The skyline problem

https://briangordon.github.io/2014/08/the-skyline-problem.html

by brianpgordon • 12 years ago

149 44 12 years ago

briangordon.github.io

This is targeted toward students who could benefit from someone hand-holding them through the design of an algorithm, but anyone can learn something. It's always good to have another potential interview question in your repertoire.

starmole• 12 years ago

In case you did not know: This is closely related to an s-buffer (s for span) in graphics. And the first approach (height map) is pretty much a z-buffer. Your silhouette there is eye distance (z) across a single scan line. One difference of how it is approached in graphics is that usually not all spans are an input array but inserted one at a time.

sdrothrock• 12 years ago

This is targeted toward students who could benefit from someone hand-holding them through the design of an algorithm

I enjoyed the article and then came back to read the comments and found this sort of insulting.

Just because something is detailed doesn't mean that it's hand-holding. It's interesting to see how other people think.

brianpgordon7• 12 years ago

I'm sorry, I didn't mean to be insulting. I thought this was a little basic for the HN crowd so I wanted to make sure I left a disclaimer to that effect so people didn't think I was being condescending by explaining every little detail. I guess it backfired.

optimusclimb• 12 years ago

Yup. I sure solve THAT problem every day. My boss comes in and is like, "well, we have the bugs from that security audit, we need to come up with a design for the feature we want to try and roll out in two quarters so we can agree on it and cut tasks/divy up work, and then we're using two many boxes for the Foo processor. I think you've never optmized that right? Why don't you have a look at that. Also, don't use any standard library functions or look anything up. When you're done with that, I have a question for you about manhole covers."

Yup, another great interview question for your repertoire to weed people out and talk about the kinds of things that come up in the day to day of being a modern software engineer.

legutierr• 12 years ago

Does a modern software engineer not need to demonstrate an ability to design efficient algorithms that solve problems that haven't been encountered before?

Or, in your mind, is the solution to every problem that a modern software engineer might encounter already found in some stackoverflow post?

cousin_it2• 12 years ago

Sure, many programming jobs don't involve any algorithm design. But the most interesting programming jobs tend to involve algorithm design.

praptak2• 12 years ago

It's always good to have another potential interview question in your repertoire.

It is a widely known "good interview question" which means it is a bad interview question.

sgeisenh• 12 years ago

If you think recursively, the n log n divide and conquer solution is pretty straightforward.

The problem becomes much more interesting when you consider optimal parallel performance. The best span you can do is log^2 n. I believe the problem can be reduced to sorting, though it is not required to explicitly sort the sequence of buildings in order to achieve the desired asymptotic performance.

cperciva• 12 years ago

I believe the problem can be reduced to sorting

Speaking of which: Given values 0 < a_i < A for some A, apply the skyline algorithm to the rectangles a_i < x < A, 0 < y < a_i. The resulting sequence of heights gives you the values a_i sorted in increasing order; thus the skyline problem cannot be solved in less than O(n log n) time.

thewarrior2• 12 years ago

How would you efficiently find the critical points ?

idunning• 12 years ago

The critical points are the left and right sides of the rectangles.

nitrogen• 12 years ago

The critical points are given as inputs to the algorithm in the form of the X coordinates of the rectangles' edges.

thewarrior2• 12 years ago

D'oh .. Missed that . Thanks.

curiousDog• 12 years ago

A regular google interview question as well!

srean• 12 years ago

I am very bad at these things when put in a spot. OTOH if you would let me pace around for a bit and stay invisible for a few minutes then its not too bad.

Havent checked for correctness, but my off the cuff attempt would be to sort the left and right edges, then use a variable for the height and a stack. Proceed left to right. If I encounter a left edge and current height is lower than stack, push height on stack and update current height. If left edge lower than current, just push its value on the stack. If I encounter a right edge that matches current height pop height from stack in to current height. If right edge lower than current height, just pop the stack. The awkward case is when the left and right edges coincide.

prawks2• 12 years ago

This is pretty much the solution given. I've encountered a similar problem before (draw a bunch of overlapping squares with differing z-values) and done it similarly. The difficulty you end up having is that a stack doesn't give you a fast way to find the maximum height (what building/square) is on top at any given point. Hence, the article suggests a max heap.

praptak2• 12 years ago

Fun fact: this is a specific case of finding the union (usually: area of) of a set of axes-oriented possibly overlapping rectangles.

bnegreve• 12 years ago

This is an instance of finding the Pareto front [1] of a set of points in a two dimensional space.

This problem is also known as "skyline queries"!

According to [2], author's bound O(n log(n)) is the best bound for d=2.

[1] http://en.wikipedia.org/wiki/Pareto_efficiency [2] http://pdf.aminer.org/000/211/201/on_the_computational_compl...

frownie• 12 years ago

is it me or the algorithm suggested at the end degenerates if the rectangles are organized as the steps of a stair and each of them extends to the rightmost position ? In that case it degenerates because the heap manipulation degenerates...

brianpgordon7• 12 years ago

I don't think so. No matter what shape the problem is in, there are always exactly n inserts and n deletes to the heap. Those are O(log n) operations, and there are 2n of them, so it's O(n log n).

wiz21• 12 years ago

Ahhh read too fast :-( Crucial sentence is "Fortunately, we know about a data structure which can keep track of an active set of integer-keyed objects and return the highest one in O(logn) time: a heap."

