Given two positive integers L and R which represents a range and two more positive integers d and K. The task is to find the count of numbers in the range where digit d occurs exactly K times.
Examples:
Input: L = 11, R = 100, d = 2, k = 1
Output: 17
Required numbers are 12, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82 and 92.
Input: L = 95, R = 1005, d = 0, k = 2
Output: 14
Prerequisites : Digit DP
Approach: 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.
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 times, we have placed digit d so far.
- 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.
- Last state is also boolean variable nonz which helps to consider the situation if are any leading zeroes in the number we are building, we don’t need to count them.
In the final recursive call, when we are at the last position if the count of digit d is equal to K, return 1 otherwise return 0.
Below is the implementation of the above approach:
C++
// CPP Program to find the count of// numbers in a range where digit d// occurs exactly K times#include <bits/stdc++.h>using namespace std;const int M = 20;// states - position, count, tight, nonzint dp[M][M][2][2];// d is required digit and K is occurrenceint d, K;// This function returns the count of// required numbers from 0 to numint count(int pos, int cnt, int tight, int nonz, vector<int> num){ // Last position if (pos == num.size()) { if (cnt == K) return 1; return 0; } // If this result is already computed // simply return it if (dp[pos][cnt][tight][nonz] != -1) return dp[pos][cnt][tight][nonz]; int ans = 0; // Maximum limit upto which we can place // digit. If tight is 1, means number has // already become smaller so we can place // any digit, otherwise num[pos] int limit = (tight ? 9 : num[pos]); for (int dig = 0; dig <= limit; dig++) { int currCnt = cnt; // Nonz is true if we placed a non // zero digit at the starting of // the number if (dig == d) { if (d != 0 || (!d && nonz)) currCnt++; } int currTight = tight; // At this position, number becomes // smaller if (dig < num[pos]) currTight = 1; // Next recursive call, also set nonz // to 1 if current digit is non zero ans += count(pos + 1, currCnt, currTight, nonz || (dig != 0), num); } return dp[pos][cnt][tight][nonz] = ans;}// Function to convert x into its digit vector and uses// count() function to return the required countint solve(int x){ vector<int> num; while (x) { num.push_back(x % 10); x /= 10; } reverse(num.begin(), num.end()); // Initialize dp memset(dp, -1, sizeof(dp)); return count(0, 0, 0, 0, num);}// Driver Code to test above functionsint main(){ int L = 11, R = 100; d = 2, K = 1; cout << solve(R) - solve(L - 1) << endl; return 0;} |
Java
// Java Program to find the count of// numbers in a range where digit d// occurs exactly K timesimport java.util.*;class Solution{static final int M = 20; // states - position, count, tight, nonzstatic int dp[][][][]= new int[M][M][2][2]; // d is required digit and K is occurrencestatic int d, K; // This function returns the count of// required numbers from 0 to numstatic int count(int pos, int cnt, int tight, int nonz, Vector<Integer> num){ // Last position if (pos == num.size()) { if (cnt == K) return 1; return 0; } // If this result is already computed // simply return it if (dp[pos][cnt][tight][nonz] != -1) return dp[pos][cnt][tight][nonz]; int ans = 0; // Maximum limit upto which we can place // digit. If tight is 1, means number has // already become smaller so we can place // any digit, otherwise num[pos] int limit = ((tight !=0)? 9 : num.get(pos)); for (int dig = 0; dig <= limit; dig++) { int currCnt = cnt; // Nonz is true if we placed a non // zero digit at the starting of // the number if (dig == d) { if (d != 0 || (d==0 && nonz!=0)) currCnt++; } int currTight = tight; // At this position, number becomes // smaller if (dig < num.get(pos)) currTight = 1; // Next recursive call, also set nonz // to 1 if current digit is non zero ans += count(pos + 1, currCnt, currTight, (dig != 0?1:0), num); } return dp[pos][cnt][tight][nonz] = ans;} // Function to convert x into its digit vector and uses// count() function to return the required countstatic int solve(int x){ Vector<Integer> num= new Vector<Integer>(); while (x!=0) { num.add(x % 10); x /= 10; } Collections.reverse(num); // Initialize dp for(int i=0;i<M;i++) for(int j=0;j<M;j++) for(int k=0;k<2;k++) for(int l=0;l<2;l++) dp[i][j][k][l]=-1; return count(0, 0, 0, 0, num);} // Driver Code to test above functionspublic static void main(String args[]){ int L = 11, R = 100; d = 2; K = 1; System.out.print( solve(R) - solve(L - 1) );}}//contributed by Arnab Kundu |
Python3
