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All entries for Wednesday 13 February 2019

## February 13, 2019

### Again and again and again

"Flow control" means all the ways you can control what your program does, both now and next. Conditionals, loops, function calls all count. Exceptions (throw, raise etc) may or may not - whether its OK to use exceptions as flow control or whether they're meant for, well, exceptional occurences (not necessarily rare, but something that can't be handled by the current piece of code) is a seriously vexed question.

## Loops

Loops are usually the first or second control option to be taught, and take two general forms, the 'for' type loop and the 'while' type loop. Different languages use different words, but the first one is meant to do something a certain number of times, and this number is known when the loop starts. The second is meant to do something 'until told to stop'. This is not a hard distinction though. A while-type loop can always mimic a for-type loop, and the reverse is also true (although sometimes considered to be inelegant and/or error prone).

Usually, these loops look like

For index = start to end {loop body} End for

and

While condition {loop body} End while

The condition can be as complicated as you like, it just has to evaluate to either True or False. It can use all the variables you might be changing in the loop, etc. So we create the other kind of loop something like this:

For index = start to max_iterations {loop body} If condition exit_loop End for

and

index = start While true //This means keep looping forever {loop body} If index > end exit_loop index = index + 1 End while

*Note that whether we get (end-start) iterations of a for loop, or (end-start+1) or (end-start-1) can vary by language, but we can easily adjust to match. *

## Recursion

The other way of doing something many times, is 'recursion'. This often gets classed as 'super advanced and difficult' for some reason, but is mostly quite simple. First though, we need to know a tiny bit about functions and how they're called.

### Scope

Scope is very, very important: every variable, function etc within a program has a scope. For variables this means "parts of your code which can use this variable (bit of memory) with this name". For functions, it means "parts of your code which can use this function with this name". There's a few subtleties beyond that, but for now, this will do.

So, what scopes are there? A variable defined inside a function is **only** usable within that function: it is scoped to that function, or has 'local scope'. A variable defined globally (outside any functions, including main) has 'global scope' and is available everywhere. Do note that a variable defined in 'main' is only available in 'main' and NOT in any functions 'main' may call, as main is a function like any other.

Most languages also have an idea of 'block scope' where 'blocks' (in C anything inside curly braces {}) can contain variable declarations, which are only available inside the block. This can cause some particularly confusing errors, such as when you try and do the following:

while i < 10 int i end while

which will not compile unless there is already a variable, called i, and the one you declare inside the loop then 'shadows' this - inside the loop i refers to one variable, outside the loop, including the loop condition line, i refers to something different. If this isn't completely clear, the following example should help:

string name = 'Nobody' int i for i = 1 to 3 string name = get_string_from_user() print i, name end for print 'You entered', name

which gets 3 names from the user an outputs something like:

1 Bill Bailey 2 Madonna 3 Engelbert Humperdinck You entered Nobody

Another tempting thing is to try

if condition then int i = 0 else long int i = 1 end if print i

which again has either an undefined 'i' or a shadowing problem. There is no way to get a different type for i using an if like this, and with good reason. i's type could only be determined in general when the program runs - so how much storage should be given for it, and can it be passed to any given function?

Function scope has one more really important thing though - each call to a function is a *new* scope. The variables you used last time **do not** keep their values.

**FORTRAN PROGRAMMERS - READ THIS!!!**

In Fortran there is one really important idea called the 'SAVE' attribute. A variable in a function (or module) can be given this, as e.g. "INTEGER, SAVE :: a" and the value of 'a' will be kept from call to call. This is very useful. * BUT *there is a catch

*So, if you do something like `INTEGER :: alpha = 0`, declaring an Integer alpha and in the same line defining it, alpha is set to zero*

**. Any variable declared and defined in a single line in a Fortran function is given the SAVE attribute.***the first time the function is called. Subsequent calls will inherit whatever value alpha had last time. This is rarely what you intended.*

**ONLY**

**Be careful!**### Call stack

When you write code to call a function, the computer has to stop what its currently doing, and enter a new scope containing only the variables available inside the function. It also has to remember where it should go back to after the function ends. This is done using the 'call stack'.

We haven't talked about 'stacks' as a data structure yet (coming soon) but we did mention here that they're a "last-in-first-out" structure where the last thing you add to the stack (think of a stack of papers or books) is the first one you take off. Each time you call a function, you add an entry to the stack, and when you return this is 'popped off' and the stack shrinks. Each entry is called a 'frame'.

The stack frame usually contains the location to return to, and also memory for all of the local variables in a function. It often also has space to hold all the parameters passed to the function and sometimes a few other bits of operational stuff. When a function is called, a frame is created with all this in, and when it returns this is destroyed.

We mentioned above that variables inside a function are available only inside it, but we didn't ask what happens if we call a function from within itself. We've seen that between calls to a function the values are 'reset' or lost, and having read the previous paragraph you probably guess that this is both because and why the stack frame gets destroyed.

Now, you might suggest that you could always make sure every call to a given function shares the same variables, but if you've ever used a function pointer you know that you can call a function without *ever* using its name at the place the call actually happens, so this isn't practical. So, each call to a function, any function, creates a stack frame containing all its local variables, and calling a function from within itself makes two, independent, sets of all the local variables, that know nothing about each other.

