1. Krakow, Jan. 9, 2008 1 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

2. Krakow, Jan. 9, 2008 2 List Scheduling 1 2 3 4 5 6 7 jobs Given a list of jobs (each with a specified processing time), assign them to processors to minimize makespan (max load) Online algorithm: assignment of a job does not depend on future jobs Goal: small competitive ratio processors time

3. Krakow, Jan. 9, 2008 3 1 2 3 4 5 6 7 processors jobs 1 2 3 4 5 6 7 makespan Greedy: Assign each job to the machine with the lightest load

4. Krakow, Jan. 9, 2008 4 1 2 3 4 5 6 7 jobs processors 1 2 3 4 5 6 7 makespan better schedule:

5. Krakow, Jan. 9, 2008 5 List Scheduling Greedy is (2-1/m)-competitive [Graham ’66] Lower bound ≈1.88 [Rudin III, Chandrasekaran’03] Best known ratio ≈1.92 [Albers ‘99] [Fleischer, Wahl ‘00] Lots of work on randomized algorithms, preemptive scheduling, …

6. Krakow, Jan. 9, 2008 6 List Scheduling on Related Machines processors 1 2 3 Related machines: machines may have different speeds  0.25  0.5  1 1 1 jobs 1 1 1

7. Krakow, Jan. 9, 2008 7  0.25  0.5  1 1 1 2 3 4 5 6 7 jobs 1 2 3 4 5 6 7 Algorithm 2PACK(L): schedule each job on the slowest machine whose load will not exceed 2L L 2L2L processors 1 2 3 Hey, the opt makespan is at most L

8. Krakow, Jan. 9, 2008 8 Lemma: If the little birdie is right (opt makespan ≤ L) then 2PACK will succeed. Proof: Suppose 2PACK fails on job h h’s length on processor 1 ≤ L, so so load of processor 1 > L r = first processor with load ≤ L (or m+1, if no such processor) 1 2 … … m L 2L2L Claim: if opt executes k on a machine in {r,r+1,…,m} then so does 2PACK optimum 2PACK r r

9. Krakow, Jan. 9, 2008 9 1 2 … … m L 2L2L so k fits on r k r r optimum 2PACK so k‘s length here ≤ L k Lemma: If the little birdie is right (opt makespan ≤ L) then 2PACK will succeed. Proof: Suppose 2PACK fails on job h h’s length on processor 1 ≤ L, so so load of processor 1 > L r = first processor with load ≤ L (or m+1, if no such processor) Claim: if opt executes k on a machine in {r,r+1,…,m} then so does 2PACK

10. Krakow, Jan. 9, 2008 10 1 2 … … m L 2L2L r r optimum 2PACK So opt’s (speed-weighted) total load on processors {1,2,…,r-1} is > (r-1)L Lemma: If the little birdie is right (opt makespan ≤ L) then 2PACK will succeed. Proof: Suppose 2PACK fails on job h h’s length on processor 1 ≤ L, so so load of processor 1 > L r = first processor with load ≤ L (or m+1, if no such processor) In other words: if 2PACK executes k on a machine in {1,2,…,r-1} then so does opt So some opt’s processor has load > L -- contradiction

11. Krakow, Jan. 9, 2008 11 Algorithm: 1. Choose d 1 < d 2 < d 3 < … (makespan estimates) Let B j = 2·( d 1 + d 2 + … + d j ) “bucket” j : time interval [B j-1, B j ] 2. j = 0 while there are unassigned jobs apply 2PACK with L = d j in bucket j if 2PACK fails on job k let j = j+1 and continue (starting with job k)

12. Krakow, Jan. 9, 2008 12 k bucket j 1 2 m … processor B1B1 B2B2 B j-1 BjBj B j+1 … k k’

13. Krakow, Jan. 9, 2008 13 Analysis: Suppose the optimal makespan is u Choose j such that d j-1 < u ≤ d j Then 2PACK will succeed in j ’th bucket (L = d j ) so algorithm’s makespan ≤ 2·(d 1 +d 2 + … + d j ) and We get ratio 8 for d j = 2 j 2  (bidding ratio)

