GVSU CIS 263

Week 7 / Day 1

Leftovers:

• Red black tree
• iterator invalidation

Heaps

Max-Heap:

• Each node is greater than its children.
• Usually a binary tree, but not necessarily.
• Insert new node at leaf and “bubble up”
• Remove: Move last leaf up and bubble down with greatest child.
  • Why greatest child?
• Can be used with a FIFO cache
• Min-Heap: Same thing with minimum at top.

Heap using array:

• Heap can implemented using an array.
• For nodes 0 - 4, draw table showing index of children
• What is the formula?
• Given a node x, what is formula for parent?
  • floor((x-1)/2)
• What is the benefit to this implementation?
  • Less metadata (e.g., pointers)
• Can we implement a BST this way? Why or why not?
  • Rotations wouldn’t be as simple as moving pointers.
• How can we use a Heap to sort?
  • Insert items into a heap, then pull them out.
• What would the running time be?
  • O(n log n). There would be n insertions, each taking time log n, followed by n removals, each taking time log n.
• Any short-cuts?
  • Just take the array and pretend it’s a max heap. The leaf nodes are, by definition, fine. Only need n/2 operations to make a valid heap.
• Notice that this is n log n, but not recursive.

Priority Queue

• Put things in order such that the “most important” stays at the front.
• Typically implemented with a heap.
• What would be running time using a “regular” vector or list?
• What is the running time using a heap?

Use of priority queues in discrete event simulation

d-Heap:

• Heaps don’t have to be binary. Suppose each node has d children.
• What is the advantage of increasing d?
  • Inserts are faster because the tree is shallower.
• What is the disadvantage?
  • Deletes are slower because you have to look for the maximum of all d children.
  • Unless d is a power of 2, the multiplication/division can be expensive.
• There is evidence that having 4 children improves performance overall.

Treap:

• BST with random secondary key.
• Add item to BST.
• Use secondaray key to percolate up; but, rotate instead of copy. You end up with a valid BST.