Superlinear Lower Bounds for Multipass Graph Processing
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1 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 1/29 Superlinear Lower Bounds for Multipass Graph Processing Krzysztof Onak IBM T.J. Watson Research Center Joint work with Venkat Guruswami (CMU)
2 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 2/29 Streaming Algorithms for Graphs Model: Input: large stream of edges Goal: minimize the amount of space and processing time per edge Allowed: randomization and small error probability Algorithm (5,4) (1,2) (4,3) (2,5) (3,1)...
3 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 2/29 Streaming Algorithms for Graphs Model: Input: large stream of edges Goal: minimize the amount of space and processing time per edge Allowed: randomization and small error probability Algorithm (5,4) (1,2) (4,3) (2,5) (3,1)... Worst-case ordering of edges (as opposed to random) The adversary knows the algorithm but not its random bits
4 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 3/29 One Pass vs. Multiple Passes Algorithm stream vs. Algorithm stream stream stream
5 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 3/29 One Pass vs. Multiple Passes Algorithm stream vs. Algorithm stream stream stream Do multiple passes make sense?
6 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 3/29 One Pass vs. Multiple Passes Algorithm stream vs. Algorithm stream stream stream Do multiple passes make sense? YES: Data on a large external storage device Sequential access often maximizes throughput
7 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 4/29 Graph Streaming Sweet-spot for graph streaming: Semi-streaming model [Muthukrishnan 2003] Allow n poly(log n) space Enough space to store vertices, but not edges
8 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 4/29 Graph Streaming Sweet-spot for graph streaming: Semi-streaming model [Muthukrishnan 2003] Allow n poly(log n) space Enough space to store vertices, but not edges General challenge: Which graph-theoretic problems admit n poly(logn) space streaming algorithms in one or a few passes? This Work: Rule out such algorithms for some basic graph problems
9 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 5/29 Our Results Undirected graphs: Problem 1: Are v and w at distance at most 2(p+1)? v w 2(p+1)?
10 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 5/29 Our Results Undirected graphs: Problem 1: Are v and w at distance at most 2(p+1)? Problem 2: Is there a perfect matching? vs.
11 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 5/29 Our Results Undirected graphs: Problem 1: Are v and w at distance at most 2(p+1)? Problem 2: Is there a perfect matching? Directed graphs: Problem 3: Is there a directed path from v to w? w v?
12 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 5/29 Our Results Undirected graphs: Problem 1: Are v and w at distance at most 2(p+1)? Problem 2: Is there a perfect matching? Directed graphs: Problem 3: Is there a directed path from v to w? Solving these graph problems in p passes requires Ω ( ) n 1+1/(2p+2) p 20 log 3/2 = n1+ω(1/p) n p O(1) bits of space (n = #vertices)
13 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 6/29 Comparison to Previous Results Known to require Ω(n 2 ) bits in one pass [Feigenbaum, Kannan, McGregor, Suri, Zhang 2004]
14 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 6/29 Comparison to Previous Results Known to require Ω(n 2 ) bits in one pass [Feigenbaum, Kannan, McGregor, Suri, Zhang 2004] Easy to prove Ω(n/p) for p passes via set disjointness
15 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 6/29 Comparison to Previous Results Known to require Ω(n 2 ) bits in one pass [Feigenbaum, Kannan, McGregor, Suri, Zhang 2004] Easy to prove Ω(n/p) for p passes via set disjointness We want n 1+Ω(1) lower bounds
16 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 6/29 Comparison to Previous Results Known to require Ω(n 2 ) bits in one pass [Feigenbaum, Kannan, McGregor, Suri, Zhang 2004] Easy to prove Ω(n/p) for p passes via set disjointness We want n 1+Ω(1) lower bounds Main challenge: embed hard problems into the space of edges not just vertices
17 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 7/29 Related Results: Shortest Path(s) Feigenbaum, Kannan, McGregor, Suri, Zhang (2005): Computing the first k layers of BFS tree in <k/2 passes requires Ω(n 1+1/k /k O(1) (logn) 1/k ) space k
18 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 7/29 Related Results: Shortest Path(s) Feigenbaum, Kannan, McGregor, Suri, Zhang (2005): Computing the first k layers of BFS tree in <k/2 passes requires Ω(n 1+1/k /k O(1) (logn) 1/k ) space Can be improved to < k passes using [Guha, McGregor 2007] k
19 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 7/29 Related Results: Shortest Path(s) Feigenbaum, Kannan, McGregor, Suri, Zhang (2005): Computing the first k layers of BFS tree in <k/2 passes requires Ω(n 1+1/k /k O(1) (logn) 1/k ) space Can be improved to < k passes using [Guha, McGregor 2007] Our problem: Fewer passes suffice k w v k? [FKMSZ 05] Here
20 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 8/29 Warmup: One-Pass Lower Bound [Feigenbaum et al. 2004]
21 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 9/29 Construction for Perfect Matching n 1 n n n 1
22 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 9/29 Construction for Perfect Matching n 1 n n n 1
23 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 9/29 Construction for Perfect Matching n 1 n n n 1
24 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 9/29 Construction for Perfect Matching n 1 n n n 1
25 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 9/29 Construction for Perfect Matching n 1 n n n 1 Stream = 1 2
26 Construction for Perfect Matching n 1 n n n 1 Stream = 1 2 Lower bound of Ω(n 2 ) via indexing Alice A[1...n 2 Bob ] x Bob s task: output A[x] Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 9/29
27 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 10/29 Construction for Shortest Path Approximation better then 5/3 requires Ω(n 2 ) space
28 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 10/29 Construction for Shortest Path Approximation better then 5/3 requires Ω(n 2 ) space v w 1 n n 1
29 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 10/29 Construction for Shortest Path Approximation better then 5/3 requires Ω(n 2 ) space v w 1 n n 1 Stream = 1 2
30 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 11/29 Stream Ordering How do we order edges in the stream?
