Parallel Algorithms for Graphs on a Very Large Number of Nodes - PowerPoint PPT Presentation

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general • Can reduce from Sparse Connectivity: Do edges span a connected graph? Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general • Can reduce from Sparse Connectivity: Do edges span a connected graph? • Conjecture: superconstant number of rounds with N 1 − Ω( 1 ) memory Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general • Can reduce from Sparse Connectivity: Do edges span a connected graph? • Conjecture: superconstant number of rounds with N 1 − Ω( 1 ) memory • Is this instance hard? vs. Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general • Can reduce from Sparse Connectivity: Do edges span a connected graph? • Conjecture: superconstant number of rounds with N 1 − Ω( 1 ) memory • Is this instance hard? (solvable in O ( log N ) rounds) vs. Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general • Can reduce from Sparse Connectivity: Do edges span a connected graph? • Conjecture: superconstant number of rounds with N 1 − Ω( 1 ) memory • Is this instance hard? (solvable in O ( log N ) rounds) vs. • Reduction: connect select vertex to all vertices with heavy edges Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

N 1 − Ω( 1 ) Space in O ( 1 ) Rounds? • Unlikely to be possible in general • Can reduce from Sparse Connectivity: Do edges span a connected graph? • Conjecture: superconstant number of rounds with N 1 − Ω( 1 ) memory • Is this instance hard? (solvable in O ( log N ) rounds) vs. • Reduction: connect select vertex to all vertices with heavy edges • This talk: algorithms with O ( N ǫ ) space per machine Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 10 / 26

Outline Model of Computation 1 Sample Algorithms and Their Limitations 2 Efficiently Estimating MST Weight 3 Computing MST in Geometric Setting 4 Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 11 / 26

Result [Ł ˛ acki, M ˛ adry, Mitrovi´ c, O., Sankowski] • Input: M edges, weights in { 1 , 2 , . . . , W } (#nodes N ≤ #edges M ) Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 12 / 26

Result [Ł ˛ acki, M ˛ adry, Mitrovi´ c, O., Sankowski] • Input: M edges, weights in { 1 , 2 , . . . , W } (#nodes N ≤ #edges M ) • Algorithm: • Computes ( 1 + ǫ ) -approximation to MST weight Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 12 / 26

Result [Ł ˛ acki, M ˛ adry, Mitrovi´ c, O., Sankowski] • Input: M edges, weights in { 1 , 2 , . . . , W } (#nodes N ≤ #edges M ) • Algorithm: • Computes ( 1 + ǫ ) -approximation to MST weight • Space per machine: � � 2 � M m + N � W for M / m = M Ω( 1 ) O m · ǫ Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 12 / 26

Result [Ł ˛ acki, M ˛ adry, Mitrovi´ c, O., Sankowski] • Input: M edges, weights in { 1 , 2 , . . . , W } (#nodes N ≤ #edges M ) • Algorithm: • Computes ( 1 + ǫ ) -approximation to MST weight • Space per machine: � � 2 � M m + N � W for M / m = M Ω( 1 ) O m · ǫ • Number of rounds: O ( log ( W /ǫ )) Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 12 / 26

Result [Ł ˛ acki, M ˛ adry, Mitrovi´ c, O., Sankowski] • Input: M edges, weights in { 1 , 2 , . . . , W } (#nodes N ≤ #edges M ) • Algorithm: • Computes ( 1 + ǫ ) -approximation to MST weight • Space per machine: � � 2 � M m + N � W for M / m = M Ω( 1 ) O m · ǫ • Number of rounds: O ( log ( W /ǫ )) • Note: No dependence on W would disprove Sparse Connectivity Conjecture Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 12 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) : • G i = graph restricted to edges of weight < i Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) : • G i = graph restricted to edges of weight < i • T i = #connected components in G i Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) : • G i = graph restricted to edges of weight < i • T i = #connected components in G i • Number of edges of weight ≥ i in MST = T i − 1 Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) : • G i = graph restricted to edges of weight < i • T i = #connected components in G i • Number of edges of weight ≥ i in MST = T i − 1 W � ⇒ weight ( MST ) = ( T i − 1 ) i = 1 Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) : • G i = graph restricted to edges of weight < i • T i = #connected components in G i • Number of edges of weight ≥ i in MST = T i − 1 W � ⇒ weight ( MST ) = ( T i − 1 ) i = 1 • C i ( v ) = size of the component of v in G i � T i = 1 / C i ( v ) v Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Approach Use techniques of Chazelle, Rubinfeld, Trevisan (2005) : • G i = graph restricted to edges of weight < i • T i = #connected components in G i • Number of edges of weight ≥ i in MST = T i − 1 W � ⇒ weight ( MST ) = ( T i − 1 ) i = 1 • C i ( v ) = size of the component of v in G i � T i = 1 / C i ( v ) v • Good approximation: • Compute sizes of small components • Replace 1 / C i ( v ) with 0 if C i ( v ) ≥ W /ǫ Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 13 / 26

