1. **Problem Statement:** Insert keys \{14, 30, 22, 10, 50, 60, 90\} into a hash table of size 8 using the hash function $$H(k) = k \bmod 8$$ and resolve collisions using linear probing. 2. **Hash Function Explanation:** The hash function calculates the index by taking the remainder when the key is divided by 8. 3. **Insertion Steps with Linear Probing:** - Insert 14: $$14 \bmod 8 = 6$$, place at index 6. - Insert 30: $$30 \bmod 8 = 6$$, collision at index 6. Linear probing checks next index 7, place 30 at index 7. - Insert 22: $$22 \bmod 8 = 6$$, collision at 6. Check 7 (occupied), then 0 (empty), place 22 at index 0. - Insert 10: $$10 \bmod 8 = 2$$, place at index 2. - Insert 50: $$50 \bmod 8 = 2$$, collision at 2. Check 3 (empty), place 50 at index 3. - Insert 60: $$60 \bmod 8 = 4$$, place at index 4. - Insert 90: $$90 \bmod 8 = 2$$, collision at 2. Check 3 (occupied), 4 (occupied), 5 (empty), place 90 at index 5. 4. **Final Hash Table:** Index: 0 1 2 3 4 5 6 7 Keys: 22 - 10 50 60 90 14 30 5. **Advantages of Linear Probing:** - Simple to implement. - Good cache performance due to sequential access. 6. **Disadvantages of Linear Probing:** - Primary clustering: long runs of occupied slots increase search time. - Performance degrades as load factor approaches 1. 7. **Advantages of Quadratic Probing:** - Reduces primary clustering by using quadratic increments. - Better distribution of keys compared to linear probing. 8. **Disadvantages of Quadratic Probing:** - More complex to implement. - May not probe all slots, leading to secondary clustering. - Requires careful choice of table size and probing function to guarantee finding empty slots.