# Select a random number from stream, with O(1) space

Given a stream of numbers, generate a random number from the stream. You are allowed to use only O(1) space and the input is in the form of a stream, so can’t store the previously seen numbers.

So how do we generate a random number from the whole stream such that the probability of picking any number is 1/n. with O(1) extra space? This problem is a variation of Reservoir Sampling. Here the value of k is 1.

1) Initialize ‘count’ as 0, ‘count’ is used to store count of numbers seen so far in stream.
2) For each number ‘x’ from stream, do following
…..a) Increment ‘count’ by 1.
…..b) If count is 1, set result as x, and return result.
…..c) Generate a random number from 0 to ‘count-1’. Let the generated random number be i.
…..d) If i is equal to ‘count – 1’, update the result as x.

## C++

 `// An efficient C++ program to randomly select  ` `// a number from stream of numbers.  ` `#include ` `#include ` `using` `namespace` `std; ` ` `  `// A function to randomly select a item  ` `// from stream, stream, .. stream[i-1]  ` `int` `selectRandom(``int` `x)  ` `{  ` `    ``static` `int` `res; ``// The resultant random number  ` `    ``static` `int` `count = 0; ``// Count of numbers visited ` `                          ``// so far in stream  ` ` `  `    ``count++; ``// increment count of numbers seen so far  ` ` `  `    ``// If this is the first element from stream,  ` `    ``// return it  ` `    ``if` `(count == 1)  ` `        ``res = x;  ` `    ``else` `    ``{  ` `        ``// Generate a random number from 0 to count - 1  ` `        ``int` `i = ``rand``() % count;  ` ` `  `        ``// Replace the prev random number with  ` `        ``// new number with 1/count probability  ` `        ``if` `(i == count - 1)  ` `            ``res = x;  ` `    ``}  ` `    ``return` `res;  ` `}  ` ` `  `// Driver Code ` `int` `main()  ` `{  ` `    ``int` `stream[] = {1, 2, 3, 4};  ` `    ``int` `n = ``sizeof``(stream) / ``sizeof``(stream);  ` ` `  `    ``// Use a different seed value for every run.  ` `    ``srand``(``time``(NULL));  ` `    ``for` `(``int` `i = 0; i < n; ++i)  ` `        ``cout << ``"Random number from first "` `<< i + 1  ` `             ``<< ``" numbers is "` `<< selectRandom(stream[i]) << endl;  ` `    ``return` `0;  ` `}  ` ` `  `// This is code is contributed by rathbhupendra `

## C

 `// An efficient C program to randomly select a number from stream of numbers. ` `#include ` `#include ` `#include ` ` `  `// A function to randomly select a item from stream, stream, .. stream[i-1] ` `int` `selectRandom(``int` `x) ` `{ ` `    ``static` `int` `res;    ``// The resultant random number ` `    ``static` `int` `count = 0;  ``//Count of numbers visited so far in stream ` ` `  `    ``count++;  ``// increment count of numbers seen so far ` ` `  `    ``// If this is the first element from stream, return it ` `    ``if` `(count == 1) ` `        ``res = x; ` `    ``else` `    ``{ ` `        ``// Generate a random number from 0 to count - 1 ` `        ``int` `i = ``rand``() % count; ` ` `  `        ``// Replace the prev random number with new number with 1/count probability ` `        ``if` `(i == count - 1) ` `            ``res  = x; ` `    ``} ` `    ``return` `res; ` `} ` ` `  `// Driver program to test above function. ` `int` `main() ` `{ ` `    ``int` `stream[] = {1, 2, 3, 4}; ` `    ``int` `n = ``sizeof``(stream)/``sizeof``(stream); ` ` `  `    ``// Use a different seed value for every run. ` `    ``srand``(``time``(NULL)); ` `    ``for` `(``int` `i = 0; i < n; ++i) ` `        ``printf``(``"Random number from first %d numbers is %d \n"``, ` `                                ``i+1, selectRandom(stream[i])); ` `    ``return` `0; ` `} `

## Java

 `//An efficient Java program to randomly select a number from stream of numbers. ` ` `  `import` `java.util.Random; ` ` `  `public` `class` `GFG  ` `{ ` `    ``static` `int` `res = ``0``;    ``// The resultant random number ` `    ``static` `int` `count = ``0``;  ``//Count of numbers visited so far in stream ` `     `  `    ``//A method to randomly select a item from stream, stream, .. stream[i-1] ` `    ``static` `int` `selectRandom(``int` `x) ` `    ``{ ` `        ``count++; ``// increment count of numbers seen so far ` `         `  `        ``// If this is the first element from stream, return it ` `        ``if` `(count == ``1``) ` `            ``res = x; ` `        ``else` `        ``{ ` `             ``// Generate a random number from 0 to count - 1 ` `            ``Random r = ``new` `Random(); ` `            ``int` `i = r.nextInt(count); ` `             `  `            ``// Replace the prev random number with new number with 1/count probability ` `            ``if``(i == count - ``1``) ` `                ``res = x; ` `        ``} ` `        ``return` `res; ` `    ``} ` `     `  `    ``// Driver program to test above function. ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `stream[] = {``1``, ``2``, ``3``, ``4``}; ` `        ``int` `n = stream.length; ` `        ``for``(``int` `i = ``0``; i < n; i++) ` `            ``System.out.println(``"Random number from first "` `+ (i+``1``) + ` `                               ``" numbers is "` `+ selectRandom(stream[i])); ` `    ``} ` `} ` `//This code is contributed by Sumit Ghosh `