14. Krakow, Jan. 9, 2008 14 Theorem: There is an 8-competitive online algorithm for list scheduling on related machines (to minimize makespan). With randomization the ratio can be improved to 2e. [Aspnes, Azar, Fiat, Plotkin, Waarts ‘06] World records: upper bound ≈ 5.828 (4.311 randomized) lower bound ≈ 2.438 (2 randomized) [Berman, Charikar, Karpinski ‘97] [Epstein, Sgall ‘00] List Scheduling on Related Machines

15. Krakow, Jan. 9, 2008 15 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

16. Krakow, Jan. 9, 2008 16 Minimum Latency Tour X = metric space P = v 1 v 2 …v h : path in X Latency of v i on P latency P (v i ) = d(v 1, v 2 ) + … + d(v i-1, v i ) (Total) latency of P =  i latency P (v i ) Minimum Latency Tour Problem: Given X, find a tour (path visiting all vertices) of minimum total latency Goal: polynomial-time approximation algorithm

17. Krakow, Jan. 9, 2008 17 Total latency = 2 + 4 + 8 + 11 + 15 = 40 A 15 E 2 D 11 C 4 B 8 F

18. Krakow, Jan. 9, 2008 18 A E D C B F Minimum k-Tour Problem: find a shortest k-tour (a path that starts and ends at v 1 and visits ≥ k different vertices) 2-tour 4-tour

19. Krakow, Jan. 9, 2008 19 Algorithm: 1. Choose d 1 < d 2 < d 3 < … 2. For each k compute the optimal k-tour T k 3. Choose p(1) < … < p(m) = n s.t. length(T p(i) ) = d i (For simplicity assume they exist) 4. Output Q = T p(1) T p(2) …T p(m) (concatenation) Denote Q = q 1 q 2 …q n (q i = first point on Q different from q 1, q 2,…,q i-1 )

20. Krakow, Jan. 9, 2008 20 v1v1 T p(1) T p(2) T p(3) Q

21. Krakow, Jan. 9, 2008 21 Lemma: S = s 1 s 2 …s n : tour with optimum latency. Then latency S (s k ) ≥ (1/2) · length(T k ) Proof: s1s1 s2s2 s3s3 sksk S T T is a k-tour, so 2·latency S (s k ) = length(T) ≥ length(T k )

22. Krakow, Jan. 9, 2008 22 Analysis: For p(j-1) < k ≤ p(j) latency S (s k ) ≥ (1/2)·length(T k ) ≥ (1/2)·length(T p(j-1) ) = d j-1 /2 q k will be visited in T p(j) (or earlier), so latency Q (q k ) ≤ d 1 +d 2 + … + d j We get ratio 8 for d j = 2 j … if we can compute k-tours efficiently !!! 2  (bidding ratio)

23. Krakow, Jan. 9, 2008 23 If X is a weighted tree, optimal k-tours can be computed in polynomial time… 10 4 5 11 12 7 6 8 2 3 9 7 7 5 Theorem: There is a polynomial-time 8-approximation algorithm for maximum latency tours on weighted trees [Blum, Chalasani, Coppersmith, Pulleyblank, Raghavan, Sudan ‘94] Can we do better?

24. Krakow, Jan. 9, 2008 24 Choose a random direction (clockwise or counter-clockwise) and traverse each T p(j) in this direction … v1v1 Tp(j)Tp(j) u Expected latency of u = d 1 +d 2 + …+ d j-1 + d j /2 We get ratio 6 for d j = 2 j

25. Krakow, Jan. 9, 2008 25 Can be extended to arbitrary spaces, with ratio 3.591 [Chauduri, Godfrey, Rao, Talwar ‘03] Theorem: There is a polynomial-time 3.591 -approximation algorithm for maximum latency tours on weighted trees [Goemans, Kleinberg ‘98] Can we do even better? Instead of d j = 2 j choose d j = c j+x, where c is the constant from the Cow Path problem and x is random in [0,1) We don’t really need randomization: choose better direction (clockwise or counter-clockwise) There are only O(n) x’s that matter, so try them all

