B-Tree Insertions, Intro to Heaps CSE 373 SP 18 - Kasey Champion B-Tree Insertions, Intro to Heaps Data Structures and Algorithms

Warm Up What operations would occur in what order if a call of get(24) was called on this b-tree? What is the M for this tree? What is the L? If Binary Search is used to find which child to follow from an internal node, what is the runtime for this get operation? 12 6 20 27 34 1 2 3 6 4 8 5 9 10 7 12 8 14 9 16 10 17 11 20 12 22 13 24 14 27 15 28 16 32 17 34 18 38 19 39 20 41 21 CSE 373 SP 18 - Kasey Champion

Administrivia 1. Midterm grades will be published by Friday 2. HW #4 is due Friday 3. 1 partner must fill out partner form by Friday 4. HW Grade Review Requests coming next week CSE 373 SP 18 - Kasey Champion

Review: B-Trees Has 3 invariants that define it 1. B-trees must have two different types of nodes: internal nodes and leaf nodes An internal node contains M pointers to children and M – 1 sorted keys. M must be greater than 2 Leaf Node contains L key-value pairs, sorted by key. 2. B-trees order invariant For any given key k, all subtrees to the left may only contain keys that satisfy x < k All subtrees to the right may only contain keys x that satisfy k >= x 3. B-trees structure invariant If n<= L, the root is a leaf If n >= L, root node must be an internal node containing 2 to M children All nodes must be at least half-full CSE 373 SP 18 - Kasey Champion

Put() for B-Trees Build a new b-tree where M = 3 and L = 3. Insert (3,1), (18,2), (14,3), (30,4) where (k,v) When n <= L b-tree root is a leaf node No space for (30,4) ->split the node Create two new leafs that each hold ½ the values and create a new internal node 3 1 18 2 wrong -> 14 3 18 <- use smallest value in larger subset as sign post 2. B-trees order invariant For any given key k, all subtrees to the left may only contain keys that satisfy x < k All subtrees to the right may only contain keys x that satisfy k >= x 3 1 14 18 2 30 4 CSE 373 SP 18 - Kasey Champion

You try! Try inserting (32, 5) and (36, 6) into the following tree 18 14 18 2 30 4 32 5 36 6 32 5 CSE 373 SP 18 - Kasey Champion

Splitting internal nodes Try inserting (15, 7) and (16, 8) into our existing tree 18 32 Make a new internal node! 3 1 14 15 7 16 8 18 2 30 4 32 5 36 6 15 7 18 Make a new internal node! 15 32 3 1 14 15 7 16 8 18 2 30 4 32 5 36 6 CSE 373 SP 18 - Kasey Champion

B-tree Run Time Time to find correct leaf Time to insert into leaf Time to split leaf Time to split leaf’s parent internal node Number of internal nodes we might have to split All up worst case runtime: Height = logm(n)log2(m) = tree traversal time Θ(L) Θ(L) Θ(M) Θ(logm(n)) Θ(L + Mlogm(n)) CSE 373 SP 18 - Kasey Champion

New Topic: Heaps CSE 373 SP 18 - Kasey Champion

Priority Queue ADT Imagine you have a collection of data from which you will always ask for the extreme value Min Priority Queue ADT removeMin() – returns the element with the smallest priority, removes it from the collection state behavior Set of comparable values - Ordered based on “priority” peekMin() – find, but do not remove the element with the smallest priority insert(value) – add a new element to the collection Max Priority Queue ADT removeMax() – returns the element with the largest priority, removes it from the collection state behavior Set of comparable values - Ordered based on “priority” peekMax() – find, but do not remove the element with the largest priority insert(value) – add a new element to the collection CSE 373 SP 18 - Kasey Champion

Implementing Priority Queue Idea Description removeMin() runtime peekMin() runtime insert() runtime Unsorted ArrayList Linear collection of values, stored in an Array, in order of insertion O(n) O(1) Unsorted LinkedList Linear collection of values, stored in Nodes, in order of insertion Sorted ArrayList Linear collection of values, stored in an Array, priority order maintained as items are added Sorted Linked List Linear collection of values, stored in Nodes, priority order maintained as items are added Binary Search Tree Hierarchical collection of values, stored in Nodes, priority order maintained as items are added AVL tree Balanced hierarchical collection of values, stored in Nodes, priority order maintained as items are added O(logn) CSE 373 SP 18 - Kasey Champion

Let’s start with an AVL tree AVLPriorityQueue<E> removeMin() – traverse through tree all the way to the left, remove node, rebalance if necessary state behavior overallRoot peekMin() – traverse through tree all the way to the left insert() – traverse through tree, insert node in open space, rebalance as necessary What is the worst case for peekMin()? What is the best case for peekMin()? Can we do something to guarantee best case for these two operations? O(logn) O(1) CSE 373 SP 18 - Kasey Champion

Binary Heap A type of tree with new set of invariants 1. Binary Tree: every node has at most 2 children 2. Heap: every node is smaller than its child 3. Structure: Each level is “complete” meaning it has no “gaps” - Heaps are filled up left to right 1 8 9 10 2 4 5 3 6 7 1 2 3 22 4 5 36 47 8 9 10 CSE 373 SP 18 - Kasey Champion

Self Check - Are these valid heaps? Binary Heap Invariants: Binary Tree Heap Complete Self Check - Are these valid heaps? INVALID INVALID VALID 1 4 5 9 8 6 7 4 3 2 5 7 3 7 8 6 9 11 10 CSE 373 SP 18 - Kasey Champion

Implementing peekMin() 2 4 7 5 8 10 9 11 13 CSE 373 SP 18 - Kasey Champion

Implementing removeMin() Removing overallRoot creates a gap Replacing with one of its children causes lots of gaps What node can we replace with overallRoot that wont cause any gaps? 2 13 4 7 4 7 5 8 10 9 5 8 10 9 11 13 11 Structure maintained, heap broken CSE 373 SP 18 - Kasey Champion

Fixing Heap – percolate down Recursively swap parent with smallest child 13 4 4 13 5 7 5 13 11 8 10 9 11 13 percolateDoen(node) { while (node.data is bigger than its children) { swap data with smaller child } CSE 373 SP 18 - Kasey Champion

Self Check – removeMin() on this tree 5 18 9 10 9 18 11 17 14 11 18 13 20 19 16 15 24 22 18 CSE 373 SP 18 - Kasey Champion

Implementing insert() Insert a node to ensure no gaps Fix heap invariant percolate UP 2 4 7 3 10 5 8 4 9 3 11 13 3 7 CSE 373 SP 18 - Kasey Champion