Heaps
๐ for Heap definition
๐refresher on Binary Trees
Binary Trees
 Trees where each node only has up to 2 children
 "Complete" tree = every level filled; last level filled from left to right
 "Full" tree = No node has 1 child.
 "Perfect" = Full + Complete
Binary Search Trees
 All left descendents <= node value < all right descendents
Did you know
 Heaps are used to execute Javascript!
 Async calls are inserted on to the heap and popped off based on priority
 Read about JS Memory Leaks
 and profiling the JS Heap
 React 16's performance also derives from moving from a stack to a pqueue
React 16
React 15
JS Heap in Chrome
Heap Definition
 Complete Binary Trees
 A type of priority queue
 Satisfies the heap property, either a MinHeap or MaxHeap.
 In a MaxHeap, if P is a parent node of C
 then P >= C
 compare vs BSTs:
 No idea if C1 < C2
 Top node is max/min value
 There are nonbinary heaps
๐ for Heap methods
๐ vs other Data Structures
BSTs
 Insert: O(lg N) avg
 Contains: O (lg N) avg
 Remove: O(lg N) avg
 getMin: O(lg N)
 in/post/preordered traversal
Heaps
 Insert: O(lg N) avg
 Contains: O (lg N) avg
 Remove: O(lg N) avg
 getMin: O(1)
 ordered traversal not possible
PQueues (Linked List)
ย
 Insert: O(N)
 Contains: O(N) avg
 Remove: O(1) avg
 getMin: O(1)
 basically insertion sort
Heaps
 Insert: O(lg N) avg
 Contains: O (lg N) avg
 Remove: O(lg N) avg
 getMin: O(1)
What else?
I can't do the full comparisons justice
Key Methods
(and their Complexity)
 peek: O(1)
 remove: O(lg N)
 insert: O(lg N)
 Internal Methods:
 Swap(i, j): O(1)
 SiftDown(i): O(lg N)  used by remove
 SiftUp(i): O(lg N)  used by insert
๐ Practice Time!
๐ pseudocode for methods
SiftUp
Example:
 Given a valid MaxHeap, insert 79 at the end (of the array/heap)
 Compare to 79's parent
 Swap up if 79 > 79's parent
 repeat until 79 < parent
 You now have a new, valid MaxHeap with 79 in it
SiftDown
Example:
 Given a valid MaxHeap, remove the max (91) and pull the last node (24) into the root
 Now the Heap is invalid, 24 is less than 76 or 85
 Swap down 24 with the bigger of the two children (85, then 53)
 Repeat until you have a valid MaxHeap again, with 91 popped off.
Practice!
It's tricky the first time
๐ Why Use A Heap
Review: Why Use A Heap
 O(1) Retrieval of Max or Min
 O(1) Space complexity (same as Trees)
 O(lg N) Insertion/Removal
Used in these Algorithms:
 Heap Sort (great big O properties)
 no O(n^2) worst case eg QuickSort
 O(1) space compared to MergeSort O(n)
 Continuous Median
 (if sparse) Dijkstra's Algorithm
๐ for End of Slides/More Resources
๐Heap Sort and Continuous Median
Heap Sort
Given an unsorted array, produce a sorted array using heap sort
ย
First try doing it on top of a complete Heap Sort implementation like here:
https://repl.it/@swyx/HeapsBasicSolution
Then practice writing from scratch
Solution
 Subdivide into unsorted (A) and an empty (B)
 Put A into a MaxHeap
 While A is not empty, pop off the root of A and unshift it into B
 B is the sorted array
ย
Notes
 Sorting done inplace
 Video
Continuous Median
Write a class that can support the following two functionalities: 1) the continuous insertion of numbers and 2) the instant (O(1) time) retrieval of the median of the numbers that have been inserted thus far.
Solution
 Class has a maxHeap, a minHeap, and a median

insert by comparing with max of minHeap or min of maxHeap
 then rebalance
 then update median

rebalance by comparing lengths
 e.g. minHeap > maxHeap
 x = minHeap.remove()
 maxHeap.insert(x)

update median
 if equal length, get average
 if unbalanced, peek longer
Dijkstra's Algo
 Find the shortest path between a and b.
 It picks the unvisited vertex with the lowest distance
 calculates the distance through it to each unvisited neighbor
 updates the neighbor's distance if smaller.
 Use a pQ to keep track of vertex with lowest distance
 LList if not sparse
 Heap if E << Vยฒ / logV
 Especially Fibonacci Heap
 see wiki for pseudocode
Other Notes
 Here

DecreaseKey is tricky
 either write it
 or keep a hash table
More Resources
 Priority Queue Reacto (all Reactos)
 algoexpert.io (code: 'impostor')
 More Heap Presentations
 Gayle McDowell (CTCI Author) on Heaps
 Heap Variants on Wikipedia
 Related DSes
 RedBlack Trees (self balancing BSTs)
Heaps  The Data Structure and How They Are Used
By Shawn Swyx Wang
Heaps  The Data Structure and How They Are Used
Heaps  The Data Structure and How They Are Used
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