1. ADT Table and Heap Ellen Walker CPSC 201 Data Structures Hiram College

2. ADT Table Represents a table of searchable items with at least one key (index) –E.g. dictionary, thesaurus, phone book Value-based operations –Insert –Delete –Retrieve Structural operations –Traverse (no specified order) –Create, destroy

3. Implementing a Table Linear data structures –Unsorted array –Sorted array –Unsorted linked list –Sorted linked list Tree –Binary search tree Other –Hash table (ch. 12)

4. Issues to consider Different structures are better for different operations –Sorted arrays and trees can be searched fastest –Linked lists and trees are easier to insert and delete –Sorted structures take longer for insertion than non-sorted structures –Consider simplicity of implementation, especially if the table is “small”

5. Restrictions Constrain Implementations If we limit traversal to traversal in sorted order, we cannot use unsorted arrays or linked lists. General rule: define the least restrictive set of operations that will satisfy needs; then choose an appropriate data structure Example: heap vs. tree for priority queue

6. Priority Queue Every item has a (numeric) priority Multiple items can have the same priority When dequeuing, the oldest item with the highest priority should be retrieved first –If all items of equal priority, then it is a queue –If all items different priority, it acts somewhat like a sorted list

7. Sorted Structure for Priority Queue Items are kept sorted in reverse order (largest first) To enqueue: insert item in place according to priority To dequeue: remove first (highest) item –First item in reverse-sorted list or array –Last item in sorted list or array –Rightmost item in binary search tree

8. Heap: A new structure A heap is a full binary tree The root of the heap is larger than any node in either the left or right subtree The left and right subtrees are both themselves heaps An empty tree is a heap (base case)

9. Heap is less restrictive Given a set of values, there are more possible heaps than there are binary search trees for the same set of values (why?) When inserting an item into a heap, we don’t have to find its exact location in the sort order We do have to make sure the heap property holds for the tree and all its subtrees We only have to worry about deleting the root

10. Heap in Array Because a heap is a full binary tree, it represents very well in an array Root is heap[0] Children of heap[k] are –heap[2*k+1] –heap[2*k+2] Values are packed into the array (no holes)

11. An example heap 21 17 5 12 9 4 3 11 10 2 21 175 12 9 43 11102

12. Deleting Remove the root (now you have a semiheap) Replace the root by the last (bottom, rightmost) element Swap root with largest child recursively until root is largest. –New root item “trickles down” a path until it finds its correct (sorted in the path) location

13. Example Deletion 175 12 9 43 1110 2 Root replaced 17 5 12 9 4311 10 2 Heap property restored

14. Inserting into a Heap Put new item at first available position at deepest level (last element in the array) If it is larger than its parent, swap them Continue swapping up the tree until the parent is larger or the new item has become the root.

15. 9 Example Insertion New item (14) at bottom 17 5 12 9 4311 10 2 Heap property restored 17 5 12 4311 10 2 14

16. Efficiency of Heap Adding –Element starts at the bottom, takes a single path from the bottom to the root (at most) –This is O(log N) because the tree is balanced (every path is <= log N + 1) Removing –Element starts at the root, takes a single path down the tree –Again, O(log N)

17. Heap Sort Begin with an array (in arbitrary order) Make the array into a valid heap –Starting at the next-to-bottom level, rearrange “triples” so the local root is largest –Essentially, this works backwards in the array While(heap is not empty) –Delete the root (and swap it with the last leaf) Since the root is largest (each time), in the end, the array will be sorted

18. Example Initial Array: –X W Z A R C D M Q E F Initial heap: X A ZW RCD MQEF

19. MaxHeap & MinHeap We’ve been looking at MaxHeap –Largest item at root –Highest number is highest priority Textbook describes MinHeap –Smallest item at root –Lowest number is highest priority The only difference in algorithm is “max” vs. “min”