and will notice that it gets close to φ = (1 + √5) / 2 ≈ 1.618033989, which makes the number of recursive calls O(φⁿ), which is much more fun than O(2ⁿ).
Really it's worse than exponential, because the size of the input is not n, it's the number of bits needed to store n i.e. log n. So as the number of bits k grow, it's growing phi^(2^k).
Big O analysis never implies size in bits. It's just often done. In this case, n is just the numerical value of the input, so I don't think this is correct.
no mention of dynamic programming for dealing with recursive functions? Dynamic programming was built for this, you don't even need to thrash the heap as much as that trampoline. Instantiate your array, make sure you set your base cases and loops so that you don't step in an `undefined` hole, and then recurse in reverse.
TCO is surprisingly commonly supported in the real world. TCO is supported in C++ compilers, JavaScript in Safari, Scala, Clojure in a way, etc.!
For instance:
"All current mainstream [C++] compilers perform tail call optimisation fairly well (and have done for more than a decade), even for mutually recursive calls" [1].
"As of July 22, 2023 Safari is the only browser that supports tail call optimization" of JavaScript [2].
"Since Clojure uses the Java calling conventions, it cannot, and does not, make the same tail call optimization guarantees. Instead, it provides the recur special operator, which does constant-space recursive looping" [3].
"The Scala compiler will automatically optimize any truly tail-recursive method. If you annotate a method that you believe is tail-recursive with the @tailrec annotation, then the compiler will warn you if the method is actually not tail-recursive" [4].
The troubles of handling call stack recursion is downstream of the lack of strong tooling for static analysis of stack usage. Of the few tools available for generating a build-time call graph for an application, almost none of them can do so in a machine-readable format. AFAIK, the state of the art here is LLVM's dot-callgraph pass, and even that emits DOT rather than something more widely adopted like CSV or JSON. Outside of that, you have to build your own thing, either via runtime profiling or a custom compiler plugin.
Most of these issues are a consequence of recursion never getting the same codification as the rest of the jmp patterns we eventually turned into control structures - eg: if, for, while, try/catch.
In the meantime, the theory of structured recursion[recursion schemes] has been developing, yet no language offers then as first class constructs. The best we get is library support. Imagine if we had to import a package to support if statements. The result? Programmers write recursive programs while navigating all the foot guns described in the article. No wonder recursion is hard to get right.
I might not be understanding what you’re saying here… but recursion compared to your if/whiles, are inherently coupled to the shape of the type they traverse where in the prior two we define data comparisons or input sizes.
Using a jmp isn’t really a call as you’d know, and information is lost that would be crucial to unwinding a recursion.
A recurse keyword would still leave the person writing the code with the decision with proving termination with base cases or trampolines — lest there is just a bunch of math that would unwind your recursion into a better bounded problem. Which is kind of what current keywords do anyway.
lol. i once interviewed with facebook and had some "senior" dev on the phone who was balking and grousing at my claim that iterative algorithms are faster than recursive ones. the minute i mentioned spatial locality, he went quiet.
more to the point, i feel like there should be a compiler switch or decorator style flag in modern languages that declare "this function is expected to optimize with tail recursion, throw a compiler or linter error at static analysis time if that doesn't work out."
Re: your first paragraph, I don't get it, what does spatial locality have to do with recursion vs. iteration? And the blanket speed claim doesn't make sense either. I feel like you're thinking of a specific algorithm or access pattern or technology and overgeneralizing to the idea that it's somehow impossible for recursion to ever match the performance or be faster?
maybe the wrong terminology, but non-tail-call-optimized recursions spray their state across space with each iteration consuming a new stack frame, where iterative algorithms live in one stack frame and can re-use temporaries. the state spraying results in consumption and spilling down the memory hierarchy, from registers through caches. i think of this as the memory hierarchy being designed to best perform when spatial locality of memory usage is maintained, but it's slightly different from what most people mean when they discuss spatial locality... maybe "cache efficiency" is the better term?
> maybe the wrong terminology, but non-tail-call-optimized recursions spray their state across space with each iteration consuming a new stack frame, where iterative algorithms live in one stack frame and can re-use temporaries
If that's what you mean then I'm afraid it sounds like you're mixing a few things up. For example, imagine depth-first search: you're going to need a stack somewhere, whether it's the CPU stack which you use via recursion, or an explicit stack you use via iteration. Iterating doesn't magically remove your need for that space and somehow collapse everything down to one stack frame. And you can reuse temporaries from the heap too, etc.
Fundamentally, there is the question of how much space you need for given algorithm, the question of what algorithm you should use in the first place, the question of whether that particular algorithm should be implemented recursively or iteratively, and the question of what is more maintainable and easier to evolve in practice.
