1. COMP 103 Priority Queues, Partially Ordered Trees and Heaps

2. 2 RECAP-TODAY RECAP Clever stuff with linked structures linked list idea  nice way to implement a Stack, or a Queue  Then we saw trees, and then Binary Search Trees (BST) adding, and finding (significance of balance) removing is “interesting” BST idea  nice way to implement a Set, Bag, or Map in-order traversals  TreeSort TODAY:  Priority queue (PQ) – variation on queue – “best” first  partially ordered tree (POT)  nice way to implement PQ  oh and we’ll also get another fast sorting algorithm!  Reading: Chapter 17 (section 17.4 - 17.5)

3. 3 Queues and priority queues  Queues: First in, first out ⇒ Oldest out first  Priority Queues: Highest priority (“best”) out first  Emergency room, 111 calls, job scheduling in factory, etc...  Operating system process scheduling  Graph/network algorithms: route planning, find shortest/cheapest path  AI search algorithms BobJackJimIan BobJackJimIan 1432 Sometimes low number means high priority!

4. 4 Implementing Priority Queues  Unsorted list (array or linked list):  Fast to enqueue (“offer”): O(1)  Slow to dequeue (“poll”): O(n) have to search for highest priority item.  Sorted list (array or linked list):  Fast to dequeue (“poll”): O(1) ‏  Slow to enqueue (“offer”): O(n) have to search for insertion point (and move items up).  Array of “Buckets” of queues for each priority:  Fast to enqueue and dequeue: O(1) ‏  Requires a small finite number of priorities.  Best choice when applicable. 1 2 3 4 5 6 7 8 9 10

5. 5 Implementing Priority Queues: preview...  So, what to do when we have unlimited priorities?  Binary Search Tree:  Keeps items in order, and allows easy insertion  Fast to enqueue: O(log n) [if balanced]  Fast to dequeue: O(log n) [if balanced] How?  But, hard to keep balanced  Partially Ordered Tree:  Fast to enqueue (“offer”): O(log n)  Fast to dequeue (“poll”): O(log n) ‏  Keeps itself balanced!  Fast to construct from unordered list!

6. 6 Partially Ordered Trees Binary Search Tree:  Binary tree  All left subtree < parent, All right subtree ≥ parent. Partially Ordered Tree Binary tree Children ≤ parent, Order of children not important Cat 19 Eel 26 Gnu 13 Fox 3 Dog 14 Bee 35 Hen 23 Ant 9 Bee 35 Eel 26 Cat 19 Dog 14 Fox 3 Ant 9 Hen 23 Gnu 13 Jay 1 Jay 1 Keep largest (or smallest) element at the root.

7. 7 Partially Ordered Tree: add  Easier to add and remove because the order is not complete.  Add:  insert at bottom rightmost  “push up” to correct position. (swapping) Bee 35 Eel 26 Cat 19 Dog 14 Fox 3 Ant 9 Hen 23 Gnu 13 Jay 1 Kea 24

8. 8 Partially Ordered Tree: remove  Easier to add and remove because the order is not complete.  Add:  insert at bottom rightmost  “push up” to correct position.  Remove:  “pull up” largest child and recurse.  But: makes tree unbalanced! Bee 35 Eel 26 Kea 24 Dog 14 Cat 19 Ant 9 Hen 23 Gnu 13 Jay 1 Fox 3

9. 9 Partially Ordered Tree: remove I  Easier to add and remove because the order is not complete.  Add:  insert at bottom rightmost  “push up” to correct position.  Remove:  “pull up” largest child of root and recurse on that subtree.  But: makes tree unbalanced! Alternative:  replace root by bottom rightmost node  “push down” to correct position (swapping)  keeps tree balanced – and complete! Bee 35 Eel 26 Kea 24 Dog 14 Cat 19 Ant 9 Hen 25 Gnu 13 Jay 1 Fox 3

10. 10 Partially Ordered Tree: remove II  Easier to add and remove because the order is not complete.  Add:  insert at bottom right  “push up” to correct position.  Remove:  “pull up” largest child and recurse.  But: makes tree unbalanced! Alternative:  replace root by bottom rightmost node  “push down” to correct position  keeps tree balanced – and complete! Eel 26 Hen 23 Kea 24 Dog 14 Cat 19 Ant 9 Fox 3 Gnu 13 Jay 1

11. 11 Partially Ordered Tree  Add: insert at bottom rightmost, swap with parent, …  Remove: replace root with bottom rightmost, swap with largest child, … But:  How do you find the bottom right?  Once you have found it, how do you find its parent to push it up? We need a tree where you can quickly get to:  the bottom right node,  children from parent,  parent from children. Bee 35 Eel 26 Cat 19 Dog 14 Fox 3 Ant 9 Hen 23 Gnu 13 Jay 1 Kea 24

