386. Lexicographical Numbers

一、题目描述

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Given an integer n, return 1 - n in lexicographical order.

For example, given 13, return: [1,10,11,12,13,2,3,4,5,6,7,8,9].

Please optimize your algorithm to use less time and space. The input size may be as large as 5,000,000.

二、分析

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Observations:
For any given digit k, the lexical order of all numbers starting with digit k looks like:

k, k10, k10^2, k10^3, ..., k10^p +1, k10^p + 2, ..., k10^(p-m) +1, ...

For example, the sequence of numbers starting with digit 1 will be like (with n = 2000):

1, 10, 100, 1000, 1001, 1002, ..., 1009, 101, 1010, 1011, 1012, ..., 1019, 102, 1020, ..., 1099, 11, 110, 1100, 1101, ...

Through the observations above, a valid sub-sequence can only be started when we encounter 1) we cannot append more zeroes to the previous number; or 2) a number that ends in consecutive '9's.

三、解答

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class Solution {
public:
    vector<int> lexicalOrder(int n) {
        vector<int> rs;
        
        int i = 1, j;
        int k;
        for(;;)
        {
            // append as many zeroes as possible to the previous number
            for(k = 0; i*pow(10,k) <= n; ++k) rs.push_back(i*pow(10,k));
            
            // count continuously until we reach a number that ends with consecutive '9's
            for(j = rs.back()+1; j <= n && (j % 10) != 0; ++j) rs.push_back(j);
            
            // backtrace
            if(j % 10 == 0)
            {
                j--;
            }
            else
            {
                j /= 10;
            }

            // find the last non-'9' digit
            while(j % 10 == 9) j /= 10;
            
            // start a new sub-sequence
            i = j+1;
            
            if(rs.size() >= n) break;
        }

        return rs;
    }
};