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Midterm test on March 10, 2016

Midterm test on March 10, 2016. We will have a 2 hour midterm test on March 10 2016 during the lecture time. CS 2468: Assignment 2 Due Week 9, Thursday (March 17) Drop a hard copy in Mail Box 75 or hand in during the lecture.

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Midterm test on March 10, 2016

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  1. Midterm test on March 10, 2016 We will have a 2 hour midterm test on March 10 2016 during the lecture time. Trees

  2. CS 2468: Assignment 2 Due Week 9, Thursday (March 17) Drop a hard copy in Mail Box 75 or hand in during the lecture • Design an algorithm which takes a root r of a binary tree (class Bnode) as input and tests if the binary tree rooted at r is a complete binary tree. • Implement your algorithm in Java • Test your Java program using examples of the following cases: • A case, where root r is NULL • A case, where root r has no children • A case, where there are 8 nodes in the tree which is a complete binary tree • A case, where there are 8 nodes in the tree which is not a complete binary tree Trees

  3. Complete Binary Trees A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible. 1 2 3 4 5 6 A binary tree of height 3 has the first p leaves at the bottom for p=1, 2, …, 8=23. Priority Queues

  4. A … B D C E F J G H Array-Based Representation of Binary Trees • nodes are stored in an array 1 2 3 • let rank(node) be defined as follows: • rank(root) = 1 • if node is the left child of parent(node), rank(node) = 2*rank(parent(node)) • if node is the right child of parent(node), rank(node) = 2*rank(parent(node))+1 4 5 6 7 10 11 Trees

  5. Array-Based Representations • How many cells can be wasted with array based representation? Trees

  6. D C A B E Linked Structure for Binary Trees • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT    B A D     C E Trees

  7. D C B A E Linked Structure for Binary Trees • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT    B A D     D C     F G Trees G F

  8. Full Binary Tree • A full binary tree: • All the leaves are at the bottom level • All nodes which are not at the bottom level have two children. • A full binary tree of height h has 2h leaves and 2h-1 internal nodes. 1 3 2 4 6 7 5 This is not a full binary tree. A full binary tree of height 2 Trees

  9. Part-D1Priority Queues Priority Queues

  10. A priority queue stores a collection of entries Each entry is a pair(key, value) Main methods of the Priority Queue ADT insert(k, x)inserts an entry with key k and value x removeMin()removes and returns the entry with smallest key Additional methods min()returns, but does not remove, an entry with smallest key size(), isEmpty() Applications: Standby flyers Auctions Stock market Priority Queue ADT Priority Queues

  11. An entry in a priority queue is simply a key-value pair Priority queues store entries to allow for efficient insertion and removal based on keys Methods: key(): returns the key for this entry value(): returns the value associated with this entry As a Java interface: /** * Interface for a key-value * pair entry **/ public interface Entry { public Object key(); public Object value(); } Entry ADT Priority Queues

  12. A comparator encapsulates the action of comparing two objects according to a given total order relation A generic priority queue uses an auxiliary comparator The comparator is external to the keys being compared When the priority queue needs to compare two keys, it uses its comparator The primary method of the Comparator ADT: compare(a, b): Returns an integer i such that i < 0 if a < b, i = 0 if a = b, and i > 0 if a > b; an error occurs if a and b cannot be compared. Comparator ADT Priority Queues

  13. Priority Queue Sorting • We can use a priority queue to sort a set of comparable elements • Insert the elements one by one with a series of insert operations • Remove the elements in sorted order with a series of removeMin operations • The running time of this sorting method depends on the priority queue implementation AlgorithmPQ-Sort(S, C) Inputsequence S, comparator C for the elements of S Outputsequence S sorted in increasing order according to C P priority queue with comparator C whileS.isEmpty () e S.removeFirst () P.insert (e, 0) whileP.isEmpty() e P.removeMin().key() S.insertLast(e) Priority Queues

  14. Implementation with an unsorted list Performance: insert takes O(1) time since we can insert the item at the beginning or end of the sequence removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key Implementation with a sorted list Performance: insert takes O(n) time since we have to find the place where to insert the item removeMin and min take O(1) time, since the smallest key is at the beginning 4 5 2 3 1 1 2 3 4 5 Sequence-based Priority Queue Priority Queues

  15. Selection-Sort • Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted list • Running time of Selection-sort: • Inserting the elements into the priority queue with ninsert operations takes O(n) time • Removing the elements in sorted order from the priority queue with nremoveMin operations takes time proportional to1 + 2 + …+ n-1 • Selection-sort runs in O(n2) time Priority Queues

  16. Selection-Sort Example Sequence S Priority Queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (7,4) .. .. .. . . . (g) () (7,4,8,2,5,3,9) Phase 2 (a) (2) (7,4,8,5,3,9) (b) (2,3) (7,4,8,5,9) (c) (2,3,4) (7,8,5,9) (d) (2,3,4,5) (7,8,9) (e) (2,3,4,5,7) (8,9) (f) (2,3,4,5,7,8) (9) (g) (2,3,4,5,7,8,9) () Running time: 1+2+3+…n-1=O(n2). No advantage is shown by using unsorted list. Priority Queues

  17. Complete Binary Trees A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible. 1 2 3 4 5 6 A binary tree of height 3 has the first p leaves at the bottom for p=1, 2, …, 8=23. Priority Queues

  18. Complete Binary Trees Once the number of nodes in the complete binary tree is fixed, the tree is fixed. For example, a complete binary tree of 15 node is shown in the slide. A CBT of 14 nodes is the one without node 8. A CBT of 13 node is the one without nodes 7 and 8. 1 2 3 4 5 6 7 8 A binary tree of height 3 has the first p leaves at the bottom for p=1, 2, …, 8=23. Priority Queues

