Coding PYTHON Randomized Algorithms

Python Programming – Select a Random Node from a Singly Linked List

The idea is to use Reservoir Sampling. Following are the steps. This is a simpler version of Reservoir Sampling as we need to select only one key.

Given a singly linked list, select a random node from linked list (the probability of picking a node should be 1/N if there are N nodes in list). You are given a random number generator.

Below is a Simple Solution
1) Count number of nodes by traversing the list.
2) Traverse the list again and select every node with probability 1/N. The selection can be done by generating a random number from 0 to N-i for i’th node, and selecting the i’th node node only if generated number is equal to 0 (or any other fixed number from 0 to N-i).

We get uniform probabilities with above schemes.

i = 1, probability of selecting first node = 1/N
i = 2, probability of selecting second node =
                   [probability that first node is not selected] * 
                   [probability that second node is selected]
                  = ((N-1)/N)* 1/(N-1)
                  = 1/N

Similarly, probabilities of other selecting other nodes is 1/N

The above solution requires two traversals of linked list.

How to select a random node with only one traversal allowed?
The idea is to use Reservoir Sampling. Following are the steps. This is a simpler version of Reservoir Sampling as we need to select only one key instead of k keys.

(1) Initialize result as first node
   result = head->key 
(2) Initialize n = 2
(3) Now one by one consider all nodes from 2nd node onward.
    (3.a) Generate a random number from 0 to n-1. 
         Let the generated random number is j.
    (3.b) If j is equal to 0 (we could choose other fixed number 
          between 0 to n-1), then replace result with current node.
    (3.c) n = n+1
    (3.d) current = current->next

Below is the implementation of above algorithm.

Python Program
# Python program to randomly select a node from singly
# linked list 
import random
# Node class 
class Node:
    # Constructor to initialize the node object
    def __init__(self, data): data = None
class LinkedList:
    # Function to initialize head
    def __init__(self):
        self.head = None
    # A reservoir sampling based function to print a
    # random node from a linkd list
    def printRandom(self):
        # If list is empty 
        if self.head is None:
        # Use a different seed value so that we don't get 
        # same result each time we run this program
        # Initialize result as first node
        result =
        # Iterate from the (k+1)th element nth element
        current = self.head 
        n = 2
        while(current is not None):
            # change result with probability 1/n
            if (random.randrange(n) == 0 ):
                result = 
            # Move to next node
            current =
            n += 1
        print "Randomly selected key is %d" %(result)
    # Function to insert a new node at the beginning
    def push(self, new_data):
        new_node = Node(new_data) = self.head
        self.head = new_node
    # Utility function to print the linked LinkedList
    def printList(self):
        temp = self.head
            temp =
# Driver program to test above function
llist = LinkedList()

Note that the above program is based on outcome of a random function and may produce different output.

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How does this work?
Let there be total N nodes in list. It is easier to understand from last node.

The probability that last node is result simply 1/N [For last or N’th node, we generate a random number between 0 to N-1 and make last node as result if the generated number is 0 (or any other fixed number]

The probability that second last node is result should also be 1/N.

The probability that the second last node is result 
          = [Probability that the second last node replaces result] X 
            [Probability that the last node doesn't replace the result] 
          = [1 / (N-1)] * [(N-1)/N]
          = 1/N

Similarly we can show probability for 3rd last node and other nodes.

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