31 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 11/29 Stream Ordering How do we order edges in the stream? Graphs = vertices + relations between them
32 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 11/29 Stream Ordering How do we order edges in the stream? Graphs = vertices + relations between them Solving most graph problems requires exploration
33 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 11/29 Stream Ordering How do we order edges in the stream? Graphs = vertices + relations between them Solving most graph problems requires exploration To prove lower bounds, create obstacles for exploration
34 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 11/29 Stream Ordering How do we order edges in the stream? Graphs = vertices + relations between them Solving most graph problems requires exploration To prove lower bounds, create obstacles for exploration One possibility: present edges in order opposite to what is suitable for exploration
35 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 12/29 Hard Instance for Multiple Passes
36 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a perfect matching? Θ(1) columns Each column Θ(n) rows
37 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a perfect matching? Θ(1) columns Each column Θ(n) rows
38 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a perfect matching? Θ(1) columns Each column Θ(n) rows
39 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a perfect matching? Θ(1) columns Each column Θ(n) rows
40 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a perfect matching? Θ(1) columns Each column Θ(n) rows
41 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a path of length 9 between red nodes? Θ(1) columns Each column Θ(n) rows
42 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 13/29 Construction for Perfect Matching Is there a path of length 6 between red nodes? Θ(1) columns Each column Θ(n) rows
43 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 14/29 Our Stream Ordering Is there a path of length 6 between red nodes?
44 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 14/29 Our Stream Ordering Is there a path of length 6 between red nodes? Stream =
45 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 14/29 Our Stream Ordering Is there a path of length 6 between red nodes? Stream = is easy in O(n) space
46 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 15/29 Streaming and Communication Protocols Assign each layer to one player Small-space streaming algorithm efficient communication protocol Goal: prove communication lower bound
47 The Proof Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 16/29
48 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 17/29 Important Problem: Pointer Chasing Definition: Input: p functions f i : [n] [n] Goal: Compute f p (f p 1 (...f 2 (f 1 (1))...)) f 1 f 2 f 3 f 4 f 5
49 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 17/29 Important Problem: Pointer Chasing Definition: Input: p functions f i : [n] [n] Goal: Compute f p (f p 1 (...f 2 (f 1 (1))...)) Two-player version: What players have: Alice Bob f 2, f 4, f 6,... f 1, f 3, f 5,... Alice speaks first
50 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 17/29 Important Problem: Pointer Chasing Definition: Input: p functions f i : [n] [n] Goal: Compute f p (f p 1 (...f 2 (f 1 (1))...)) Two-player version: What players have: Alice speaks first Alice Bob f 2, f 4, f 6,... f 1, f 3, f 5,... Nisan, Wigderson (1993): Computing in less then p = Θ(1) messages of communication requires Ω(n) communication
51 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 17/29 Important Problem: Pointer Chasing Definition: Input: p functions f i : [n] [n] Goal: Compute f p (f p 1 (...f 2 (f 1 (1))...)) p-player version: What players have: Player 1 Player 2... Playerp 1 Player p f p f p 1... f 2 f 1 Each round: players speak in order Player 1 through Player p
52 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 17/29 Important Problem: Pointer Chasing Definition: Input: p functions f i : [n] [n] Goal: Compute f p (f p 1 (...f 2 (f 1 (1))...)) p-player version: What players have: Player 1 Player 2... Playerp 1 Player p f p f p 1... f 2 f 1 Each round: players speak in order Player 1 through Player p Guha, McGregor (2007): Computing in less then p = Θ(1) rounds requires Ω(n) communication
53 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 18/29 FKMSZ Lower Bound for BFS Their Problem: Compute p levels of BFS tree from v
54 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 18/29 FKMSZ Lower Bound for BFS Their Problem: Compute p levels of BFS tree from v Sketch of their proof: Take communication lower bound for pointer chasing
55 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 18/29 FKMSZ Lower Bound for BFS Their Problem: Compute p levels of BFS tree from v Sketch of their proof: Take communication lower bound for pointer chasing Apply direct sum theorem of [Jain, Radhakrishnan, Sen (2003)]: Solving k instances requires k times more communication
56 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 18/29 FKMSZ Lower Bound for BFS Their Problem: Compute p levels of BFS tree from v Sketch of their proof: Take communication lower bound for pointer chasing Apply direct sum theorem of [Jain, Radhakrishnan, Sen (2003)]: Solving k instances requires k times more communication Show: Computing a level p BFS tree in graph of degree k = n Θ(1/p) enables solving k instances of pointer chasing.