# Python Program to find the count of# numbers in a range where digit d# occurs exactly K timesM = 20# states - position, count, tight, nonzdp = []# d is required digit and K is occurrenced, K = None, None# This function returns the count of# required numbers from 0 to numdef count(pos, cnt, tight, nonz, num: list): # Last position if pos == len(num): if cnt == K: return 1 return 0 # If this result is already computed # simply return it if dp[pos][cnt][tight][nonz] != -1: return dp[pos][cnt][tight][nonz] ans = 0 # Maximum limit upto which we can place # digit. If tight is 1, means number has # already become smaller so we can place # any digit, otherwise num[pos] limit = 9 if tight else num[pos] for dig in range(limit + 1): currCnt = cnt # Nonz is true if we placed a non # zero digit at the starting of # the number if dig == d: if d != 0 or not d and nonz: currCnt += 1 currTight = tight # At this position, number becomes # smaller if dig < num[pos]: currTight = 1 # Next recursive call, also set nonz # to 1 if current digit is non zero ans += count(pos + 1, currCnt, currTight, (nonz or dig != 0), num) dp[pos][cnt][tight][nonz] = ans return dp[pos][cnt][tight][nonz]# Function to convert x into its digit vector and uses# count() function to return the required countdef solve(x): global dp, K, d num = [] while x: num.append(x % 10) x //= 10 num.reverse() # Initialize dp dp = [[[[-1, -1] for i in range(2)] for j in range(M)] for k in range(M)] return count(0, 0, 0, 0, num)# Driver Codeif __name__ == "__main__": L = 11 R = 100 d = 2 K = 1 print(solve(R) - solve(L - 1))# This code is contributed by# sanjeev2552 |
C#
// C# Program to find the count of// numbers in a range where digit d// occurs exactly K timesusing System;using System.Collections.Generic; class GFG{ static readonly int M = 20; // states - position, count, tight, nonz static int [,,,]dp= new int[M, M, 2, 2]; // d is required digit and K is occurrence static int d, K; // This function returns the count of // required numbers from 0 to num static int count(int pos, int cnt, int tight, int nonz, List<int> num) { // Last position if (pos == num.Count) { if (cnt == K) return 1; return 0; } // If this result is already computed // simply return it if (dp[pos, cnt, tight, nonz] != -1) return dp[pos, cnt, tight, nonz]; int ans = 0; // Maximum limit upto which we can place // digit. If tight is 1, means number has // already become smaller so we can place // any digit, otherwise num[pos] int limit = ((tight != 0) ? 9 : num[pos]); for (int dig = 0; dig <= limit; dig++) { int currCnt = cnt; // Nonz is true if we placed a non // zero digit at the starting of // the number if (dig == d) { if (d != 0 || (d == 0 && nonz != 0)) currCnt++; } int currTight = tight; // At this position, number becomes // smaller if (dig < num[pos]) currTight = 1; // Next recursive call, also set nonz // to 1 if current digit is non zero ans += count(pos + 1, currCnt, currTight, (dig != 0 ? 1 : 0), num); } return dp[pos, cnt, tight, nonz] = ans; } // Function to convert x into its // digit vector and uses count() // function to return the required count static int solve(int x) { List<int> num = new List<int>(); while (x != 0) { num.Add(x % 10); x /= 10; } num.Reverse(); // Initialize dp for(int i = 0; i < M; i++) for(int j = 0; j < M; j++) for(int k = 0; k < 2; k++) for(int l = 0; l < 2; l++) dp[i, j, k, l]=-1; return count(0, 0, 0, 0, num); } // Driver Code public static void Main() { int L = 11, R = 100; d = 2; K = 1; Console.Write( solve(R) - solve(L - 1) ); }}// This code is contributed by Rajput-JI |
Javascript
<script>// JavaScript Program to find the count of// numbers in a range where digit d// occurs exactly K timesvar M = 20;// states - position, count, tight, nonzvar dp = Array.from(Array(M), ()=>Array(M));// d is required digit and K is occurrencevar d, K;// This function returns the count of// required numbers from 0 to numfunction count( pos, cnt, tight, nonz, num){ // Last position if (pos == num.length) { if (cnt == K) return 1; return 0; } // If this result is already computed // simply return it if (dp[pos][cnt][tight][nonz] != -1) return dp[pos][cnt][tight][nonz]; var ans = 0; // Maximum limit upto which we can place // digit. If tight is 1, means number has // already become smaller so we can place // any digit, otherwise num[pos] var limit = (tight ? 9 : num[pos]); for (var dig = 0; dig <= limit; dig++) { var currCnt = cnt; // Nonz is true if we placed a non // zero digit at the starting of // the number if (dig == d) { if (d != 0 || (!d && nonz)) currCnt++; } var currTight = tight; // At this position, number becomes // smaller if (dig < num[pos]) currTight = 1; // Next recursive call, also set nonz // to 1 if current digit is non zero ans += count(pos + 1, currCnt, currTight, nonz || (dig != 0?1:0), num); } return dp[pos][cnt][tight][nonz] = ans;}// Function to convert x into its digit vector and uses// count() function to return the required countfunction solve(x){ var num = []; while (x) { num.push(x % 10); x = parseInt(x/10); } num.reverse(); for(var i =0; i<M; i++) for(var j =0; j<M; j++) dp[i][j] = Array.from(Array(2), ()=>Array(2).fill(-1)) return count(0, 0, 0, 0, num);}// Driver Code to test above functionsvar L = 11, R = 100;d = 2, K = 1;document.write( solve(R) - solve(L - 1));</script> |
Time Complexity: O(logR*k) where R and k are the given input.
Auxiliary Space: O(M2) where M is the given constant.
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