Given a range represented by two positive integers L and R and a positive integer K. Find the count of numbers in the range where the number does not contain more than K non zero digits.
Input : L = 1, R = 1000, K = 3 Output : 1000 Explanation : All the numbers from 1 to 1000 are 3 digit numbers which obviously cannot contain more than 3 non zero digits. Input : L = 9995, R = 10005 Output : 6 Explanation : Required numbers are 10000, 10001, 10002, 10003, 10004 and 10005
Prerequisites : Digit DP
There can be two approaches to solve this type of problem, one can be a combinatorial solution and other can be a dynamic programming based solution. Below is a detailed approach of solving this problem using a digit dynamic programming approach.
Dynamic Programming Solution : Firstly, if we are able to count the required numbers upto R i.e. in the range [0, R], we can easily reach our answer in the range [L, R] by solving for from zero to R and then subtracting the answer we get after solving for from zero to L – 1. Now, we need to define the DP states.
- Since we can consider our number as a sequence of digits, one state is the position at which we are currently in. This position can have values from 0 to 18 if we are dealing with the numbers upto 1018. In each recursive call, we try to build the sequence from left to right by placing a digit from 0 to 9.
- Second state is the count which defines the number of non zero digits, we have placed in the number we are trying to build.
- Another state is the boolean variable tight which tells the number we are trying to build has already become smaller than R, so that in the upcoming recursive calls we can place any digit from 0 to 9. If the number has not become smaller, maximum limit of digit we can place is digit at current position in R.
In the final recursive call, when we are at the last position if the count of non zero digits is less than or equal to K, return 1 otherwise return 0.
Below is the implementation of the above approach.
Time Complexity : O(18 * 18 * 2 * 10), if we are dealing with the numbers upto 1018
- Count numbers in range L-R that are divisible by all of its non-zero digits
- Count numbers in range such that digits in it and it's product with q are unequal
- Count Numbers with N digits which consists of odd number of 0's
- Count Numbers with N digits which consists of even number of 0’s
- Count total number of N digit numbers such that the difference between sum of even and odd digits is 1
- Program to find count of numbers having odd number of divisors in given range
- Total numbers with no repeated digits in a range
- Count of Numbers in Range where first digit is equal to last digit of the number
- Count numbers with same first and last digits
- Count Unary Numbers in a Range
- Count factorial numbers in a given range
- Count the numbers divisible by 'M' in a given range
- Count different numbers that can be generated such that there digits sum is equal to 'n'
- Count of n digit numbers whose sum of digits equals to given sum
- Count numbers formed by given two digit with sum having given digits
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