2 Push and Pull Source Sink Push

Transcription

1 1 Plaing Push vs Pull: Models and Algorithms for Disseminating Dnamic Data in Networks. R.C. Chakinala, A. Kumarasubramanian, Kofi A. Laing R. Manokaran, C. Pandu Rangan, R. Rajaraman

2 Push and Pull 2 Source Sink

3 Push and Pull 2 Push Source Sink

4 Push and Pull 2 Push Source push Sink

5 Push and Pull 2 Push Source push Sink Pull

6 Push and Pull 2 Push Source push Sink Pull quer

7 Push and Pull 2 Push Source push Sink response Pull quer

8 Push and Pull 2 Push Source push Sink response Pull quer Mied Store

9 Push and Pull 2 Push Source push Sink response Pull quer Mied Store

10 Push and Pull 2 Push Source push Sink response Pull quer Mied Store

11 Push and Pull 2 Push Source push Sink response Pull quer Mied Store

12 A Simple Eample Using average source and sink frequencies. 3 Sources Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

13 A Simple Eample Using average source and sink frequencies. 3 Sources Push Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

14 A Simple Eample Using average source and sink frequencies. 3 Sources Push Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

15 A Simple Eample Using average source and sink frequencies. 3 Sources Push Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

16 A Simple Eample Using average source and sink frequencies. 3 Sources Pull Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

17 A Simple Eample Using average source and sink frequencies. 3 Sources Pull Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

18 A Simple Eample Using average source and sink frequencies. 3 Sources Pull Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

19 A Simple Eample Using average source and sink frequencies. 3 Sources Pull Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

20 A Simple Eample Using average source and sink frequencies. 3 Sources Pull Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

21 A Simple Eample Using average source and sink frequencies. 3 Sources Pull Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

22 A Simple Eample Using average source and sink frequencies. 3 Sources Mied Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

23 A Simple Eample Using average source and sink frequencies. 3 Sources Mied Store Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

24 A Simple Eample Using average source and sink frequencies. 3 Sources Mied Store Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

25 A Simple Eample Using average source and sink frequencies. 3 Sources Mied Store Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

26 A Simple Eample Using average source and sink frequencies. 3 Sources Mied Store Sink Aggregation No Aggregation Push 4 4 Pull 6 7 Mied Mied is Best 2 < 2 < 4 3 < 2 < 4

27 General Problem High Level 4 INPUTS: Graph G = (V, E) with: cost of updating set of stores: SetC : V Powerset(V ) R + Source Set P V, Sink Set Q V For ever source i P, a source frequenc p i For ever sink j Q, a sink frequenc q j For ever sink j Q, an interest set I j

28 General Problem High Level 4 INPUTS: Graph G = (V, E) with: cost of updating set of stores: SetC : V Powerset(V ) R + Source Set P V, Sink Set Q V For ever source i P, a source frequenc p i For ever sink j Q, a sink frequenc q j For ever sink j Q, an interest set I j OUTPUTS: For ever source i P, a Push set P i For ever sink j Q, a Pull Set Q j Intersection requirement: i I j P i Qj. MINIMIZE: total cost of push-updates, queries and responses: i Pp i SetC(i, P i )+ j Qq j SetC(j, Q j )+ j Q q j RespC(j)

29 Routing Cost Models 5 Multicast Cost Eample Definition Multicast 3 Steiner tree cost Unicast 4 Sum of path costs (non-)metric Distance function Broadcast Model 5 Breadth first tree cost to depth r

30 Routing Cost Models 5 Unicast Cost Eample Definition Multicast 3 Steiner tree cost Unicast 4 Sum of path costs (non-)metric Distance function Broadcast Model 5 Breadth first tree cost to depth r

31 Routing Cost Models 5 Controlled Broadcast Cost Eample Definition Multicast 3 Steiner tree cost Unicast 4 Sum of path costs (non-)metric Distance function Broadcast Model 5 Breadth first tree cost to depth r

32 Related Work 6 FeedTree: RSS via P2P Multicast, [Sandler et al., IPTPS 05] Web Caching applications Combs, Needles and Hastacks Paper, [Liu et al. SENSYS 04] Data Gerrmandering, [Bagchi et al. T.A. TKDE] Minimum Cost 2-spanners: [Dodis & Khanna STOC 99] and [Kortsarz & Peleg SICOMP 98] Multicommodit facilit location, [Ravi & Sinha SODA 04] Classical Theor Problems Facilit Location Steiner Tree (including Group Steiner Tree)

