# Applying a permutation to a vector, part 1

Suppose you have a vector `indices` of N integers that is a permutation of the numbers 0 through N − 1. Suppose you also have a vector `v` of N objects. The mission is to apply the permutation to the vector. If we let `v2` represent the contents of the vector at the end of the operation, the requirement is that `v2[i] = v[indices[i]]` for all `i`.

For example, if the objects are strings, and the vector of objects is `{ "A", "B", "C" }` and the vector of integers is `{ 1, 2, 0 }`, then this means the output vector is `{ "B", "C", "A" }`.

This sounds like something that would be in the standard library, but I couldn't find it, so I guess we'll have to write it ourselves. (If you find it in the standard library, let me know!)

Let's start with the easy version: You don't update the vector in place. Instead, you produce a new vector. Solving this version is fairly straightforward, because all you have to do is write the problem statement!

```template<typename T>
std::vector<T>
apply_permutation(
const std::vector<T>& v,
const std::vector<int>& indices)
{
std::vector<T> v2(v.size());
for (size_t i = 0; i < v.size(); i++) {
v2[i] = v[indices[i]];
}
return v2;
}
```

Now we get to the analysis, which is where the fun is.

The above algorithm assumes that `T` is default-constructible, because we create a vector `v2` of default-constructed `T` objects, and then replace them one by one with copies of the real `T` objects from `v`. I wrote it that way to make the solution match the problem statement, making its correctness easy to verify by inspection. You can even write a unit test for this function, so that you can verify that the transformations we are going to perform don't break the fundamental algorithm.

First, let's get rid of the requirement that there be a default constructor for `T`. All we'll require is that `T` be copy-constructible.

```template<typename T>
std::vector<T>
apply_permutation(
const std::vector<T>& v,
const std::vector<int>& indices)
{
std::vector<T> v2;
v2.reserve(v.size());
for (size_t i = 0; i < v.size(); i++) {
v2.push_back(v[indices[i]]);
}
return v2;
}
```

But still, we're copying the elements around. Let's refine the problem statement so that instead of returning a new vector, we mutate the vector in place. That way, it'll work with movable objects, too.¹

```template<typename T>
void
apply_permutation(
std::vector<T>& v,
const std::vector<int>& indices)
{
std::vector<T> v2;
v2.reserve(v.size());
for (size_t i = 0; i < v.size(); i++) {
v2.push_back(std::move(v[indices[i]]));
}
v = std::move(v2);
}
```

This version is basically the same thing as before, except that we move the objects around instead of copying them. But this kind of misses the point of the exercise: We didn't really update the vector in place. We created a second vector and then swapped it in. Can we do it without having to create a second vector? After all, that vector might be large (either because `T` is large or N is large, or both). Actually, let's go even further: Can we do it in O(1) space?

At this point, I needed to pull out a piece of paper and do some sketching. My first idea² led nowhere, but eventually I found something that worked: The idea is that we look at `indices[0]` to see what item should be moved into position zero. We move that item in, but we save the item that was originally there. Then we look at what item should be moved into the place that was just vacated, and move that item into that place. Repeat until the item that was originally in position zero is the one that gets placed.

This completes one cycle, but there could be multiple cycles in the permutation, so we need to look for the next slot that hasn't yet been processed. We'll have to find a place to keep track of which slots have already been processed, and we cannot allocate a vector for it, because that would be O(N) in space.

So we pull a trick: We use the indices vector to keep track of itself.

```template<typename T>
void
apply_permutation(
std::vector<T>& v,
std::vector<int>& indices)
{
for (size_t i = 0; i < indices.size(); i++) {
T t{std::move(v[i])};
auto current = i;
while (i != indices[current]) {
auto next = indices[current];
v[current] = std::move(v[next]);
indices[current] = current;
current = next;
}
v[current] = std::move(t);
indices[current] = current;
}
}
```

Okay, what just happened here?

We start by "picking up" the `i`'th element (into a temporary variable `t`) so we can create a hole into which the next item will be moved. Then we walk along the cycle using `current` to keep track of where the hole is, and `next` to keep track of the item that will move into the hole. If the next item is not the one in our hand, then move it in, and update `current` because the hole has moved to where the next item was. When we finally cycle back and discover that the hole should be filled with the item in our hand, we put it there.

