5.7. Priority Queues

  • File: PriorityQueue.ml

Recall our main motivation for studying binary heaps: efficient retrieval of an element with maximal/minimal key in an array, without re-sorting it from scratch between the changes. A data structure that allows for efficient retrieval of an element with the highest/lowest key is called a priority queue. In this section, we will design a priority queue based on the implementation of the heaps we already have.

The priority queue will be implemented by a dedicated data type and a number of operations, all residing within the following functor:

module PriorityQueue(C: CompareAndPrint) = struct

  (* To be filled *)


A carrier of a priority queue (i.e., a container for its elements) will be, of course, an array. Therefore, a priority queue may only hold as many elements as is the size of the array.

We introduce a small encoding tweak, which will be very helpful for accounting for the fact that the array might be not fully filled, allowing the priority queue to grow (as more elements are added to it) and shrink (as the elements are extracted). Let us add the following definitions into the body of PriorityQueue:

module COpt = struct
  type t = C.t option

  let comp x y = match x, y with
    | Some a, Some b -> C.comp a b
    | None, Some _ -> -1
    | Some _, None -> 1
    | None, None -> 0

  let pp x = match x with
    | Some x -> C.pp x
    | None -> "None"

module H = Heaps(COpt)
(* Do no inline, just include *)
open H

The module COpt “lifts” the pretty-printer and the comparator of a given module C (of signature CompareAndPrint), to the elements of type option. Specifically, if an element is None, it is strictly smaller than any Some-like elements. As you can guess, the None elements will denote the “empty space” in our priority queue.

5.7.1. Creating Priority Queues

The queue is represented by an OCaml record of the following shape (also to be added to the module):

type heap = {
  heap_size : int ref;
  arr : H.t array

The records in OCaml are similar to those in C and are simply collections of named values (referred to as record fields). Specifically, the record type heap pairs the carrier array arr of elements of type H.t (i.e., C.t lifted to an option), and the dedicated “heap threshold” heap_size to determine which part of arr serves as a heap.

The following two functions allow to create an empty priority queue of a given size, and also turn an array into a priority queue (by effectively building a heap out of it):

let mk_empty_queue size =
  assert (size >= 0);
  {heap_size = ref 0;
   arr = Array.make size None}

(* Make a priority queue from an array *)
let mk_queue a =
  let ls = List.map (fun e -> Some e) (to_list a) in
  let a' = list_to_array ls in
  build_max_heap a';
  {heap_size = ref (Array.length a);
   arr = a'}

Finally, the following construction allows to print out the contents of a priority queue by reusing the functor ArrayPrinter defined at the beginning of this chapter:

module P = ArrayPrinter(COpt)

let print_heap h =
  P.print_array h.arr

5.7.2. Operations on Priority Queues

The first and the simplest operation on a priority queue h is to take its highest-ranked element (i.e., the one with the greatest priority, expressed by means of its key value):

let heap_maximum h = (h.arr).(0)

The next operation allows not just look at, but also extract (i.e., fetch and remove) the maximal element from the priority queue:

let heap_extract_max h =
  if !(h.heap_size) < 1 then None
    let a = h.arr in
    let max = a.(0) in
    a.(0) <- a.(!(h.heap_size) - 1);
    a.(!(h.heap_size) - 1) <- None;
    h.heap_size := !(h.heap_size) - 1;
    max_heapify !(h.heap_size) h.arr 0;

The way heap_extract_max works for a non-empty heap is by taking its maximal element, and then putting one of the smallest elements (a.(!(h.heap_size) - 1)) to its place, reducing the heap size and restoring the heap shape via already familiar procedure max_heapify applied to the first element in the array (which is the only heap offender after swapping).