33 Our Results 7 Multicast Model Eact Tree Algorithm (Distributed) General Graphs O(log n)-approimation NP-Completeness

34 Our Results 7 Multicast Model Eact Tree Algorithm (Distributed) General Graphs O(log n)-approimation NP-Completeness Unicast Model Nonmetric Case O(log n)-approimation Identical Interest Sets / Metric Case O(1)-Approimation NP-Completeness

35 Our Results 7 Multicast Model Eact Tree Algorithm (Distributed) General Graphs O(log n)-approimation NP-Completeness Unicast Model Nonmetric Case O(log n)-approimation Identical Interest Sets / Metric Case O(1)-Approimation NP-Completeness Controlled Broadcast Model A Polnomial LP solution A Combinatorial solution

36 The Multicast Model With Aggregation 8 want the following A push subtree T i for each source i A pull subtree T j for each sink j Whenever j is interested in i (i I j ), T i T j. Total cost of all trees (summing edge weights in each tree) is minimized. For Trees: Basic idea: for each edge, compute minimum possible cost for connectedness of trees. Claim: Global optimum consists of this solution at ever edge.

37 The Multicast Model 9 Indicator variable uvi sas whether uv T i (push tree i) uvj indicates uv T j (pull tree j) z uvij indicates i I j and uv P (T i T j, j) arbitrar m ij is average response frequenc Minimize Objective function p i uv uvi + uv Ec q j uv uvj + j Q uv Ec i P i P j Q m ij c uv z uvij uv E

38 Multicast Model An Eact (Distributed) Tree Algorithm 10 G is a tree T = (V, E) MinC(T i T j, j) is sum of edge weights on shortest path P (T i, j) For edge uv, let S uv be largest subtree containing u but not v Note S vu = V \ S uv Substituting V = S uv S vu, we obtain two smmetric terms (eg.): c uv p i uvi + q j uvj + m ij z uvij i S uv j S vu j S vu uv E i S uv Claim: Global optimum minimizes [...] independentl!

39 Tree Algorithm Diagram 11 a b c z Interest sets: {, z} want {a, b, c}; wants onl a.

40 Tree Algorithm Diagram 11 a b c z Interest sets: {, z} want {a, b, c}; wants onl a.

41 Tree Algorithm Diagram 11 a b c v w z Interest sets: {, z} want {a, b, c}; wants onl a. Question: What is the minimum we can pa on edge vw?

42 Tree Algorithm Diagram 11 a b a b c z c v w z Interest sets: {, z} want {a, b, c}; wants onl a. Question: What is the minimum we can pa on edge vw?

43 Bipartite Minimum Weight Verte Cover 12 a b c z

44 Bipartite Minimum Weight Verte Cover 12 a b c z Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R)

45 Bipartite Minimum Weight Verte Cover 12 a s b z t c Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R)

46 Bipartite Minimum Weight Verte Cover 12 s a b c z t Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R)

47 Bipartite Minimum Weight Verte Cover 12 A B s a b c z t Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R)

48 Bipartite Minimum Weight Verte Cover 12 a b c z Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R) Application: Set A = P vw and B = Q vw

49 Bipartite Minimum Weight Verte Cover 12 a b c z Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R) Application: Set A = P vw and B = Q vw { ij (i, j) X vw } for response costs.

50 Bipartite Minimum Weight Verte Cover 12 a b c z Well Known: For bipartite G vw = (A B, E), MWVC P (Ma flow). Find min cut R, to get MWVC C vw = (A \ R) (B R) Application: Set A = P vw and B = Q vw { ij (i, j) X vw } for response costs. Lemma 1. For each arc e = vw, the MWVC weight of G vw is the minimum value paid for vw in an optimal solution.

51 Tree Algorithm Tiebreaking 13 a b c z Long chain, no sources or sinks.

52 Tree Algorithm Tiebreaking 13 a a b z a b z a b z b c c c c z Long chain, no sources or sinks. Identical Bipartite graph problem

53 Tree Algorithm Tiebreaking 13 a a b z a b z a b z b c c c c z Long chain, no sources or sinks. Identical Bipartite graph problem Suppose man possible MWVCs (eg a + b + c = a + z = + z). How to break MWVC ties?

