I have just started learning about time complexities and am currently reading Logarithmic Complexity. Here's an example of a piece of code which is $O(\log n)$:

def intToStr(i):
    '''Assumes i is a nonnegative int Returns the string representation of i'''

    digits = '0123456789'
    if i == 0:
        return '0'
    result = ''
    while i > 0:
        result = digits[i % 10] + result
        i = i // 10
    return result

So this just gives is the str of an integer. intToStr(1243) returns '1243'. I completely understand that the above function is of logarithmic complexity.

In continuation,

def addDigits(n):

    '''Assumes n is a nonnegative int
    Returns the sum of the digits in n'''
    stringRep = intToStr(n)
    val = 0
    for c in stringRep:
        val += int(c)
    return val

My book says that: (emphasis added)

The complexity of converting $n$ to a string is $O(\log n)$, and intToStr returns a string of length $O(\log n)$. The for loop will be executed $O(\operatorname{len}(\mathit{stringRep}))$ times, i.e., $O(\log n)$ times. Putting it all together, and assuming that a character representing a digit can be converted to an integer in constant time, the program will run in time proportional to $O(\log n) + O(\log n)$, which makes it $O(\log n)$.

Can someone please explain how intToStr returns a string of length $O(\log n)$? Doesn't it just return a string of length $n$, where $n$ is the length of the integer that we pass?


The length of the binary representation of a natural number $n$ is roughly $\log_2 n$. As an example, the number represented by the binary string $10^{n-1}$ of length $n$ is $2^n$.

Your sources are misleading. Usually $n$ is reserved for the input length or a related quantity, not the input value. If the input to a function is an integer $m$, then the input length is only $\sim \log_2 |m|$. In particular, intToStr actually runs in linear time in the input length, rather than logarithmic time.

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  • $\begingroup$ Hi. I just edited my post to add the complete paragraph, do you still think that it is incorrect? $\endgroup$ – R Doe. Jun 12 '19 at 13:00
  • $\begingroup$ It is correct, just highly misleading. For a similar example, see cs.stackexchange.com/questions/88398/…. $\endgroup$ – Yuval Filmus Jun 12 '19 at 13:02
  • $\begingroup$ Can we say that addDigits() is of linear complexity but as it calls intToStr() ( which is of log complexity ), the whole program is of logarithmic complexity? $\endgroup$ – R Doe. Jun 12 '19 at 13:16
  • $\begingroup$ intToStr has linear complexity. $\endgroup$ – Yuval Filmus Jun 12 '19 at 13:20
  • $\begingroup$ WHAT!? i = i // 10 boils down to the number of times we can use integer division to divide i by 10 before getting a result of 0. So the complexity must be O(log(i))! Is that explanation incorrect? $\endgroup$ – R Doe. Jun 12 '19 at 13:24

To answer your question literally, yes, the code does just return a string of length $n$ where $n$ is the length of the integer that we pass. And this is the right way to think about it.

Your source, though, is using $n$ to denote the value of the integer, not its length. This is an unusual thing to do and it is, in my opinion, a very bad idea when teaching the basics of algorithm analysis. This is a point that students find fundamentally confusing, and your book seems to be contributing to that confusion, instead of reducing it.

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