357. Count Numbers with Unique Digits

本文介绍了一种算法,用于计算在给定非负整数n的情况下,所有具有唯一数字的数的数量,范围从0到10^n-1。提供了一个具体的例子:当n等于2时,返回91个这样的数,排除了所有重复数字的情况。

Given a non-negative integer n, count all numbers with unique digits, x, where 0 ≤ x < 10n.

Example:

Given n = 2, return 91. (The answer should be the total numbers in the range of 0 ≤ x < 100, excluding [11,22,33,44,55,66,77,88,99])


#include <iostream>
using namespace std;
class Solution {
public:
    int countNumbersWithUniqueDigits(int n) {
    int result = 10;
    int dig = 9;
        if(n == 0)
return 1;
else if(n == 1)
return 10;

else
{
   for(int i=2;i<=n;i++)
{
dig = dig*(9-i+2);
result = result + dig;
}
return result; 
}
    }
};

A-1 Intersection Set of Prime Factors 分数 20 作者 陈越 单位 浙江大学 Given a positive integer n. Select two distinct digits from the decimal repersentation(十进制表示)of n, we obtain another integer m. What is the size of the intersection set(交集)of the prime factors of n and m? For example, given n=623457198, its prime factor set is A = {2, 3, 7, 13, 380621}. Swapping 2 and 9 gives us m=693457128, of which the prime factor set is B = {2, 3, 7, 13, 109, 971}. Then the intersection set of A and B is {2, 3, 7, 13}, with 4 factors. Input Specification: Each input file contains one test case, which gives a positive integer n (10<n≤10 9 ). It is guaranteed that there are at least 2 distinct digits in n. Output Specification: Swap any pair of digits in n to obtain m, you are supposed to find the m with the largest intersection set of the prime factors of n and m. Output in a line the number of the prime factors in the intersection set, together with m. The numbers must be separated by 1 space, and there must be no extra space at the beginning or the end of the line. In case such an m is not unique, output the one with the smallest value. Sample Input: 623457198 Sample Output: 4 123457698 Hint: There are two m's with 4 common factors. Besides the one given in the problem description, we can also swap 6 and 1 to obtain 123457698. This number has a prime factor set {2, 3, 7, 13, 23, 29, 113}, and so the intersection set is also {2, 3, 7, 13}. This number is in the ouput because it is smaller than 693457128.(c++)
07-28
翻译并用 latex 渲染: First, let's see how many zebra-Like numbers less than or equal to 1018 exist. It turns out there are only 30 of them, and based on some zebra-like number zi , the next one can be calculated using the formula zi+1=4⋅zi+1 . Then, we have to be able to quickly calculate the zebra value for an arbitrary number x . Since each subsequent zebra-like number is approximately 4 times larger than the previous one, intuitively, it seems like a greedy algorithm should be optimal: for any number x , we can determine its zebra value by subtracting the largest zebra-like number that does not exceed x , until x becomes 0 . Let's prove the correctness of the greedy algorithm: Assume that y is the smallest number for which the greedy algorithm does not work, meaning that in the optimal decomposition of y into zebra-like numbers, the largest zebra-like number zi that does not exceed y does not appear at all. If the greedy algorithm works for all numbers less than y , then in the decomposition of the number y , there must be at least one number zi−1 . And since y−zi−1 can be decomposed greedily and will contain at least 3 numbers zi−1 , we will end up with at least 4 numbers zi−1 in the decomposition. Moreover, there will be at least 5 numbers in the decomposition because 4⋅zi−1<zi , which means it is also less than y . Therefore, if the fifth number is 1 , we simply combine 4⋅zi−1 with 1 to obtain zi ; otherwise, we decompose the fifth number into 4 smaller numbers plus 1 , and we also combine this 1 with 4⋅zi−1 to get zi . Thus, the new decomposition of the number y into zebra-like numbers will have no more numbers than the old one, but it will include the number zi — the maximum zebra-like number that does not exceed y . This means that y can be decomposed greedily. We have reached a contradiction; therefore, the greedy algorithm works for any positive number. Now, let's express the greedy decomposition of the number x in a more convenient form. We will represent the decomposition as a string s of length 30 consisting of digits, where the i -th character will denote how many zebra numbers zi are present in this decomposition. Let's take a closer look at what such a string might look like: si∈{0,1,2,3,4} ; if si=4 , then for any j<i , the character sj=0 (this follows from the proof of the greedy algorithm). Moreover, any number generates a unique string of this form. This is very similar to representing a number in a new numeric system, which we will call zebroid. In summary, the problem has been reduced to counting the number of numbers in the interval [l,r] such that the sum of the digits in the zebroid numeral system equals x . This is a standard problem that can be solved using dynamic programming on digits. Instead of counting the suitable numbers in the interval [l,r] , we will count the suitable numbers in the intervals [1,l] and [1,r] and subtract the first from the second to get the answer. Let dp[ind][sum][num_less_m][was_4] be the number of numbers in the interval [1,m] such that: they have ind+1 digits; the sum of the digits equals sum ; num_less_m=0 if the prefix of ind+1 digits of the number m is lexicographically greater than these numbers, otherwise num_less_m=1 ; was_4=0 if there has not been a 4 in the ind+1 digits of these numbers yet, otherwise was_4=1 . Transitions in this dynamic programming are not very difficult — they are basically appending a new digit at the end. The complexity of the solution is O(log2A) , if we estimate the number of zebra-like numbers up to A=1018 as logA .
08-26
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