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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 – Traverse (no specified order) – Create, destroy 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) 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” 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 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 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 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) 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 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) An example heap • 21 17 5 12 9 4 3 11 10 2 21 17 12 11 10 5 9 4 2 3 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 Example Deletion 2 17 12 11 17 5 9 4 10 Root replaced 12 3 11 2 5 9 4 3 10 Heap property restored 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. Example Insertion 17 17 12 11 2 9 4 10 14 5 11 3 14 New item (14) at bottom 5 2 12 4 10 3 9 Heap property restored 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) 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 Example • Initial Array: – XWZARCDMQEF • Initial heap: X W Z A M R Q E C F D 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”