1. Computer Science Elementary Data Structures, Dictionaries CS 330: Algorithms Elementary Data Structures, Dictionaries Gene Itkis

2. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis2 Primitive DS  Linked lists (singly- and doubly- linked)  “Stretchable”, efficient, but…  Sequential access  Arrays  Even more efficient, but not “stretchable”  Random Access

3. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis3 Simple Data Structures SStacks llllast inserted QQueues ffffirst inserted PPriority Queues hhighest priority Methods for all:  isEmpty()  size()  insert()  insert(elem)  remove()  which element ?

4. Computer Science Dictionaries by key Search/Retrieve by key

5. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis5 Dictionary Examples  Phone directory  Name is key; phone # is the info  Reverse lookup: phone # is key  Student records  Possible keys: name, ID#  Credit cards DB; PKI Certificate Revocation Lists (CRLs)  E.g. check that the given credit card / certificate is valid  Extra: “Authenticate” the result  ETC.

6. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis6 Dictionary Interface  isEmpty()  size()  find(key)  insert()  insert(elem)  remove()  remove(key) Dynamic

7. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis7 Implementations  Simple/naïve  Lists  Better implementations Hash Tables  Probabilistic methods, Expected values Ordered trees  “Good enough” approximations; Augmenting (  order stats )  Other Skip-lists

8. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis8 Naïve: Lists  Unordered List  insert & delete  O(1) – fast  find  O(n) – slow  Ordered list  Linked-List  find & insert  O(n) – slow  Array  find O(lg n)  Binary search: O(lg n) – pretty fast  insert  O(n) – slow

9. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis9 Ordered Trees  Order  x< y, z : min at root – heap (  )  y < x < z  y < x < z : search   Depth  Shallow  Balanced  There might be exceptions:  e.g., Leftist heaps  “Strong” balance  Approximate   E.g., depth of leaves within factor of 2 (R-B trees) heap x z y

10. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis10 Ordered Trees  Searching  Easy  Insert/Delete  Naïve: destroys balance  Fix balance  How?  AVL trees Nodes keep children heights Rotate when needed: when children heights are >1 apart  Red-Black2-3-4  Red-Black / 2-3-4 trees “Almost balanced”

11. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis11 Hash tables  Example  Find professors by office number  Find tools in a tool-box  Might not work for everyone  Idea  “Figure” info location from the key

12. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis12 Hash Tables: Idea  Hash function  H(key)=i  If it works…  find  O(1)  insert & delete  O(1)  Problem?  Collisions:  H(key’)=H(key) … … key H Hash table i key info key’

13. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis13 Hash Table: Issues  Good Hash functions  Minimize collision chances  “Random looking”  Collision Resolution  Chaining  Open Addressing  Linear Probing: d=1  Quadratic Probing: d=i 2  Double Hashing: d=H2(key’) key info … … key H Hash table key’ key’ info key info key’ info d key” key” info d

14. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis14 Collision Resolution Methods Comparison  Chaining  Requires extra space  As with linked list  Stretchable  Can degenerate to linked-list search  Open Addressing  No extra space needed   Has size limit  Also can degenerate to unordered list search  Perfect Hashing