These are all separate questions, but you're conflating them. If your iteration uses constant space but your recursion doesn't, that's because you're not implementing the same algorithm. You're implementing a different algorithm that achieves the same original goal you had. Of course one algorithm might beat the other, that's no surprise.
let me try arguing it this way: it is impossible for a non-tail-call-optimized recursion to use constant space. the stack grows with O(depth) and therefore memory usage does as well. a consequence of this is that the finite storage lru caches and registers are polluted with one-time-use variables which reduces their availability for actually useful caching.
it IS possible for a loop to use constant space. pre-allocate temporaries and inline any function calls. everything lives in one stack frame. simply iterating the loop itself does not come with fixed space overheads from allocating new stack frames.
i suppose one thing i should mention, i am thinking of extreme optimization use cases where the entire data structure fits in cache (or close to it). think like in-cache tries or similar. if you're hitting main memory with each iteration anyway, it doesn't really matter.
Yes, and a high-level scripting language can in some specific circumstances be made to run code faster than a low-level compiled language. Generally, it's the other way around though.
> more to the point, i feel like there should be a compiler switch or decorator style flag in modern languages that declare "this function is expected to optimize with tail recursion, throw a compiler or linter error at static analysis time if that doesn't work out."
Scala and Kotlin have that. Other modern languages probably do as well.
> Each call branches into two more calls, so the total number of calls grows as O(2ⁿ).
Well, no, not really. If anyone bothers counting how many recursive calls are actually made, the result is far from powers of two:
A curious person will then calculate the actual ratio: and will notice that it gets close to φ = (1 + √5) / 2 ≈ 1.618033989, which makes the number of recursive calls O(φⁿ), which is much more fun than O(2ⁿ).For instance:
"All current mainstream [C++] compilers perform tail call optimisation fairly well (and have done for more than a decade), even for mutually recursive calls" [1].
"As of July 22, 2023 Safari is the only browser that supports tail call optimization" of JavaScript [2].
"Since Clojure uses the Java calling conventions, it cannot, and does not, make the same tail call optimization guarantees. Instead, it provides the recur special operator, which does constant-space recursive looping" [3].
"The Scala compiler will automatically optimize any truly tail-recursive method. If you annotate a method that you believe is tail-recursive with the @tailrec annotation, then the compiler will warn you if the method is actually not tail-recursive" [4].
[1]: https://stackoverflow.com/a/34129
[2]: https://stackoverflow.com/a/37224563
[3]: https://stackoverflow.com/a/34097339
[4]: https://stackoverflow.com/a/3114245
And read..
Proper Tail Calls were implemented behind experimental flags but never shipped by default — the flags were later removed.
In the meantime, the theory of structured recursion[recursion schemes] has been developing, yet no language offers then as first class constructs. The best we get is library support. Imagine if we had to import a package to support if statements. The result? Programmers write recursive programs while navigating all the foot guns described in the article. No wonder recursion is hard to get right.
Using a jmp isn’t really a call as you’d know, and information is lost that would be crucial to unwinding a recursion.
A recurse keyword would still leave the person writing the code with the decision with proving termination with base cases or trampolines — lest there is just a bunch of math that would unwind your recursion into a better bounded problem. Which is kind of what current keywords do anyway.
more to the point, i feel like there should be a compiler switch or decorator style flag in modern languages that declare "this function is expected to optimize with tail recursion, throw a compiler or linter error at static analysis time if that doesn't work out."
If that's what you mean then I'm afraid it sounds like you're mixing a few things up. For example, imagine depth-first search: you're going to need a stack somewhere, whether it's the CPU stack which you use via recursion, or an explicit stack you use via iteration. Iterating doesn't magically remove your need for that space and somehow collapse everything down to one stack frame. And you can reuse temporaries from the heap too, etc.
Fundamentally, there is the question of how much space you need for given algorithm, the question of what algorithm you should use in the first place, the question of whether that particular algorithm should be implemented recursively or iteratively, and the question of what is more maintainable and easier to evolve in practice.
These are all separate questions, but you're conflating them. If your iteration uses constant space but your recursion doesn't, that's because you're not implementing the same algorithm. You're implementing a different algorithm that achieves the same original goal you had. Of course one algorithm might beat the other, that's no surprise.
it IS possible for a loop to use constant space. pre-allocate temporaries and inline any function calls. everything lives in one stack frame. simply iterating the loop itself does not come with fixed space overheads from allocating new stack frames.
i suppose one thing i should mention, i am thinking of extreme optimization use cases where the entire data structure fits in cache (or close to it). think like in-cache tries or similar. if you're hitting main memory with each iteration anyway, it doesn't really matter.
Scala and Kotlin have that. Other modern languages probably do as well.