Priority Queues With Distinct Elements

August 9, 2016

My solution uses pairing heaps for the priority queue and sets implemented as a binary search tree to “pre-qualify” inserts and guarantee distinctness of the items in the priority queue. Here’s the priority queue, stolen from a previous exercise:

; priority queue -- pairing heaps

(define pq-empty (list))

(define pq-empty? null?)

(define (pq-first pq)
  (if (null? pq)
      (error 'pq-first "can't extract minimum from null queue")
      (car pq)))

(define (pq-merge lt? p1 p2)
  (cond ((null? p1) p2)
        ((null? p2) p1)
        ((lt? (car p2) (car p1))
          (cons (car p2) (cons p1 (cdr p2))))
        (else (cons (car p1) (cons p2 (cdr p1))))))

(define (pq-insert lt? x pq)
  (pq-merge lt? (list x) pq))

(define (pq-merge-pairs lt? ps)
  (cond ((null? ps) '())
        ((null? (cdr ps)) (car ps))
        (else (pq-merge lt? (pq-merge lt? (car ps) (cadr ps))
                            (pq-merge-pairs lt? (cddr ps))))))

(define (pq-rest lt? pq)
  (if (null? pq)
      (error 'pq-rest "can't delete minimum from null queue")
      (pq-merge-pairs lt? (cdr pq))))

(define (list->pq lt? xs)
  (let loop ((xs xs) (pq pq-empty))
    (if (null? xs) pq
      (loop (cdr xs) (pq-insert lt? (car xs) pq)))))

(define (pq->list lt? pq)
  (let loop ((pq pq) (xs '()))
    (if (pq-empty? pq) (reverse xs)
      (loop (pq-rest lt? pq) (cons (pq-first pq) xs)))))

Pairing heaps are my favorite implementation of the priority queue data structure; they are simple to implement, and perform quite well in practice, but their time complexity has defied analysis. Next we have a simple (unbalanced) binary search tree, with deletion implemented by tracking down the left spine of the right child of the node to be deleted to find the in-order successor, placing the in-order successor in the node of the item to be deleted, and then recursively deleting the in-order successor from the right child. This algorithm differs from the deletion in our previous implementation of binary search trees, which recursively rotates the node to be deleted down the tree until it reaches the bottom, where it is simply discarded:

; binary search tree

(define bst-empty (list))

(define bst-empty? null?)

(define (bst-member? lt? item bst)
  (cond ((bst-empty? bst) #f)
        ((lt? item (car bst))
          (bst-member? lt? item (cadr bst)))
        ((lt? (car bst) item)
          (bst-member? lt? item (caddr bst)))
        (else #t)))

(define (bst-insert lt? item bst)
  (cond ((bst-empty? bst)
          (list item (list) (list)))
        ((lt? item (car bst))
          (list (car bst)
                (bst-insert lt? item (cadr bst))
                (caddr bst)))
        ((lt? (car bst) item)
              (list (car bst)
              (cadr bst)
              (bst-insert lt? item (caddr bst))))
        (else bst)))

(define (bst-successor bst)
  (cond ((bst-empty? bst) bst-empty)
        ((bst-empty? (cadr bst)) bst)
        (else (bst-successor (cadr bst)))))

(define (bst-delete-root lt? bst)
  (cond ((and (bst-empty? (cadr bst))
              (bst-empty? (caddr bst)))
        ((bst-empty? (cadr bst)) (caddr bst))
        ((bst-empty? (caddr bst)) (cadr bst))
        (else (let ((new-root (car (bst-successor (caddr bst)))))
                (list new-root (cadr bst)
                      (bst-delete lt? new-root (caddr bst)))))))

(define (bst-delete lt? item bst)
  (cond ((bst-empty? bst) bst)
        ((lt? item (car bst))
          (list (car bst)
                (bst-delete lt? item (cadr bst))
                (caddr bst)))
        ((lt? (car bst) item)
          (list (car bst)
                (cadr bst)
                (bst-delete lt? item (caddr bst))))
        (else (bst-delete-root lt? bst))))

It would be better to use either a hash table or a balanced tree instead of a naive binary search tree, but we won’t bother. Feel free to do so if you wish. Note that hash tables are less convenient than trees for this application because using a hash table requires the programmer to provide both less-than and equal-to predicates.

Now it is easy to implement the priority queue with distinct elements. The only difference to the standard priority queue is that insertion first checks if the item is already in the queue, by querying the the binary search tree, then inserts the item only if it is not already present; insertion and deletion both update the binary search tree. The distinct priority queue is represented as a three-slot list containing the less-than predicate, priority queue, and binary search tree:

; distinct priority queue

(define (make-dpq lt?) (list lt? pq-empty bst-empty))

(define (dpq-empty? dpq) (pq-empty? (cadr dpq)))

(define (dpq-first dpq)
  (if (dpq-empty? dpq)
      (error 'dpq-first "can't extract minimum from null queue")
      (pq-first (cadr dpq))))

(define (dpq-insert item dpq)
  (if (bst-member? (car dpq) item (caddr dpq))
      (list (car dpq)
            (pq-insert (car dpq) item (cadr dpq))
            (bst-insert (car dpq) item (caddr dpq)))))

