Trees and Codes

1. a d e b c Trees and Codes Trees

2. Binary Trees and Prefix Codes • Imagine that you have an alphabet of symbols • e.g. A, B, …, Z, a, b, …, z, ‘ ’, ‘.’ • We wish to represent a string of these symbols as a string of bits • e.g. “This is a string of characters” becomes “01100111000101010111001” Trees

3. Method 1: Use a fixed number for each symbol • Map A -> 1, B -> 2, … • “This is a string of characters” becomes a string of numbers • “20,34,35,…” • I required commas to separate the characters! • Use a fixed width, padded with leading 0’s instead • “020034035…” Trees

4. Method 1: Fixed width • How wide do my characters need to be • Using a string of bits • How many bits are needed to represent the largest character? • ceil(log2(n)) bits • Use that many bits for each character • This is the system used within the computer with 8 bits for each ASCII code Trees

5. Method II: Prefix codes • For each symbol, we’ll use a code with a special property • No code is the prefix of any other code • How does this work? • Decoding: • We read in the codes one bit at a time • When we have a code we recognise, it must be the end of a symbol • It cannot be part of a longer symbol because no code is the prefix of another code Trees

6. Method II: Prefix codes • Example • A:00, B:010, C:011, D:10, E:11 • ADBECABADE • 00100101101100010001011 • A D B E C A B A D E • The string decodes Trees

7. Making a Prefix Code • We want the code to be efficient • No strings longer than necessary • No wasted strings • A code is a set of strings of binary digits, such that no string corresponding to one symbol is the prefix of a string corresponding to another symbol • In a tree, leaf nodes have no children • No path from the root to a leaf is the prefix of a path from the root to another node Trees

8. Binary Trees and Prefix Codes • Binary trees are in one to one correspondence with Prefix Codes • A:00, B:010, C:011, D:10, E:11 0 1 0 1 0 1 A D E 0 1 B C Trees

9. Prefix Trees • Binary Trees • The left child corresponds to 0, the right to 1 • Each leaf contains a symbol • The code for a symbol corresponds to the path from the root to the leaf containing that symbol Trees

10. Encoding and Decoding • Imagine the encoder and decoder running in parallel • Encoding • Start from the root • While you are not at the symbol’s leaf • If the symbol you wish to send is a left decendant, send 0 and move to your left child, else send 1 and move to your right child • Decoding • Start from the root • While you are not at a leaf • Read a bit. If it is 0 then move to your left chile, else move to your right child Trees

11. Encoding:ACD Decoding: Encoding and Decoding: ACD A D E A D E B C B C Trees

12. Encoding:ACD Decoding: Encoding and Decoding: ACD 0 A D E A D E B C B C Trees

13. Encoding:ACD Decoding: Encoding and Decoding: ACD A D E A D E B C B C Trees

14. Encoding:ACD Decoding: Encoding and Decoding: ACD 0 A D E A D E B C B C Trees

15. Encoding:CD Decoding:A Encoding and Decoding: ACD A D E A D E B C B C Trees

16. Encoding:CD Decoding:A Encoding and Decoding: ACD A D E A D E B C B C Trees

17. Encoding:CD Decoding:A Encoding and Decoding: ACD 0 A D E A D E B C B C Trees

18. Encoding:CD Decoding:A Encoding and Decoding: ACD A D E A D E B C B C Trees

19. Encoding:CD Decoding:A Encoding and Decoding: ACD 1 A D E A D E B C B C Trees

20. Encoding:CD Decoding:A Encoding and Decoding: ACD A D E A D E B C B C Trees

21. Encoding:CD Decoding:A Encoding and Decoding: ACD 1 A D E A D E B C B C Trees

22. Encoding:D Decoding:AC Encoding and Decoding: ACD A D E A D E B C B C Trees

23. Encoding:D Decoding:AC Encoding and Decoding: ACD A D E A D E B C B C Trees

24. Encoding:D Decoding:AC Encoding and Decoding: ACD 1 A D E A D E B C B C Trees

25. Encoding:D Decoding:AC Encoding and Decoding: ACD A D E A D E B C B C Trees

26. Encoding:D Decoding:AC Encoding and Decoding: ACD 1 A D E A D E B C B C Trees

27. Encoding: Decoding:ACD Encoding and Decoding: ACD A D E A D E B C B C Trees

28. Back to Method I: Balanced Tree • Method I was to used fixed length code words • Each path from the root to a leaf is the same length: a balanced tree • Balanced trees are good for worst case path length. Are they good for coding? • Yes, if you assume the worst case • But we can normally do better… Trees

29. Statically optimal codes • Want common symbols to have short codes • This will make uncommon symbols have longer codes • In a tree with a fixed number of leave/symbols, moving one leaf/symbol closer to the root will move others further away Trees

30. Huffman codes • From Shannon’s information theory, The optimal static code assigns -log2(p) bits to a symbol that occurs with probability p • It is possible to make a Huffman code tree with this property • Will look at this later in the course Trees

31. Adaptive Codes • As long as the same change is made in both sending and receiving trees/codes, there is no reason why the tree/code must remain static • Send a character using the initial tree • Update the tree using that character • Can also be updated in the receiver as it already has the character • Send the next character Trees

32. Encoding:ACD Decoding: Encoding and Decoding: ACD A D E A D E B C B C Trees

33. Encoding:CD Decoding:A Encoding and Decoding: ACD 00 A D E A D E B C B C Trees

34. Encoding:CD Decoding:A Encoding and Decoding: ACD Make same change in both trees: Rotate A’s parent A A B C D E B C D E Trees

35. Encoding:CD Decoding:A Encoding and Decoding: ACD A A B C D E B C D E Trees

36. Encoding:D Decoding:CA Encoding and Decoding: ACD 101 A A B C D E B C D E Trees

37. Encoding:D Decoding:CA Encoding and Decoding: ACD A A B B C C D E D E Trees

38. Encoding:D Decoding:CA Encoding and Decoding: ACD A A B B C C D E D E Trees

39. Encoding: Decoding:CAD Encoding and Decoding: ACD A A 1110 B B C C D E D E Trees

40. Splay Trees do Rotations after Every Operation • new operation: splay • splaying moves a node to the root using rotations • right rotation • makes the left child x of a node y into y’s parent; y becomes the right child of x • left rotation • makes the right child y of a node x into x’s parent; x becomes the left child of y y x a right rotation about y a left rotation about x y T1 x T3 x y T3 T1 T2 T2 y T1 x T3 T3 T2 T1 T2 (structure of tree above x is not modified) (structure of tree above y is not modified) Trees

41. “x is aleft-left grandchild” means x is a left child of its parent, which is itself a left child of its parent • p is x’s parent; g is p’s parent Splaying: start with node x is x a left-left grandchild? is x the root? zig-zig yes stop right-rotate about g, right-rotate about p yes no is x a right-right grandchild? zig-zig is x a child of the root? no left-rotate about g, left-rotate about p yes is x a right-left grandchild? yes zig-zag is x the left child of the root? left-rotate about p, right-rotate about g no yes is x a left-right grandchild? zig zig zig-zag yes right-rotate about p, left-rotate about g right-rotate about the root left-rotate about the root yes Trees

42. Visualizing the Splaying Cases zig-zag x z z z y y T4 T1 y T1 T2 T3 T4 T4 x T3 x T2 T3 T1 T2 zig-zig y zig x T4 x T1 x y w T2 y z T3 w T1 T2 T3 T4 T3 T4 T1 T2 Trees