Lecture 7 Greedy Algorithms Interval Scheduling There are

Lecture 7 Greedy Algorithms

Interval Scheduling • There are n meeting requests, meeting i takes time (si, ti) • Cannot schedule two meeting together if their intervals overlap. • Goal: Schedule as many meetings as possible. • Example: Meetings (1, 3), (2, 4), (4, 5), (4, 6), (6, 8) • Solution: 3 meetings ((1, 3), (4, 5), (6, 8))

Recap: Algorithm • Always try to schedule the meeting that ends earliest Schedule 1. Sort the meetings according to end time 2. For i = 1 to n 3. If meeting i does not conflict with previous meetings 4. Schedule meeting i

Proof of correctness in pictures I have scheduled the most number of possible meetings! ……

I have scheduled the most number of possible meetings! …… …… No you are wrong! There is a way to schedule more meetings.

Can you show me how to do that? Here are the first two meetings I scheduled. …… …… Yes, those are fine. I made the same decisions.

and this is my third meeting i 3 …… j 3 …… Wait, you can’t do that. You need to schedule meeting j 3

My meeting i 3 ends before your meeting j 3, so if you swap j 3 to i 3, you are still OK right? i 3 …… ij 33 …… I guess you are right. Fine I will do that.

But isn’t this the same thing? You could have swapped to my 5 th meeting. …… …… OK you got me on the 3 rd meeting. But you made another mistake on the 5 th meeting!

May rounds later……

…… …… OK, you can keep all of your meetings, but I could still schedule a meeting jk+1!

That doesn’t make any sense. If there is such a meeting, I must consider it after ik, and I would have scheduled that meeting! …… …… OK, you win.

The Encoding Problem (Hoffman Tree) • Problem: Given a long string with n different characters in alphabet, find a way to encode these characters into binary codes that minimizes the length. • Example: “aababc”, n = 3, alphabet = {a, b, c} • If a = ‘ 0’, b = ’ 11’, c = ‘ 10’, then the string can be encoded as ‘ 001101110’.

Why don’t just use ASCII? • Different letters appear with very different frequency. • Using an encoding of varying length can save space. • Now of course memory/disk are cheap, but some communications can be expensive.

What kind of encodings are allowed? • Bad example: If a = ‘ 0’, b = ’ 01’, c = ‘ 1’ • Given encoding 00010011, it can be ‘aababc’, but can also be ‘aaacabc’ (and others) • Want an encoding that allow unique decoding. • Sufficient condition: Prefix free encoding • Definition: No code should be a prefix of another code. • (In previous example, encoding of a is a prefix for encoding of b. )

Prefix-free encoding vs. Trees 0 a = 00 b = 01 c = 110 d = 111 0 a 1 1 1 b 0 c Character = Leaves of the Tree Encoding = Path from root to the character Prefix free = no characters for intermediate nodes 1 d

Cost of the Tree 0 0 a Cost = sum of Frequency * Depth = 5*2 + 10*2 + 3*3 + 6*3 = 57 1 1 1 b 0 5 10 c 3 1 d 6

Hoffman Tree problem • Given a long string with n different characters in alphabet, find a way to encode these characters into binary codes that minimizes the length. • Given an alphabet of n characters, character i appears wi times, find an encoding tree with minimum cost.

Algorithm Hoffman Tree 1. REPEAT 2. Select two characters with smallest frequencies 3. Merge them into a new character, whose frequency is the sum 4. UNTIL there is only one character abcd 24 acd 14 ac 8 a b c d 5 10 3 6

Proof Sketch • Use induction: Assume algorithm finds the optimal solution for alphabet of size n, we want to prove that it works for alphabet of size n+1. • Base Case: n = 1 is trivial. (cost = 0 in this case)

Induction Step i j First merged i, j (characters with lowest frequency) i j i, j may be in different branches Goal: Convince OPT that it is OK to merge i, j first.

Induction Step i j j i After ALG and OPT both merged i, j, it reduces to a problem of size n! Induction hypothesis ALG is optimal, OPT cannot be better.