My question is the same as this but with a catch: How many substrings of length m can be formed from a string of length n?

To find the number of possible substrings of length C in a parent string of length P I can use this formula:

$$P - C + 1$$

For example, for a parent string of length $5$ and a substring of length $3$, there are $5 - 3 + 1 = 3$ different positions the substring can take in its parent.

The catch:
The problem is, if we ask this formula how many substrings of length 3 fit in a parent string of length 1, we get $1 - 3 + 1 = -1$, instead of the expected $0$.

Is there a simple function that will give $0$ when $P < C$ without using an if statement?


So, you want $\max(P-C+1,0)$? In general $$\max(x,y)=\frac{x+y+|x-y|}2.$$ Therefore $$\max(P-C+1,0)=\frac{P-C+1+|P-C+1|}2.$$

But really, there is nothing wrong with case-by-case definitions!


You can, but it will generally involve cheating by using a function that has some kind of implicit "if" in it. For example, we could use the Heaviside step function which is defined as:

$$H(x) = \begin{cases} 1 & \mbox{if } x \gt 0 \\ \frac{1}{2} & \mbox{if } x = 0 \\ 0 & \mbox{if } x \lt 0\end{cases}$$

So you just multiply your value $S$ by $H(S)$, so that for negative values it becomes 0.

You can do the same thing with absolute values by noting that

$$x + \left|x\right| = \begin{cases} 2x & \mbox{if } x \geq 0 \\ 0 & \mbox{if } x \lt 0\end{cases}$$

So you just take your expression, add its absolute value, and divide by two, giving $\frac{1}{2}\left(P - C + 1 + \left|P - C + 1\right|\right)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.