Representing a Set

• Consider the set of integers from 0 to 7

  {0, 1, 2, 3, 4, 5, 6, 7}

• How could we represent this set in C?

• How could we represent subsets of this set?

• How could we perform set operations?

Using Bits for a Set

• Each bit of an integer can be used considered a separate boolean value
• Let the bit in position n represent whether or or not n is in the set
• Example Set: {1, 3, 6, 7}
• Binary: 11001010
• Example Set: {5}
• Binary: 00100000
• Singleton sets can be simply created by shifting, can we combine them?

Set Operations

Counting Bits

• How could we count the number of items are in a set?
• Basically how many bits are set to 1?

Counting by Looping

int numBitsSet(short n) {
    int i;
    int count = 0;

    for(i = 0; i < 16; i++) {
        if(n & (1 << i)) count++;
    }
    return count;
}

Counting with Shifting and Masking

int numBitsSet(short n) {
  int value = n;
  value = (value & 0x5555) + 
          ((value >> 1) & 0x5555);
  value = (value & 0x3333) + 
          ((value >> 2) & 0x3333);
  value = (value & 0x0f0f) + 
          ((value >> 4) & 0x0f0f);
  value = (value & 0x00ff) + 
          ((value >> 8) & 0x00ff);
  return value;
}

Counting with Shifting and Masking

Counting with Shifting and Masking

Let’s try it with n = 0xF1A2 which is 1111 0001 1010 0010 which has 8 bits set

value = 
  (value & 0x5555) 
  + 
  ((value >> 1) & 0x5555);

Counting with Shifting and Masking

value = 
  (value & 0x5555) 
  + 
  ((value >> 1) & 0x5555);

Counting with Shifting and Masking

value = 
    (value & 0x5555) 
    + 
    ((value >> 1) & 0x5555);

Counting with Shifting and Masking

value = 
    (value & 0x3333) 
    +
    ((value >> 2) & 0x3333);

Counting with Shifting and Masking

value = 
    (value & 0x3333) 
    +
    ((value >> 2) & 0x3333);

Counting with Shifting and Masking

value = 
    (value & 0x3333) 
    +
    ((value >> 2) & 0x3333);

Counting with Shifting and Masking

value = 
    (value & 0x0f0f) 
    +
    ((value >> 4) & 0x0f0f);

Counting with Shifting and Masking

value = 
    (value & 0x0f0f) 
    +
    ((value >> 4) & 0x0f0f);

Counting with Shifting and Masking

value = 
    (value & 0x0f0f) 
    +
    ((value >> 4) & 0x0f0f);

Counting with Shifting and Masking

value = 
    (value & 0x00ff) 
    +
    ((value >> 8) & 0x00ff);

Counting with Shifting and Masking

value = 
    (value & 0x00ff) 
    +
    ((value >> 8) & 0x00ff);

Counting with Shifting and Masking

value = 
    (value & 0x00ff) 
    +
    ((value >> 8) & 0x00ff);