Knowledge States: A Tool in Randomized Online Algorithms

Transcription

1 : A Tool in Rndomized Online Algorithms Center for the Advnced Study of Algorithms School of Computer Science University of Nevd, Ls Vegs ADS 2007 couthors: Lwrence L. Lrmore, John Nog, Rüdiger Reischuk supported by NSF grnt CCR

2 Online Problems offline: ll input dt is completely vilble before the lgorithm strts.

3 Online Problems offline: ll input dt is completely vilble before the lgorithm strts. online: input dt rrives piece t time the lgorithm must mke decision without knowledge of the entire input.

4 Online Problems offline: ll input dt is completely vilble before the lgorithm strts. online: input dt rrives piece t time the lgorithm must mke decision without knowledge of the entire input. online problems: resource lloction in operting systems, network routing, robotics, dt-structuring, distributed computing, scheduling...

5 Theme of the Tlk It is difficult to construct good rndomized online lgorithms Use of work functions in the context of rndomized online lgorithms

6 Theme of the Tlk It is difficult to construct good rndomized online lgorithms Use of work functions in the context of rndomized online lgorithms Give distributionl descriptions of lgorithms

7 Theme of the Tlk It is difficult to construct good rndomized online lgorithms Use of work functions in the context of rndomized online lgorithms Give distributionl descriptions of lgorithms Use the concept of forgiveness

8 Theme of the Tlk It is difficult to construct good rndomized online lgorithms Use of work functions in the context of rndomized online lgorithms Give distributionl descriptions of lgorithms Use the concept of forgiveness Give two new results: better thn 2-comptitive 2-server lgorithm in cross polytope spces optimlly competitive k-pging with only O(k) memory

9 Ppers Online Computtion Equitble Revisited. Bein, Lrmore, Nog, ESA 2007 A Rndomized Algorithm for Two Servers in Cross Polytope Spces. Bein, Iwm, Kwhr, Lrmore, Orvec, WAOA 2007 : A Tool for Rndomized Online Algorithms. Bein, Lrmore, Reischuk, HICSS 2008 Knowledge Stte Algorithms: Rndomiztion with Limited Informtion. Bein, Lrmore, Reischuk, Arxiv 2007 for the Cching Problem in Shred Memory Multiprocessor Systems. Bein, Lrmore, Reischuk, IJFCS, to pper Bein, Lrmore. Trckless nd Limited-Bookmrk Algorithms for Pging. Bein, Lrmore, ACM SIGACT News, 2004 Tutoril t bein/tutoril

10 Pging Online Computtion Universe of pges in slow memory Fst memory cn hold k pges Requests = r 1 r 2...r n for pges hve to be served Hit (no cost) or Miss (cost 1) Fst Memory Fst Memory b f h j k x b f j k Eject h Slow Memory Request x Slow Memory Solution cn be described s sequence of configurtions

11 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost 1 r

12 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r 1

13 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r 2

14 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r 2

15 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r 3

16 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r 3

17 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r4

18 The CNN Problem News crew in Mnhttn (streets nd venues) event sequence = r 1 r 2,...,r n Event cn be seen either horizontlly or verticlly Solution cn be described s sequence configurtions Gol: minimize totl movement cost r4

19 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost Server 1 Server 2

20 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Server 1 r 1 Server 2 Gol: minimize totl movement cost

21 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost Server 1 r 1 Server 2

22 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost Server 1 r 2 Server 2

23 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost Server 1 r 2 Server 2

24 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost Server 1 r3 Server 2

25 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost r3 Server 2 Server 1

26 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost r3 Server 2 Server 1 r 4

27 The 2-Server Problem 2 servers in metric spce M request sequence = r 1 r 2,...,r n online: decision must be mde before r i+1 is reveled Gol: minimize totl movement cost Server 1 r 4 Server 2

28 Configurtions for the 2-Server Problem the loctions of the two servers is clled configurtion solution cn be described s sequence of configurtions the movement cost is the trnsporttion distnce between configurtions Server 1 r3 r 2 r 1 Server 2 r 4

29 Configurtions for the 2-Server Problem the loctions of the two servers is clled configurtion solution cn be described s sequence of configurtions the movement cost is the trnsporttion distnce between configurtions Server 1 r3 r 2 r 1 Server 2 r 4

