1. CISC220 Fall 2009 James Atlas Nov 13: Heap Implementations, Graphs

2. Heaps Specialized Tree Often are Binary Heaps Satisfy Heap Property –if B is a child node of A, then key(A) ≥ key(B) [in a Max-Heap] –if B is a child node of A, then key(A)  key(B) [in a Min-Heap]

3. Binary Max Heap

4. Heap operations insert: –adding a new key to the heap delete-max or delete-min: –removing the root node increase-key or decrease-key: –updating a key within a max- or min-heap, respectively merge: –joining two heaps to form a valid new heap containing all the elements of both.

5. Heap Insert 1.Add the element on the bottom level of the heap. 2.Compare the added element with its parent; if they are in the correct order, stop. 3.If not, swap the element with its parent and return to the previous step.

6. Heap Insert (example) 2 1 3

7. Heap Delete Replace root with last element Find max (or min) between root and children Swap if child > root, recursively do find max on subtree now rooted at new child

8. Heap Delete (example) 12

9. Implementations Array

10. Operation Running Times OperationBinary findMaxO(1) deleteMaxO(log n) insertO(log n) decreaseKeyO(log n) mergeO(n)

11. Seven Bridges of Königsberg solved by Leonhard Euler c1735 Find a path that crosses all bridges exactly once

12. Seven Bridges of Königsberg (cont’)

13. vertex edge Undirected Degree at each node is 3

14. Implementation/Storage Details How do we represent a graph?

15. List Structures V = {A, B, C, D, E} E = {{A, B}, {A, D}, {C, E}, {D, E}} Incidence List –E = {{A, B}, {A, D}, {C, E}, {D, E}} Adjacency List –A = {{A, B, D}, {B, A}, {C, E}, {D, A, E}, {E, C, D}} A B E D C

16. Matrix Structures V = {A, B, C, D, E} E = {{A, B}, {A, D}, {C, E}, {D, E}} Adjacency Matrix A B E D C ABCDE A11010 B11000 C00101 D10011 E00111