Graph & Geometry Problems in Data Streams

Transcription

1 Graph & Geometry Problems in Data Streams 2009 Barbados Workshop on Computational Complexity Andrew McGregor

2 Introduction Models: Graph Streams: Stream of edges E = {e 1, e 2,..., e m } describe a graph G on n nodes. Estimate properties of G. Geometric Streams: Stream of points X = {p 1, p 2,..., p m } from some metric space (X, d). Estimate properties of X.

3 Introduction Models: Graph Streams: Stream of edges E = {e 1, e 2,..., e m } describe a graph G on n nodes. Estimate properties of G. Geometric Streams: Stream of points X = {p 1, p 2,..., p m } from some metric space (X, d). Estimate properties of X. Notes: Õ is our friend: we ll hide dependence on polylog(m, n) terms. Assume that p i can be stored in Õ(1) space and d(p i, p j ) can be calculated if both p i and p j are stored in memory. Theory isn t as cohesive but we get to cherry-pick results...

4 Counting Triangles Matching Clustering Graph Distances

5 Outline Counting Triangles Matching Clustering Graph Distances

6 Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ɛ) with probability 1 δ given promise that T 3 > t.

7 Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ɛ) with probability 1 δ given promise that T 3 > t. Warm-Up What s an algorithm using O(ɛ 2 (n 3 /t) log δ 1 ) space?

8 Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ɛ) with probability 1 δ given promise that T 3 > t. Warm-Up What s an algorithm using O(ɛ 2 (n 3 /t) log δ 1 ) space? Theorem Ω(n 2 ) space required to determine if t = 0 (with δ = 1/3).

9 Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ɛ) with probability 1 δ given promise that T 3 > t. Warm-Up What s an algorithm using O(ɛ 2 (n 3 /t) log δ 1 ) space? Theorem Ω(n 2 ) space required to determine if t = 0 (with δ = 1/3). Theorem (Sivakumar et al. 2002) Õ(ɛ 2 (nm/t) 2 log δ 1 ) space is sufficient.

10 Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ɛ) with probability 1 δ given promise that T 3 > t. Warm-Up What s an algorithm using O(ɛ 2 (n 3 /t) log δ 1 ) space? Theorem Ω(n 2 ) space required to determine if t = 0 (with δ = 1/3). Theorem (Sivakumar et al. 2002) Õ(ɛ 2 (nm/t) 2 log δ 1 ) space is sufficient. Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t) log δ 1 ) space is sufficient.

11 Lower Bound Theorem Ω(n 2 ) space required to determine if T 3 0 when δ = 1/3.

12 Lower Bound Theorem Ω(n 2 ) space required to determine if T 3 0 when δ = 1/3. Reduce from set-disjointness: Alice has n n binary matrix A, Bob has n n binary matrix B. Is A ij = B ij = 1 for some (i, j)? Needs Ω(n 2 ) bits of communication [Razborov 1992].

13 Lower Bound Theorem Ω(n 2 ) space required to determine if T 3 0 when δ = 1/3. Reduce from set-disjointness: Alice has n n binary matrix A, Bob has n n binary matrix B. Is A ij = B ij = 1 for some (i, j)? Needs Ω(n 2 ) bits of communication [Razborov 1992]. Consider graph G = (V, E) with V = {v 1,..., v n, u 1,..., u n, w 1,..., w n } and E = {(v i, u i ) : i [n]}

14 Lower Bound Theorem Ω(n 2 ) space required to determine if T 3 0 when δ = 1/3. Reduce from set-disjointness: Alice has n n binary matrix A, Bob has n n binary matrix B. Is A ij = B ij = 1 for some (i, j)? Needs Ω(n 2 ) bits of communication [Razborov 1992]. Consider graph G = (V, E) with V = {v 1,..., v n, u 1,..., u n, w 1,..., w n } and E = {(v i, u i ) : i [n]} Alice runs algorithm on G and edges {(u i, w j ) : A ij = 1}.

