PPT-Binary Heaps

COL 106 Shweta Agrawal and Amit Kumar 2 Revisiting FindMin Application Find the smallest or highest priority item quickly Operating system needs to schedule jobs according to priority instead of FIFO. ii The height or depth of a binary tree is the maxi mum depth of any node or 1 if the tree is empty Any binary tree can have at most 2 nodes at depth Easy proof by induction EFINITION A complete binary tree of height is a binary tree which contain Data . Structures. Self-Adjusting. Data . Structures. 2. Lists. [D.D. . Sleator. , R.E. . Tarjan. , . Amortized Efficiency of List Update Rules. , Proc. 16. th. Annual ACM Symposium on Theory of Computing, 488-492, 1984]. Heaps Is this a heap? Why or why not? Is this a heap? Why or why not? Heap OperationsAs mentioned earlier, there are two basic operations to be Inserting an item into the heapFinding and removing th Numbering of a tree’s nodes for storage in a arrayIf we use the number of a node as an array Index, this technique gives us an orderin which we can store tree nodes In an array. The tree may be e Data . Structures. Self-Adjusting. Data . Structures. 2. Lists. [D.D. . Sleator. , R.E. . Tarjan. , . Amortized Efficiency of List Update Rules. , Proc. 16. th. Annual ACM Symposium on Theory of Computing, 488-492, 1984]. Data . Structures. Self-Adjusting. Data . Structures. 2. Lists. [D.D. . Sleator. , R.E. . Tarjan. , . Amortized Efficiency of List Update Rules. , Proc. 16. th. Annual ACM Symposium on Theory of Computing, 488-492, 1984]. A heap is a binary tree.. A heap is best implemented in sequential representation (using an array).. Two important uses of heaps are: . (. i. ) efficient implementation of priority queues. (ii) sorting -- . Data . Structures. Self-Adjusting. Data . Structures. 2. Lists. [D.D. . Sleator. , R.E. . Tarjan. , . Amortized Efficiency of List Update Rules. , Proc. 16. th. Annual ACM Symposium on Theory of Computing, 488-492, 1984]. Reference. : Chapter . 2, Algorithms,4rd . Edition, Robert . Sedgewick. , . Kevin Wayne . . Outline. Priority Queue. Binary Heaps. Implementation and demo. HeapSort. 7/11/17. Prioriry Queue. 2. Example 1: Scheduling. A binary min-heap allows the operations of push and pop to occur . in an average case of . Q. (1). and . Q. (. ln. (. n. )). time, respectively. Merging two binary min-heaps, however, is an . Q. (. Prof. . Neary. Based on slides from previous iterations of this course. Today’s Topics. Review of Min Heaps. Introduction of Left-. ist. Heaps. Merge Operation. Heap Operations. Review of Heaps. Min Binary Heap. A binary . heap. . is a binary tree (2 or fewer subtrees for each node). A . heap is structured so that the node with the most relevant data is the root node, the next most relevant as the children of the root, etc. Cynthia Lee. CS106X. Topics:. Priority Queue. Linked List implementation. Heap . data . structure implementation. TODAY’S TOPICS . NOT. ON THE MIDTERM. 2. Some priority queue implementation options. Definition of a . d. -ary min heap . Implementation as a complete tree. Examples of binary, ternary, quaternary, and quinary min heaps. Properties. Relative speeds. Optimal choice is a quaternary heap.