Queries for decimal values of subarrays of a binary array
Last Updated :
28 Mar, 2023
Given a binary array arr[], we to find the number represented by the subarray a[l..r]. There are multiple such queries.
Examples:
Input : arr[] = {1, 0, 1, 0, 1, 1};
l = 2, r = 4
l = 4, r = 5
Output : 5
3
Subarray 2 to 4 is 101 which is 5 in decimal.
Subarray 4 to 5 is 11 which is 3 in decimal.
Input : arr[] = {1, 1, 1}
l = 0, r = 2
l = 1, r = 2
Output : 7
3
A Simple Solution is to compute decimal value for every given range using simple binary to decimal conversion. Here each query takes O(len) time where len is length of range.
An Efficient Solution is to do per-computations, so that queries can be answered in O(1) time.
The number represented by subarray arr[l..r] is arr[l]*2^{r-l} + arr[l+1]*2^{r - l - 1} ..... + arr[r]*2^{r-r}
- Make an array pre[] of same size as of given array where pre[i] stores the sum of arr[j]*2^{n - 1 - j} where j includes each value from i to n-1.
- The number represented by subarray arr[l..r] will be equal to (pre[l] - pre[r+1])/2^{n-1-r} .pre[l] - pre[r+1] is equal to arr[l]*2^{n - 1 - l} + arr[l+1]*2^{n - 1 - l - 1} +......arr[r]*2^{n - 1 - r} . So if we divide it by 2^{n - 1 - r} , we get the required answer
Flowchart
Flowchart
Implementation:
C++
// C++ implementation of finding number
// represented by binary subarray
#include <bits/stdc++.h>
using namespace std;
// Fills pre[]
void precompute(int arr[], int n, int pre[])
{
memset(pre, 0, n * sizeof(int));
pre[n - 1] = arr[n - 1] * pow(2, 0);
for (int i = n - 2; i >= 0; i--)
pre[i] = pre[i + 1] + arr[i] * (1 << (n - 1 - i));
}
// returns the number represented by a binary
// subarray l to r
int decimalOfSubarr(int arr[], int l, int r,
int n, int pre[])
{
// if r is equal to n-1 r+1 does not exist
if (r != n - 1)
return (pre[l] - pre[r + 1]) / (1 << (n - 1 - r));
return pre[l] / (1 << (n - 1 - r));
}
// Driver Function
int main()
{
int arr[] = { 1, 0, 1, 0, 1, 1 };
int n = sizeof(arr) / sizeof(arr[0]);
int pre[n];
precompute(arr, n, pre);
cout << decimalOfSubarr(arr, 2, 4, n, pre) << endl;
cout << decimalOfSubarr(arr, 4, 5, n, pre) << endl;
return 0;
}
Java
// Java implementation of finding number
// represented by binary subarray
import java.util.Arrays;
class GFG {
// Fills pre[]
static void precompute(int arr[], int n, int pre[])
{
Arrays.fill(pre, 0);
pre[n - 1] = arr[n - 1] * (int)(Math.pow(2, 0));
for (int i = n - 2; i >= 0; i--)
pre[i] = pre[i + 1] + arr[i] * (1 << (n - 1 - i));
}
// returns the number represented by a binary
// subarray l to r
static int decimalOfSubarr(int arr[], int l, int r,
int n, int pre[])
{
// if r is equal to n-1 r+1 does not exist
if (r != n - 1)
return (pre[l] - pre[r + 1]) / (1 << (n - 1 - r));
return pre[l] / (1 << (n - 1 - r));
}
// Driver code
public static void main(String[] args)
{
int arr[] = { 1, 0, 1, 0, 1, 1 };
int n = arr.length;
int pre[] = new int[n];
precompute(arr, n, pre);
System.out.println(decimalOfSubarr(arr,
2, 4, n, pre));
System.out.println(decimalOfSubarr(arr,
4, 5, n, pre));
}
}
// This code is contributed by Anant Agarwal.
