVector.fromArray(SPECIES, b, i);
	    var vc = va.add(vb);
	    vc.intoArray(c, i);
	}

An Issue?

Question. What if we don’t want to apply an operation to all entries in a Vector?

E.g., conditional assignment:

for (int i = 0; i < a.length; i++) {
    if (b[i] > 0) {
        c[i] = a[i] + b[i];
    } else {
        c[i] = a[i];
    }
}

A Vector Solution

To apply an operation (say, add) only to some lanes:

1. store a vector of Boolean “flags”
   • this vector is a vector mask
2. only apply the operation for the lanes where the mask is true
   • hardware supports vector masking!
   • avoids conditional statements (which tend to slow execution)

Masking Example, In Pictures

Set c[i] = a[i] + b[i] if b[i] > 0 and c[i] = a[i] otherwise

Masking in Java

• VectorMask<Float> datatype: think Vector of Booleans

• masked arithmetic operations:

    public final FloatVector add(Vector<Float> v,
                                 VectorMask<Float> m)
  

“Adds this vector to a second input vector, selecting lanes under the control of a mask. This is a masked lane-wise binary operation which applies the primitive addition operation (+) to each pair of corresponding lane values. For any lane unset in the mask, the primitive operation is suppressed and this vector retains the original value stored in that lane. This method is also equivalent to the expression lanewise(ADD, v, m).”

Creating and Using a Mask

A VectorMask<Float> that is true when b[i] > 0:

var va = FloatVector.fromArray(SPECIES, a, i);
var vb = FloatVector.fromArray(SPECIES, b, i);
var bMask = vb.compare(VectorOperators.GT, 0);
var vc = va.add(vb, bMask);

Examle: Hamming Weights

Definition. Given an int a, the Hamming weight of a is the number of 1s in the binary representation of a.

Hamming Weights via Bitwise &

Bitwise & operator, a & b:

How to determine if bit i is 0 or 1?

Computing the Hamming Weight

Idea. For bits i = 1...32, check if bit i is 1

• if so, increment a count

int val;
int idx = 1;
int weight = 0;
for (int j = 0; j < INT_LENGTH; j++) {
    if ((val & idx) != 0) {
        weight++;
    }
    idx = idx << 1;
}
// weight is the Hamming weight of val

Question

Want to compute Hamming weight of an array of ints…

for (int j = 0; j < INT_LENGTH; j++) {
    if ((val & idx) != 0) {
        weight++;
    }
    idx = idx << 1;
}

How could we vectorize this method?

Vectorization Idea

1. create vector va from array of values
2. create vector vb initialized to all 0
   • this will store Hamming weights
3. int idx has 1 in exactly one bit position
4. iterate over bits idx:
   • create mask that is 1 if ith bit of va lane is 1
   • use mask to add 1 to corresponding lanes

Vectorized Code

var va = IntVector.fromArray(SPECIES, a, i);
var vb = IntVector.broadcast(SPECIES, 0);
int idx = 1;
for (int j = 0; j < INT_LENGTH; j++) {
    var bitMask = va.and(idx).eq(0).not();
    vb = vb.add(1, bitMask);
    idx = idx << 1;
}
vb.intoArray(b, i);

HammingWeight.java Demo

Benchmarking Notes

To give “accurate” measure of efficiency:

• test running time of method for many invocations
• run several invocations before starting timing
  • “warm up” primes hardware with correct instructions

Benchmarking Demo

Lab 03: Mandelbrot Set

Draw this picture as quickly as possible!

Next Week

1. Mandelbrot set definition
2. Thread pools and executers
   • handling threads more elegantly