这是一道面试题。计算 [1, N] 范围内具有唯一数字(十进制)的所有数字。
显而易见的解决方案是测试范围内的每个数字是否唯一。我们还可以生成所有具有唯一数字的数字(作为排列)并测试它们是否在范围内。
现在我想知道是否有针对这个问题的DP(动态规划)解决方案。
最佳答案
我在想:
Number of unique digits numbers 1-5324
= Number of unique digits numbers 1-9
+ Number of unique digits numbers 10-99
+ Number of unique digits numbers 100-999
+ Number of unique digits numbers 1000-5324
所以:
f(n) = Number of unique digits numbers with length n.
f(1) = 9 (1-9)
f(2) = 9*9 (1-9 * 0-9 (excluding first digit))
f(3) = 9*9*8 (1-9 * 0-9 (excluding first digit) * 0-9 (excluding first 2 digits))
f(4) = 9*9*8*7
将以上所有内容相加,直到 N 减去 1 的位数为止。
那么你只需要做 Number of unique digits numbers 1000-5324
和:
Number of unique digits numbers 1000-5324
= Number of unique digits numbers 1000-4999
+ Number of unique digits numbers 5000-5299
+ Number of unique digits numbers 5300-5319
+ Number of unique digits numbers 5320-5324
所以:
N = 5324
If N[0] = 1, there are 9*8*7 possibilities for the other digits
If N[0] = 2, there are 9*8*7 possibilities for the other digits
If N[0] = 3, there are 9*8*7 possibilities for the other digits
If N[0] = 4, there are 9*8*7 possibilities for the other digits
If N[0] = 5
If N[1] = 0, there are 8*7 possibilities for the other digits
If N[1] = 1, there are 8*7 possibilities for the other digits
If N[1] = 2, there are 8*7 possibilities for the other digits
If N[1] = 3
If N[2] = 0, there are 7 possibilities for the other digits
If N[2] = 1, there are 7 possibilities for the other digits
If N[2] = 2
If N[3] = 0, there is 1 possibility (no other digits)
If N[3] = 1, there is 1 possibility (no other digits)
If N[3] = 2, there is 1 possibility (no other digits)
If N[3] = 3, there is 1 possibility (no other digits)
上面是这样的:
uniques += (N[0]-1)*9!/(9-N.length+1)!
for (int i = 1:N.length)
uniques += N[i]*(9-i)!/(9-N.length+1)!
// don't forget N
if (hasUniqueDigits(N))
uniques += 1
你真的不需要 DP,因为上面的应该足够快了。
编辑:
上面实际上需要稍微复杂一点(上面的 N[2] = 2 和 N[3] = 2 是无效的)。它需要更像:
binary used[10]
uniques += (N[0]-1)*9!/(9-N.length+1)!
used[N[0]] = 1
for (int i = 1:N.length)
uniques += (N[i]-sum(used 0 to N[i]))*(9-i)!/(9-N.length+1)!
if (used[N[i]] == 1)
break
used[N[i]] = 1
// still need to remember N
if (hasUniqueDigits(N))
uniques += 1
关于algorithm - 计算给定范围内具有唯一数字的所有数字,我们在Stack Overflow上找到一个类似的问题: https://stackoverflow.com/questions/15155165/