15 Lower Bound Theorem Ω(n 2 ) space required to determine if T 3 0 when δ = 1/3. Reduce from set-disjointness: Alice has n n binary matrix A, Bob has n n binary matrix B. Is A ij = B ij = 1 for some (i, j)? Needs Ω(n 2 ) bits of communication [Razborov 1992]. Consider graph G = (V, E) with V = {v 1,..., v n, u 1,..., u n, w 1,..., w n } and E = {(v i, u i ) : i [n]} Alice runs algorithm on G and edges {(u i, w j ) : A ij = 1}. Bob continues running algorithm on edges {(v i, w j ) : B ij = 1}.

16 Lower Bound Theorem Ω(n 2 ) space required to determine if T 3 0 when δ = 1/3. Reduce from set-disjointness: Alice has n n binary matrix A, Bob has n n binary matrix B. Is A ij = B ij = 1 for some (i, j)? Needs Ω(n 2 ) bits of communication [Razborov 1992]. Consider graph G = (V, E) with V = {v 1,..., v n, u 1,..., u n, w 1,..., w n } and E = {(v i, u i ) : i [n]} Alice runs algorithm on G and edges {(u i, w j ) : A ij = 1}. Bob continues running algorithm on edges {(v i, w j ) : B ij = 1}. T 3 > 0 iff A ij = B ij = 1 for some (i, j).

17 First Algorithm Theorem (Sivakumar et al. 2002) Õ(ɛ 2 (nm/t 3 ) 2 log δ 1 ) space is sufficient.

18 First Algorithm Theorem (Sivakumar et al. 2002) Õ(ɛ 2 (nm/t 3 ) 2 log δ 1 ) space is sufficient. Given stream of edges induce stream of node-triples: edge (u, v) gives rise to {u, v, w} for w V \ {u, v}

19 First Algorithm Theorem (Sivakumar et al. 2002) Õ(ɛ 2 (nm/t 3 ) 2 log δ 1 ) space is sufficient. Given stream of edges induce stream of node-triples: edge (u, v) gives rise to {u, v, w} for w V \ {u, v} Consider F k = (freq. of {u,v,w}) k and note F 0 F 1 F 2 = T 1 T 2 T 3 where T i is the set of node-triples having exactly i edges in the induced subgraph.

20 First Algorithm Theorem (Sivakumar et al. 2002) Õ(ɛ 2 (nm/t 3 ) 2 log δ 1 ) space is sufficient. Given stream of edges induce stream of node-triples: edge (u, v) gives rise to {u, v, w} for w V \ {u, v} Consider F k = (freq. of {u,v,w}) k and note F 0 F 1 F 2 = T 1 T 2 T 3 where T i is the set of node-triples having exactly i edges in the induced subgraph. T 3 = F 0 3F 1 /2 + F 2 /2 so good approx. for F 0, F 1, F 2 suffice.

21 Second Algorithm Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t 3 ) log δ 1 ) space is sufficient.

22 Second Algorithm Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t 3 ) log δ 1 ) space is sufficient. Pick an edge e i = (u, v) uniformly at random from the stream.

23 Second Algorithm Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t 3 ) log δ 1 ) space is sufficient. Pick an edge e i = (u, v) uniformly at random from the stream. Pick w uniformly at random from V \ {u, v}

24 Second Algorithm Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t 3 ) log δ 1 ) space is sufficient. Pick an edge e i = (u, v) uniformly at random from the stream. Pick w uniformly at random from V \ {u, v} If e j = (u, w), e k = (v, w) for j, k > i exist return 1; else 0.

25 Second Algorithm Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t 3 ) log δ 1 ) space is sufficient. Pick an edge e i = (u, v) uniformly at random from the stream. Pick w uniformly at random from V \ {u, v} If e j = (u, w), e k = (v, w) for j, k > i exist return 1; else 0. Lemma Expected outcome of algorithm is T 3 3m(n 2).

