CompSci 201 Tree Traversals & Heaps

Transcription

1 CompSci 201 Tree Traversals & Heaps Jeff Forbes March 28, /28/18 CompSci 201, Spring 2018, Heaps 1

2 R is for R Programming language of choice in Stats Random From Monte-Carlo to [0,1) Recursion Base case of wonderful stuff Refactoring Better not different 11/1/17 Compsci 201, Fall 2017, Tree, Tries, Tradoffs 2

3 Plan for the Week Announcements Autocomplete Mentored tutorial sessions No instructor office hours on Friday Tree Traversals Heaps for Priority Queues List & Tree Practice 3/28/18 CompSci 201, Spring 2018, Heaps 3

4 Balanced Trees In average case, trees are O(log n) for searching in a binary search tree If tree is full (see picture) or complete (last row being filled in) we get O(log n) Insertion/Deletion algorithms that "adjust" Ensure trees good in worst-case AVL tree, Red-Black Tree Similar, Red-Black in TreeMap 2-3 Trees or Trees Leaves in file system? 3/28/18 CompSci 201, Spring 2018, Heaps 4

5 Printing a search tree in order When is root printed? After left subtree, before right subtree. void visit(treenode t){ if (t!= null) { visit(t.left); System.out.println(t.info); visit(t.right); } } giraffe llama tiger elephant jaguar monkey Inorder traversal hippo leopard pig 3/28/18 CompSci 201, Spring 2018, Heaps 5

6 Tree traversals Inorder visits search tree in order Visit left-subtree, process root, visit rightsubtree elephant, giraffe, jaguar, llama, monkey, tiger Navigate following nodes/links? Visit on passing under Second time by node giraffe llama tiger elephant jaguar monkey 3/28/18 CompSci 201, Spring 2018, Heaps 6

7 Tree traversals Preorder good for reading/writing trees Process root, then Visit left-subtree, visit right-subtree llama, giraffe,elephant jaguar, tiger, monkey Navigate following nodes/links? Visit on passing-by to left First time by node giraffe llama tiger elephant jaguar monkey 3/28/18 CompSci 201, Spring 2018, Heaps 7

8 Tree traversals Post order good for deleting tree Visit left-subtree, visit right-subtree, process root elephant, jaguar, giraffe, monkey, tiger, llama Navigate following nodes/links? Visit on passing up llama giraffe tiger elephant jaguar monkey 3/28/18 CompSci 201, Spring 2018, Heaps 8

9 Writing Tree, Serializing Tree Conventional wisdom: need two traversals to uniquely identify tree 3/28/18 CompSci 201, Spring 2018, Heaps 9

10 Given Inorder and Preorder Suppose we have a search tree, we know in-order Are there more trees with these values? Are these search trees? 3/28/18 CompSci 201, Spring 2018, Heaps 10

11 Construct tree given pre-order? Given 7, 4, 1, 3, 6, 13, 8, 10, 14 Knowing it's a search tree: what's root? Left child of root? Right child of root? If we implement the algorithm developed, what will complexity be? How do we reason about this? T(n) = What are sub-problems? What work before we do subproblems? 3/28/18 CompSci 201, Spring 2018, Heaps 11

12 Label null nodes, pre-order is unique What is the tree (not a search tree) with pre-order 5, 9, x, 4, 7, x, x, x, 2, 8, x, 3, x, x, 6, 1, x, x, 10, x, x What is root? Why? What's the left-subtree, how do we "stop" scanning the traversal? Given a tree of N nodes, how many x's are there? How do reason about this? Is this traversal "wasteful" in storing nulls? 3/28/18 CompSci 201, Spring 2018, Heaps 12

13 Priority! Applications of Priority Queues Shortest Path: Google Maps to Internet Routing Event based simulation: Predicting collisions Best-first search, game-playing, AI Java code below sorts list. How? Why? PriorityQueue<Integer> pq = new PriorityQueue<Integer>(); // add all elements from list to pq for (int elem : list) pq.add(elem); for (int index = 0; pq.size() > 0; index++) // remove minimum remaining element list[index] = pq.poll(); 3/28/18 CompSci 201, Spring 2018, Heaps 13