### My First Recursive function is My First Recursive function is My First....

So what is recursion then? "**Recursion** occurs when a thing is defined in terms of itself or of its type." (Wikipedia, Recursion) For a function, recursion means having the function call itself. In maths, the factorial function of a number which is the product of all positive integers up to it. So we see immediately that factorial(n) is n*factorial(n-1), which we'd code up something like this:

function factorial(integer n) return factorial(n-1)*n end function

We can pretty immediately see a problem there though - how does the chain ever end? We need something which is not recursive or we'll go on calling forever. This is called the 'base case' and for factorial its obvious from how we said 'positive numbers'. The function above won't stop when n = 1 and it should. So what we actually want is:

function factorial(integer n) if n > 1 return factorial(n-1)*n else return 1 end if end function

Follow this by hand, on paper, for a starting n of say 4. We enter factorial(4), which enters factorial(3), which enters... until we reach factorial(1), which immediately returns '1' to the layer above, factorial(2), which multiplies this by '2' to get '2' and returns this to factorial(3) and so on.

### Each Call is Its Own Scope

Remember when looking at recursive functions that each layer of call is a separate scope, with a separate copy of any variables you may define. Anything which needs to go between the layers has to be passed as an argument.

## Step by step by step

So we have these two ideas that both let us keep going until we reach some condition, namely recursion and a while loop: what's the difference. There isn't one. Anything you can do with a loop you can do with recursion and vice versa. There are differences in elegance, and often one is a better choice, but not more. In fact some functional languages don't have any concept of the loop, relying solely on recursion. Mostly, elegantly recursive problems are better written as 'while' type loops and rarely as 'for' type loops, because the base case is the same as the loop-stop condition. Some recursive problems, usually those involving trees, are very hard to do elegantly with a loop.

### Induction

Most of programming is about working out the sequence of steps to get from A to B, so that what you actually code is just a series of things, one after the other. Sometimes these steps are completely independent, and sometimes they aren't but they always (in a single-threaded program) run one after the other. We're always having to think about things, not in terms of the big picture, but just in terms of getting from here to there, ignoring how to get here in the first place.

In maths, one of the simplest methods of proof is called 'induction' which is closely related to recursion. Rather than try and prove the 'whole of a thing' we say 'if it was true for a smaller thing, can we show it's true for the next larger thing?' and then we say 'can we prove its true for the smallest thing?'. If we can do both of these, we've shown its true. As used here a common example is climbing a ladder. We say 'can we climb onto the bottom rung?' and we say 'can we climb onto the next higher rung than we're on?' and if so, we can climb any ladder.

Sometimes, a slightly stronger assumption is used where we instead say 'if we have reached every rung below and including the one we're on, can we reach the next one'. This is actually equivalent, but is sometimes a more useful phrasing.

Proofs by induction only work if it doesn't matter which rung we're on to climb to the next one. We don't ever have to reach rung 53 to know we can reach rung 54. **If you have a problem which is easy to think about this way, then it is a prime candidate for programming recursively.** The step is always the part going from 'here' to the next 'there', and the base case is how you get to the first 'here'.

## Problems with Recursion

### Stack Overflow

The 'call stack' we've been talking about is the inspiration for the programming forum Stack Overflow which is probably the most encountered error when programming recursively. Each function call creates a call stack frame and there is a limit to how much memory is available for this. If you forget or mis-program the base case, your recursive function never stops calling itself, until it has filled the call stack and your program crashes horribly with a stack overflow.

The other common way to get a stack overflow is creating large temporaries inside functions since these are all part of the stack. Hopefully more on that soon.

### Function Parameters

Secondly, recursion can be a bit tricky to actually set up. Our factorial function was nice and simple, with a single parameter, and a single return value. But what if we have more than one parameter? For instance, a binary search can be done nicely in recursive fashion. Each step is about deciding which half, upper or lower, our target is in, and passing only this half on to the next step, and the base case is when this has length 1. Here though, you want to pass at least two items - the segment of list, and the target value, and you want to return either true or false, or the index the target was found at. If returning the index as an offset into the passed segment, you then have to adjust this at each step so that you end up with the index in the original, complete list. This can get mucky.

### Excess Work

The other common example used for a recursive operation is the Fibonacci sequence where the nth value is the sum of the (n-1)th and the (n-2)th. Usually, the first two values are 1, so the sequence goes 1, 1, 2, 3, 5, 8, 13 etc. It's not hard to write a recursive version of this, e.g.

function fibonacci(n) if n eq 1 or n eq 2 return 1 else return fibonacci(n-1) + fibonacci(n-2) end if end

but if we work through this on paper for say n =5 we find that we calculate the n=4 case once, the n=3 case twice, the n=2 case three times and the n=1 case twice.

A Python version of this and my model answers to the challenges are here.

Challenge: what's the rule in general for how many times fib(m) is called for each m < n?

Challenge 2: rewrite this recursively with exactly n-1 calls to calculate fibonacci(n). Hint after the post, or solutions at the Github link above.

If we're not careful, we might never notice all the extra work we're doing, which we could avoid. In this case, there's a big hint that something is funny at the point where we put two values of n into our base case.

## Challenge 3:

Some image sharing sites now try to use a few real words to create memorable random urls. If you're given n lists of words, can you write loop based and recursive variants of the code to create every combination of one word from each list? Your code should work for any value of 'n'.

For example [large, small] [radiant, lame] [picnic, bobcat] should give (order not important) large-radiant-picnic, small-lame-bobcat, small-radiant-bobcat etc etc.

Small hint below the post

## Moral of this Post

The takeaway from this one is that recursion isn't scary if you just think about getting from here to there, pretending all the business to get 'here' has been dealt with. Never mind the rest of the ladder, just think about the next rung. This is one of the vital skills to develop as a programmer, on every scale. Break things down into manageable steps and then build them into a program.

Keep scrolling for the hints....

Hint 1: can you return both the (n-1) and the (n-2) values?

Hint 2: the list created by combining lists 1 and 2 is itself a list of words. 3 lists is just 2 lists, and then another list.