26 Second Algorithm Theorem (Buriol et al. 2006) Õ(ɛ 2 (nm/t 3 ) log δ 1 ) space is sufficient. Pick an edge e i = (u, v) uniformly at random from the stream. Pick w uniformly at random from V \ {u, v} If e j = (u, w), e k = (v, w) for j, k > i exist return 1; else 0. Lemma Expected outcome of algorithm is T 3 3m(n 2). Repeat O(ɛ 2 (mn/t) log δ 1 ) times in parallel and scale average up by 3m(n 2).

27 Outline Counting Triangles Matching Clustering Graph Distances

28 Maximum Weight Matching Problem Stream of weighted edges (e, w e ): Find M E that maximizes e M w e such that no two edges in M share an endpoint.

29 Maximum Weight Matching Problem Stream of weighted edges (e, w e ): Find M E that maximizes e M w e such that no two edges in M share an endpoint. Warm-Up An easy 2 approx. for unweighted case in Õ(n) space?

30 Maximum Weight Matching Problem Stream of weighted edges (e, w e ): Find M E that maximizes e M w e such that no two edges in M share an endpoint. Warm-Up An easy 2 approx. for unweighted case in Õ(n) space? Theorem = approx. in Õ(n) space.

31 Maximum Weight Matching Problem Stream of weighted edges (e, w e ): Find M E that maximizes e M w e such that no two edges in M share an endpoint. Warm-Up An easy 2 approx. for unweighted case in Õ(n) space? Theorem = approx. in Õ(n) space. Improved to [Mariano 07] and [Sarma et al. 09].

32 Maximum Weight Matching Problem Stream of weighted edges (e, w e ): Find M E that maximizes e M w e such that no two edges in M share an endpoint. Warm-Up An easy 2 approx. for unweighted case in Õ(n) space? Theorem = approx. in Õ(n) space. Improved to [Mariano 07] and [Sarma et al. 09]. Open Problem Prove a lower bound or a much better algorithm!

33 An Algorithm At all times maintain a matching M, initially M =.

34 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M

35 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M If w e (1 + γ)(w e + w e ), M M {e} \ {e, e }

36 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M If w e (1 + γ)(w e + w e ), M M {e} \ {e, e } For the analysis we use the following definitions to describe the execution of the algorithm:

37 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M If w e (1 + γ)(w e + w e ), M M {e} \ {e, e } For the analysis we use the following definitions to describe the execution of the algorithm:

38 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M If w e (1 + γ)(w e + w e ), M M {e} \ {e, e } For the analysis we use the following definitions to describe the execution of the algorithm: An edge e kills an edge e if e was removed when e arrives.

39 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M If w e (1 + γ)(w e + w e ), M M {e} \ {e, e } For the analysis we use the following definitions to describe the execution of the algorithm: An edge e kills an edge e if e was removed when e arrives. We say an edge is a survivor if it s in the final matching.

40 An Algorithm At all times maintain a matching M, initially M =. On seeing e = (u, v), suppose e = (u, u 1 ), e = (v, u 2 ) M If w e (1 + γ)(w e + w e ), M M {e} \ {e, e } For the analysis we use the following definitions to describe the execution of the algorithm: An edge e kills an edge e if e was removed when e arrives. We say an edge is a survivor if it s in the final matching. For survivor e, the trail of the dead is T (e) = C 1 C 2..., where C 0 = {e} and C i = e C i 1 {edges killed by e }

41 Analysis Lemma Let S be set of survivors and w(s) be weight of final matching. 1. w(t (S)) w(s)/γ 2. opt (1 + γ) (w(t (S)) + 2w(S)) Approximation factor is 1/γ γ and γ = 1/ 2 gives result.

42 Analysis Lemma Let S be set of survivors and w(s) be weight of final matching. 1. w(t (S)) w(s)/γ 2. opt (1 + γ) (w(t (S)) + 2w(S)) Approximation factor is 1/γ γ and γ = 1/ 2 gives result. Proof.