(define (dpq-rest dpq)
  (if (dpq-empty? dpq)
      (error 'dpq-rest "can't delete minimum from null queue")
      (list (car dpq)
            (pq-rest (car dpq) (cadr dpq))
            (bst-delete (car dpq) (dpq-first dpq) (caddr dpq)))))

(define (dpq-enlist dpq) (pq->list (car dpq) (cadr dpq)))

There’s a lot of code shown above, but most if it was stolen from previous exercises, so it’s not as bad as it looks. Here are some examples:

> (define dpq (make-dpq <)) > (set! dpq (dpq-insert 3 dpq))
> (set! dpq (dpq-insert 4 dpq))
> (set! dpq (dpq-insert 1 dpq))
> (set! dpq (dpq-insert 5 dpq))
> (set! dpq (dpq-insert 3 dpq))
> (set! dpq (dpq-insert 2 dpq))
> (set! dpq (dpq-insert 5 dpq))
> (set! dpq (dpq-insert 1 dpq))
> (set! dpq (dpq-insert 2 dpq))
> (set! dpq (dpq-insert 4 dpq))
> (dpq-enlist dpq)
(1 2 3 4 5)
> (dpq-first dpq)
> (set! dpq (dpq-rest dpq))
> (set! dpq (dpq-rest dpq))
> (dpq-first dpq)
> (set! dpq (dpq-rest dpq))
> (set! dpq (dpq-rest dpq))
> (dpq-first dpq)
> (set! dpq (dpq-rest dpq))
> (dpq-enlist dpq)

Each item 1, 2, 3, 4 and 5 was inserted twice into the priority queue, but was output only once, as desired. You can run the program at

Pages: 1 2

5 Responses to “Priority Queues With Distinct Elements”

  1. Daniel said

    Here’s a solution in Java.

    It extends Java’s built-in priority queue and overrides *add* and *remove*.

    import java.util.Arrays;
    import java.util.HashSet;
    import java.util.PriorityQueue;
    import java.util.Set;
    public class PriorityQueueDistinct<E> extends PriorityQueue<E> {
      private Set<E> elements = new HashSet<>();
      public boolean add(E e) {
        if (elements.contains(e)) return false;
        boolean added = super.add(e); 
        if (added) elements.add(e);
        return added;
      public boolean remove(Object o) {
        boolean removed = super.remove(o);
        if (removed) elements.remove(o);
        return removed;
      public static void main(String[] args) {
        PriorityQueueDistinct<Integer> priorityQueue = new PriorityQueueDistinct<>();
        // Add 0, 1, 2, 3, 4 and output the priority queue.
        for (int i = 0; i < 5; ++i) priorityQueue.add(i);
        // Add 0, 1, 2, 3, 4 again and output the priority queue.
        // Output should remain the same as above.
        for (int i = 0; i < 5; ++i) priorityQueue.add(i);

    Here’s the output from the test in the main method.

    [0, 1, 2, 3, 4]
    [0, 1, 2, 3, 4]
  2. Globules said

    Here is a Haskell version that uses the “priority search queue” provided by the psqueues package.

    -- There is an existing data structure called the "priority search queue" that
    -- allows us to easily implement this.  It's a priority queue that also supports
    -- efficient searching.  In addition to a priority and a value, each item also
    -- has a search key.  The implementation we use is OrdPSQ, provided by the
    -- psqueues package.
    -- Our new function, insert', makes use of the alter function provided by
    -- OrdPSQ.  This is sort of a Swiss army knife function that allows us to
    -- insert, delete or modify items associated with a given key.  Insert' simply
    -- uses its second argument as both the key and the value.
    -- A fuller implementation would wrap OrdPSQ in a new type and provide its own
    -- API (e.g. hiding the keys) to prevent other functions from screwing up the
    -- "distinct items" property that we want to maintain.
    import Control.Arrow ((>>>))
    import Control.Monad (mplus)
    import Data.List (unfoldr)
    import Data.OrdPSQ
    -- Insert value v with priority p into the queue, only if the value is absent.
    insert' :: (Ord v, Ord p) => p -> v -> OrdPSQ v p v -> OrdPSQ v p v
    insert' p v = snd . alter update v
      where update pv = ((), pv `mplus` Just (p, v))
    -- Demonstrate our new insertion function.
    demo :: OrdPSQ String Int String -> OrdPSQ String Int String
    demo = insert' 3 "nematode" >>>
           insert' 2 "axolotl"  >>>
           insert' 1 "zebra"    >>>
           insert' 0 "axolotl"  -- This won't be added because the value already
                                -- exists.  So "axolotl" will still appear after
                                -- "zebra" when sorted by priority.
    -- Sort the values of a queue in order of priority by repeatedly extracting the
    -- minimum priority item.
    sortByPriority :: (Ord k, Ord p) => OrdPSQ k p v -> [v]
    sortByPriority = unfoldr step
      where step = fmap (\(_, _, v, q') -> (v, q')) . minView
    main :: IO ()
    main = print $ sortByPriority $ demo empty
    $ ./pqdist 
  3. zreo15 said

    a bit late but….how about using a heap it acts as a priority queue and we can add an extra condition to check for duplicates and skipping over them

  4. […] a little bit of time. The second change is more important; we use the distinct priority queue of a previous exercise to eliminate duplicate candidates, which greatly reduced the amount of computation needed to find […]

  5. r. clayton said

    A solution in Racket.

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