30 Configurtions for the 2-Server Problem the loctions of the two servers is clled configurtion solution cn be described s sequence of configurtions the movement cost is the trnsporttion distnce between configurtions Server 1 r 2 r3 Server 2 r 4

31 Configurtions for the 2-Server Problem the loctions of the two servers is clled configurtion solution cn be described s sequence of configurtions the movement cost is the trnsporttion distnce between configurtions r3 Server 2 Server 1 r 4

32 Configurtions for the 2-Server Problem the loctions of the two servers is clled configurtion solution cn be described s sequence of configurtions the movement cost is the trnsporttion distnce between configurtions Server 1 r 4 Server 2

33 Online Algorithms 1 Algorithm A is t some initil configurtion 0

34 Online Algorithms 1 Algorithm A is t some initil configurtion 0 2 Requests: = r 1,..., r n.

35 Online Algorithms 1 Algorithm A is t some initil configurtion 0 2 Requests: = r 1,..., r n. 3 At time (t 1), A is t configurtion t 1.

36 Online Algorithms 1 Algorithm A is t some initil configurtion 0 2 Requests: = r 1,..., r n. 3 At time (t 1), A is t configurtion t 1. 4 A hs to serve r t not knowing r t+1,...

37 Online Algorithms 1 Algorithm A is t some initil configurtion 0 2 Requests: = r 1,..., r n. 3 At time (t 1), A is t configurtion t 1. 4 A hs to serve r t not knowing r t+1,... 5 A chooses configurtion t.

38 Online Algorithms 1 Algorithm A is t some initil configurtion 0 2 Requests: = r 1,..., r n. 3 At time (t 1), A is t configurtion t 1. 4 A hs to serve r t not knowing r t+1,... 5 A chooses configurtion t. 6 A incurs cost( t 1, r t, t ).

39 Online Algorithms 1 Algorithm A is t some initil configurtion 0 2 Requests: = r 1,..., r n. 3 At time (t 1), A is t configurtion t 1. 4 A hs to serve r t not knowing r t+1,... 5 A chooses configurtion t. 6 A incurs cost( t 1, r t, t ). If A uses rndomiztion in bullet 5 then A is clled rndomized online lgorithm.

40 2-Server Exmple Exmple: k = 2 nd = xyxyz y 2 1 z x 2 y z x y z x cost = 7 x y x y z Work Function Algorithm (WFA) y x y x y x z z z

41 The Adversry: Optiml Cost Function on configurtions: Dynmic progrmming The optiml cost of being there then y 2 x x 1 y x optcost = 4 = min lst workfunction Given request sequence ρ z z y x 2 x y 2 y ω ρ () = min cost of serving ρ nd ending in configurtion X 1 4 z z y x z y 2 x y x z z

42 Support of Work Functions ω({y, z}) ω({x, z}) ω({x, y}) initil request x request y request x request y request z S X supports ω if for ny b X there exists some S such tht ω(b) = ω() +, b.

43 Support of Work Functions ω({y, z}) ω({x, z}) ω({x, y}) initil request x request y request x request y request z S X supports ω if for ny b X there exists some S such tht ω(b) = ω() +, b. A resonble lgorithm will move to configurtions in the support.

44 Support of Work Functions ω({y, z}) ω({x, z}) ω({x, y}) initil request x request y request x request y request z S X supports ω if for ny b X there exists some S such tht ω(b) = ω() +, b. A resonble lgorithm will move to configurtions in the support. WFA moves for request r from configurtion to configurtion b such tht, b + ω(b) is minimized.

45 Competitiveness For request sequence = r 1, r 2,... consider cost A ( ): the cost on chieved by A cost opt ( ): the cost on chieved by opt We sy tht A is C-competitive if for ech sequence we hve Ecost A ( ) C cost opt ( ) + K - Exmple: cost WFA (xyxyz) = 7 WFA is 2-competitive cost opt (xyxyz) 4

46 The Distributionl Model A rndomized lgorithm cn be viewed s determistic lgorithm on distributions. X = ll configurtions π is distribution on X. Algorithm A is t some initil configurtion 0.

