VIPSolutions•You have a hash table of size 13. •You insert records with the following keys in the following order: 33, 17, 21, 30, 20, 46, 7 •Draw the hash table after the insertions using chaining to resolve collisions? •Draw the hash table after the insertions using linear probing to resolve collisions? •Draw the hash table after the insertions using quadratic probing to resolve collisions? •If there are keys for which a hash location cannot be found (within 5 collision resolution attempts), do not include that key in the drawing. IN STEP 1 GIVE THE INTRODUCTION OF THE CONCEPT AND GIVE ANSWER FOR EACH PART OF THE QUESTION IN EACH DIFFERENT STEP WITH CLEAR EXPLANATION AND IN THE FINAL STEP GIVE THE WHOLE FINAL ANSWER IN JUST VERY FEW SENTENCES AND MOREOVER I NEED COMPLETE AND CLEAR ANSWER at last explain what we did in each step in just few sentences
Question:
•You have a hash table of size 13. •You insert records with the following keys in the following order: 33, 17, 21, 30, 20, 46, 7 •Draw the hash table after the insertions using chaining to resolve collisions? •Draw the hash table after the insertions using linear probing to resolve collisions? •Draw the hash table after the insertions using quadratic probing to resolve collisions? •If there are keys for which a hash location cannot be found (within 5 collision resolution attempts), do not include that key in the drawing. IN STEP 1 GIVE THE INTRODUCTION OF THE CONCEPT AND GIVE ANSWER FOR EACH PART OF THE QUESTION IN EACH DIFFERENT STEP WITH CLEAR EXPLANATION AND IN THE FINAL STEP GIVE THE WHOLE FINAL ANSWER IN JUST VERY FEW SENTENCES AND MOREOVER I NEED COMPLETE AND CLEAR ANSWER at last explain what we did in each step in just few sentences
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Asked by: Test User 100221
Created at: 2025-05-09 09:10:47
Asked by: Test User 100221
Created at: 2025-05-09 09:10:47
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Certainly! Here’s a step-by-step, well-explained answer to your hash table question:
---
### **Step 1: Introduction to the Concept**
A **hash table** is a data structure that stores key-value pairs and uses a hash function to compute an index (hash value) into an array of buckets or slots. Collisions happen when two keys hash to the same index. We can resolve collisions with various strategies. Here, we’ll consider:
1. **Chaining**: Store multiple elements at the same index using a linked list.
2. **Linear Probing**: If a collision occurs, check the next slot (sequentially) until an empty slot is found.
3. **Quadratic Probing**: Resolve collisions by checking i² positions away from the original hash index.
Assume the hash function is:
**h(key) = key mod 13**
Table size = 13
Keys to insert (in order): **33, 17, 21, 30, 20, 46, 7**
---
### **Step 2: Chaining**
For each key, compute the hash:
- 33 mod 13 = **7**
- 17 mod 13 = **4**
- 21 mod 13 = **8**
- 30 mod 13 = **4**
- 20 mod 13 = **7**
- 46 mod 13 = **7**
- 7 mod 13 = **7**
Let’s insert each key:
| Index | Chaining (list at slot) |
|-------|------------------------|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | 17 → 30 |
| 5 | |
| 6 | |
| 7 | 33 → 20 → 46 → 7 |
| 8 | 21 |
| 9 | |
| 10 | |
| 11 | |
| 12 | |
**Explanation:**
- 33, 20, 46, 7 all hash to 7, so they chain at index 7.
- 17 and 30 hash to 4, so they chain at index 4.
---
### **Step 3: Linear Probing**
Insert each key, moving to the next available slot if needed:
- 33 → 33 mod 13 = **7** (empty, insert)
- 17 → 17 mod 13 = **4** (empty, insert)
- 21 → 21 mod 13 = **8** (empty, insert)
- 30 → 30 mod 13 = **4** (occupied, check 5: empty, insert)
- 20 → 20 mod 13 = **7** (occupied, check 8: occupied, check 9: empty, insert)
- 46 → 46 mod 13 = **7** (occupied, check 8: occupied, check 9: occupied, check 10: empty, insert)
- 7 → 7 mod 13 = **7** (occupied, check 8: occupied, check 9: occupied, check 10: occupied, check 11: empty, insert)
| Index | Value |
|-------|-------|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | 17 |
| 5 | 30 |
| 6 | |
| 7 | 33 |
| 8 | 21 |
| 9 | 20 |
| 10 | 46 |
| 11 | 7 |
| 12 | |
**Explanation:**
Each collision is resolved by moving one slot forward until an empty slot is found.
---
### **Step 4: Quadratic Probing**
Probing sequence: (h + i²) mod 13, for i = 0, 1, 2, 3, 4 (5 attempts max).
Insert each key:
- 33 → 33 mod 13 = **7** (empty, insert)
- 17 → 17 mod 13 = **4** (empty, insert)
- 21 → 21 mod 13 = **8** (empty, insert)
- 30 → 30 mod 13 = **4** (occupied)
- i=1: (4+1) mod 13 = 5 (empty, insert)
- 20 → 20 mod 13 = **7** (occupied)
- i=1: (7+1) mod 13 = 8 (occupied)
- i=2: (7+4) mod 13 = 11 (empty, insert)
- 46 → 46 mod 13 = **7** (occupied)
- i=1: (7+1) mod 13 = 8 (occupied)
- i=2: (7+4) mod 13 = 11 (occupied)
- i=3: (7+9) mod 13 = 3 (empty, insert)
- 7 → 7 mod 13 = **7** (occupied)
- i=1: (7+1) mod 13 = 8 (occupied)
- i=2: (7+4) mod 13 = 11 (occupied)
- i=3: (7+9) mod 13 = 3 (occupied)
- i=4: (7+16) mod 13 = (7+3) mod 13 = 10 (empty, insert)
| Index | Value |
|-------|-------|
| 0 | |
| 1 | |
| 2 | |
| 3 | 46 |
| 4 | 17 |
| 5 | 30 |
| 6 | |
| 7 | 33 |
| 8 | 21 |
| 9 | |
| 10 | 7 |
| 11 | 20 |
| 12 | |
**Explanation:**
Each collision is resolved using increasing quadratic steps from the original index.
---
### **Step 5: Final Summary**
**In summary:**
- With **chaining**, all keys hash to their base slot and are chained if collisions occur.
- With **linear probing**, keys are placed in the next open slot sequentially.
- With **quadratic probing**, keys are placed using quadratic increments, which distributes keys more evenly and avoids primary clustering.
---
### **Step 6: What We Did**
**Step-by-step:**
- Introduced hash tables and collision resolution strategies.
- Used the given hash function and keys to fill the table for each collision strategy.
- Explained each insertion and collision resolution process.
- Summarized the results and difference among methods in a concise way.
---
**Final Answer in Few Sentences:**
We created hash tables of size 13 using chaining, linear probing, and quadratic probing. Chaining stored colliding keys in linked lists at their hash index. Linear probing stored colliding keys in the next available slot. Quadratic probing used quadratic steps to find an open slot. All keys were successfully inserted in all three methods within 5 attempts per key.
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