1. CS106X – Programming Abstractions in C++ Cynthia Bailey Lee CS2 in C++ Peer Instruction Materials by Cynthia Bailey Lee is licensed under a Creative Commons Attribution-NonCommercial- ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org.Cynthia Bailey LeeCreative Commons Attribution-NonCommercial- ShareAlike 4.0 International Licensehttp://peerinstruction4cs.org

2. Today’s Topics: Map Interface, continued 1. Hashing: collisions  The birthday problem 2. Hashing: live coding 3. Tree traversals (time permiting) 2

3. Hash collisions  We may need to worry about hash collisions  Hash collision:  Two keys a, b, a≠b, have the same hash code index (i.e. hash(a) = hash(b))

4. Hash key collisions  We can NOT overwrite the value the way we would if it really were the same key  Use a linked list chain extending from each hash table array entry to allow multiple distinct keys to share a single hash table location

5. Map Interface: hash-map.h … private: struct node { Key key; Value value; node *next; }; node **buckets; int numBuckets; int count; int hash(const Key& key) const; }; #include "hash-map-impl.h“

6. HashMap

7. TRUE or FALSE: We don’t actually need to store the keys in the nodes—we only really need the key to find which index, then store the value there A. TRUE B. FALSE

8. Hash key collisions & Big-O of HashMap  If there are no collisions, find/add/remove are all O(1) —just compute the key and go!  Two factors for ruining this magical land of instantaneous lookup: 1. Too-small table (worst case = 1) 2. Hash function doesn’t produce a good spread int awfulHashFunction(char* input, int size) { return 4; }  Find/add/remove all O(n) worst case HTTP :// XKCD. COM /221/

9. One simple, common approach  Hash table (array) size, numBuckets, is some prime number  Keep numBuckets big enough that at most 70% of the slots are filled (resize as needed)  Somehow convert your data to an int and then int array_index = input % numBuckets;  Can we still get super unlucky? YES  Hashmap analysis of O(1) is “average case”  For a nicely designed hash: 99.9% of the time

10. The Birthday Problem  Commonly mentioned fact of probability theory:  In a group of n people, what are the odds that 2 people have the same birthday?  (assume birthdays are uniformly distributed across 365 days, which is wrong in a few ways)  For n ≥ 23, it is more likely than not  For n ≥ 57, >99% chance  Means that hash tables almost certainly have at least one collision http://commons.wikimedia.org/wiki/File:Birthday_Paradox.svg http://commons.wikimedia.org/wiki/File:Birthday_Paradox.svg Creative Commons share-alike license.

11. (Live coding of hash-map.h)

12. Quickly Back to Trees: Tree Traversals!

13. What does this print? void traverse(Node *node) { if (node != NULL) { cout key << " "; traverse(node->left); traverse(node->right); } A. A B C D E F B. A B D E C F C. D B E F C A D. D E B F C A E. Other/none/more A BC DEF

14. What does this print? void traverse(Node *node) { if (node != NULL) { traverse(node->left); cout key << " "; traverse(node->right); } A. A B C D E F B. A B D E C F C. D B E F C A D. D E B F C A E. Other/none/more A BC DEF

15. How can we get this to print our ABCs in order? void traverse(Node *node) { if (node != NULL) { cout key << " "; traverse(node->left); traverse(node->right); } A. You can’t—we already tried moving the cout to all 3 possible places. B. You can but you use a stack instead of recursion. C. You can but you use a queue instead of recursion. D. Other/none/more A BC DEF

16. That prints the ABC nodes in order, but notice that our ABC tree isn’t a BST. How do we print BST nodes in order? void traverse(Node *node) { if (node != NULL) { traverse(node->left); traverse(node->right); } A. Use the BFS (queue instead of recursion). B. Use the DFS and put cout first. C. Use the DFS and put cout between. D. Use the DFS and put cout last. E. Other/none/more

17. We can play the same game with non-alpha characters as keys: What does this print? void traverse(Node *node) { if (node != NULL) { traverse(node->left); traverse(node->right); cout key << " "; } A. * + / 3 4 8 2 B. * + 3 4 / 2 C. 3 + 4 * 8 / 2 D. 3 4 + 8 2 / * E. Other/none/more * +/ 3428

18. Remember our old friend:

19. Practical uses of inorder/preorder/postorder traversals void traverse(Node *node) { if (node != NULL) { traverse(node->left); traverse(node->right); cout key << " "; } To calculate the answer: postorder To print this in human-friendly format: inorder * +/ 3428