Collision Handling Using Open Addressing Linear Probing in DSA C# - Time & Space Complexity

Time Complexity: Collision Handling Using Open Addressing Linear Probing

O(n)

Understanding Time Complexity

We want to understand how long it takes to find or insert items when collisions happen in a hash table using linear probing.

How does the time grow as the table fills up?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public int LinearProbingSearch(int[] table, int key) {
    int size = table.Length;
    int index = key % size;
    int startIndex = index;
    while (table[index] != -1 && table[index] != key) {
        index = (index + 1) % size;
        if (index == startIndex) return -1; // full loop
    }
    return table[index] == key ? index : -1;
}
    

This code searches for a key in a hash table using linear probing to handle collisions.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that checks each slot in the table one by one.
• How many times: Up to the size of the table in the worst case, because it may check every slot once.

How Execution Grows With Input

As the table size grows, the number of checks can grow up to the table size if many slots are filled.

Input Size (n)	Approx. Operations
10	Up to 10 checks
100	Up to 100 checks
1000	Up to 1000 checks

Pattern observation: The number of operations can grow linearly with the table size in the worst case.

Final Time Complexity

Time Complexity: O(n)

This means in the worst case, the search or insert operation may check every slot once before finding the key or an empty slot.

Common Mistake

[X] Wrong: "Linear probing always finds the key in constant time because it just moves one step at a time."

[OK] Correct: If the table is nearly full or many collisions happen, it may need to check many slots, making the time grow with the table size.

Interview Connect

Understanding how collision handling affects search time helps you explain trade-offs in hash tables clearly and confidently.

Self-Check

"What if we used quadratic probing instead of linear probing? How would the time complexity change?"

Practice

1. What is the main idea behind Linear Probing in collision handling for hash tables?

easy

A. Use a linked list at each slot to store multiple elements.

B. If a collision occurs, check the next slot sequentially until an empty slot is found.

C. Double the size of the hash table when a collision happens.

D. Replace the existing element in the slot with the new one.

Solution

1. Step 1: Understand collision handling methods

   Linear probing is a method where if a collision happens, the algorithm checks the next slot in the array sequentially.

2. Step 2: Compare options with linear probing definition

   If a collision occurs, check the next slot sequentially until an empty slot is found. correctly describes checking the next slot sequentially until an empty slot is found, which matches linear probing.

3. Final Answer:

   If a collision occurs, check the next slot sequentially until an empty slot is found. -> Option B

4. Quick Check:

   Linear probing = check next slot sequentially [OK]

Hint: Linear probing means try next slot one by one [OK]

Common Mistakes:

• Confusing linear probing with chaining
• Thinking linear probing replaces existing elements
• Assuming table size doubles on collision

2. Which of the following is the correct way to calculate the next index in linear probing for a hash table of size tableSize when a collision occurs at index?

easy

A. nextIndex = (index * 2) % tableSize;

B. nextIndex = (index - 1) % tableSize;

C. nextIndex = (index + 1) % tableSize;

D. nextIndex = (index + 3) % tableSize;

Solution

1. Step 1: Recall linear probing formula

   Linear probing moves to the next slot by adding 1 to the current index and taking modulo with table size to wrap around.

2. Step 2: Match formula with options

   nextIndex = (index + 1) % tableSize; uses (index + 1) % tableSize, which is the correct formula for linear probing.

3. Final Answer:

   nextIndex = (index + 1) % tableSize; -> Option C

4. Quick Check:

   Linear probing next index = (index + 1) % size [OK]

Hint: Next index = (current index + 1) mod table size [OK]

Common Mistakes:

• Multiplying index instead of adding
• Subtracting instead of adding
• Adding a number other than 1

3. Consider a hash table of size 5 using linear probing. The hash function is h(key) = key % 5. Insert keys in this order: 7, 12, 17. What will be the final state of the hash table (index: value)?

medium

A. {0: null, 1: null, 2: 7, 3: 12, 4: 17}

B. {0: 12, 1: 17, 2: 7, 3: null, 4: null}

C. {0: null, 1: 7, 2: null, 3: 12, 4: 17}

D. {0: null, 1: 7, 2: 17, 3: 12, 4: null}

Solution

1. Step 1: Calculate initial indices for each key

   7 % 5 = 2, 12 % 5 = 2, 17 % 5 = 2. All keys initially map to index 2.

2. Step 2: Insert keys using linear probing

   Insert 7 at index 2 (empty). Insert 12: index 2 occupied, try index 3 (empty), insert 12. Insert 17: index 2 and 3 occupied, try index 4 (empty), insert 17.

3. Final Answer:

   {0: null, 1: null, 2: 7, 3: 12, 4: 17} -> Option A

4. Quick Check:

   Keys collide at 2, placed at 2,3,4 sequentially [OK]

Hint: All keys hash to same index, place next free slot [OK]

Common Mistakes:

• Placing all keys at index 2 ignoring collisions
• Not wrapping around the table
• Inserting keys in wrong order

4. The following C# code tries to insert keys into a hash table using linear probing. What is the bug?

int tableSize = 5;
int[] hashTable = new int[tableSize];
for (int i = 0; i < tableSize; i++) hashTable[i] = -1;

int Hash(int key) => key % tableSize;

void Insert(int key) {
    int index = Hash(key);
    while (hashTable[index] != -1) {
        index = (index + 1);
    }
    hashTable[index] = key;
}

Insert(7);
Insert(12);
Insert(17);

medium

A. The hash function is incorrect for linear probing.

B. The Insert method should use recursion instead of a loop.

C. The initial hash table values should be 0, not -1.

D. The code does not wrap index back to 0 when it reaches the end of the table.

Solution

1. Step 1: Analyze the index update in the loop

   The code increments index by 1 on collision but does not use modulo to wrap around the table size.

2. Step 2: Identify the consequence of missing modulo

   Without modulo, index can go beyond array bounds causing runtime error or incorrect behavior.

3. Final Answer:

   The code does not wrap index back to 0 when it reaches the end of the table. -> Option D

4. Quick Check:

   Linear probing requires modulo to wrap index [OK]

Hint: Always use modulo to wrap index in linear probing [OK]

Common Mistakes:

• Forgetting modulo causes index out of range
• Changing hash function unnecessarily
• Thinking initial values must be zero

5. You have a hash table of size 7 using linear probing. The hash function is h(key) = key % 7. You insert keys: 10, 24, 31, 17, 3. Then you delete key 24 by marking its slot as empty. Now you want to search for key 31. What problem might occur and how to fix it?

hard

A. Search may stop early at the empty slot of deleted key 24, missing key 31; fix by using a special 'deleted' marker instead of empty.

B. Search will always find key 31 without issues; no fix needed.

C. Key 31 will be overwritten by key 24; fix by rehashing all keys after deletion.

D. The hash function is wrong; fix by changing it to key % 5.

Solution

1. Step 1: Understand deletion effect in linear probing

   Deleting by marking slot empty breaks the probe chain, so search may stop early and miss keys inserted after the deleted slot.

2. Step 2: Identify correct deletion method

   Use a special 'deleted' marker to indicate slot is free for insertion but search should continue probing.

3. Final Answer:

   Search may stop early at the empty slot of deleted key 24, missing key 31; fix by using a special 'deleted' marker instead of empty. -> Option A

4. Quick Check:

   Deleted slots need special marker to continue search [OK]

Hint: Use 'deleted' marker, not empty slot, to avoid search errors [OK]

Common Mistakes:

• Marking deleted slots as empty breaks search
• Assuming search always finds keys after deletion
• Changing hash function unnecessarily