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ADT Table and Heap

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

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ADT Table and Heap

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  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 17 5 12 4 3 9 11 10 2

  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 2 17 17 5 12 5 12 4 3 9 11 4 3 9 11 10 2 10 Root replaced 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. Example Insertion 17 17 12 14 5 5 11 12 4 3 11 4 3 9 14 2 9 2 10 10 New item (14) at bottom Heap property restored

  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 W Z A R C D M Q E F

  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”

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