But then again. If I read well, he wants to 1st sort the points, 2 iterate over them and using the heap. So we have n log n for the sort and n * n log n for the heap use.. Basically (1+n) n log n (or i'm still tired ? :-))

signa11• 12 years ago

if we think about it somewhat, it becomes apparent, that it would take linear time to merge one building to the skyline, and also two skylines together.

merging skylines together gets us more bang-for-the-buck. and is trivially done with scanning them from left->right, match x-coordinates, adjust heights...

all of this is just divide-and-conquer, and usual recurrence rules apply i.e. T(n) = 2T(n/2)+O(n), which means T(n) = O(n lg n)

michaelmior3• 12 years ago

My first thought was to dump the rectangles into an interval tree. Also O(n log n), but less space efficient since it always requires O(n) space regardless of the structure of the problem.

brianpgordon7• 12 years ago

Are you sure that it's n log n? I think that the pyramid shaped configuration defeats this strategy by forcing the interval tree to report something on the order of n intervals with every query. So there are n queries on the interval tree, and each query takes log(n) + n time.

Also, something interesting to note that I was thinking about yesterday is that you don't actually need to use an interval tree for this. You can use a range tree instead, which is simpler (to my mind). You only need an interval tree to do a certain kind of query which we never need to do:

https://briangordon.github.io/images/range-interval.png

michaelmior3• 12 years ago

I'm not sure what you mean by the pyramid-shaped configuration. In any case, n queries taking log(n) + n time each is O(n log n). I think you're right that a range tree should work fine.

infinotize• 12 years ago

To interviewers who have given this problem: how do candidates typically perform? I would expect a lot of people to come up with some variant of an O(n^2) solution on the spot, but probably not get much further without some help, in an interview where you realistically have ~30 min to solve a problem.

Of course, this being HN, there are already several comments stating how the optimal solution is obvious.

drostie• 12 years ago

I'm also interested in how this varies by different languages.

For example, a programmer decently knowledgeable in JS will probably start with the "pre-data-structure-change" solution presented in the article, because if you're pretty good with JS then you know how events work and you probably start with, "okay, I'm going to scan left to right and emit an event on each new edge; that will look like this."

    var events = [];
    buildings.forEach(function (x) {
        events.push([x.l, 'add', x.h]);
        events.push([x.r, 'rem', x.h]);
    });
    events.sort(function (x, y) { 
        return x[0] - y[0];
    });

If you're lucky, they'll then say, "okay, I now need a height-stack where I can query the largest height, so..."

    function max(arr){
        return Math.max.apply(null, arr);
    }
    var height_stack = [], out;
    out = events.map(function (e) {
        var x = e[0], y = e[2], cmd = e[1];
        switch (cmd) {
        case "add":
            height_stack.push(y);
            return [x, max(height_stack);
        case "rem": 
            height_stack.splice(height_stack.indexOf(y), 1);
            return [x, max(height_stack)];
        default:
            throw new TypeError("No such command: " + cmd);
        }
    });

(There are a couple of other ways to implement max() above, depending on style; you could use .reduce(max_fn) for example, where max_fn is Math.max truncated to exactly two arguments.)

But I wouldn't expect them to come up with binary heaps a la http://eloquentjavascript.net/1st_edition/appendix2.html . I guess the best you could expect the candidate to do is to say, "then I Google someone who has done public-domain binary heaps and steal their code."

mpalme• 12 years ago

The problem es essentially to map from an indiscrete space to a discrete one. The solution that comes into my mind is a simple sweep-line algorithm:

1. Take all the start x-coordinates of a buildings (=start events).

2. Take all the end x-coodinates of the buildings (=end events).

3. Sort all the events the by the value of the coordinate.

4. Now walk over all the events and keep count of how many there currently are (or in this case: how high the building is) by adding one for a start event and substracting one for a end event.

http://en.wikipedia.org/wiki/Sweep_line_algorithm

prawks2• 12 years ago

A closely analogous problem which I have had to implement is the problem of drawing n overlapping events with varying duration on a timeline, coloring the event based on how many are underneath it.

Personally I find that a simpler problem to visualize, however it deals with almost the same information. Rather than what height to draw, you're keeping a count of how many events are underneath. You increment at each leading "critical point" and decrement at each ending one.

The problem presented here is nice because it requires thought on both the algorithm as well as the data structure. Very nice article.

tomjen3• 12 years ago

Why not just sort the rectangles by the left side position, then walk through the list, drawing a line from the current position to either the right of the next rectangle (in case that one is taller) or to the right of the current rectangle, in case it is not?

This even makes it easy to parallize it because you can use the quick sort algorithm to split the rectangles in two non-overlapping sets and once your two cores have made the two parts of the skyline you only need to update the line between them.

Retric• 12 years ago

Widths are not fixed. A tall skinny building might be drawn over a short wide one.

Also, you end up as N^2 degenerate case if the buildings form a pyramid.

arikrak2• 12 years ago

The animations are cool and helpful. How were they created?

brianpgordon7• 12 years ago

I used the trial of Edge Animate:

https://creative.adobe.com/products/animate

It's a really interesting tool. Here are the animation source files if you want to play around with them:

https://dl.dropboxusercontent.com/u/14282009/skyline.zip

arikrak2• 12 years ago

Thanks, I'll check it out!

CharlesC• 12 years ago

Got this as an interview question at Blizzard. Failed pretty spectacularly. I kept getting close but then the interviewer would jump in with "Well think about it this way...what if blah blah blah?"

The skyline problem

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