The way we remember that this slot has been filled with its final item is by setting its index to a sentinel value. One option would be to use −1, since that is an invalid index, and we can filter it out in our loop.

But instead, I set the index to itself. That makes the item look like a single-element cycle, and processing a single-element cycle has no effect, since nothing moves anywhere. This avoids having an explicit sentinel test. We merely chose our sentinel to be a value that already does nothing.

Now, one thing about this algorithm is that processing a single-element cycle does require us to pick up the element, and then put it back down. That's two move operations that could be avoided. Fixing that is easy: Don't pick up the item until we know that we will need to vacate the space.

```template<typename T>
void
apply_permutation(
std::vector<T>& v,
std::vector<int>& indices)
{
for (size_t i = 0; i < indices.size(); i++) {
if (i != indices[current]) {
T t{std::move(v[i])};
auto current = i;
while (i != indices[current]) {
auto next = indices[current];
v[current] = std::move(v[next]);
indices[current] = current;
current = next;
}
v[current] = std::move(t);
indices[current] = current;
}
}
}
```

Another trick we can use to avoid having to have a temporary item `t` is realizing that instead of holding the item in your hand, you can just put the item in the hole.

```template<typename T>
void
apply_permutation(
std::vector<T>& v,
std::vector<int>& indices)
{
using std::swap; // to permit Koenig lookup
for (size_t i = 0; i < indices.size(); i++) {
auto current = i;
while (i != indices[current]) {
auto next = indices[current];
swap(v[current], v[next]);
indices[current] = current;
current = next;
}
indices[current] = current;
}
}
```

Note that since the item that used to be in our hand is now in the hole, we don't need the manual last step of moving the item from our hand to the hole. It's already in the hole!

Okay, so is it better to move or to swap? This is an interesting question. We'll pick this up next time.

¹ Note that by making this operate on movable objects, we lose the ability to operate on objects which are not MoveAssignable. This is not a significant loss of amenity, however, because most of the algorithms in the standard library which rearrange elements within a container already require Move­Assignable, and many of them also require Swappable.

² My first idea was to start with the first position, swap the correct item into that first position, and update the indices so that what remained was a permutation of the remaining items that preserved the intended final result. I could then repeat the process with each subsequent position, and when I had finished walking through the entire vector, every item would be in the correct final position. I figured I could use the last position as a sort of scratchpad, similar to the "hole" we ended up using in our final algorithm. I struggled for a while trying to find the correct cleverly-chosen two-way or three-way swap among the current position, the last position, and somebody that would get me to the desired exit condition. I got hung up on this line of investigation and had to walk away from the problem for a little while, and then come back to it with a fresh mind.

I mention this because a lot of the time, these articles which explain how to solve a problem don't talk much about the attempted solutions that didn't work. Instead, they give the impression that the author got it right the first time. In reality, the author got it wrong the first few times, too. The reason you don't see a lot of writing about the failures is that the usual order of operations is (1) find solution, (2) write about it. You usually find the solution first, and only then do you start writing about it. As a result, you don't have very good documentation for your failures.

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1. Ionel POP says:

It is worth noting that when the vector is updated in place (2nd part of the post), you dropped the const-ness on indices. It may not be the expected behavior for the caller.

1. camhusmj38 says:

You could drop the reference as well as the const to copy the indices parameter (or make a copy in the body of the function if you can't change the signature). Technically, the time to copy the index vector would depend on n but if we assume that the objects in v are more heavy weight than ints that is a small cost.

2. Tramboi says:

If you want to do proper modern C++, the indices should be vector::size_type.
Of course proper modern C++ is boring and too verbose, so I'd settle for a size_t :)

1. Billy O'Neal says:

If you haven't customized the allocator then size_type must be size_t, since that's what the default type for allocator_traits<A>::size_type.

That said I think the purpose of this example was assuming the set of indices to use came from some other data source that speaks ints -- as long as you know that all of the ints will be in the valid range of the vector using size_t instead here buys you nothing. Note that the loop that iterates over the vector itself is already (correctly) using size_t.

1. Tramboi says:

That was just a comment about how convoluted and verbose the standard library is :)

3. Billy O'Neal says:

As far as I am aware there is no standard library facility for this -- if you squint the proposed algorithm looks like selection sort. You could write something like a zip iterator and a predicate that orders the resulting pair by the indices vector and feed that to std::sort. On the other hand, I suspect on real data apply_permutation, despite being technically O(n²) worst case, will perform better.