12. 12 Heap:  A complete, partially ordered, binary tree complete = every level full, except bottom, where nodes are to the left  Implemented in an array using breadth-first order Bee 35 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 Cat 4 Bee 35 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 Cat 4 0137894265

13. 13 Heap  We can compute the index of parent and children of a node:  the children of node i are at (2i+1) and (2i+2)  the parent of node i is at (i-1)/2  Bottom right node is last element used.  There are no gaps! Bee 35 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 Cat 4 0137894265 Bee 35 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 Cat 4

14. 14 Heap: add Insert at bottom of tree and push up:  Put new item at end: 10  Compare with parent: (10-1)/2 = 4 ⇒ Fox/7  If larger than parent, swap  Compare with parent: (4-1)/2 = 1 ⇒ Kea/19  If larger than parent, swap  Compare with parent: (1-1)/2 = 0 ⇒ Bee/35 Bee 35 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 Cat 4 0137894265101112 10 Pig 21 11

15. 15 Heap: remove  Remove item at 0:  Move last item to 0  Find largest child 2  0+1 = 1, 2  0+2 = 2  If smaller than largest child, swap  Find largest child 2  2+1 = 5, 2  2+2 = 6  If smaller than largest child, swap  Find largest child 2  5+1 = 11 : No such child Bee 35 Eel 26 Pig 21 Dog 14 Kea 19 Ant 9 Hen 23 Gnu 13 Jay 1 Cat 4 0137894265101112 10 Fox 7 11

16. 16 HeapQueue public class HeapQueue extends AbstractQueue { private List data = new ArrayList (); private Comparator comp; public HeapQueue (Comparator c) { comp = c; } public boolean isEmpty() { return data.isEmpty(); } public int size () { return data.size(); } public E peek () { if (isEmpty()) return null; else return data.get(0); } Use ArrayList, not array, so it handles resizing. Comparator must be designed so that it compares the priority values (not the items) Note to whoever teaches this next: I think it’d be better to present this in terms of ordinary array notation, because it’s easier to see what it’s doing, and leave it to them as an exercise to reimplement it using ArrayList once they understand it. I think the coding used for pushup and pushdown (ie writing things like data.set(child, data.set(parent, data.get(child)));) is revolting and should not be inflicted up poor struggling first year students! I’ve changed it to use a method which at least makes the intention clear. Also, this code doesn’t distinguish between the values in the priority queue and their priorities. It can be made to work IF the comparator you use can extract a priority from an object, but if that is what is intended, it needs to be explained. I worked all this out too late to fix it this time round, so and adding this note to help whoever does it next time. Lindsay

17. 17 HeapQueue: offer and poll public boolean offer(E value) { if (value == null)return false; else { data.add(value); pushup(data.size()-1); return true; } public E poll() { if (isEmpty())return null; if (data.size() == 1)return data.remove(0); else { E ans = data.get(0); data.set(0, data.remove(data.size()-1)); pushdown(0); return ans; } add at the end of the array move last element into root

18. 18 HeapQueue: pushup private void pushup(int child) { if (child == 0) return; int parent = (child-1)/2; // compare with value at parent and swap if parent smaller if (comp.compare(data.get(parent), data.get(child)) < 0) { swap(data, child, parent); pushup(parent); } private void swap(List data, int from, int to) data.set(child, data.set(parent, data.get(child))); Bee 35 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 Tui 22 0137894265 recurse up the tree...

19. 19 HeapQueue: pushdown private void pushdown(int parent) { int largeCh = 2*parent+1; int otherCh = largeCh+1; // check if any children if (largeCh >= data.size()) return; // find largest child if (otherCh < data.size() && comp.compare(data.get(largeCh), data.get(otherCh)) < 0 ) largeCh = otherCh; // compare with largest child, and swap if smaller if (comp.compare(data.get(parent), data.get(largeCh)) < 0) { swap(data, largeCh, parent); pushdown(largeCh); } Cat 4 Eel 26 Kea 19 Dog 14 Fox 7 Ant 9 Hen 23 Gnu 13 Jay 1 0137894265 recurse down the tree...

20. 20 HeapQueue: Analysis  Cost of offer: = cost of pushup = O(log(n)) log(n) comparisons, 2 log(n) assignments  Cost of poll: = cost of pushdown = O(log(n)) 2 log(n) comparisons, 2 log(n) assignments  Conclusion: HeapQueue is always fast!!