In a week of cool algorithms, this one in java.util.PriorityQueue was the most interesting.

If you step through the code adding and remove-ing elements, you'll see the array represented by PriorityQueue.queue changing but it doesn't represent the order that elements come out of the queue. The documentation says the "Priority queue [is] represented as a balanced binary heap".

What is a binary heap?

"Definition: A binary heap is a collection of keys arranged in a complete heap-ordered binary tree, represented in level order in an array (not using the first entry).

"In a binary heap, the keys are stored in an array such that each key is guaranteed to be larger than (or equal to) two additional keys, and so forth.

"Definition: A binary tree is heap-ordered if the key in each node is larger than or equal to the keys in that node's two children (if any). Equivalently, the key in each node of a heap-ordered binary tree is smaller than or equal to the key in that node's parent (if any). Moving up from any node, we get a nondecreasing sequence of keys; moving down from any node, we get a nonincreasing sequence of keys. In particular... the largest key in a heap-ordered binary tree is found at the root." - Algorithms, Sedgewick & Wayne.

So, a tree that looks like this:

[Graphic taken from the Wikipedia entry]

maps isomorphically to an array, in this case 100, 19, 36, 17, 3, 25, 1, 2, 7. The rule is (from comments in PriorityQueue):

And "if multiple elements are tied for least value, the head is one of those elements -- ties are broken arbitrarily."

One last thing, the order of child nodes is irrelevant.

If you step through the code adding and remove-ing elements, you'll see the array represented by PriorityQueue.queue changing but it doesn't represent the order that elements come out of the queue. The documentation says the "Priority queue [is] represented as a balanced binary heap".

What is a binary heap?

"Definition: A binary heap is a collection of keys arranged in a complete heap-ordered binary tree, represented in level order in an array (not using the first entry).

"In a binary heap, the keys are stored in an array such that each key is guaranteed to be larger than (or equal to) two additional keys, and so forth.

"Definition: A binary tree is heap-ordered if the key in each node is larger than or equal to the keys in that node's two children (if any). Equivalently, the key in each node of a heap-ordered binary tree is smaller than or equal to the key in that node's parent (if any). Moving up from any node, we get a nondecreasing sequence of keys; moving down from any node, we get a nonincreasing sequence of keys. In particular... the largest key in a heap-ordered binary tree is found at the root." - Algorithms, Sedgewick & Wayne.

So, a tree that looks like this:

[Graphic taken from the Wikipedia entry]

maps isomorphically to an array, in this case 100, 19, 36, 17, 3, 25, 1, 2, 7. The rule is (from comments in PriorityQueue):

*the two children of queue[n] are queue[2*n+1] and queue[2*(n+1)]*And "if multiple elements are tied for least value, the head is one of those elements -- ties are broken arbitrarily."

One last thing, the order of child nodes is irrelevant.