14 How are PriorityQueues implemented? Heap is an array-based implementation of a binary tree used for implementing priority queues, supports: insert, findmin, deletemin: complexities? Using array minimizes storage (no explicit pointers), faster too --- children are located by index/position in array Heap is a binary tree with shape property, heap/value property 1. shape: tree filled at all levels (except perhaps last) and filled left-to-right (complete binary tree) 2. value: each node has value smaller than both children 3/28/18 CompSci 201, Spring 2018, Heaps 14

15 Using an array for a Heap Store node values in array beginning at index 1 Could also store starting at index 0 For node with index k left child: index 2*k right child: index 2*k+1 parent: index k/2 Why is this structure conducive for maintaining heap shape? What about value property? Is the heap a search tree? Where is minimal node? /28/18 CompSci 201, Spring 2018, Heaps 15

16 Heap questions Where is minimal element? Root, why? Where is maximal element? What is complexity of find max in a min-heap? Why? Where is second smallest element? Why? 3/28/18 CompSci 201, Spring 2018, Heaps 16

17 Adding values to heap To maintain heap shape, must add new value in left-to-right order of last level could violate heap property move value up if too small Change places with parent if heap property violated stop when parent is smaller stop when root is reached Pull parent down, swapping isn t necessary (optimization) bubble 8 up insert /28/18 CompSci 201, Spring 2018, Heaps 17

18 Heap add implementation ArrayList<Integer> list void add(arraylist<integer> list, int elt) { // add elt to heap in mylist list.add(elt); int loc = list.size(); } while (1 < loc && elt < list.get(loc / 2)) { list.set(loc, list.get(loc/2)); // go to parent loc = loc / 2; } // What s true here? list.set(loc, elt); 3/28/18 CompSci 201, Spring 2018, Heaps 18

19 Removing extremal element Where is minimal element? If we remove it, what changes, shape/property? How can we maintain shape? last element moves to root What property is violated? After moving last element, subtrees of root are heaps, why? Move root down (pull child up) does it matter where? When can we stop re-heaping? Less than both children Reach a leaf /28/18 CompSci 201, Spring 2018, Heaps 19

20 Priority Queue implementations Implementing priority queues: average and worst case Insert Getmin Insert Getmin average (peek) worst (delete) Unsorted ArrayList O(1) O(n) O(1) O(n) Sorted ArrayList O(n) O(1) O(n) O(1)/ O(n) Heap O(log n) O(1) O(log n) O(log n) Balanced binary search tree O(log n) O(log n)/ O(1) O(log n) O(log n) Heap has O(1) find-min (no delete) and O(n) build heap 3/28/18 CompSci 201, Spring 2018, Heaps 20

21 Problem 1 1. Keep track of top M of N elements (N >> M). Big Oh if PriorityQueue implemented with Heap? 3/28/18 CompSci 201, Spring 2018, Heaps 21

22 Problem 2 2. Determine the number of elements < target in heap lesscount(heap,13) 3 lesscount(heap,8) 1 O(k) where k is #of elements less than target What is you had helper method that returns # elements in subheap rooted/starting at index that are less than target? int lesscount(int[] heap, int index, int target) What s the initial call to helper method? 3/28/18 CompSci 201, Spring 2018, Heaps 22

23 New topics for Test 2 Algorithm analysis Analyzing recursive algorithms with recurrences Analyzing algorithms in terms of multiple variables (e.g., for topkmatches in Autocomplete) Data structures Union-find Stacks, Queues, & Priority Queues Linked Structures Linked lists Binary Trees BSTs Tries Recursion Identifying base & recursive cases Grid recursion Recursive structures Sorting Comparators & Comparable Percolation & Autocomplete Everything in lecture, discussion, and assignments Muddy topics? 3/28/18 CompSci 201, Spring 2018, Heaps 23