Wednesday, November 24, 2010

Three uses for a binary heap

Lately I have been interviewing for jobs, so doing a lot of whiteboard programming, and binary heaps keep arising in the solutions to these interview problems. There is nothing new or remarkable about these applications (binary heaps and their uses are covered in any undergraduate algorithms class), but I thought I would write them down because they are cute, and in the hope that they might be useful to someone else who (like me) gets by most days as a working programmer with no algorithm fancier than quicksort or binary search.

Binary heaps

Here’s a signature for a binary heap module Heap:

module type OrderedType = 
  type t 
  val compare : t -> t -> int 
module type S = sig 
  type elt 
  type t 
  val make : unit -> t 
  val add : t -> elt -> unit 
  val peek_min : t -> elt option 
  val take_min : t -> elt 
  val size : t -> int 
module Make (O : OrderedType) : S with type elt = O.t 

We start with a signature for ordered types (following the Set and Map modules in the standard library), so we can provide a type-specific comparison function.

From an ordered type we can make a heap which supports adding elements, peeking the smallest element (None if there are no elements) without removing it, removing and returning the smallest element (raising Not_found if the heap is empty), and returning the number of elements.

We’ll work out the asymptotic running times of the algorithms below, so it will be useful to know that the worst-case running time of the add and take_min functions is O(log n) where n is the number of elements in the heap.

Finding the k smallest elements in a list

Here’s a simple one. To find the smallest element in a list, we could sort the list then take the first element in the sorted list, at a cost of O(log n). Or we could just take a pass over the list keeping a running minimum, at a cost of O(n).

What if we want the k smallest elements? Again, we could sort the list, but if k < n we can do better by generalizing the single-pass solution. The idea is to keep the k smallest elements we’ve seen so far in a binary heap. For each element in the list we add it to the heap, then (if there were already k elements in the heap) remove the largest element in the heap, leaving the k smallest.

The running time is O(n log k) since we do an add and a take_min in a heap of size k for each of n elements in the list. Here’s the code:

let kmin (type s) k l = 
  let module OT = struct 
    type t = s 
    let compare e1 e2 = compare e2 e1 
  end in 
  let module H = Heap.Make(OT) in 
  let h = H.make () in 
    (fun e -> 
       H.add h e; 
       if H.size h > k 
       then ignore (H.take_min h)) 
  let rec loop mins = 
    match H.peek_min h with 
      | None -> mins 
      | _ -> loop (H.take_min h :: mins) in 
  loop [] 

Here we make good use of OCaml 3.12’s new feature for explicitly naming type variables in a polymorphic function to make a structure matching OrderedType. The heap has the same element type as the list, but we reverse the comparison since we want to remove the largest rather than smallest element from the heap in the loop. At the end of kmin we drain the heap to build a list of the k smallest elements.

Merging k lists

Suppose we want to merge k lists. We could merge them pairwise until there is only one list, but that would take k - 1 passes, for a worst-case running time of O(n * (k - 1)). Instead we can merge them all in one pass, using a binary heap so we can find the next smallest element of k lists in O(log k) time, for a running time of O(n log k). Here’s the code:

let merge (type s) ls = 
  let module OT = struct 
    type t = s list 
    let compare e1 e2 = 
      compare (List.hd e1) (List.hd e2) 
  end in 
  let module H = Heap.Make(OT) in 
  let h = H.make () in 
  let add = function 
    | [] -> () 
    | l -> H.add h l in 
  List.iter add ls; 
  let rec loop () = 
    match H.peek_min h with 
      | None -> [] 
      | _ -> 
          match H.take_min h with 
            | [] -> assert false 
            | m :: t -> 
                add t; 
                m :: loop () in 
  loop () 

We store the lists in the heap, and compare them by comparing their head element (we’re careful not to put an empty list in the heap). When we take the smallest list from the heap, its head becomes the next element in the output list, and we return its tail (if it is not empty) to the heap.

Computing a skyline

The next problem was told to me in terms of computing the skyline of a set of buildings. A building has a height and a starting and ending x-coordinate; buildings may overlap. The skyline of a set of buildings is a list of (x, y) pairs (in ascending x order), describing a sequence of horizontal line segments (each starting at (x, y) and ending at the subsequent x), such that at any x there is no space between the line segment and the tallest building. (Here’s another description with diagrams.)

I googled a bit to see what this is useful for, and didn’t find much. One application is to extract a monophonic line from polyphonic music, where x is time and height is some metric on notes, like pitch or volume. It might be useful for searching data which is only intermittently applicable—say, to compute a schedule over time of the nearest open restaurant.

The algorithm scans the building start and end points in ascending x order, keeping the “active” buildings (those which overlap the current x) in a binary heap. The height of the skyline can only change at a building start or end point. We can determine the tallest building at a point by calling peek_min on the heap.

