1. Better Algorithms for Better Computers
Shuchi Chawla
3/18/02
references

2. An Optimal Algorithm for List Update
Shuchi Chawla
3/18/02
List factoring slide
Uses of update…

3. The List Update Problem
Query for cereal
bananas
chocolate
bread
cereal
milk
eggs
apples = No = No = No =
Yes
Maintain an unordered linked list of items
Query for ith item takes i time 1

4. The List Update Problem
Query for cereal
bananas
chocolate
bread
cereal
milk
eggs
apples = No = No = No =
Yes
Maintain an unordered linked list of items
Query for ith item takes (i-1) time
Goal: Minimize time spent 2

5. The List Update Problem
Query for cereal
bananas
chocolate
bread
cereal
milk
eggs
apples =
Yes
What can the algorithm do?
Change positions of items in between accesses
– at some cost! 3

6. Moving items between accesses
Query for cereal
bananas
cereal
chocolate
bread
milk
eggs
apples
What can the algorithm do?
Change positions of items in between accesses
– at some cost! 4

7. Moving items between accesses
Query for cereal
bananas
cereal
chocolate
bread
milk
eggs
apples
Move “up” accessed items for FREE!
Other movements allowed at cost 1 per operation
New Goal: minimize total cost
What can the algorithm do? 5

8. Why are we studying this?
Lists are a widely used data structure
Algorithms for lists can be generalized to other problems such as Binary Search Trees, Paging
Eg. List Update:
Order elements by frequency or occurrence, most frequent in front
Search Trees:
Make a “Huffman code” tree based on frequency counts 6

9. Why are we studying this?
Lists are a widely used data structure
Algorithms for lists can be generalized to other problems such as Binary Search Trees, Paging
Goals:
Develop algorithm for List Update
Extend algorithm to a general strategy that provably works for any online data structure
Give detailed example of trees 7

10. Outline of the talk.. Rules of the game The algorithm
Extensions and Conclusions 8

11. Outline of the talk.. Rules of the game The algorithm
Online algorithms – the scenario
Examples of known list update algorithms
Formal definition of Competitive Ratio
Previously known results
The algorithm
Extensions and Conclusions 9

12. Online Algorithms.. The input is specified in steps
Compared against the best “offline” algorithm
The input is specified in steps
At every step, algorithm makes a move
Move should be good even with respect to future input!
knows future input
Constrain the algorithm: cannot change state of the system
Static Alg 10

13. Back to the Grocery List
Query for cereal
bananas
cereal
chocolate
bread
milk
eggs
apples
Which items should we move to incur a minimum total cost?
Suggestions?
Frequency Counts
Move To Front 11

14. Move To Front MTF is always at most twice worse than OPTb c d d c b a
d (3)
b (1)
c (2)
a (0)
d (3) d a b c a d c b
b (3)
c (3)
a (3)
MTF’s Total cost = 12
OPT’s Total cost = 6
MTF is always at most twice worse than OPT
FC achieves the same 12

15. Framework for measuring Alg’s performance
We compare ratio of Alg’s cost to Opt’s cost:
If: On input sequence S, Opt incurs cost O
And: On S, Alg incurs cost at most O
Alg’s “Competitive Ratio” is 
Formal Definition: Competitive Ratio = maxinputs (our cost)/(cost of OPT)
Next slide: what does opt look like
For distributions, opt is a static alg
Comparing to static algs makes our task easier
Worst case analysis 13

16. Worst –Case Analysis Adversary is all powerful
Can figure out which input is worst for Alg
Our ratio holds for all input sequences
Adversary cannot force us to perform worse 14

17. Can we do better than twice Opt?
Adversary knows the state of the algorithm
She always picks the last element
Even best SOPT has cost less than n/2 more than half the time
No Algorithm can do better than twice the optimal
Deterministic 15

18. Performance of randomized algorithms
Deterministic Competitive Ratio = maxinputs (our cost)/(cost of OPT)
Randomized Competitive Ratio = maxinputs Erandom bits(our cost)/(cost of OPT)
“Worst Case Expected Ratio” 16

19. Expected Worst Case Ratio
For every input
good performance with high probability
Adversary cannot force Alg to perform badly with probability 1 17

20. What’s known so far
No deterministic algorithm can perform better than a competitive ratio of 2.
For randomized algorithms, the best known algorithm achieves a 1.6 competitive ratio. 18

21. Our result Algorithm with (1+) competitive ratio
Give a general strategy that achieves (1+) ratio for any data structure 19

22. Outline of the talk.. Rules of the game The algorithm
Extensions and Conclusions 20

23. Outline of the talk.. Rules of the game The algorithm
Machine learning alg: Expert’s advice
Apply Expert’s advice to list update
Neat idea: List Factoring
Algorithm for a two-element list
Extend it to n-element list
Extensions and Conclusions 21