(As an aside, this is a classic time/memory tradeoff -- if you have O(n) extra memory you can have an O(n) algorithm, which is optimal)

1. The function looks O(n²) at first glance, but it is O(n) because each item moves at most once.

1. Billy O'Neal says:

Oh derp, that's right.

2. Marco says:

Ok, it's O(n) regarding moves/copies, but it's O(n2) for the index searching stuff. Right?

1. Zarat says:

Probably not, it should be O(n) as claimed, but the argument for it will be more complicated. The point is that the inner loop modifies itself on the first iteration. When the outer loop visits an already processed slot there will be no inner loop (just a constant number of checks to determine the fact), so the algorithm should be O(n) and not O(n²)

2. Billy O'Neal says:

I suppose you could also build a divide and conquer algorithm out of std::rotate:
1. Find middle of the resulting vector. O(1)
2. Search indices vector for element that goes there O(n)
3. std::rotate that element into the middle == 2 O(n) (one O(n) for the elements, one O(n) for the indices array)
4. Recursively do the same on left and right sides.

This produces an algorithm which takes T(n) = 2T(n/2) + O(3n + 1) = 2T(n/2) + O(n), master theorem says O(n lg n). But again, probably worse constant factors than apply_permutation, and this is substantially more complex to code up :)

4. There's a permutation_iterator in Boost, which permits creating an iterator pair over the permuted items, which can either be used directly or used to reify the collection into a new vector.

5. Arash Partow says:

Out of curiosity what happens when ‘v.size() > indices.size()’ ?

1. IInspectable says:

Even if just out of curiosity, asking for the behavior of an algorithm outside its specification isn't very insightful. The rules are simple: Don't operate the algorithm outside its designated limits.

2. French Guy says:

v.size() > indices.size() doesn't cause problems (because the algorithm iterates over indices): the elements from 0 to indices.size() - 1 will be moved (according to indices), and the remaining elements will stay in place (as if indices[i] was i for all i from indices.size() to v.size() - 1).

The real problem scenario is if indices.size() > v.size(): that will cause out-of-bounds accesses. If you're lucky (the language you're using checks for them), such accesses will cause exceptions. If you're not, expect nasal demons.

1. Not sure what you're trying to say. `next_permutation` solves a different problem.

6. Pseudonym says:

Well that's nice, but what I'd really like is:

template void apply_permutation(std::vector& v, const std::vector& indices);

1. Pseudonym says:

Curse you HTML markup. What I meant, of course, is that "indices" should be const.

2. kme says:

So just copy 'indices' at the start then. It's still O(n).

7. Micha says:

In your last iteration you swap the element to the whole, but than, manually swap the indices, how about using swap twice:

while (i != indices[current]) {
std::swap(x[i], x[indices[current]]);
std::swap(indices[i], indices[indices[current]]);
current = indices[current];
}

8. Pietro Gagliardi (andlabs) says:

It's spending time on things like the second half of this post and then sitting back, looking at this function or this entire source file and thinking "wow, look at me go with this ingenious code, I'm so great" that makes programming and computer science so fun. Bonus points if you then think to yourself "...but I wonder if other people will get it; heck I wonder if *I* will get it six months from now, even with the best comments in the world" =P

1. Zarat says:

Bonus points are only applied if you actually add the comments and not only think about them.

9. laonianren says:

It is easier to permute an array given the inverse permutation. The couple of times I faced your problem it was simpler to change the source function to produce an inverse permutation, rather than trying to apply the normal permutation it was producing. Obviously this isn't always an option.

10. xix says:

Almost anything can be attacked as a loss of amenity, and almost anything can be defended as not a significant loss of amenity, which seems to signify that one should appreciate the significance of significant.

11. Peter Taylor says:

There is a compromise position between having const indices (the ideal) and destroying the contents of indices (the given implementation) without sacrificing the O(1) space requirement. Since we know that the indices will be non-negative, we can use the bitwise negation operator (~) to generate sentinels, and then do a post-permutation pass through indices to apply ~ a second time and restore the original values. It's not thread-safe, but you can't have everything.