Implementation • Reachability sets R v for each node v : • Set of W /ǫ nodes accessible via cheapest edges Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 14 / 26

Implementation • Reachability sets R v for each node v : • Set of W /ǫ nodes accessible via cheapest edges • Initially: collect cheapest incident edges Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 14 / 26

Implementation • Reachability sets R v for each node v : • Set of W /ǫ nodes accessible via cheapest edges • Initially: collect cheapest incident edges • Repeat O ( log ( W /ǫ )) times: Ask nodes u on R v for their R u and update Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 14 / 26

Implementation • Reachability sets R v for each node v : • Set of W /ǫ nodes accessible via cheapest edges • Initially: collect cheapest incident edges • Repeat O ( log ( W /ǫ )) times: Ask nodes u on R v for their R u and update • O ( log ( W /ǫ )) updates suffice to explore useful nodes up to distance W /ǫ Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 14 / 26

Implementation • Reachability sets R v for each node v : • Set of W /ǫ nodes accessible via cheapest edges • Initially: collect cheapest incident edges • Repeat O ( log ( W /ǫ )) times: Ask nodes u on R v for their R u and update • O ( log ( W /ǫ )) updates suffice to explore useful nodes up to distance W /ǫ • Use QuickSort-like sorting algorithm of Goodrich, Sitchinava, Zhang (2011) to organize communication Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 14 / 26

Outline Model of Computation 1 Sample Algorithms and Their Limitations 2 Efficiently Estimating MST Weight 3 Computing MST in Geometric Setting 4 Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 15 / 26

Geometric Setting Input: set of points in low dimensional metric space Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 16 / 26

Geometric Setting Input: set of points in low dimensional metric space 7 9 14 8 11 10 • Points induce a weighted graph Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 16 / 26

Geometric Setting Input: set of points in low dimensional metric space 7 9 14 8 11 10 • Points induce a weighted graph • Graph problems to consider: • Minimum Spanning Tree • Earth Mover Distance • Transportation Problem • Travelling Salesman Problem • . . . Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 16 / 26

Result [Andoni, Nikolov, O., Yaroslavtsev 2014] • Input: N points in low dimensional metric space • Example: R 2 • Generalizes to bounded doubling dimension Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 17 / 26

Result [Andoni, Nikolov, O., Yaroslavtsev 2014] • Input: N points in low dimensional metric space • Example: R 2 • Generalizes to bounded doubling dimension • Algorithm: • Computes ( 1 + ǫ ) -approximate MST Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 17 / 26

Result [Andoni, Nikolov, O., Yaroslavtsev 2014] • Input: N points in low dimensional metric space • Example: R 2 • Generalizes to bounded doubling dimension • Algorithm: • Computes ( 1 + ǫ ) -approximate MST • Space per machine: roughly O ( N / m ) (as long as it fits subproblems) Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 17 / 26

Result [Andoni, Nikolov, O., Yaroslavtsev 2014] • Input: N points in low dimensional metric space • Example: R 2 • Generalizes to bounded doubling dimension • Algorithm: • Computes ( 1 + ǫ ) -approximate MST • Space per machine: roughly O ( N / m ) (as long as it fits subproblems) • Number of rounds: O ( 1 ) Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 17 / 26

Result [Andoni, Nikolov, O., Yaroslavtsev 2014] • Input: N points in low dimensional metric space • Example: R 2 • Generalizes to bounded doubling dimension • Algorithm: • Computes ( 1 + ǫ ) -approximate MST • Space per machine: roughly O ( N / m ) (as long as it fits subproblems) • Number of rounds: O ( 1 ) • Running time: near-linear Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 17 / 26

Random Gridding We reuse the Arora-Mitchell approach: Apply a randomly shifted grid Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 18 / 26

Random Gridding We reuse the Arora-Mitchell approach: Apply a randomly shifted grid Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 18 / 26

Random Gridding We reuse the Arora-Mitchell approach: Apply a randomly shifted grid Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 18 / 26

Random Gridding We reuse the Arora-Mitchell approach: Apply a randomly shifted grid Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 18 / 26

Random Gridding We reuse the Arora-Mitchell approach: Apply a randomly shifted grid Key property: cell of side ∆ separates points x and y w.p. O ( 1 ) · ρ ( x , y ) ∆ Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 18 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Using Random Gridding Typical usage: Recursive dynamic program for approximately solving problem Can partially isolate what happens inside a cell Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 19 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them and repeat Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them and repeat • Pass up ǫ 2 ∆ -covering with information about connected components Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them and repeat • Pass up ǫ 2 ∆ -covering with information about connected components Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them and repeat • Pass up ǫ 2 ∆ -covering with information about connected components Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Our Algorithm • Connect points closer than ǫ · diam ( S ) arbitrarily 100 · N • Sub-solution for cell of side ∆ : ǫ 2 ∆ -covering with induced components • Combining sub-solutions: Truncated version of Kruskal’s algorithm 1. Find two closest clusters 2. If their distance less than ǫ ∆ , connect them and repeat • Pass up ǫ 2 ∆ -covering with information about connected components • Expected cost of solution: optimum · ( 1 + ǫ · #levels ) Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 20 / 26

Select Implementation Details • Merge N Ω( 1 ) × N Ω( 1 ) cells at once Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 21 / 26

Select Implementation Details • Merge N Ω( 1 ) × N Ω( 1 ) cells at once • Sub-solutions for all subcells should fit on a single machine Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 21 / 26

Select Implementation Details • Merge N Ω( 1 ) × N Ω( 1 ) cells at once • Sub-solutions for all subcells should fit on a single machine • Use sorting [Goodrich, Sitchinava, Zhang 2011] for grouping points and subcells that are close Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 21 / 26

Select Implementation Details • Merge N Ω( 1 ) × N Ω( 1 ) cells at once • Sub-solutions for all subcells should fit on a single machine • Use sorting [Goodrich, Sitchinava, Zhang 2011] for grouping points and subcells that are close • Near-linear time: • Relax Kruskal’s algorithm • Efficient nearest neighbor data structure [Krauthgamer, Lee 2004], [Cole, Gottlieb 2006] Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 21 / 26

Lower Bounds for MST • Natural questions to ask: • Can generalize to unbounded dimension? • Can compute exact solution? Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 22 / 26

Lower Bounds for MST • Natural questions to ask: • Can generalize to unbounded dimension? • Can compute exact solution? • Query complexity: • Model: distance queries • Our algorithm can be adapted to arbitrary bounded doubling dimensional metric in this model • Lower bound: N Ω( 1 ) rounds Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 22 / 26

Lower Bounds for MST • Natural questions to ask: • Can generalize to unbounded dimension? • Can compute exact solution? • Query complexity: • Model: distance queries • Our algorithm can be adapted to arbitrary bounded doubling dimensional metric in this model • Lower bound: N Ω( 1 ) rounds • We give a conditional lower bound based on Sparse Connectivity Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 22 / 26

Reduction In constant number of rounds: Computing exact MST in ℓ d ∞ for d = 100 log N ⇒ deciding Sparse Connectivity Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 23 / 26

Reduction In constant number of rounds: Computing exact MST in ℓ d ∞ for d = 100 log N ⇒ deciding Sparse Connectivity Construction: • For each vertex, pick a random vector v i in {− 1 , + 1 } d • For each edge e = ( i , j ) , add point f ( e ) = v i + v j Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 23 / 26

Reduction In constant number of rounds: Computing exact MST in ℓ d ∞ for d = 100 log N ⇒ deciding Sparse Connectivity Construction: • For each vertex, pick a random vector v i in {− 1 , + 1 } d • For each edge e = ( i , j ) , add point f ( e ) = v i + v j Distances (whp.): • Adjacent edges: � f ( e ) − f ( e ′ ) � ∞ ≤ 2 • Non-adjacent edges: � f ( e ) − f ( e ′ ) � ∞ = 4 Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 23 / 26

Reduction In constant number of rounds: Computing exact MST in ℓ d ∞ for d = 100 log N ⇒ deciding Sparse Connectivity Construction: • For each vertex, pick a random vector v i in {− 1 , + 1 } d • For each edge e = ( i , j ) , add point f ( e ) = v i + v j Distances (whp.): • Adjacent edges: � f ( e ) − f ( e ′ ) � ∞ ≤ 2 • Non-adjacent edges: � f ( e ) − f ( e ′ ) � ∞ = 4 MST weight: • Connected: ≤ 2 ( M − 1 ) • Not connected: ≥ 2 M Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 23 / 26

Other Results [Andoni, Nikolov, O., Yaroslavtsev 2014] • Algorithm for approximating Earth-Mover Distance • A new way of partitioning the instance into subproblems • Resolves an open question of Sharathkumar & Agarwal (2012) about the transportation problem: First near-linear time algorithm Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 24 / 26

Summary • Main goal: Efficient algorithms for the Massive Parallel Computation Model Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 25 / 26

Summary • Main goal: Efficient algorithms for the Massive Parallel Computation Model • Important efficiency measure: number of rounds When can it be made O ( 1 ) with low memory? Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 25 / 26

Summary • Main goal: Efficient algorithms for the Massive Parallel Computation Model • Important efficiency measure: number of rounds When can it be made O ( 1 ) with low memory? • Well known obstacle: Sparse Connectivity Krzysztof Onak (IBM Research) Parallel Algorithms for Graphs on a Very Large. . . 25 / 26