1. Data Structures –An Introduction Comp Sci Club 12 June 2014

2. After CS III AP… • First of all – Congratulations on finishing CS III • There is much more to Comp Sci • (A lot more than I thought when I took the AP Exam!)

3. Overview (for Today’s Meeting) • Data Structures – some background • Binary Trees – BSTs, methods(), uses, etc. • *Types of Binary Trees • *Heaps, *HeapSort • Conclusion

4. Data Structures • Simple, efficient ways of organizing data • (Goes far Beyond the ‘Array’ and ‘ArrayList’ concepts taught in CS III!)

5. Data Structures • Abstract Data Type – more generalized forms of these structures (e.g. Queue, Stack, LinkedList, etc.) • Data Structure – how programmers/software developers implement these data types (ArrayQueue, LinkedStack/ArrayStack, Singly/DoublyLinkedList) • However… • There are some concepts which are more ambiguous, and can be considered part-“data type” and part-“data structure” • And, we will see a few of these in this presentation  http://cs.lmu.edu/~ray/notes/dtds/

6. The Binary (Search) Tree • Organization ofTreeNodeobjects: each ‘level’ of this structure has up to 2N Nodes. • Each Node has a reference to value; left, and right • Several different kinds of binary trees; a very common one is the Binary Search Tree • All Nodes left of root have VALUES less than the root; All Nodes right of root have VALUES greater than the root. • This principle applies to every Node, in fact.

7. The Binary (Search) Tree • http://www.cs.cmu.edu/~adamchik/15-121/lectures/Trees/trees.html • BST represents hierarchy; allows very efficient searching or inserting; and can be manipulated easily Root Node → J F B H M T Y Leaves

8. Binary Trees – Specific Types* • Full Binary Tree – every node has zero or two ‘children’ • Perfect Binary Tree – all leaf nodes lie on same level • Complete Binary Tree – every level is filled completely, except perhaps the last level (more detail later) http://en.wikipedia.org/wiki/Binary_tree http://www.cs.cmu.edu/~adamchik/15-121/lectures/Trees/trees.html http://www.scribd.com/doc/17116770/Types-of-Binary-Tree

9. Infinite Binary Trees – A Digression • Stern-Brocot Tree – Mathematical model for the set of all positive rational numbers (known since the 1850s!) • Wikipedia: the cardinalities of these numbers (the nodes themselves = א0; ways of permuting them = c) • There are numerous interesting facts about this model – see cut-the-knort.org and other sites for more info http://en.wikipedia.org/wiki/Binary_tree http://www.cut-the-knot.org/blue/Stern.shtml

10. Heaps

11. A Heap* Disambiguation: I am not talking about THE Heap, which is considered the source for all dynamic data (i.e., how data is stored in an Object rather than a primitive type.) ….What do I mean then?

12. A Heap* • Special Binary Tree – a complete binary tree • Bottom row (‘level’) fills from the left • Allows simple implementations (Array duality) • Max-heap property – root node is greater than all others; min-heap property – root node is less than all others • Technically, the heap can be seen as a data structure for a PriorityQueue • Queue, you may recall, is a built-in Java interface, modeling ‘FIFO’ • PriorityQueue – more specific type that assigns a priority to each task/value, removing the highest Priority first

13. Heaps* • Percolation Up/Down Method of adding/deleting a Node to/from a Heap. Involves swapping the last/final Node with the Root Node, then comparing the new ‘last’ with its parent Nodes to find where it should be. • https://www.cs.auckland.ac.nz/~jmor159/PLDS210/heaps.html • http://www.cs.cmu.edu/~adamchik/15-121/lectures/Binary%20Heaps/heaps.html

14. HeapSort! • Relatively efficient kind of sort • O (n log (n)) [for a worst case, this is not too bad] • Collects all elements into heap, sorts them inside the Heap, then remove each while updating the structure • In-place algorithm – uses small, constant memory • Somewhat complicated process • Make a heap out of the given info // Can be max or min! • Remove the root Node; replace it with the final Node in the heap • Perform comparisons (from left to right) to “rebalance” or reconfigure the heap again (method siftUp() or siftDown()) • Repeat steps 2-3 until there are no Nodes remaining • http://stackoverflow.com/questions/8938375/an-intuitive-understanding-of-heapsort • http://en.wikipedia.org/wiki/Heapsort • https://www.cs.auckland.ac.nz/~jmor159/PLDS210/heapsort.html

15. Animated Simulations • http://www.ee.ryerson.ca/~courses/coe428/sorting/heapsort.html • http://en.wikipedia.org/wiki/Heapsort#mediaviewer/File:Heapsort-example.gif

16. Thank you Very Much! • This should be posted on http://mthcompsci.wordpress.com/within a week • Many thanks to all of you for a great year of CS Club • I exhort you to continue next year • (March competition!) • (Recruiting more people?) • Good luck on Final Exams; have a great summer 