Binary Heaps

Heapsort in phases

1. Rearrange the array elements into max heap order 2. Repeatedly move the maximum element to its final sorted place toward the end of the array, and heapify the remaining elements

VLC from Huffman's algorithm

With prefix property such that there is no other encoding with a smaller expected code length; left branch - 0; right branch - 1;

Priority Queue

chooses the next element to delete based on priority

Binary Heap Deletion

delete and return the element with the extreme (max/min) priority; maintain the complete shape by replacing the root value with the value in the lowest, right-most leaf, then delete that leaf; swap values as necessary on a root-to-leaf path to maintain the partial order

Binary Heap Insertion

insert the new element in the one and only one location that will maintain the complete shape; swap values as necessary on a leaf-to-root path to maintain the partial order

Priority

some value associated with the element that could represent importance, cost, or some other problem-specific concept

Text compression

stores the same information in fewer bytes

Expected Code Length

want code tree with minimum; weighted average of all possible character code lengths

Transform into total order

Repeatedly move the maximum element to its final sorted place toward the end of the array, and heapify the remaining elements

Way we map a hierarch onto a linear array

Store the root at index 0; For a node stored at index i: Left child: 2i +1 Right child: 2i + 2 Parent: (i-1)/2

Extreme priority

The element with the most is returned by the remove operation (whether max or min depends on queue set up)

Implementing a priority queue

Binary heap-- usually array; Add- O(logN), remove- O(logN),peek- O(1); things are lined up in priority order vs time order; heap implies array

Code trees

Binary trees in which the leaves contain the characters to be coded, interior nodes are just place-holders; root of every subtree is annotated with the cumulative frequency of all its descendent leaves

Binary heap

Complete binary tree; obeys a partial order property (max or min heap)

Character codes from code trees

Generated by root to leaf traversals-- no code is the prefix to another because the path for each letter is different

Huffman's algorithm code tree

Generated code tree with an expected code length that is at least as small as any other code tree that could be generated

Huffman's algorithm

Generates a variable-length encoding for a given alphabet of the purposes of data compression (rather then using defaults)

Heapsort

In-place comparison sort with O(N log N) time complexity; Works in two phases: the initial arbitrary order of the array is transformed into a partial order, then the partial order is transformed into a total order

Generating a variable-length code

Must have a prefix property- code for one character can't be a prefix of another character's code

Variable-length codes

Number of bits per character determined by the char's relative frequency of occurrence (determined by first pass); Most frequent characters use fewest bits

Array-based implementation

Acceptable space efficient, easy traversal (parent and children determined from multiplication and division);

ASCII

American Standard Code for Information Interchange; includes English alphabet, punctuation, digits, and "control" characters; fixed length code

Arrange into max heap order

Beginning with the lowest, right-most parent and continuing to the root, heapify each subtree