Python3
# implementation of finding number
# represented by binary subarray
from math import pow
# Fills pre[]
def precompute(arr, n, pre):
pre[n - 1] = arr[n - 1] * pow(2, 0)
i = n - 2
while(i >= 0):
pre[i] = (pre[i + 1] + arr[i] *
(1 << (n - 1 - i)))
i -= 1
# returns the number represented by
# a binary subarray l to r
def decimalOfSubarr(arr, l, r, n, pre):
# if r is equal to n-1 r+1 does not exist
if (r != n - 1):
return ((pre[l] - pre[r + 1]) /
(1 << (n - 1 - r)))
return pre[l] / (1 << (n - 1 - r))
# Driver Code
if __name__ == '__main__':
arr = [1, 0, 1, 0, 1, 1]
n = len(arr)
pre = [0 for i in range(n)]
precompute(arr, n, pre)
print(int(decimalOfSubarr(arr, 2, 4, n, pre)))
print(int(decimalOfSubarr(arr, 4, 5, n, pre)))
# This code is contributed by
# Surendra_Gangwar
C#
// C# implementation of finding number
// represented by binary subarray
using System;
class GFG {
// Fills pre[]
static void precompute(int[] arr, int n, int[] pre)
{
for (int i = 0; i < n; i++)
pre[i] = 0;
pre[n - 1] = arr[n - 1] * (int)(Math.Pow(2, 0));
for (int i = n - 2; i >= 0; i--)
pre[i] = pre[i + 1] + arr[i] * (1 << (n - 1 - i));
}
// returns the number represented by
// a binary subarray l to r
static int decimalOfSubarr(int[] arr, int l, int r,
int n, int[] pre)
{
// if r is equal to n-1 r+1 does not exist
if (r != n - 1)
return (pre[l] - pre[r + 1]) / (1 << (n - 1 - r));
return pre[l] / (1 << (n - 1 - r));
}
// Driver code
public static void Main()
{
int[] arr = { 1, 0, 1, 0, 1, 1 };
int n = arr.Length;
int[] pre = new int[n];
precompute(arr, n, pre);
Console.WriteLine(decimalOfSubarr(arr,
2, 4, n, pre));
Console.WriteLine(decimalOfSubarr(arr,
4, 5, n, pre));
}
}
// This code is contributed by vt_m.
PHP
<?php
// PHP implementation of finding number
// represented by binary subarray
// Fills pre[]
function precompute(&$arr, $n, &$pre)
{
$pre[$n - 1] = $arr[$n - 1] * pow(2, 0);
for ($i = $n - 2; $i >= 0; $i--)
$pre[$i] = $pre[$i + 1] + $arr[$i] *
(1 << ($n - 1 - $i));
}
// returns the number represented by
// a binary subarray l to r
function decimalOfSubarr(&$arr, $l, $r, $n, &$pre)
{
// if r is equal to n-1 r+1 does not exist
if ($r != $n - 1)
return ($pre[$l] - $pre[$r + 1]) /
(1 << ($n - 1 - $r));
return $pre[$l] / (1 << ($n - 1 - $r));
}
// Driver Code
$arr = array(1, 0, 1, 0, 1, 1 );
$n = sizeof($arr);
$pre = array_fill(0, $n, NULL);
precompute($arr, $n, $pre);
echo decimalOfSubarr($arr, 2, 4, $n, $pre) . "\n";
echo decimalOfSubarr($arr, 4, 5, $n, $pre) . "\n";
// This code is contributed by ita_c
?>
JavaScript
<script>
// Javascript implementation of finding number
// represented by binary subarray
// Fills pre[]
function precompute(arr, n, pre)
{
for (let i = 0; i < n; i++)
pre[i] = 0;
pre[n - 1] = arr[n - 1] * (Math.pow(2, 0));
for (let i = n - 2; i >= 0; i--)
pre[i] = pre[i + 1] + arr[i] *
(1 << (n - 1 - i));
}
// returns the number represented by
// a binary subarray l to r
function decimalOfSubarr(arr, l, r,n, pre)
{
// if r is equal to n-1 r+1 does not exist
if (r != n - 1)
return (pre[l] - pre[r + 1]) / (1 << (n - 1 - r));
return pre[l] / (1 << (n - 1 - r));
}
// Driver code
let arr = [1, 0, 1, 0, 1, 1];
let n = arr.length;
let pre = new Array(n)
precompute(arr, n, pre);
document.write(decimalOfSubarr(arr,2, 4, n, pre)+"<br>");
document.write(decimalOfSubarr(arr, 4, 5, n, pre));
</script>
Time complexity: O(n)
Auxiliary Space: O(n)
Efficient approach :
traverse the array from the given start index to end index, multiplying each binary digit with the corresponding power of 2 and adding the result.
