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

Time Complexity: Collision Handling Using Open Addressing Linear Probing

O(n)

Understanding Time Complexity

When we use open addressing with linear probing to handle collisions in a hash table, we want to know how long it takes to insert or find an item.

We ask: How does the time grow as the table fills up?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

class HashTable:
    def __init__(self, size):
        self.size = size
        self.table = [None] * size

    def insert(self, key):
        index = key % self.size
        while self.table[index] is not None:
            index = (index + 1) % self.size
        self.table[index] = key

This code inserts keys into a hash table using linear probing to find the next empty slot when a collision happens.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that checks for the next empty slot when a collision occurs.
• How many times: In the worst case, it can check almost all slots if the table is nearly full.

How Execution Grows With Input

When the table is mostly empty, the loop runs very few times. As the table fills, the loop runs more times to find an empty slot.

Input Size (n)	Approx. Operations
10 (10% full)	About 1 check per insert
100 (50% full)	Several checks per insert
1000 (90% full)	Many checks, close to 1000 in worst case

Pattern observation: The more full the table, the longer it takes to find an empty slot, growing roughly linearly with the number of filled slots.

Final Time Complexity

Time Complexity: O(n)

This means in the worst case, the time to insert or search grows linearly with the number of items in the table.

Common Mistake

[X] Wrong: "Insertion always takes constant time because hashing is constant time."

[OK] Correct: When collisions happen and the table is full or nearly full, linear probing must check many slots, making insertion slower.

Interview Connect

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

Self-Check

"What if we changed linear probing to quadratic 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. Double the size of the hash table when a collision occurs.

B. If a slot is occupied, check the next slot sequentially until an empty one is found.

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

D. Rehash all elements using a new hash function.

Solution

1. Step 1: Understand linear probing concept

   Linear probing handles collisions by checking the next slot in order if the original slot is taken.

2. Step 2: Compare with other methods

   Other methods like chaining use linked lists, resizing, or rehashing, which are different from linear probing.

3. Final Answer:

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

4. Quick Check:

   Linear probing = sequential slot check [OK]

Hint: Linear probing means try next slot until free [OK]

Common Mistakes:

• Confusing linear probing with chaining
• Thinking linear probing resizes the table
• Assuming rehashing is part of linear probing

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

easy

A. (i + 2) % m

B. (i * 2) % m

C. (i + 1) % m

D. (i - 1) % m

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.

2. Step 2: Apply modulo for wrap-around

   Using modulo ensures the index wraps around to the start if it reaches the end of the table.

3. Final Answer:

   (i + 1) % m -> Option C

4. Quick Check:

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

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

Common Mistakes:

• Adding 2 instead of 1 for next index
• Multiplying index instead of adding
• Not using modulo for wrap-around

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

medium

A. [None, None, 7, 12, 17]

B. [7, 12, 17, None, None]

C. [None, 7, 12, 17, None]

D. [7, None, 12, 17, None]

Solution

1. Step 1: Calculate initial positions

   h(7) = 7 % 5 = 2, h(12) = 12 % 5 = 2, h(17) = 17 % 5 = 2.

2. Step 2: Insert keys using linear probing

   Insert 7 at index 2. For 12, index 2 is taken, so try index 3. Insert 12 at index 3. For 17, indices 2 and 3 are taken, so insert at index 4.

3. Final Answer:

   [None, None, 7, 12, 17] -> Option A

4. Quick Check:

   Keys placed at 2, 3, 4 with linear probing [OK]

Hint: Check hash index, then next free slot sequentially [OK]

Common Mistakes:

• Placing all keys at same index ignoring collisions
• Not probing to next slot after collision
• Miscomputing hash values

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

def insert(table, key):
    m = len(table)
    index = key % m
    while table[index] is not None:
        index = (index + 1) % m
    table[index] = key

hash_table = [None]*5
insert(hash_table, 5)
insert(hash_table, 10)
insert(hash_table, 15)
insert(hash_table, 20)
insert(hash_table, 25)
insert(hash_table, 30)

medium

A. Infinite loop if table is full because no stop condition in while loop.

B. No bug; code works correctly for all insertions.

C. Wrong hash function used; should be key // m.

D. Table size should be dynamic, not fixed.

Solution

1. Step 1: Analyze insertion loop

   The while loop continues probing until it finds an empty slot but has no check for full table.

2. Step 2: Identify problem when table is full

   If all slots are occupied, the loop cycles endlessly, causing infinite loop.

3. Final Answer:

   Infinite loop if table is full because no stop condition in while loop. -> Option A

4. Quick Check:

   Full table causes infinite loop without exit [OK]

Hint: Check for full table to avoid infinite loop [OK]

Common Mistakes:

• Assuming hash function is wrong
• Ignoring infinite loop possibility
• Thinking dynamic size fixes this bug

5. You have a hash table of size 7 using linear probing. Insert keys [14, 21, 28, 35, 42] using hash function h(k) = k % 7. After all insertions, what is the index of key 42 in the table?

hard

A. 5

B. 0

C. 6

D. 4

Solution

1. Step 1: Calculate initial hash indices

   h(14)=0, h(21)=0, h(28)=0, h(35)=0, h(42)=0 (all keys hash to index 0).

2. Step 2: Insert keys using linear probing

   Insert 14 at 0. For 21, index 0 taken, try 1. Insert 21 at 1. For 28, indices 0 and 1 taken, insert at 2. For 35, insert at 3. For 42, insert at 4.

3. Step 3: Find index of key 42

   Key 42 is at index 4.

4. Final Answer:

   4 -> Option D

5. Quick Check:

   42 placed at first free slot after collisions [OK]

Hint: All keys collide; count steps from initial index [OK]

Common Mistakes:

• Assuming key stays at initial hash index
• Not probing sequentially after collisions
• Miscounting index after multiple collisions