Finding an error in hash function when resizing table

Question

While preparing for an exam I came across a question about hash tables. I am given a table of length 11 with the following hash function:

h(k,i) = ( k mod 13 + i * (1 + k mod 7) ) mod 11

The hash table is then resized to size 12. So the new hash function becomes:

h'(k,i) = ( k mod 13 + i * (1 + k mod 7) ) mod 12

Which problems occur?

Answer 1

The problem is that the hash function becomes worse.

In the first case, the distribution of different combinations of k and i is very even among the 11 hash bins. In the second case, the distribution is not so even - particularly, the number of combinations of k and i for which the result of the hash function will be 0 is noticably higher.

Of course, during an exam, one would probably have to argue why it is this way. It's somehow related to

• k mod 13 being a value between 0 and 12
• k mod 7 being a value between 0 and 6 (which divides 12)
• maybe, somehow: 11 is a prime number and 12 has many divisors...

but (at least for me) it is hard to find a convincing reasoning that goes beyond these trivial insights. Maybe you have another idea based on that.

import java.util.LinkedHashMap;
import java.util.Map;


public class HashTest
{
    public static void main(String[] args)
    {
        int maxK = 30;
        int maxI = 30;
        System.out.println(computeFrequencies(h0, maxK, maxI));
        System.out.println(computeFrequencies(h1, maxK, maxI));
    }

    private static Map<Integer, Integer> computeFrequencies(
        Hash hash, int maxK, int maxI)
    {
        Map<Integer, Integer> frequencies = 
            new LinkedHashMap<Integer, Integer>();
        for (int k=0; k<maxK; k++)
        {
            for (int i=0; i<maxI; i++)
            {
                int value = hash.compute(k, i);
                Integer count = frequencies.get(value);
                if (count == null)
                {
                    count = 0;
                }
                frequencies.put(value, count+1);
            }
        }
        return frequencies;
    }

    private static interface Hash
    {
        int compute(int k, int i);
    }

    private static final Hash h0 = new Hash()
    {
        @Override
        public int compute(int k, int i)
        {
            return ((k % 13) + i * (1 + (k % 7))) % 11;
        }
    };
    private static final Hash h1 = new Hash()
    {
        @Override
        public int compute(int k, int i)
        {
            return ((k % 13) + i * (1 + (k % 7))) % 12;
        }
    };

}