26. Krakow, Jan. 9, 2008 26 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

27. Krakow, Jan. 9, 2008 27 k-Clustering X = metric space For C  X, diameter(C) = maximum distance between points in C k-Clustering Problem: Given k, partition X into k disjoint clusters C 1,…,C k to minimize the maximum diameter(C j ) Offline: approximable with ratio 2 [Gonzales ‘85] [Hochbaum, Shmoys ‘85] lower bound of 2 for polynomial algorithms (unless P = NP) [Feder, Greene ‘88] [Bern, Eppstein ‘96]

28. Krakow, Jan. 9, 2008 28 E D C B F A G H 3-Clustering with maximum diameter 5 k=3

29. Krakow, Jan. 9, 2008 29 E D C B F A G H 3-Clustering with maximum diameter 3 k=3

30. Krakow, Jan. 9, 2008 30 Incremental k-Clustering Problem: Maintain k-clustering when points in X arrive online allowed operations:  add point to a cluster  merge clusters  create a new singleton cluster Goal: online competitive algorithm (polynomial-time)

31. Krakow, Jan. 9, 2008 31 D C G diameter = 0 A k=3

32. Krakow, Jan. 9, 2008 32 D C G A E diameter = 2 k=3

33. Krakow, Jan. 9, 2008 33 D C G A E H diameter = 3 k=3

34. Krakow, Jan. 9, 2008 34 D C G A E H diameter = 3 k=3

35. Krakow, Jan. 9, 2008 35 Notation and terminology: Algorithm’s clusters C 1,C 2,…,C k’ with k’ ≤ k in each C i fix a center o i radius of C i = max distance between x  C i and o i diameter of C i ≤ 2 · (radius of C i ) CiCi oioi radius

36. Krakow, Jan. 9, 2008 36 Procedure CleanUp(z). Goal: merge some clusters C 1,C 2,…,C k’ so that afterwards all inter-center distances are > z 1.Find a maximal set J of clusters with all inter- center distances > z 2. for each cluster C a  J choose C b  J with d(o a,o b ) ≤ z merge C a into C b (with center o b )

37. Krakow, Jan. 9, 2008 37 Lemma: If the max radius before CleanUp is h then after CleanUp it is ≤ h+z. Proof: follows from the ∆ inequality z

38. Krakow, Jan. 9, 2008 38 Lemma: If the max radius before CleanUp is h then after CleanUp it is ≤ h+z. Proof: follows from the ∆ inequality z v z h

39. Krakow, Jan. 9, 2008 39 Algorithm: 1. Choose d 1 < d 2 < d 3 < … 2. Initially C 1,C 2,…,C k are singletons (first k points) (Assume min distance between these points is > 1) 3. j  1 4. repeat when a new point x arrives if d(x,o i ) ≤ d j for some i add x to C i else if k’ < k k’  k’+1; C k’  {x} else create a temporary cluster C k+1  {x} while there k+1 clusters j  j+1 do CleanUp with z = d j (merge clusters) checkpoint j Invariant: inter-center distance > d j

40. Krakow, Jan. 9, 2008 40 Example: k = 4

41. Krakow, Jan. 9, 2008 41 Example: k = 4 d j+1 d j+2

42. Krakow, Jan. 9, 2008 42 Analysis: At checkpoint j : Before clean-up k+1 clusters with inter-center distances > d j-1 so opt diameter > d j-1 After clean-up max radius ≤ d 1 + d 2 + … + d j so max diameter ≤ 2·(d 1 + d 2 + … + d j ) We get ratio 8 for d j = 2 j 2  (bidding ratio)

43. Krakow, Jan. 9, 2008 43 Theorem: There is an 8-competitive online algorithm for incremental clustering (ratio 2e with randomization). [Charikar, Chekuri, Feder, Motwani ‘06] Other results: upper bound 6 (4.33 randomized) (not polynomial-time) lower bound 2.414 (2 randomized) Incremental Clustering

44. Krakow, Jan. 9, 2008 44 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering 8. Other scheduling problems: List scheduling, related machines with preemption Scheduling with min-sum criteria Non-clairvoyant scheduling 9. Hierarchical clustering 10. Load balancing 11. Online algorithms for set cover (combined with primal-dual) …. Doubling Method Applications: