Data Streams & Communication Complexity Lecture 2: Graph - PowerPoint PPT Presentation

Data Streams & Communication Complexity Lecture 2: Graph Spanners, Sparsifiers, & Sketches Andrew McGregor, UMass Amherst 1/25

Graph Streams ◮ Consider a stream of m edges � e 1 , e 2 , . . . . . . , e m � defining a graph G with nodes V = [ n ] and E = { e 1 , . . . , e m } 2/25

Graph Streams ◮ Consider a stream of m edges � e 1 , e 2 , . . . . . . , e m � defining a graph G with nodes V = [ n ] and E = { e 1 , . . . , e m } ◮ Semi-streaming: What can we compute with O ( n · polylog n ) space? 2/25

Outline Spanners and Distances Sparsifiers and Cuts Sketches and Dynamic Graphs Connectivity k -Connectivity Minimum Cut 3/25

Outline Spanners and Distances Sparsifiers and Cuts Sketches and Dynamic Graphs Connectivity k -Connectivity Minimum Cut 4/25

Graph Distances ◮ Goal: Approximate length of the shortest path d G ( u , v ) between a pair of nodes u , v ∈ G , 5/25

Graph Distances ◮ Goal: Approximate length of the shortest path d G ( u , v ) between a pair of nodes u , v ∈ G , Definition An α -spanner of graph G is a subgraph H such that for any nodes u , v , d G ( u , v ) ≤ d H ( u , v ) ≤ α d G ( u , v ) . 5/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F ◮ F ← ∅ 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F ◮ F ← ∅ ◮ For each edge ( u , v ), if u and v aren’t connected in F , F ← F ∪ { ( u , v ) } 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F ◮ F ← ∅ ◮ For each edge ( u , v ), if u and v aren’t connected in F , F ← F ∪ { ( u , v ) } ◮ Analysis: 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F ◮ F ← ∅ ◮ For each edge ( u , v ), if u and v aren’t connected in F , F ← F ∪ { ( u , v ) } ◮ Analysis: ◮ F has the same number of connected components as G 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F ◮ F ← ∅ ◮ For each edge ( u , v ), if u and v aren’t connected in F , F ← F ∪ { ( u , v ) } ◮ Analysis: ◮ F has the same number of connected components as G ◮ F has at most n − 1 edges. 6/25

Warm-Up: Connectivity ◮ Goal: Compute the number of connected components. ◮ Algorithm: Maintain a spanning forest F ◮ F ← ∅ ◮ For each edge ( u , v ), if u and v aren’t connected in F , F ← F ∪ { ( u , v ) } ◮ Analysis: ◮ F has the same number of connected components as G ◮ F has at most n − 1 edges. ◮ Thm: Can count connected components in O ( n log n ) space. 6/25

Spanners ◮ Algorithm: 7/25

Spanners ◮ Algorithm: ◮ H ← ∅ . 7/25

Spanners ◮ Algorithm: ◮ H ← ∅ . ◮ For each edge ( u , v ), if d H ( u , v ) ≥ 2 t , H ← H ∪ { ( u , v ) } 7/25

Spanners ◮ Algorithm: ◮ H ← ∅ . ◮ For each edge ( u , v ), if d H ( u , v ) ≥ 2 t , H ← H ∪ { ( u , v ) } ◮ Analysis: 7/25

Spanners ◮ Algorithm: ◮ H ← ∅ . ◮ For each edge ( u , v ), if d H ( u , v ) ≥ 2 t , H ← H ∪ { ( u , v ) } ◮ Analysis: ◮ Distances increase by at most a factor 2 t − 1 since an edge ( u , v ) is only forgotten if there’s already a detour of length at most 2 t − 1. 7/25

Spanners ◮ Algorithm: ◮ H ← ∅ . ◮ For each edge ( u , v ), if d H ( u , v ) ≥ 2 t , H ← H ∪ { ( u , v ) } ◮ Analysis: ◮ Distances increase by at most a factor 2 t − 1 since an edge ( u , v ) is only forgotten if there’s already a detour of length at most 2 t − 1. ◮ Lemma: H has O ( n 1+1 / t ) edges since all cycles have length ≥ 2 t + 1. 7/25

Spanners ◮ Algorithm: ◮ H ← ∅ . ◮ For each edge ( u , v ), if d H ( u , v ) ≥ 2 t , H ← H ∪ { ( u , v ) } ◮ Analysis: ◮ Distances increase by at most a factor 2 t − 1 since an edge ( u , v ) is only forgotten if there’s already a detour of length at most 2 t − 1. ◮ Lemma: H has O ( n 1+1 / t ) edges since all cycles have length ≥ 2 t + 1. Theorem Can (2 t − 1) -approximate all distances using only O ( n 1+1 / t ) space. 7/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. 8/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. ◮ Let d = 2 m / n be average degree of H . 8/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. ◮ Let d = 2 m / n be average degree of H . ◮ Let J be the graph formed by removing all nodes with degree less than d / 2. 8/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. ◮ Let d = 2 m / n be average degree of H . ◮ Let J be the graph formed by removing all nodes with degree less than d / 2. Note J � = ∅ because < n ( d / 2) = m edges are removed. 8/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. ◮ Let d = 2 m / n be average degree of H . ◮ Let J be the graph formed by removing all nodes with degree less than d / 2. Note J � = ∅ because < n ( d / 2) = m edges are removed. ◮ Grow a BFS of depth t from an arbitrary node in J . 8/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. ◮ Let d = 2 m / n be average degree of H . ◮ Let J be the graph formed by removing all nodes with degree less than d / 2. Note J � = ∅ because < n ( d / 2) = m edges are removed. ◮ Grow a BFS of depth t from an arbitrary node in J . ◮ Because a) no cycles of length less than 2 t + 1 and b) all degrees in J are at least d / 2, number of nodes at t -th level of BFS is at least ( d / 2 − 1) t = ( m / n − 1) t 8/25

Proof of Lemma Lemma A graph H on n nodes with no cycles of length ≤ 2 t has O ( n 1+1 / t ) edges. ◮ Let d = 2 m / n be average degree of H . ◮ Let J be the graph formed by removing all nodes with degree less than d / 2. Note J � = ∅ because < n ( d / 2) = m edges are removed. ◮ Grow a BFS of depth t from an arbitrary node in J . ◮ Because a) no cycles of length less than 2 t + 1 and b) all degrees in J are at least d / 2, number of nodes at t -th level of BFS is at least ( d / 2 − 1) t = ( m / n − 1) t ◮ But ( m / n − 1) t ≤ | J | ≤ n and therefore m ≤ n + n 1+1 / t . 8/25

Outline Spanners and Distances Sparsifiers and Cuts Sketches and Dynamic Graphs Connectivity k -Connectivity Minimum Cut 9/25

Cuts and Sparsifiers ◮ Goal: Approximate capacity C G ( S ) of any cut ( S , V \ S ) in G . 10/25

Cuts and Sparsifiers ◮ Goal: Approximate capacity C G ( S ) of any cut ( S , V \ S ) in G . Definition An α -sparsifier of graph G is a weighted subgraph H such that for any cut ( S , V \ S ), C G ( S ) ≤ C H ( S ) ≤ α C G ( S ) . where C G and C H is the capacity of the cut in G and H respectively. 10/25

Cuts and Sparsifiers ◮ Goal: Approximate capacity C G ( S ) of any cut ( S , V \ S ) in G . Definition An α -sparsifier of graph G is a weighted subgraph H such that for any cut ( S , V \ S ), C G ( S ) ≤ C H ( S ) ≤ α C G ( S ) . where C G and C H is the capacity of the cut in G and H respectively. Theorem (Batson, Spielman, Srivastava) Exists offline algorithm A returning (1 + ǫ ) -sparsifier with O ( n ǫ − 2 ) edges. 10/25

Cuts and Sparsifiers ◮ Goal: Approximate capacity C G ( S ) of any cut ( S , V \ S ) in G . Definition An α -sparsifier of graph G is a weighted subgraph H such that for any cut ( S , V \ S ), C G ( S ) ≤ C H ( S ) ≤ α C G ( S ) . where C G and C H is the capacity of the cut in G and H respectively. Theorem (Batson, Spielman, Srivastava) Exists offline algorithm A returning (1 + ǫ ) -sparsifier with O ( n ǫ − 2 ) edges. ◮ Idea: Use A as a black box to recursively sparsify graph stream. 10/25

Basic Properties of Sparsifiers Lemma If H 1 and H 2 are α -sparsifiers of G 1 and G 2 . Then H 1 ∪ H 2 is an α -sparsifier of G 1 ∪ G 2 . 11/25

Basic Properties of Sparsifiers Lemma If H 1 and H 2 are α -sparsifiers of G 1 and G 2 . Then H 1 ∪ H 2 is an α -sparsifier of G 1 ∪ G 2 . Lemma If J is an α -sparsifiers of H and H is an α -sparsifier of G. Then J is an α 2 -sparsifier of G. 11/25

Stream Sparsification ◮ Divide stream into segments G 1 , G 2 , . . . each of t = O ( n ǫ − 2 ) edges. 12/25

Stream Sparsification ◮ Divide stream into segments G 1 , G 2 , . . . each of t = O ( n ǫ − 2 ) edges. ◮ Consider binary tree over segments G=G1 ∪ G2 ∪ G3 ∪ G4 ∪ G5 ∪ G6 ∪ G7 ∪ G8 G1 ∪ G2 ∪ G3 ∪ G4 G5 ∪ G6 ∪ G7 ∪ G8 G1 ∪ G2 G3 ∪ G4 G5 ∪ G6 G7 ∪ G8 G1 G2 G3 G4 G5 G6 G7 G8 12/25