54 Tree Algorithm Tiebreaking 13 a a b z a b z a b z b c c c c z Long chain, no sources or sinks. Identical Bipartite graph problem Suppose man possible MWVCs (eg a + b + c = a + z = + z). How to break MWVC ties? Defn: In bipartite G = (A B, E), an MWVC is A-maimum if it has maimum weight in A.

55 A Consistent Tiebreaking Solution 14 Defn: In bipartite G = (A B, E), an MWVC is A-maimum if it has maimum weight in A.

56 A Consistent Tiebreaking Solution 14 Defn: In bipartite G = (A B, E), an MWVC is A-maimum if it has maimum weight in A. A E A 1 B 1 B A 2 B 2 Lemma 2. Let G = (A B, E), let A 1, A 2 A and let B 1, B 2 B. If A 1 B 1 and A 2 B 2 are both A-maimum minimum weight verte covers,...

57 A Consistent Tiebreaking Solution 14 Defn: In bipartite G = (A B, E), an MWVC is A-maimum if it has maimum weight in A. A E B A 1 B 1 A 2 B 2 Lemma 2. Let G = (A B, E), let A 1, A 2 A and let B 1, B 2 B. If A 1 B 1 and A 2 B 2 are both A-maimum minimum weight verte covers,... then A 1 = A 2 and B 1 = B 2. Unique solution per edge!

58 Tree Algorithm Structural Continuit 15 a b a b c z c v w z Interest sets recall: {, z} want {a, b, c}; wants onl a.

59 Tree Algorithm Structural Continuit 15 a b u c v w z Interest sets recall: {, z} want {a, b, c}; wants onl a. What about G uv? Clearl different.

60 Tree Algorithm Structural Continuit 15 a a b z b u c v w z Interest sets recall: {, z} want {a, b, c}; wants onl a. What about G uv? Clearl different.

61 Tree Algorithm Structural Continuit 15 a b u a b z a b c z c v w z Interest sets recall: {, z} want {a, b, c}; wants onl a. What about G uv? Clearl different.

62 Tree Algorithm Structural Continuit 15 a b u a b z a b c z c v w z Interest sets recall: {, z} want {a, b, c}; wants onl a. What about G uv? Clearl different. Are push trees, pull trees and response paths connected? Lemma 3. If we compute push-maimum MWVC for ever edge, then Push and Pull subtrees are connected.

63 Yes!!! a b u a b Structural Continuit Solution a z b z c 16 c v w z

64 Yes!!! a b Structural Continuit Solution a z b z c 16

65 Yes!!! a b a b z Structural Continuit Solution a z b z c 16 c

66 Yes!!! Structural Continuit Solution 16 a b c z

67 Yes!!! Structural Continuit Solution A B 16 a b c z Lemma 4. Let uvw be two consecutive edges, let A be the set of push nodes in G uv, and let B be the set of (non-push) nodes in G vw.

68 Yes!!! Structural Continuit Solution A A 1 B 1 B 16 a b c z Lemma 4. Let uvw be two consecutive edges, let A be the set of push nodes in G uv, and let B be the set of (non-push) nodes in G vw. Let A 1, B 1 be parts of push-maimum MWVC of G vw in A, B resp., and

69 Yes!!! Structural Continuit Solution A A B 1 B 1 16 a b z A 2 B 2 c Lemma 4. Let uvw be two consecutive edges, let A be the set of push nodes in G uv, and let B be the set of (non-push) nodes in G vw. Let A 1, B 1 be parts of push-maimum MWVC of G vw in A, B resp., and A 2, B 2 be parts of push-maimum MWVC of G uv in A, B resp.

70 Yes!!! Structural Continuit Solution A A 2 B 1 B 16 a b z A 1 B 2 c Lemma 4. Let uvw be two consecutive edges, let A be the set of push nodes in G uv, and let B be the set of (non-push) nodes in G vw. Let A 1, B 1 be parts of push-maimum MWVC of G vw in A, B resp., and A 2, B 2 be parts of push-maimum MWVC of G uv in A, B resp. then A 1 A 2 and B 1 B 2. Push/Pull subtrees, Response paths are connected!

71 Tree Algorithm 17 for each directed edge uv construct the graph G uv find its canonical minimum cut C uv for all i P uv if i C uv then include uv in T i for all j Q vu if j C uv then include uv in T j for all (i, j) X uv if ij C uv then include uv in P (T i, j)

72 Distributed Implementation 18 Global All-to-all echange of sets of push nodes frequencies, pull nodes frequencies and interest sets. Locall, each edge solves both its directions independentl. Use the solution to push and pull information Notes: Cost of first phase small compared to third. For small sets of distinct values, communication improved.

73 Multicast Model General Graph Approimation algorithm 19 Reduction from Min Steiner Tree; NP-hard to approimate within 96/95. Chlebìk & Chlebìkovà SWAT 02 Theorem 1. There is an epected O(log n)-approimation for the Multicast problem in general graphs.

74 Multicast Model General Graph Approimation algorithm 19 Reduction from Min Steiner Tree; NP-hard to approimate within 96/95. Chlebìk & Chlebìkovà SWAT 02 Theorem 1. There is an epected O(log n)-approimation for the Multicast problem in general graphs. We use the following: Theorem 2 (Fakcharoenphol et al. STOC 03). The distribution over tree metrics resulting from (their) algorithm O(log n)-probabilisticall approimates the metric d.

75 General Graph Approimation algorithm ctd. 20 Bound Derivation Choose T randoml from distribution of metric-spanning trees. Project structures in G into T. Obtain feasible solution for T. OP T (T ) O(log n) OP T (G). Approimation Algorithm Solve T eactl using our algorithm. Project structures in T into G. Obtain feasible solution for G. ALG(G) 2 OP T (T )

76 General Graph Approimation algorithm ctd. 20 Bound Derivation Choose T randoml from distribution of metric-spanning trees. Project structures in G into T. Obtain feasible solution for T. OP T (T ) O(log n) OP T (G). Approimation Algorithm Solve T eactl using our algorithm. Project structures in T into G. Obtain feasible solution for G. ALG(G) 2 OP T (T ) O(log n) OP T (G).

77 Multicast Model Hardness 21 Multicast problem with(out) aggregation: eas reduction from Min Steiner tree. Arbitrar node becomes low-freq source Rest become high-freq Sink nodes Each interested in Source

78 Multicast Model Hardness 21 Multicast problem with(out) aggregation: eas reduction from Min Steiner tree. Arbitrar node becomes low-freq source Rest become high-freq Sink nodes Each interested in Source

79 Multicast Model Hardness 21 Multicast problem with(out) aggregation: eas reduction from Min Steiner tree. Arbitrar node becomes low-freq source Rest become high-freq Sink nodes Each interested in Source Min Steiner Tree NP-hard to approimate within 96/95. Chlebìk & Chlebìkovà SWAT 02

80 The Unicast Model 22

81 The Unicast Model 22 Given (non-)metric distances d uv for ever pair (u, v) V V. SetC(u, S) = k S d uk find push-sets P i and pull-sets Q j that minimize total communication cost: p i d ik + q j d kj + q j RespC(j), i P k P i j Q k Q j j Q and satisfies: for all i I j, P i Q j where RespC(j) = { SetC(j, Qj ) (aggregation model) i I j MinC(P i Q j, j) otherwise.

82 Unicast Model with Aggregation An Integer Program 23 Replace response cost b doubling sink frequencies ik = 1 means i pushes to k kj = 1 means j pulls from k r ijk = 1 means i talks to j through k. Minimize: p i d ik ik + q j d kj kj i P j Q k V k V subject to r ijk ik r ijk kj k r ijk 1, where ik, kj, r ijk {0, 1}.

83 Unicast Model with Aggregation Nonmetric Case via Randomized Rounding 24 Convert to LP: Use 0 instead of {0, 1} Solve and discard values 1/n 2 and scale b n/(n 1) Round values up to powers of 1/2, obtain (, ỹ, z) For node k and 0 p < 2 log n, define X pk as i such that ik 1/2 p. p, k: with probabilit min{1, (log n)/2 p } add k to P i and Q j for all i X pk and j Y pk. Theorem 3. With high probabilit, solution is feasible, with cost O(log n) OP T LP.