47 The Distributionl Model A rndomized lgorithm cn be viewed s determistic lgorithm on distributions. X = ll configurtions π is distribution on X. Algorithm A is t some initil configurtion 0. Requests: = r 1,..., r n.

48 The Distributionl Model A rndomized lgorithm cn be viewed s determistic lgorithm on distributions. X = ll configurtions π is distribution on X. Algorithm A is t some initil configurtion 0. Requests: = r 1,..., r n. At time (t 1), A is t distribution π t 1.

49 The Distributionl Model A rndomized lgorithm cn be viewed s determistic lgorithm on distributions. X = ll configurtions π is distribution on X. Algorithm A is t some initil configurtion 0. Requests: = r 1,..., r n. At time (t 1), A is t distribution π t 1. A hs to serve r t not knowing r t+1,...

50 The Distributionl Model A rndomized lgorithm cn be viewed s determistic lgorithm on distributions. X = ll configurtions π is distribution on X. Algorithm A is t some initil configurtion 0. Requests: = r 1,..., r n. At time (t 1), A is t distribution π t 1. A hs to serve r t not knowing r t+1,... A chooses deterministiclly distribution π t.

51 Cost in the Distributionl Model 1 2 y x z cost 0, 1/4trnsport cost 2, 1/4 trnsport x y x z y x z cost 2, 1/4 trnsport y x z z 3 4 Totl cost: 1 Trnsportion Cost = 1 The cost incured by moving from one distribution to the next is clculted by moving mss long trnsportion problem. The trnsporttion problem hs the Monge property.

52 Work Function Guided 1 b c _ 1 2 _ 1 2 b 3 d c 3 3 b 4 e d c b Problem: The support grows without bound.

53 Forgiveness Online Computtion Lower the work function on selective configurtions

54 Forgiveness Online Computtion Lower the work function on selective configurtions d 3 c 3 3 b 4 e d c b 1 e d c b

55 Forgiveness Online Computtion Lower the work function on selective configurtions d 3 c 3 3 b 4 e d c b e 1 d c Work functions re now estimtors b

56 Ls Vegs Online Computtion Algorithm is constructed using the mixed model of online computtion b c _ 1 2 _ b d c c 3 3 d 1 1 b d c b _ 2_ 3 0 b 1 d c b

57 The Rndomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Rghvn, Snir, 90]

58 The Rndomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Rghvn, Snir, 90] Not known to be best possible.

59 The Rndomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Rghvn, Snir, 90] Not known to be best possible. Lower Bound: 1 + e [Chrobk, Lrmore, Lund, Reingold, 97]

60 The Rndomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Rghvn, Snir, 90] Not known to be best possible. Lower Bound: 1 + e [Chrobk, Lrmore, Lund, Reingold, 97] Line: [Brtl, Chrobk, Lrmore, 98]

61 2-Server Problem: M 24 M 24 consists of ll metric spces such tht All distnces re 1 or 2. d(x, y) + d(x, z) + d(y, z) 4

62 Why M 24? Online Computtion A step in the direction of the gol (better thn 2-competitive rndomized lgorithm for the 2-server problem).

63 Why M 24? Online Computtion A step in the direction of the gol (better thn 2-competitive rndomized lgorithm for the 2-server problem). Allows simple exmple of the knowledge stte method.

64 Why M 24? Online Computtion A step in the direction of the gol (better thn 2-competitive rndomized lgorithm for the 2-server problem). Allows simple exmple of the knowledge stte method. An interesting clss in its own right, generlizing the octhedron.

65 Exctly Three Infinite Fmilies of Convex Regulr Polytopes (Ludwig Schläfli, 1852) Infinite Fmily of Regulr Polytopes Grph Clss Metric Spce Clss 3 d 4 d Regulr Simplices Complete Grphs Uniform Spces Cross Polytopes Circulnt Grphs = Orthoplices Ci 1,...,n 1 (2n) M 24 Hypercubes Hypercubes Hmming Metric Spces

66 Wht is Knowledge Stte? Knowledge stte k = (ω, π): ω : X R is the estimtor. π is distribution on X. 7/8 0 x z 1/8 1/2 y π(x, y) is the probbility we re t {x, y}. ω(x, y) is the estimted unpid cost of the dversry if it is t {x, y}.

