Simplify Complexities

Premise

A hashmap has a time complexity of O(1) for all operations.

Problem

You have to find a constant time method to uniformly random selection of a key in a Hash map.

Assumptions

1. The map can grow to memory size.
2. You can use any readymade hash map.

Solution

I’m discussing the pseudo code for a Python solution. All operations work in constant time except remove().

randMap = {}

def insert(k, v):
    randMap[k] = v    

The random must pick a random key. But how? We can store all the keys in a list and then run random() on the list indexes.

randMap = {}
indexMap = {}
keyList = []
last = 0
def insert(k, v):
    randMap[k] = v
    keyList.append(k)
    last += 1
    
    # Maintain index of each key
    indexMap[k] = len(keyList)    

def getRandom():
    start = 0
    end = len(keyList) - 1
    randomIndex = random(start, end)
    key = keyList[randomIndex]
    
    return key

How do you delete a key?

This is the crux of the problem. Deletion of a key would need us to change the index array too. That means we need to shift all elements of the array and complexity would shoot to O(n).

def remove(k):
    # Remove from the hash map
    randMap.pop(k, None)
    
    # Remove from key list
    index = indexMap[k]
    
    # This is O(n) operation
    keyList.remove(index)

    # Pop the index too
    indexMap.pop(k) 

How to improve deletion?

1. Delete the entry and mark the entry invalid. The probability distribution does not change, however with more and more deletions, you have a sparse array. Thus you would need multiple getRandom() to get a valid key.
2. Move the last element to the deleted element. Adjust the index of the last element.

def remove(k):
    # Remove from the hash map
    randMap.pop(k, None)

    index = indexMap[k]
    indexMap[last] = index
    
    # This is O(1) operation
    keyList[index] = keyLast[last]

    # Pop the index too
    indexMap.pop(k)
    last -= 1 

This is a constant time solution.