Implementation :
C++
#include <bits/stdc++.h>
using namespace std;
// function to find the decimal equivalent of the subarray
int decimalOfSubarr(int arr[], int l, int r, int n) {
int ans = 0, p = 1; // initialize ans and p variables to 0 and 1 respectively
for (int i = r; i >= l; i--) { // loop through the subarray from r to l
ans += arr[i] * p; // add the current bit multiplied by its corresponding power of 2 to the ans
p *= 2; // update the power of 2 for the next bit
}
return ans; // return the decimal equivalent of the subarray
}
int main() {
int arr[] = {1, 0, 1, 0, 1, 1}; // initialize the input array
int n = sizeof(arr) / sizeof(arr[0]); // calculate the size of the array
// output the decimal equivalents of the subarrays [2, 4] and [4, 5] of the input array
cout << decimalOfSubarr(arr, 2, 4, n) << endl; // Output: 5
cout << decimalOfSubarr(arr, 4, 5, n) << endl; // Output: 3
return 0; // indicate successful program execution
}
//this code is contributed by bhardwajji
Java
import java.util.*;
public class Main {
// function to find the decimal equivalent of the subarray
public static int decimalOfSubarr(int[] arr, int l, int r, int n) {
int ans = 0, p = 1; // initialize ans and p variables to 0 and 1 respectively
for (int i = r; i >= l; i--) { // loop through the subarray from r to l
ans += arr[i] * p; // add the current bit multiplied by its corresponding power of 2 to the ans
p *= 2; // update the power of 2 for the next bit
}
return ans; // return the decimal equivalent of the subarray
}
public static void main(String[] args) {
int[] arr = {1, 0, 1, 0, 1, 1}; // initialize the input array
int n = arr.length; // calculate the size of the array
// output the decimal equivalents of the subarrays [2, 4] and [4, 5] of the input array
System.out.println(decimalOfSubarr(arr, 2, 4, n)); // Output: 5
System.out.println(decimalOfSubarr(arr, 4, 5, n)); // Output: 3
}
}
Python3
# Python equivalent of the above Java code
def decimalOfSubarr(arr, l, r, n):
ans = 0
p = 1
# loop through the subarray from r to l
for i in range(r, l-1, -1):
ans += arr[i] * p # add the current bit multiplied by its corresponding power of 2 to the ans
p *= 2 # update the power of 2 for the next bit
return ans # return the decimal equivalent of the subarray
# driver code
arr = [1, 0, 1, 0, 1, 1] # initialize the input array
n = len(arr) # calculate the size of the array
# output the decimal equivalents of the subarrays [2, 4] and [4, 5] of the input array
print(decimalOfSubarr(arr, 2, 4, n)) # Output: 5
print(decimalOfSubarr(arr, 4, 5, n)) # Output: 3
JavaScript
function decimalOfSubarr(arr, l, r, n) {
let ans = 0
let p = 1
// loop through the subarray from r to l
for (let i = r; i >= l; i--) {
ans += arr[i] * p // add the current bit multiplied by its corresponding power of 2 to the ans
p *= 2 // update the power of 2 for the next bit
}
return ans // return the decimal equivalent of the subarray
}
// driver code
let arr = [1, 0, 1, 0, 1, 1] // initialize the input array
let n = arr.length // calculate the size of the array
// output the decimal equivalents of the subarrays [2, 4] and [4, 5] of the input array
console.log(decimalOfSubarr(arr, 2, 4, n)) // Output: 5
console.log(decimalOfSubarr(arr, 4, 5, n)) // Output: 3
C#
// C# implementation of finding number
// represented by binary subarray
using System;
class GFG {
// returns the number represented by
// a binary subarray l to r
static int decimalOfSubarr(int[] arr, int l, int r,
int n)
{
int ans = 0, p = 1; // initialize ans and p variables to 0 and 1 respectively
for (int i = r; i >= l; i--) { // loop through the subarray from r to l
ans += arr[i] * p; // add the current bit multiplied by its corresponding power of 2 to the ans
p *= 2; // update the power of 2 for the next bit
}
return ans; // return the decimal equivalent of the subarray
}
// Driver code
public static void Main()
{
int[] arr = { 1, 0, 1, 0, 1, 1 };
int n = arr.Length;
Console.WriteLine(decimalOfSubarr(arr,
2, 4, n));
Console.WriteLine(decimalOfSubarr(arr,
4, 5, n));
}
}
// This code is contributed by shubhamrajput6156
Time complexity: O(r-l+1), which is equivalent to the length of the subarray
Auxiliary Space: O(1), as we are not using any additional data structures.
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Hereâs the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem