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Matching Size and Matrix Rank Estimation in Data Streams Sepehr Assadi University of Pennsylvania Joint work with Sanjeev Khanna (Penn), and Yang Li (Penn) Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar The Streaming Model

  1. Estimating Maximum Matching Size Question. Is matching size estimation strictly easier than finding an approximate matching? Yes! Theorem There is a randomized algorithm that outputs an – -approximate estimate of maximum matching size in: Â O ( n/ – 2 ) space in insertion-only streams. Â O ( n 2 / – 4 ) space in dynamic streams. In constrast, to find an – -approximate matching, the space necessary is: Ω ( n/ – ) in insertion-only streams. Ω ( n 2 / – 3 ) in dynamic streams. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  2. Algorithms for – -Estimation of Matching Size The main ingredient of our algorithms is the following sampling lemma: Lemma (Vertex Sampling Lemma) Let H be a subgraph of G obtained by sampling each vertex independently w.p. 1 / – . Define: µ G : the maximum matching size in G , µ H : the maximum matching size in H . Then, w.h.p., µ G – 2 Æ µ H Æ 2 µ G – . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  3. Algorithms for – -Estimation of Matching Size The main ingredient of our algorithms is the following sampling lemma: Lemma (Vertex Sampling Lemma) Let H be a subgraph of G obtained by sampling each vertex independently w.p. 1 / – . Define: µ G : the maximum matching size in G , µ H : the maximum matching size in H . Then, w.h.p., µ G – 2 Æ µ H Æ 2 µ G – . Therefore, maximum matching size in H is an – -estimation for the maximum matching size in G . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  4. Proof by Picture Any graph G with a maximum matching size of µ G looks as follows: A matching of size µ G G between the blue vertices. L R No edges between the green vertices. µ G Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  5. Proof by Picture The vertex sampled graph H then look as follows: A matching of size µ G / – 2 H between the blue vertices L R ∆ µ H Ø µ G / – 2 . = All edges are incident on µ G / – blue vertices µ G / – = ∆ µ H Æ 2 µ G / – . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  6. An – -Estimation Algorithm To distinguish between graphs with maximum matching of size Ø k and o ( k/ – ) : Sample each vertex in G w.p. 1 / – to obtain H . 1 Test whether H has a matching of size at least Ω ( k/ – 2 ) or not. 2 Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  7. An – -Estimation Algorithm To distinguish between graphs with maximum matching of size Ø k and o ( k/ – ) : Sample each vertex in G w.p. 1 / – to obtain H . 1 Test whether H has a matching of size at least Ω ( k/ – 2 ) or not. 2 Can be implemented in: O ( k/ – 2 ) = Â Â O ( n/ – 2 ) in insertion-only streams. O ( k 2 / – 4 ) = Â Â O ( n 2 / – 4 ) in dynamic streams. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  8. Estimating Maximum Matching Size Matching size estimation is indeed easier than finding an approximate matching! Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  9. Estimating Maximum Matching Size Matching size estimation is indeed easier than finding an approximate matching! Question. Is it possible to achieve an arbitrary good estimation of matching size in sub-quadratic space? Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  10. Estimating Maximum Matching Size Matching size estimation is indeed easier than finding an approximate matching! Question. Is it possible to achieve an arbitrary good estimation of matching size in sub-quadratic space? Question. In general, what is the space-approximation tradeo ff for matching size estimation? Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  11. Estimating Maximum Matching Size Matching size estimation is indeed easier than finding an approximate matching! Question. Is it possible to achieve an arbitrary good estimation of matching size in sub-quadratic space? Question. In general, what is the space-approximation tradeo ff for matching size estimation? We make progress on each of these questions. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  12. Our Main Results Near-optimal approximation of maximum matching size may require almost quadratic space even in insertion-only streams. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  13. Our Main Results Near-optimal approximation of maximum matching size may require almost quadratic space even in insertion-only streams. Theorem Any randomized (1 + Á ) -approximate estimation of maximum matching size requires: RS ( n ) · n 1 − O ( Á ) space in insertion-only streams. n 2 − O ( Á ) space in dynamic streams. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  14. Our Main Results Near-optimal approximation of maximum matching size may require almost quadratic space even in insertion-only streams. Theorem Any randomized (1 + Á ) -approximate estimation of maximum matching size requires: RS ( n ) · n 1 − O ( Á ) space in insertion-only streams. n 2 − O ( Á ) space in dynamic streams. RS ( n ) denotes the maximum number of edge-disjoint induced matchings of size Θ ( n ) in an n -vertex graph: [ Fischer et al., 2002 ] n Ω (1 / log log n ) Æ RS ( n ) Æ n/ log n [ Fox et al., 2015 ] Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  15. Our Main Results We further establish the first non-trivial lower bound for super-constant approximation of matching size. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  16. Our Main Results We further establish the first non-trivial lower bound for super-constant approximation of matching size. Theorem Any randomized – -approximate estimate of maximum matching size requires Ω ( n/ – 2 ) in dynamic streams. Furthermore, even if we restrict to sparse graphs with arboricity O ( – ) , Ω ( Ô n/ – 2 . 5 ) space is necessary. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  17. Our Main Results We further establish the first non-trivial lower bound for super-constant approximation of matching size. Theorem Any randomized – -approximate estimate of maximum matching size requires Ω ( n/ – 2 ) in dynamic streams. Furthermore, even if we restrict to sparse graphs with arboricity O ( – ) , Ω ( Ô n/ – 2 . 5 ) space is necessary. There is an active line of research on estimating matching size of bounded arboricity graphs in graph streams [Chitnis et al., 2016] [Bury and Schwiegelshohn, 2015] [Esfandiari et al., 2015] [McGregor and Vorotnikova, 2016b] [Cormode et al., 2016] [McGregor and Vorotnikova, 2016a] . . . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  18. Schatten p -Norms Given an n ◊ n matrix A , for any p œ [0 , Œ ) : Schatten p -norm of A is the p -th frequency moment of vector of singular values ( ‡ 1 , . . . , ‡ n ) of A . A n B 1 /p ÿ ‡ p Î A Î p := i i =1 Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  19. Schatten p -Norms Given an n ◊ n matrix A , for any p œ [0 , Œ ) : Schatten p -norm of A is the p -th frequency moment of vector of singular values ( ‡ 1 , . . . , ‡ n ) of A . A n B 1 /p ÿ ‡ p Î A Î p := i i =1 Î A Î 0 = Rank of A . Î A Î 1 = Trace norm of A . Î A Î 2 = Frobenius norm of A . Î A Î ∞ = Operator norm of A . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  20. Schatten p -Norms in Data Streams Question. What is the space complexity of (1 + Á ) -approximating the Schatten p -norm of a matrix which its entries are revealed in a data stream? Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  21. Schatten p -Norms in Data Streams Question. What is the space complexity of (1 + Á ) -approximating the Schatten p -norm of a matrix which its entries are revealed in a data stream? Previous work: For p = 0 , Ω ( n 1 − g ( Á ) ) space is necessary [Bury and Schwiegelshohn, 2015]. For p œ (0 , Œ ) \ 2 Z , Ω ( n 1 − g ( Á ) ) space is necessary [Li and Woodru ff , 2016]. For p œ 2 Z \ { 0 } , Ω ( n 1 − 2 /p ) space is necessary [Li and Woodru ff , 2016] (and is su ffi cient for sparse matrices). Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  22. Schatten p -Norms in Data Streams Question. What is the space complexity of (1 + Á ) -approximating the Schatten p -norm of a matrix which its entries are revealed in a data stream? Previous work: For p = 0 , Ω ( n 1 − g ( Á ) ) space is necessary [Bury and Schwiegelshohn, 2015]. For p œ (0 , Œ ) \ 2 Z , Ω ( n 1 − g ( Á ) ) space is necessary [Li and Woodru ff , 2016]. For p œ 2 Z \ { 0 } , Ω ( n 1 − 2 /p ) space is necessary [Li and Woodru ff , 2016] (and is su ffi cient for sparse matrices). We answer this question for the case of rank computation. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  23. Matrix Rank Computation in Data Streams It is well-known that computing maximum matching size of a graph is equivalent to computing the rank of the (symbolic) Tutte matrix. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  24. Matrix Rank Computation in Data Streams It is well-known that computing maximum matching size of a graph is equivalent to computing the rank of the (symbolic) Tutte matrix. As a corollary, all our lower bounds for matching size estimation also extend to the matrix rank computation problem. In particular, An Ω ( n 2 − O ( Á ) ) space lower bound for (1 + Á ) -estimation of rank in dense matrices. Ω ( Ô n ) space lower bound for any polylog( n ) -estimation of An  rank in sparse matrices. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  25. An Ω ( n 2 ≠ O ( Á ) ) Lower Bound for Dynamic Streams Theorem Any randomized (1 + Á ) -approximate estimation of maximum matching size requires Ω ( n 2 − O ( Á ) ) space in dynamic streams. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  26. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound Consider the following two-player one-way communication problem. MaxMatching : Alice is given a matching M on vertices V . 1 Bob is given a collection of edges E B on vertices V . 2 Alice sends a single message to Bob and Bob outputs an 3 estimation of maximum matching size in G ( V, M fi E B ) . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  27. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound Consider the following two-player one-way communication problem. MaxMatching : Alice is given a matching M on vertices V . 1 Bob is given a collection of edges E B on vertices V . 2 Alice sends a single message to Bob and Bob outputs an 3 estimation of maximum matching size in G ( V, M fi E B ) . CC( MaxMatching ): minimum length message to solve this problem with probability, say, 2 / 3 . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  28. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound Consider the following two-player one-way communication problem. MaxMatching : Alice is given a matching M on vertices V . 1 Bob is given a collection of edges E B on vertices V . 2 Alice sends a single message to Bob and Bob outputs an 3 estimation of maximum matching size in G ( V, M fi E B ) . CC( MaxMatching ): minimum length message to solve this problem with probability, say, 2 / 3 . Fact. CC( MaxMatching ) Æ space complexity of any streaming algorithm for estimating maximum matching size. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  29. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  30. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) Alice is given a random subset of size n/ 2 from a fixed perfect matching between L and R . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  31. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) Alice is given a random subset of size n/ 2 from a fixed perfect matching between L and R . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  32. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) Alice is given a random subset of size n/ 2 from a fixed perfect matching between L and R . Bob is given a matching of size n/ 2 incident on R . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  33. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) Alice is given a random subset of size n/ 2 from a fixed perfect matching between L and R . Bob is given a matching of size n/ 2 incident on R . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  34. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) Alice is given a random subset of size n/ 2 from a fixed perfect matching between L and R . Bob is given a matching of size n/ 2 incident on R . Yes case: Each edge of Bob’s matching is incident on even number of Alice’s matching = ∆ MaxMatching = 3 n/ 4 . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  35. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) Alice is given a random subset of size n/ 2 from a fixed perfect matching between L and R . Bob is given a matching of size n/ 2 incident on R . No case: Each edge of Bob’s matching is incident on odd number of Alice’s matching = ∆ MaxMatching = n/ 2 . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  36. Previous Approaches: An n 1 ≠ O ( Á ) Lower Bound [Bury and Schwiegelshohn, 2015]: CC( MaxMatching ) = Ω ( n 1 − O ( Á ) ) . Proof Sketch. (for Á = 1 / 2 ) A better than 3 / 2 -approximation distinguishes between the two cases. Distinguishing between the two cases requires Ω ( Ô n ) communication by a reduction from the boolean hidden matching problem of [Gavinsky et al., 2007]. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  37. Our Approach A natural idea to boost the previous lower bound: Instead of one matching M , provide Alice with t independently 1 chosen matchings M 1 , . . . , M t . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  38. Our Approach A natural idea to boost the previous lower bound: Instead of one matching M , provide Alice with t independently 1 chosen matchings M 1 , . . . , M t . Provide Bob with a single set E B of edges as before. 2 Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  39. Our Approach A natural idea to boost the previous lower bound: Instead of one matching M , provide Alice with t independently 1 chosen matchings M 1 , . . . , M t . Provide Bob with a single set E B of edges as before. 2 “Ask” Alice and Bob to solve the MaxMatching problem for 3 a uniformly at random chosen matching M j ı and E B (the index j ı is unknown to Alice). Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  40. Our Approach A natural idea to boost the previous lower bound: Instead of one matching M , provide Alice with t independently 1 chosen matchings M 1 , . . . , M t . Provide Bob with a single set E B of edges as before. 2 “Ask” Alice and Bob to solve the MaxMatching problem for 3 a uniformly at random chosen matching M j ı and E B (the index j ı is unknown to Alice). The hope is that communication complexity of this problem is now Ø t · CC( MaxMatching ). Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  41. Our Approach There are three main obstacles in implementing this idea: The matchings M 1 , . . . , M t should be supported on Θ ( n ) 1 vertices as opposed to trivial Θ ( t · n ) vertices. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  42. Our Approach There are three main obstacles in implementing this idea: The matchings M 1 , . . . , M t should be supported on Θ ( n ) 1 vertices as opposed to trivial Θ ( t · n ) vertices. The matchings should be chosen independently even though 2 they are supported on the same set of Θ ( n ) vertices. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  43. Our Approach There are three main obstacles in implementing this idea: The matchings M 1 , . . . , M t should be supported on Θ ( n ) 1 vertices as opposed to trivial Θ ( t · n ) vertices. The matchings should be chosen independently even though 2 they are supported on the same set of Θ ( n ) vertices. The reduction should ensure that Alice and Bob indeed need to 3 solve the j ı -th embedded instance. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  44. Our Approach There are three main obstacles in implementing this idea: The matchings M 1 , . . . , M t should be supported on Θ ( n ) 1 vertices as opposed to trivial Θ ( t · n ) vertices. The matchings should be chosen independently even though 2 they are supported on the same set of Θ ( n ) vertices. The reduction should ensure that Alice and Bob indeed need to 3 solve the j ı -th embedded instance. (1) + (2) = ∆ Ruzsa-Szemer´ edi graphs (RS graphs). Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  45. Our Approach There are three main obstacles in implementing this idea: The matchings M 1 , . . . , M t should be supported on Θ ( n ) 1 vertices as opposed to trivial Θ ( t · n ) vertices. The matchings should be chosen independently even though 2 they are supported on the same set of Θ ( n ) vertices. The reduction should ensure that Alice and Bob indeed need to 3 solve the j ı -th embedded instance. (1) + (2) = ∆ Ruzsa-Szemer´ edi graphs (RS graphs). RS graphs + (3) = ∆ characterization of dynamic streaming algorithms via simultaneous communication complexity. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  46. Our Approach There are three main obstacles in implementing this idea: The matchings M 1 , . . . , M t should be supported on Θ ( n ) 1 vertices as opposed to trivial Θ ( t · n ) vertices. The matchings should be chosen independently even though 2 they are supported on the same set of Θ ( n ) vertices. The reduction should ensure that Alice and Bob indeed need to 3 solve the j ı -th embedded instance. (1) + (2) = ∆ Ruzsa-Szemer´ edi graphs (RS graphs). RS graphs + (3) = ∆ characterization of dynamic streaming algorithms via simultaneous communication complexity. Formalizing the lower bound = ∆ a direct-sum style argument using information complexity. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  47. Ruzsa-Szemer´ edi Graphs Definition ( ( r, t ) -RS graphs) A graph G ( V, E ) whose edges can be partitioned into t induced matchings of size r each. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  48. Ruzsa-Szemer´ edi Graphs Definition ( ( r, t ) -RS graphs) A graph G ( V, E ) whose edges can be partitioned into t induced matchings of size r each. Example. A (2 , 4) -RS graph 1 on 8 vertices: Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  49. Ruzsa-Szemer´ edi Graphs Definition ( ( r, t ) -RS graphs) A graph G ( V, E ) whose edges can be partitioned into t induced matchings of size r each. Example. A (2 , 4) -RS graph 1 on 8 vertices: Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  50. Ruzsa-Szemer´ edi Graphs Definition ( ( r, t ) -RS graphs) A graph G ( V, E ) whose edges can be partitioned into t induced matchings of size r each. Example. A (2 , 4) -RS graph 1 on 8 vertices: Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  51. Ruzsa-Szemer´ edi Graphs Definition ( ( r, t ) -RS graphs) A graph G ( V, E ) whose edges can be partitioned into t induced matchings of size r each. Example. A (2 , 4) -RS graph 1 on 8 vertices: Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  52. Ruzsa-Szemer´ edi Graphs Definition ( ( r, t ) -RS graphs) A graph G ( V, E ) whose edges can be partitioned into t induced matchings of size r each. Example. A (2 , 4) -RS graph 1 on 8 vertices: Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  53. Ruzsa-Szemer´ edi Graphs We are typically interested in RS graphs with large values of r and t as a function of n . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  54. Ruzsa-Szemer´ edi Graphs We are typically interested in RS graphs with large values of r and t as a function of n . How dense a graph with many large induced matching can be? Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  55. Ruzsa-Szemer´ edi Graphs We are typically interested in RS graphs with large values of r and t as a function of n . How dense a graph with many large induced matching can be? Theorem ([Fischer et al., 2002]) There exists an ( r, t ) -RS graph on n vertices with t = n Ω (1 / log log n ) induced matchings of size r = (1 ≠ Á ) · n/ 4 . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  56. Ruzsa-Szemer´ edi Graphs We are typically interested in RS graphs with large values of r and t as a function of n . How dense a graph with many large induced matching can be? Theorem ([Fischer et al., 2002]) There exists an ( r, t ) -RS graph on n vertices with t = n Ω (1 / log log n ) induced matchings of size r = (1 ≠ Á ) · n/ 4 . Theorem ([Alon et al., 2012]) There exists an ( r, t ) -RS graph on n vertices with t = n 1+ o (1) induced matchings of size r = n 1 − o (1) . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  57. A Simple Lower Bound for Insertion-Only Streams Let G 1 be an ( r, t ) -RS bipartite graph on n vertices on each side. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  58. A Simple Lower Bound for Insertion-Only Streams Let G 1 be an ( r, t ) -RS bipartite graph on n vertices on each side. To Alice, we give random subset of size r/ 2 from each induced matchings M 1 , . . . , M t of G 1 . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  59. A Simple Lower Bound for Insertion-Only Streams Let G 1 be an ( r, t ) -RS bipartite graph on n vertices on each side. To Alice, we give random subset of size r/ 2 from each induced matchings M 1 , . . . , M t of G 1 . Choose M j ı uniformly at random. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  60. A Simple Lower Bound for Insertion-Only Streams Let G 1 be an ( r, t ) -RS bipartite graph on n vertices on each side. To Alice, we give random subset of size r/ 2 from each induced matchings M 1 , . . . , M t of G 1 . Choose M j ı uniformly at random. To Bob we give the following input: I A matching between vertices not in M j ı and a new set of vertices. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  61. A Simple Lower Bound for Insertion-Only Streams Let G 1 be an ( r, t ) -RS bipartite graph on n vertices on each side. To Alice, we give random subset of size r/ 2 from each induced matchings M 1 , . . . , M t of G 1 . Choose M j ı uniformly at random. To Bob we give the following input: I A matching between vertices not in M j ı and a new set of vertices. I A graph E B over the set of vertices in M j ı . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  62. A Simple Lower Bound for Insertion-Only Streams Size of the maximum matching in this graph: 2( n ≠ r ) + MaxMatching ( M j ı , E B ) Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  63. A Simple Lower Bound for Insertion-Only Streams Size of the maximum matching in this graph: 2( n ≠ r ) + MaxMatching ( M j ı , E B ) For r = Θ ( n ) , Alice and Bob need to solve MaxMatching ( M j ı , E B ) for (1 + Á ) -approximation. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  64. A Simple Lower Bound for Insertion-Only Streams Size of the maximum matching in this graph: 2( n ≠ r ) + MaxMatching ( M j ı , E B ) For r = Θ ( n ) , Alice and Bob need to solve MaxMatching ( M j ı , E B ) for (1 + Á ) -approximation. To solve this for an unknown matching M j ı , the message length must be Ø t · CC( MaxMatching ). Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  65. A Simple Lower Bound for Insertion-Only Streams Main limitation of this approach: Requires r = Θ ( n ) = ∆ t = RS ( n ) . RS ( n ) maybe as large as n/ log n . However, best known bound for RS ( n ) is only n Ω (1 / log log n ) . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  66. A Simple Lower Bound for Insertion-Only Streams Main limitation of this approach: Requires r = Θ ( n ) = ∆ t = RS ( n ) . RS ( n ) maybe as large as n/ log n . However, best known bound for RS ( n ) is only n Ω (1 / log log n ) . We bypass the r = Θ ( n ) limitation in dynamic streams using the characterization result of [Li et al., 2014, Ai et al., 2016] in terms of the simultaneous communication complexity. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  67. The Simultaneous Communication Model The input is partitioned between k players P (1) , . . . , P ( k ) . Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  68. The Simultaneous Communication Model The input is partitioned between k players P (1) , . . . , P ( k ) . There exists an additional party called the referee. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

  69. The Simultaneous Communication Model The input is partitioned between k players P (1) , . . . , P ( k ) . There exists an additional party called the referee. Players P (1) , . . . , P ( k ) simultaneously send a message to the referee who outputs the answer. Sepehr Assadi (Penn) JHU Algorithms and Complexity Seminar

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