Graph & Geometry Problems in Data Streams 2009 Barbados Workshop - PowerPoint PPT Presentation

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

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 .

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: ◮ ˜ O is our friend: we’ll hide dependence on polylog( m , n ) terms. ◮ Assume that p i can be stored in ˜ O (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. . .

Counting Triangles Matching Clustering Graph Distances

Outline Counting Triangles Matching Clustering Graph Distances

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

Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ǫ ) with probability 1 − δ given promise that T 3 > t. Warm-Up What’s an algorithm using O ( ǫ − 2 ( n 3 / t ) log δ − 1 ) space?

Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ǫ ) with probability 1 − δ given promise that T 3 > t. Warm-Up What’s an algorithm using O ( ǫ − 2 ( n 3 / t ) log δ − 1 ) space? Theorem Ω( n 2 ) space required to determine if t = 0 (with δ = 1 / 3 ).

Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ǫ ) with probability 1 − δ given promise that T 3 > t. Warm-Up What’s an algorithm using O ( ǫ − 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) O ( ǫ − 2 ( nm / t ) 2 log δ − 1 ) space is sufficient. ˜

Triangles Problem Given a stream of edges, estimate the number of triangles T 3 up to a factor (1 + ǫ ) with probability 1 − δ given promise that T 3 > t. Warm-Up What’s an algorithm using O ( ǫ − 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) O ( ǫ − 2 ( nm / t ) 2 log δ − 1 ) space is sufficient. ˜ Theorem (Buriol et al. 2006) ˜ O ( ǫ − 2 ( nm / t ) log δ − 1 ) space is sufficient.

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

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].

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 ] }

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 } .

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 } .

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 ).

First Algorithm Theorem (Sivakumar et al. 2002) O ( ǫ − 2 ( nm / T 3 ) 2 log δ − 1 ) space is sufficient. ˜

First Algorithm Theorem (Sivakumar et al. 2002) O ( ǫ − 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 }

First Algorithm Theorem (Sivakumar et al. 2002) O ( ǫ − 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   1 1 1   T 1   = F 1 1 2 3 T 2      F 2 1 4 9 T 3 where T i is the set of node-triples having exactly i edges in the induced subgraph.

First Algorithm Theorem (Sivakumar et al. 2002) O ( ǫ − 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   1 1 1   T 1   = F 1 1 2 3 T 2      F 2 1 4 9 T 3 where T i is the set of node-triples having exactly i edges in the induced subgraph. ◮ T 3 = F 0 − 3 F 1 / 2 + F 2 / 2 so good approx. for F 0 , F 1 , F 2 suffice.

Second Algorithm Theorem (Buriol et al. 2006) ˜ O ( ǫ − 2 ( nm / T 3 ) log δ − 1 ) space is sufficient.

Second Algorithm Theorem (Buriol et al. 2006) ˜ O ( ǫ − 2 ( nm / T 3 ) log δ − 1 ) space is sufficient. ◮ Pick an edge e i = ( u , v ) uniformly at random from the stream.

Second Algorithm Theorem (Buriol et al. 2006) ˜ O ( ǫ − 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 }

Second Algorithm Theorem (Buriol et al. 2006) ˜ O ( ǫ − 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.

Second Algorithm Theorem (Buriol et al. 2006) ˜ O ( ǫ − 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 T 3 Expected outcome of algorithm is 3 m ( n − 2) .

Second Algorithm Theorem (Buriol et al. 2006) ˜ O ( ǫ − 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 T 3 Expected outcome of algorithm is 3 m ( n − 2) . ◮ Repeat O ( ǫ − 2 ( mn / t ) log δ − 1 ) times in parallel and scale average up by 3 m ( n − 2).

Outline Counting Triangles Matching Clustering Graph Distances

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.

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 ˜ O ( n ) space?

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 ˜ O ( n ) space? Theorem √ 2 = 5 . 83 . . . approx. in ˜ 3 + 2 O ( n ) space.