## Python3

 `# An efficient python3 program  ` `# to randomly select a number ` `# from stream of numbers. ` `import` `random ` ` `  `# A function to randomly select a item ` `# from stream, stream, .. stream[i-1] ` `def` `selectRandom(x): ` `     `  `    ``# The resultant random number ` `    ``res ``=` `0``; ` `     `  `    ``# Count of numbers visited  ` `    ``# so far in stream ` `    ``count ``=` `0``; ` ` `  `    ``# increment count of numbers  ` `    ``# seen so far ` `    ``count ``+``=` `1``; ` ` `  `    ``# If this is the first element  ` `    ``# from stream, return it ` `    ``if` `(count ``=``=` `1``): ` `        ``res ``=` `x; ` `    ``else``: ` `         `  `        ``# Generate a random number  ` `        ``# from 0 to count - 1 ` `        ``i ``=` `random.randrange(count); ` ` `  `        ``# Replace the prev random number  ` `        ``# with new number with 1/count  ` `        ``# probability ` `        ``if` `(i ``=``=` `count ``-` `1``): ` `            ``res ``=` `x; ` `    ``return` `res; ` ` `  `# Driver Code ` `stream ``=` `[``1``, ``2``, ``3``, ``4``]; ` `n ``=` `len``(stream); ` ` `  `# Use a different seed value  ` `# for every run. ` `for` `i ``in` `range` `(n): ` `    ``print``(``"Random number from first"``,  ` `         ``(i ``+` `1``), ``"numbers is"``,  ` `          ``selectRandom(stream[i])); ` ` `  `# This code is contributed by mits `

## C#

 `// An efficient C# program to randomly  ` `// select a number from stream of numbers. ` `using` `System; ` ` `  `class` `GFG  ` `{ ` `    ``// The resultant random number ` `    ``static` `int` `res = 0;  ` `     `  `    ``// Count of numbers visited ` `    ``// so far in stream ` `    ``static` `int` `count = 0;  ` `     `  `    ``// A method to randomly select  ` `    ``// a item from stream,  ` `    ``// stream, .. stream[i-1] ` `    ``static` `int` `selectRandom(``int` `x) ` `    ``{ ` `        ``// increment count of ` `        ``// numbers seen so far ` `        ``count++;  ` `         `  `        ``// If this is the first  ` `        ``// element from stream,  ` `        ``// return it ` `        ``if` `(count == 1) ` `            ``res = x; ` `        ``else` `        ``{ ` `            ``// Generate a random number  ` `            ``// from 0 to count - 1 ` `            ``Random r = ``new` `Random(); ` `            ``int` `i = r.Next(count); ` `             `  `            ``// Replace the prev random  ` `            ``// number with new number  ` `            ``// with 1/count probability ` `            ``if``(i == count - 1) ` `                ``res = x; ` `        ``} ` `        ``return` `res; ` `    ``} ` `     `  `// Driver Code ` `public` `static` `void` `Main() ` `{ ` `        ``int``[] stream = {1, 2, 3, 4}; ` `        ``int` `n = stream.Length; ` `        ``for``(``int` `i = 0; i < n; i++) ` `            ``Console.WriteLine(``"Random number from "` `+  ` `                              ``"first {0} numbers is {1}"` `, ` `                          ``i + 1, selectRandom(stream[i])); ` `    ``} ` `} ` ` `  `// This code is contributed by mits `

## PHP

 ` `

Output:

```Random number from first 1 numbers is 1
Random number from first 2 numbers is 1
Random number from first 3 numbers is 3
Random number from first 4 numbers is 4
```

Auxiliary Space: O(1)

How does this work
We need to prove that every element is picked with 1/n probability where n is the number of items seen so far. For every new stream item x, we pick a random number from 0 to ‘count -1’, if the picked number is ‘count-1’, we replace the previous result with x.

To simplify proof, let us first consider the last element, the last element replaces the previously-stored result with 1/n probability. So the probability of getting the last element as the result is 1/n.

Let us now talk about the second last element. When the second last element processed the first time, the probability that it replaced the previous result is 1/(n-1). The probability that the previous result stays when the nth item is considered is (n-1)/n. So the probability that the second last element is picked in the last iteration is [1/(n-1)] * [(n-1)/n] which is 1/n.

Similarly, we can prove for third last element and others.

References:
Reservoir Sampling

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