67 A Closer Look Online Computtion 0 1 1/ /8 0 x 0 3/2 1/8 1/2 y Bxyz z The estimtor nd distribution re defined for ll configurtions but chrcterized by their vlues only on the support If X S, then π() = 0. ω() = min b S {ω(b) +, b }

68 for the 2-Server Problem 7_ 6 1_ 0 4 z 1_ 0 4 z x Axz x y 0 2 x _ z Dxyz _5 1 1_ x 1 z 7 12 _ Gxz 1_ 0 2 x z 1_ 2 0 _ x _ y Bxyz 5 12 _ z Exz 7_ 4 1_ 4 Hxz z Cz 5 12 _ 19 _ 24 0 z z x _ _ x Fxz 5 _ 3 24 _ z x x 8 _3 0 Up to symmetry, there re 8 knowledge sttes of competitive lgorithm for M 2,4. z _ z 10

69 There re Numerous Moves. Here is One. z _7 6 x 4 0 y 0 4 _ z Dxyz 0 2 _ x Intermedite Stte W _ 1 _ _ x _ 1 y _ 1 z _ _ z _ x 1_ 3 1_ 3 1_ 3 z _ x _ Gzx _ z 0 2 _ x 0 y 4 _7 6 z 0 4 _ Dyzx 2 0 x _7 6 _ x y _ z 4 0 Dyzx x

70 One Move of the Algorithm Strt t stndrd knowledge stte over (x, y, z).

71 One Move of the Algorithm Strt t stndrd knowledge stte over (x, y, z). Red request r.

72 One Move of the Algorithm Strt t stndrd knowledge stte over (x, y, z). Red request r. Updte the estimtor.

73 One Move of the Algorithm Strt t stndrd knowledge stte over (x, y, z). Red request r. Updte the estimtor. Move the distribution.

74 One Move of the Algorithm Strt t stndrd knowledge stte over (x, y, z). Red request r. Updte the estimtor. Move the distribution. Ls Vegs. Rndomly pick subsequent.

75 One Move of the Algorithm Strt t stndrd knowledge stte over (x, y, z). Red request r. Updte the estimtor. Move the distribution. Ls Vegs. Rndomly pick subsequent. We re t stndrd knowledge stte over (x, y, r), (y, x, r), (x, z, r), (z, x, r), (y, z, r), or (z, y, r).

76 Behviorl Version: The Wirefrme Algorithm dxyz x z y _ z dyzx _ z x y _ z dyzx _ z x y _ z request x z x y dyzx z z x y dyzx z _ x _ x _ x _ x

77 Results for the 2-Server Problem in M 2,4 2-Server Problem on M 2,4, C = 7 4

78 Results for the 2-Server Problem in M 2,4 2-Server Problem on M 2,4, C = Server Problem on M 2,4, C =

79 Results for the 2-Server Problem in M 2,4 2-Server Problem on M 2,4, C = Server Problem on M 2,4, C = This is optiml for M 2,4.

80 Results for the 2-Server Problem in M 2,4 2-Server Problem on M 2,4, C = Server Problem on M 2,4, C = This is optiml for M 2,4. Uniform spces i.e. pging, the optiml competitiveness is C = 1.5.

81 Results for the 2-Server Problem in M 2,4 2-Server Problem on M 2,4, C = Server Problem on M 2,4, C = This is optiml for M 2,4. Uniform spces i.e. pging, the optiml competitiveness is C = 1.5. Open: better thn 2-competitive rndomized lgorithm for 2 servers in generl spces.

82 Pging Online Computtion Pging: k-server problem in uniform spces CNN Amzon Yhoo..requesting NYTimes UNLV Sndi

83 History of k-pging [Fit, Krp, Luby, McGeoch, Sletor, Young, 1991] Lower bound, H k = k i=1 1/i (2H k 1)-competitive lgorithm RMARK RMARK uses O(n) memory

84 History of k-pging [Fit, Krp, Luby, McGeoch, Sletor, Young, 1991] Lower bound, H k = k i=1 1/i (2H k 1)-competitive lgorithm RMARK RMARK uses O(n) memory [McGeoch, Sletor, 1991] H k -competitive lgorithm PARTITION unbounded memory