The following auxiliary function heap_increase_key is somewhat dual to max_heapify. It inserts an element key into a position i, assuming that its key is larger than what’s currently at that position. It then restores the heap property (which might be broken if the parents in the chain are smaller) by “walking up” the chain of parents and performing swaps until the correct order is restored:

let heap_increase_key h i key =
  let a = h.arr in
  let c = comp key (a.(i)) >= 0 in
  if not c then (
    Printf.printf "A new key is smaller than current key!";
    assert false);
  a.(i) <- key;
  let j = ref i in
  while !j > 0 && comp (snd (H.parent a (!j))) a.(!j) < 0 do
    let pj = fst (H.parent a (!j)) in
    swap a !j pj;
    j := pj

Question: What is the complexity of heap_increase_key?

Finally, the function max_heap_insert implements an insertion of a new element elem into a priority heap h:

let max_heap_insert h elem =
  let hs = !(h.heap_size) in
  if hs >= Array.length h.arr
  then raise (Failure "Maximal heap capacity reached!");
  h.heap_size := hs + 1;
  heap_increase_key h hs (Some elem)

It only succeeds in the case if there is still vacant space in the queue (i.e., at the end of the array), which can be determined by examining the heap_size field of h. If the space permits, the limit heap_size is increased. Since we know that None used to be installed to the vacant place (which is an invariant maintained by means of heap_size), we can simply install the new element Some elem (which is guaranteed to be larger than None as per our defined comparator) and let the heap rebalance using heap_increase_key.

Given the complexity of max_heap_insert, it is easy to show that the complexity of element insertion is \(O(\log n)\). This brings us to an important property of priority queues implemented by means of heaps:

Complexity of priority queue operations

For a priority queue of size \(n\),

  • Finding the largest element has complexity \(O(1)\),
  • Extraction of the largest element has complexity \(O(\log n)\),
  • Insertion of a new element has complexity \(O(\log n)\).

5.7.3. Working with Priority Queues

Let us see a priority queue in action. We start by creating it from a randomly generated array:

module PQ = PriorityQueue(KV)
open PQ

let q = mk_queue (
 [|(6, "egkbs"); (4, "nugab"); (4, "xcwjg");
   (4, "oxfyr"); (4, "opdhq"); (0, "huiuv");
   (0, "sbcnl"); (2, "gzpyp"); (4, "hymnz");
   (2, "yxzro")|]);;

Let us see what’s inside:

# q;;
- : PQ.heap =
{heap_size = {contents = 10};
 arr =
  [|Some (6, "egkbs"); Some (4, "nugab"); Some (4, "xcwjg");
    Some (4, "oxfyr"); Some (4, "opdhq"); Some (0, "huiuv");
    Some (0, "sbcnl"); Some (2, "gzpyp"); Some (4, "hymnz");
    Some (2, "yxzro")|]}

We can proceed by checking the maximum:

# heap_maximum q;;
- : PQ.H.t = Some (6, "egkbs")

(* It is indeed a heap! *)
#  PQ.H.is_heap q.arr;;
- : bool = true

Let us extract several maximum elements:

# heap_extract_max q;;
- : PQ.H.t option = Some (6, "egkbs")
# heap_extract_max q;;
- : PQ.H.t option = Some (4, "nugab")
# heap_extract_max q;;
- : PQ.H.t option = Some (4, "oxfyr")
# heap_extract_max q;;
- : PQ.H.t option = Some (4, "hymnz")

Is it still a heap?:

# q;;
- : PQ.heap =
{heap_size = {contents = 6};
 arr =
  [|Some (4, "opdhq"); Some (2, "yxzro"); Some (4, "xcwjg");
    Some (0, "sbcnl"); Some (2, "gzpyp"); Some (0, "huiuv");
    None; None; None; None|]}
#  PQ.H.is_heap q.arr;;
- : bool = true

Finally, let us insert a new element and check whether it is still a heap:

# max_heap_insert q (7, "abcde");;
- : unit = ()
# q;;
- : PQ.heap =
{heap_size = {contents = 7};
 arr =
  [|Some (7, "abcde"); Some (2, "yxzro"); Some (4, "opdhq");
    Some (0, "sbcnl"); Some (2, "gzpyp"); Some (0, "huiuv");
    Some (4, "xcwjg"); None; None; None|]}
# heap_maximum q;;
- : PQ.H.t = Some (7, "abcde")