43 Analysis Lemma Let S be set of survivors and w(s) be weight of final matching. 1. w(t (S)) w(s)/γ 2. opt (1 + γ) (w(t (S)) + 2w(S)) Approximation factor is 1/γ γ and γ = 1/ 2 gives result. Proof. 1. Consider e S: (1+γ)w(T (e)) = i 1 (1+γ)w(C i ) i 0 w(c i ) = w(t (e))+w(e)

44 Analysis Lemma Let S be set of survivors and w(s) be weight of final matching. 1. w(t (S)) w(s)/γ 2. opt (1 + γ) (w(t (S)) + 2w(S)) Approximation factor is 1/γ γ and γ = 1/ 2 gives result. Proof. 1. Consider e S: (1+γ)w(T (e)) = i 1 (1+γ)w(C i ) i 0 w(c i ) = w(t (e))+w(e) 2. Can charge the weights of edges in opt to the S T (S) such that each edge e T (S) is charged at most (1 + γ)w(e) and each edge e S is charged at most 2(1 + γ)w(e).

45 Outline Counting Triangles Matching Clustering Graph Distances

46 k-center Problem Given a stream of distinct points X = {p 1,..., p n } from a metric space (X, d), find the set of k points Y X that minimizes: max min d(p i, y) i y Y

47 k-center Problem Given a stream of distinct points X = {p 1,..., p n } from a metric space (X, d), find the set of k points Y X that minimizes: max min d(p i, y) i y Y Warm-Up Find 2-approx. if you re given opt. Find (2 + ɛ)-approx. if you re given that a opt b

48 k-center Problem Given a stream of distinct points X = {p 1,..., p n } from a metric space (X, d), find the set of k points Y X that minimizes: max min d(p i, y) i y Y Warm-Up Find 2-approx. if you re given opt. Find (2 + ɛ)-approx. if you re given that a opt b Theorem (Khuller and McCutchen 2009, Guha 2009) (2 + ɛ) approx. for metric k-center in Õ(kɛ 1 log ɛ 1 ) space.

49 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt.

50 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a

51 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a Say instantiation goes bad if it tries to open (k + 1)-th center

52 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a Say instantiation goes bad if it tries to open (k + 1)-th center Suppose instantiation with guess l goes bad when processing (j + 1)-th point

53 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a Say instantiation goes bad if it tries to open (k + 1)-th center Suppose instantiation with guess l goes bad when processing (j + 1)-th point Let q1,..., q k be centers chosen so far.

54 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a Say instantiation goes bad if it tries to open (k + 1)-th center Suppose instantiation with guess l goes bad when processing (j + 1)-th point Let q1,..., q k be centers chosen so far. Then p1,..., p j are all at most 2l from a q i.

55 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a Say instantiation goes bad if it tries to open (k + 1)-th center Suppose instantiation with guess l goes bad when processing (j + 1)-th point Let q1,..., q k be centers chosen so far. Then p1,..., p j are all at most 2l from a q i. Optimum for {q 1,..., q k, p j+1,..., p n } is at most opt + 2l.

56 k-center: Algorithm and Analysis Consider first k + 1 points: this gives a lower bound a on opt. Instantiate basic algorithm with guesses l 1 = a, l 2 = (1 + ɛ)a, l 3 = (1 + ɛ) 2 a,... l 1+t = O(ɛ 1 )a Say instantiation goes bad if it tries to open (k + 1)-th center Suppose instantiation with guess l goes bad when processing (j + 1)-th point Let q1,..., q k be centers chosen so far. Then p1,..., p j are all at most 2l from a q i. Optimum for {q 1,..., q k, p j+1,..., p n } is at most opt + 2l. Hence, for an instantiation with guess 2l/ɛ only incurs a small if we use {q 1,..., q k, p j+1,..., p n } rather than {p 1,..., p n }.

57 Outline Counting Triangles Matching Clustering Graph Distances

58 Distance Estimation Problem Stream of unweighted edges E defines a shortest path graph metric d G : V V N. For u, v V, estimate d G (u, v).