When we hit a start point we add the building to the heap; for an end point we do nothing (the heap has no operation to remove an element). So we may have inactive buildings in the heap. We remove them lazily—before checking the height of the highest building, we call take_min to remove any higher inactive buildings.

The worst-case running time is O(n log n), since we do some heap operations for each building, and we might end up with all the buildings in the heap.

Here’s the code:

type building = int * int * int (* x0, x1, h *) 
let skyline bs = 
  let module OT = struct 
    type t = int * building 
    let compare (x1, _) (x2, _) = compare x1 x2 
  end in 
  let module Events = Heap.Make(OT) in 
  let events = Events.make () in 
    (fun ((x0,x1,_) as b) -> 
       Events.add events (x0, b); 
       Events.add events (x1, b)) 
  let module OT = struct 
    type t = building 
    let compare (_,_,h1) (_,_,h2) = compare h2 h1 
  end in 
  let module Heights = Heap.Make(OT) in 
  let heights = Heights.make () in 
  let rec loop last = 
    match Events.peek_min events with 
      | None -> [] 
      | _ -> 
          let (x, (x0,_,h as b)) = Events.take_min events in 
          if x = x0 then Heights.add heights b; 
          while (match Heights.peek_min heights with 
                   | Some (_,x1,_) -> x1 <= x 
                   | _ -> false) do 
            ignore (Heights.take_min heights) 
          let h = 
            match Heights.peek_min heights with 
              | Some (_,_,h) -> h 
              | None -> 0 in 
          match last with 
            | Some h' when h = h' -> loop last 
            | _ -> (x, h) :: loop (Some h) in 
  loop None 

We use a second heap events to store the “events” (the start and end points of all the buildings), in order to process them in ascending x order. (This use is not dynamic—we do not add new elements to the heap while processing them—so we could just as well use another means of sorting the points.) In this heap we store the x coordinate and the building (we can tell whether we have a start or end point by comparing the x coordinate to the building’s start point), and compare elements by comparing just the x coordinates.

The main heap heights stores buildings, and we compare them by comparing heights (reversed, so peek_min peeks the tallest building). While there are still events, we add the building to heights if the event is a start point, clear out inactive buildings, then return the pair (x, y) where x is the point we’re processing and y is the height of the tallest active building. Additionally we filter out adjacent pairs with the same height; these can arise when a shorter building starts or ends while a taller building is active.

Implementing binary heaps

The following implementation is derived from the one in Daniel B├╝nzli’s React library (edited a little bit for readability). The Wikipedia article on binary heaps explains the standard technique well, so I won’t repeat it.

The only piece of trickiness is the use of Obj.magic 0 for unused elements of the array, so we can grow it by doubling the size rather than adding a single element each time, and thereby amortize the cost of blitting the old array.

module Make (O : OrderedType) : S with type elt = O.t = 
  type elt = O.t 
  type t = { mutable arr : elt array; mutable len : int } 
  let make () = { arr = [||]; len = 0; } 
  let compare h i1 i2 = h.arr.(i1) h.arr.(i2) 
  let swap h i1 i2 = 
    let t = h.arr.(i1) in 
    h.arr.(i1) <- h.arr.(i2); 
    h.arr.(i2) <- t 
  let rec up h i = 
    if i = 0 then () 
      let p = (i - 1) / 2 in 
      if compare h i p < 0 then begin 
        swap h i p; 
        up h p 
  let rec down h i = 
    let l = 2 * i + 1 in 
    let r = 2 * i + 2 in 
    if l >= h.len then () 
      let child = 
        if r >= h.len then l 
        else if compare h l r < 0 then l else r in 
      if compare h i child > 0 then begin 
        swap h i child; 
        down h child 
  let add h e = 
    if h.len = Array.length h.arr 
    then begin 
      let len = 2 * h.len + 1 in 
      let arr' = Array.make len (Obj.magic 0) in 
      Array.blit h.arr 0 arr' 0 h.len; 
      h.arr <- arr' 
    h.arr.(h.len) <- e; 
    up h h.len; 
    h.len <- h.len + 1 
  let peek_min h = 
    match h.len with 
      | 0 -> None 
      | _ -> Some h.arr.(0) 
  let take_min h = 
    match h.len with 
      | 0 -> raise Not_found 
      | 1 -> 
          let m = h.arr.(0) in 
          h.arr.(0) <- (Obj.magic 0); 
          h.len <- 0; 
      | k -> 
          let m = h.arr.(0) in 
          let k = k - 1 in 
          h.arr.(0) <- h.arr.(k); 
          h.arr.(k) <- (Obj.magic 0); 
          h.len <- k; 
          down h 0; 
  let size h = h.len 

(Complete code is here.)

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