  19. A heap is a binary tree storing keys at its nodes and satisfying the following properties: Heap-Order: for every node v other than the root,key(v)key(parent(v)) Complete Binary Tree: let h be the height of the heap for i = 0, … , h - 1, there are 2i nodes of depth i at depth h- 1, the internal nodes are to the left of the external nodes The last node of a heap is the rightmost node of depth h Root has the smallest key Heaps 2 5 6 9 7 last node A full binary without the last few nodes at the bottom on the right. Priority Queues

  20. Height of a Heap • Theorem: A heap storing nkeys has height O(log n) Proof: (we apply the complete binary tree property) • Let h be the height of a heap storing n keys • Since there are 2i keys at depth i=0, … , h - 1 and at least one key at depth h, we have n1 + 2 + 4 + … + 2h-1 + 1=2h. • Thus, n2h, i.e., hlog n depth keys 0 1 1 2 h-1 2h-1 h 1 Priority Queues

  21. Heaps and Priority Queues • We can use a heap to implement a priority queue • We store a (key, element) item at each internal node • We keep track of the position of the last node • For simplicity, we show only the keys in the pictures (2, Sue) (5, Pat) (6, Mark) (9, Jeff) (7, Anna) Priority Queues

  22. Method insertItem of the priority queue ADT corresponds to the insertion of a key k to the heap The insertion algorithm consists of three steps Create the node z (the new last node). Store k at z Put z as the last node in the complete binary tree. Restore the heap-order property, i.e., upheap (discussed next) Insertion into a Heap 2 5 6 z 9 7 insertion node 2 5 6 z 9 7 1 Priority Queues

  23. Upheap • After the insertion of a new key k, the heap-order property may be violated • Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node • Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k • Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6 Priority Queues

  24. An example of upheap 1 2 2 3 4 5 6 7 2 3 9 10 11 14 15 16 17 2 7 Different entries may have the same key. Thus, a key may appear more than once. Priority Queues

  25. Removal from a Heap • Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap • The removal algorithm consists of three steps • Replace the root key with the key of the last node w • Remove w • Restore the heap-order property .e., downheap(discussed next) 2 5 6 w 9 7 last node 7 5 6 w 9 new last node Priority Queues

  26. Downheap • After replacing the root key with the key k of the last node, the heap-order property may be violated • Algorithm downheap restores the heap-order property by swapping key k along a downward path (always use the child with smaller key) from the root • Downheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k • Since a heap has height O(log n), downheap runs in O(log n) time 7 5 5 6 7 6 9 9 Priority Queues

  27. An example of downheap 2 17 17 4 2 3 17 4 5 9 6 7 17 9 10 11 14 15 16 Priority Queues

  28. A priority queue stores a collection of entries Each entry is a pair(key, value) Main methods of the Priority Queue ADT insert(k, x) inserts an entry with key k and value x O(log n) removeMin()removes and returns the entry with smallest key O(log n) Additional methods min()returns, but does not remove, an entry with smallest key O(1) size(), isEmpty() O(1) Running time of Size(): when constructing the heap, we keep the size in a variable. When inserting or removing a node, we update the value of the variable in O(1) time. isEmpty(): takes O(1) time using size(). (if size==0 then …) Priority Queue ADT using a heap Priority Queues

  29. Consider a priority queue with n items implemented by means of a heap the space used is O(n) methods insert and removeMin take O(log n) time methods size, isEmpty, and min take time O(1) time Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time The resulting algorithm is called heap-sort Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort Heap-Sort Priority Queues

  30. 2 5 6 9 7 0 1 2 3 4 5 Array-based Heap Implementation • We can represent a heap with n keys by means of an array of length n +1 • For the node at rank i • the left child is at rank 2i • the right child is at rank 2i +1 • Links between nodes are not explicitly stored • The cell of at rank 0 is not used • Operation insert corresponds to inserting at rank n +1 • Operation removeMin corresponds to removing at rank n • Yields in-place heap-sort • The parent of node at rank i is i/2 2 5 6 9 7 Priority Queues

  31. Build a heap from an array of n elements Priority Queues

  32. Build a heap from an array of n elements Priority Queues

  33. Build a heap from an array of n elements Priority Queues

  34. Build a heap from an array of n elements Priority Queues

  35. Build a heap from an array of n elements The total running for BUILD-MAX-HEAP(A) is O(n), where A has n elements. Priority Queues

  36. Build a heap from an array of n elements Priority Queues

  37. Build a heap from an array of n elements Priority Queues

  38. Build a heap from an array of n elements Priority Queues

  39. Analysis • We visualize the worst-case time of a downheap with a proxy path that goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path) • Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n) • Thus, bottom-up heap construction runs in O(n) time • Bottom-up heap construction is faster than n successive insertions and speeds up the first phase of heap-sort Priority Queues

  40. Instead of removeMin(), we can do removeMax() If we always want to choose the one with maximum key value, we can have a max heap (instead of Min heap) Just change the definition of heap order and everything is smilar. Max heap vs Min heap Priority Queues

  41. Adaptable heap has one more operation: replace(location: of x, key) Algorithm: k=x.key X.key=key If (k>key) then upheap If (k<key) then downheap Adaptable heap Priority Queues

  42. 4 11 6 12 5 13 3 14 7 15 1 16 2 17 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 11/6 12/7 13/4 14/1 15/3 16/2 17/5 How to know the location of a heap : Array-based rep of the complete binary tree T[] Key value Rank of the node location id-name of a node Binary Search Trees

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