59 Distance Estimation Problem Stream of unweighted edges E defines a shortest path graph metric d G : V V N. For u, v V, estimate d G (u, v). Definition An α-spanner of a graph G = (V, E) is a subgraph H = (V, E ) such that for all u, v, d G (u, v) d H (u, v) αd G (u, v)

60 Distance Estimation Problem Stream of unweighted edges E defines a shortest path graph metric d G : V V N. For u, v V, estimate d G (u, v). Definition An α-spanner of a graph G = (V, E) is a subgraph H = (V, E ) such that for all u, v, d G (u, v) d H (u, v) αd G (u, v) Warm-Up 2t 1 spanner using Õ(n 1+1/t ) space.

61 Distance Estimation Problem Stream of unweighted edges E defines a shortest path graph metric d G : V V N. For u, v V, estimate d G (u, v). Definition An α-spanner of a graph G = (V, E) is a subgraph H = (V, E ) such that for all u, v, d G (u, v) d H (u, v) αd G (u, v) Warm-Up 2t 1 spanner using Õ(n 1+1/t ) space. Theorem (Elkin 2007) 2t 1 stretch spanner using Õ(n 1+1/t ) space with constant update time.

62 Towards better results if you re allowed mulitple passes... Problem Can we get better approximation for d G (u, v) with multiple passes?

63 Towards better results if you re allowed mulitple passes... Problem Can we get better approximation for d G (u, v) with multiple passes? Warm-Up Find d G (u, v) exactly in Õ(n 1+γ ) space and Õ(n 1 γ ) passes.

64 Towards better results if you re allowed mulitple passes... Problem Can we get better approximation for d G (u, v) with multiple passes? Warm-Up Find d G (u, v) exactly in Õ(n 1+γ ) space and Õ(n 1 γ ) passes. Theorem O(k) approx in Õ(n) space with O(n 1/k ) passes.

65 Towards better results if you re allowed mulitple passes... Problem Can we get better approximation for d G (u, v) with multiple passes? Warm-Up Find d G (u, v) exactly in Õ(n 1+γ ) space and Õ(n 1 γ ) passes. Theorem O(k) approx in Õ(n) space with O(n 1/k ) passes. Theorem (via Thorup, Zwick 2006) (1 + ɛ) approx in Õ(n) space with n O(log ɛ 1 )/ log log n passes.

66 Ramsey Partition Approach Definition (Mendel, Naor 2006) Ramsey Partition P is a random partion of metric space. Each cluster has diameter at most and for t /8, Pr(B X (x, t) P ) ( ) BX (x, /8) 16t/ B X (x, ) ( ) 1 16t/. n Can construct in stream model in Õ(n) space and O( ) passes.

67 Ramsey Partition Approach Definition (Mendel, Naor 2006) Ramsey Partition P is a random partion of metric space. Each cluster has diameter at most and for t /8, Pr(B X (x, t) P ) ( ) BX (x, /8) 16t/ B X (x, ) ( ) 1 16t/. n Can construct in stream model in Õ(n) space and O( ) passes. Algorithm 1. Sample beacons b 1,..., b n 1 1/k including s and t from V

68 Ramsey Partition Approach Definition (Mendel, Naor 2006) Ramsey Partition P is a random partion of metric space. Each cluster has diameter at most and for t /8, Pr(B X (x, t) P ) ( ) BX (x, /8) 16t/ B X (x, ) ( ) 1 16t/. n Can construct in stream model in Õ(n) space and O( ) passes. Algorithm 1. Sample beacons b 1,..., b n 1 1/k including s and t from V 2. Repeat O(n 1/k log n) times: 2.1 Create RP with diameter kn 1/k and consider t n 1/k. 2.2 For each beacon, add -weighted edge to center of its cluster.

69 Summary: We looked at some nice problems, our curiousity is piqued, and now we want to start finding more problems to solve. Thanks!