85 History of k-pging [Fit, Krp, Luby, McGeoch, Sletor, Young, 1991] Lower bound, H k = k i=1 1/i (2H k 1)-competitive lgorithm RMARK RMARK uses O(n) memory [McGeoch, Sletor, 1991] H k -competitive lgorithm PARTITION unbounded memory [Achliopts, Chrobk, Nog, 2000] H k -competitive lgorithm EQUITABLE O(k 2 log k) memory

86 History of k-pging [Fit, Krp, Luby, McGeoch, Sletor, Young, 1991] Lower bound, H k = k i=1 1/i (2H k 1)-competitive lgorithm RMARK RMARK uses O(n) memory [McGeoch, Sletor, 1991] H k -competitive lgorithm PARTITION unbounded memory [Achliopts, Chrobk, Nog, 2000] H k -competitive lgorithm EQUITABLE O(k 2 log k) memory [Bein, Lrmore, Nog, 2007] H k -competitive lgorithm EQUITABLE2 O(k) memory

87 Bookmrks Online Computtion A b b c 1_ 2 B cb c b c b b A cb bookmrked A trckless lgorithm (i.e. n lgorithm tht does not use ny bookmrks) cnnot hve optiml competitiveness. [Bein, Fleischer, Lrmore, 00] _ 1 2

88 An Optimlly Competitive Algorithm for 2-Pging Algorithm K 2 : 1 c b c 2 b b c 1 c b 2 b b c d d d b b b c Competitiveness: C K2 = 3 2

89 Models of Online Computtion Behviorl Distributionl b 1_ c 1_ 2 2 c c 1 b 1_ 2 c b c b 1 1 c c c 1_ 2 1 b b

90 Work Functions nd Offset Functions d d c 0 c d c 2 2 b b 1 c b c b c b d 0 d c d c offset 1 offset b c b c b

91 Offset Function Guided Algorithm EQUITABLE [Achliopts, Chrobk, Nog, 2000] The lgorithm is descibed using the distributionl model The lgorithm s distribution mss is only on the support of the work funtion b 1 c _ 2 1 _ 2 1 b 3 d c 3 3 b 4 e d c d 3 c 3 3 b b 4 e d c b 1 e d b c

92 Ls Vegs Online Computtion Algorithm is constructed using the mixed model of online computtion b c _ 1 2 _ b d c c 3 3 d 1 1 b d c b _ 2_ 3 0 b 1 d c b

93 Behviorl Trnsltion _ 2 1 _ 2 1 c b d c b d c b c b d 1 d b c 1 d c b d b c 1 d c b d c b _ _2 3

94 Knowledge Stte Description of K 2 Work functions re denoted using the br nottion. A tuple T is in the support: t lest i members of T re to the left of the i th br. 0 A b 1, 1 new (c) 1/2 B c b new (d) 1/3, 1 C d bc 1/3 1/3 1/3 0 0 A d A d b 0 A d c

95 The Cse k = 3 0 A 5/6 B 5/3 1/5, 1 1, 1 1, 1 D new new new b c bcd bcde b, c or d 0, 1/3 b, c, d or e c 0, 1/2 1/2 C 0, 1/2 d or e b cd new 0, 3/4 b cde 0, 1 E 1 Ls Vegs Step c or d 0, 1/2 e or d 0, 3/4 5/4 F new 1, 1 bc de c or b 0, 1/4 new 1/ 6, 1 Ls Vegs Step Ls Vegs Step 0 1/2 P bcdef +6/5 0 Q bc def +1 R bcd ef+5/6 1/10 1/10 1/10 1/6 1/6 1/6 1 A b c A b d A e f C C b cd b cf C c df A b c 1/2 1/2 1/2 0

96 Equitble, Equitble2 For generl k: EQUITABLE forgives to cone fter O(k 2 log k) bookmrks

97 Equitble, Equitble2 For generl k: EQUITABLE forgives to cone fter O(k 2 log k) bookmrks EQUITABLE forgives to work function with smll support (but not cone) fter O(k) bookmrks

98 Further Reserch The 2-server problem for generl spces

99 Further Reserch The 2-server problem for generl spces CNN: Deterministic: lower bound: [Koutsoupis, Tylor, 2005] Deterministic: upper bound: 10 5, 879 [Sitters Stougie 2005] Rndomized: Open