24. Predicting from Experts’ advice
Predicting the stock market
mistake
Expert#1
Expert#2
Expert#3
Neighbor’s dog
True value 1 1 1
1/2
1/2 1
1/4
1/8
1/16 1 2 5
Who should we believe?
Goal: Perform as well as the best expert 22

25. Predicting from Experts’ advice
Initialize all weights to 1
Decrease weights on a mistake
Take a weighted vote for prediction
Can also choose an expert probabilistically using weights as probability distribution
#mistakes < (1+) mexpert + O(log N)
any  > 0
# of experts
mistakes of best expert 23

26. Expert’s advice algorithm achieves a (1+) ratio wrt best expert
That’s what we want for lists! – thinking of static lists as our experts 24

27. Applying Experts to lists?
Goal: (1+) wrt best static list
Experts analysis already does that!
Use all static lists as experts!
Assign weights according to costs incurred by experts
After every access, pick next static list using the right probability
Perform transpositions moving from one list to another
Split into two slides 25

28. Applying Experts to lists?
Use all static lists as experts!
Can anyone see a problem with that?
Too many lists! – exponential in number - n!
Requires exponential running time per query – to update the weight of every static list
Split into two slides 26

29. Experts for List Update
Weight for every static list – too much computation
The BIG idea
Assign weights to every item in list and still make an experts style analysis work! 27

30. Candidate algorithm Initialize wi for item i
If ith element accessed, update wi
Order elements using some rule based on wi
Need to analyze probability of getting a particular static list 28

31. Reality check…
still awake? 29

32. Outline of the talk.. Rules of the game The algorithm
Machine learning alg: Expert’s advice
Apply Expert’s advice to list update
Neat idea: List Factoring
Algorithm for a two-element list
Extend it to n-element list
Extensions and Conclusions 30

33. Idea #2: List Factoring [Borodin ElYaniv]
Under certain conditions:
Ratio for two-element list holds for general list
Only need to analyze two-element list
Condition:
The relative order of x and y in the list does not depend upon accesses to others
Needs to be worked on 31

34. List Factoring – Move To Front4 7 2 9 9 4 7 2 7 4 9 2 4 9 7 32

35. List Factoring – Move To Front4 7 2 9 9 4 7 9 9 4 7 2 2 4 9 7
9 and 4 retain the same order – LFL applies
Can analyze performance over second list 33

36. Algorithm for two element list
List – (x,y)
Experts – (x,y) and (y,x)
weights – wx, wy – correspond to the respective experts!
Algorithm:
Initialize wx, wy to rx & ry R [1..1/]
If x accessed, wx <- wx+1 else wy <- wy+1
Always keep the element with higher weight in front 34

37. Have – algorithm with (1+) ratio for 2- element list
Need – Algorithm for n-element list
- satisfying LFL 35

38. Efficient Experts algorithm for List Update
Select ri R [1..1/] for element i
Initialize wi <- ri
If ith element accessed, wi <- wi+1
Order elements in decreasing order of weight
Points to note:
Random initialization  probabilistic picking of expert
Enables us to satisfy LFL by using a deterministic rule for ordering 36

39. Efficient Experts algorithm for List Updateb c d
 = 0.1 2 4 9 7 37

40. Efficient Experts algorithm for List Updateb
 = 0.1 7 4 2 9
d(0) b c a d 7 4 2 10
Cost of experts per cycle = 6
Cost of Opt per cycle = 6
c(2) b c a d 7 5 2 10
b(3) b c a d 7 5 3 10
a(1) b c a d 8 5 3 10 38

41. Efficient Experts algorithm for List Update
(1+) static competitive for any  > 0
“nearly optimal”
[Kalai Vempala’02] give a similar result for trees 39

42. Outline of the talk.. Rules of the game The algorithm
Extensions and Conclusions 40

43. Outline of the talk.. Rules of the game The algorithm
Extensions and Conclusions
Comparing our alg with “dynamic” algorithms
Combining good static and dynamic algorithms
Lessons to be learnt 41

44. Extensions Allowing Opt to move items between accesses
Dynamic (as opposed to static)
Comparing against “Dynamic” algorithms
Our alg could be far worse than Dynamic Opt
Previously known algorithms perform as well as a ratio of 1.6 42

45. Combining Static & Dynamic optimality
A has constant static ratio, B has constant dynamic ratio
Combine the two to get the best of both
Apply Experts again
Technical difficulties
Cannot estimate weights – running both simultaneously defeats our purpose
Experts maintain state - Huge cost of switching 43

46. Conclusions
First ever application of Experts advice to online data structures
Achieve a static ratio of (1+)
Settles the list update problem – no better can be done
Application to other online problems –
Naïve application is quite computationally intensive – problem structure should be exploited
Constant dynamic ratio for trees – still open!
State more clearly – first app of experts, settles the question on lists, still open-trees 44

47. Questions? 45