84 Unicast Model with Aggregation Nonmetric Case via Randomized Rounding Proof 25 Since i X log( rijk )k and j Y log( rijk )k, Pr[k P i Q j ] min{1, r ijk log(n)}. Clearl Pr[k P i ] p:i X pk (log n)/2 p = 2 ik log n min{1, 2 ik log n}. Pr[P i Q j = ] = k (1 r ijk log n) e P k r ijk log n 1/n 2. Define r.v. C i as push cost for i, and r.v. C ik takes value d ik with probabilit min{1, 2 ik log n}. Chernoff-Hoeffding: w.h.p. k C ik O(log n) k d ik ik. Summing over all sources, sinks gives cost bound w.h.p.

85 Unicast Model with Aggregation Uniform Interests, Metric Case O(1)-Approimation 26 Overview Applies for Identical/Disjoint Interest Sets Uses same Integer Program. Deterministic Rounding with Filtering Technique Lin & Vitter IPL 92, Shmos et al STOC 97, Ravi & Sinha SODA 04

86 Unicast Model with Aggregation Uniform Interest Sets in Metric Case Intro 27 Basic definitions Optimal solution to the LP is (,, r ). LP gives cost lower bounds C i = k d ik ik and C j = k d kj kj 2*(1/2) C = 6.5 j j 3*(1/2) 4*(1)

87 Unicast Model with Aggregation Uniform Interest Sets in Metric Case Intro 27 Basic definitions Optimal solution to the LP is (,, r ). LP gives cost lower bounds C i = k d ik ik and C j = k d kj kj For node u, r > 0, define B u (r) = {v : d uv r}. Let 1 < α < β. Clearl B j (C j ) B j(αc j ) B j(βc j ) j

88 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

89 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

90 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

91 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

92 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

93 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

94 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

95 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

96 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost.

97 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost. Define push sets: P i = {i} {l i } {j : j S and C j C i} and pull sets: Q j = {j} {l j } {i : i S and C i < C j }. j

98 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm 28 Choose leaders: nodes with disjoint β-balls, b nondecreasing cost. Define push sets: P i = {i} {l i } {j : j S and C j C i} and pull sets: Q j = {j} {l j } {i : i S and C i < C j }. Intersection guarantee: For each i P and j Q, P i Q j. m i j

99 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm Proof Relative distance limits total push etent: For i P, α > 1, ik 1/α k / B i (αc i ) 29 j C j 4C j

100 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm Proof Relative distance limits total push etent: For i P, α > 1, ik 1/α k / B i (αc i ) Derive Approimation Ratio. Recall: P i = {i} {l i } {j : j S and C j C i} Cost to i s leader l i : 2βC i 29 j l j

101 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm Proof Relative distance limits total push etent: For i P, α > 1, ik 1/α k / B i (αc i ) Derive Approimation Ratio. Recall: P i = {i} {l i } {j : j S and C j C i} Cost to i s leader l i : 2βC i Cost to (other) leaders S i : 29 C i j S i (d ij αc j ) k B j (αc j ) r ijk i j

102 Unicast Model with Aggregation Uniform Interest Set / Metric Algorithm Proof Relative distance limits total push etent: For i P, α > 1, ik 1/α k / B i (αc i ) Derive Approimation Ratio. Recall: P i = {i} {l i } {j : j S and C j C i} Cost to i s leader l i : 2βC i Cost to (other) leaders S i : 29 C i j S i (d ij αc j ) k B j (αc j ) r ijk [1 ] j S i d ij [1 β] α 1 α j S i d ij. = (β α)(α 1) αβ α = 1.69 and β = 2.86 obtains approimation.

103 Conclusions and Open Problems 30 Nonuniform Packet Lengths Multicast: General Graphs; Can O(log n) UB be improved to O(1)? Nonmetric Unicast: Derandomizing O(log n) algorithm. Close gap O(1) LB vs O(log n) UB gap Metric Unicast Case Improving the bound for Uniform Interest sets. Non-uniform interest sets (UB and/or Hardness) Dnamic Graphs Frequenc, Position and Topolog changes

104 Thank You! 31 Chakinala, Kumarasubramanian, and Manokaran: partial support generous gift from Northeastern Universit alumnus Madhav Anand. Rajaraman, partial support NSF grant IIS Laing, partial support the Mellon Foundation and the Facult Research Awards Committee of Tufts Universit, while visiting Northeastern Universit.