57 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 18/29 FKMSZ Lower Bound for BFS Their Problem: Compute p levels of BFS tree from v Sketch of their proof: Take communication lower bound for pointer chasing Apply direct sum theorem of [Jain, Radhakrishnan, Sen (2003)]: Solving k instances requires k times more communication Show: Computing a level p BFS tree in graph of degree k = n Θ(1/p) enables solving k instances of pointer chasing. Our problem: Only need to check if BFS trees intersect Seems hard to infer full tree from this
58 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing 1: 2: 3: 4: 5: 6: 7: f 4 f 3 f 2 f 1 g 1 g 2 g 3 g 4 P 4 P 3 P 2 P 1 P 5 P 6 P 7 P 8
59 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing Problem to solve: Is the result the same? 1: 2: 3: 4: 5: 6: 7: f 4 f 3 f 2 f 1 g 1 g 2 g 3 g 4 P 4 P 3 P 2 P 1 P 5 P 6 P 7 P 8
60 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing Problem to solve: Is the result the same? (If some function maps Ω(logn) elements to one element, also say YES) 1: 2: 3: 4: 5: 6: 7: f 4 f 3 f 2 f 1 g 1 g 2 g 3 g 4 P 4 P 3 P 2 P 1 P 5 P 6 P 7 P 8
61 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing Problem to solve: Is the result the same? (If some function maps Ω(logn) elements to one element, also say YES) Three Steps: 1. IC µ,1/n 2(BBB) Ω(n) (µ = uniform distribution)
62 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing Problem to solve: Is the result the same? (If some function maps Ω(logn) elements to one element, also say YES) Three Steps: 1. IC µ,1/n 2(BBB) Ω(n) (µ = uniform distribution) 2. IC µk,1/(2n )( k 2 i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n
63 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing Problem to solve: Is the result the same? (If some function maps Ω(logn) elements to one element, also say YES) Three Steps: 1. IC µ,1/n 2(BBB) Ω(n) (µ = uniform distribution) 2. IC µk,1/(2n )( k 2 i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n Implies: CC 1/10 ( k i=1 BBB) Ω(kn)
64 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 19/29 Proof Overview Problem BBB (Basic Building Block): 2p players with two instances of pointer chasing Problem to solve: Is the result the same? (If some function maps Ω(logn) elements to one element, also say YES) Three Steps: 1. IC µ,1/n 2(BBB) Ω(n) (µ = uniform distribution) 2. IC µk,1/(2n )( k 2 i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n Implies: CC 1/10 ( k i=1 BBB) Ω(kn) 3. CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n O(1/p)
65 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p)
66 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p) Want: Protocol for k i=1 BBB using protocol for BFS intersection
67 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p) Want: Protocol for k i=1 BBB using protocol for BFS intersection Stack k instances of BBB on top of each other f 4 f 3 f 2 f 1 g 1 g 2 g 3 g 4 1: 2: 3: 4: 5: 6: 7: P 4 P 3 P 2 P 1 P 5 P 6 P 7 P 8 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29
68 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p) Want: Protocol for k i=1 BBB using protocol for BFS intersection Stack k instances of BBB on top of each other f 4 f 3 f 2 f 1 g 1 g 2 g 3 g 4 1: 2: 3: 4: 5: 6: 7: P 4 P 3 P 2 P 1 P 5 P 6 P 7 P 8 Gives instance of BFS tree intersection, but pointers from two different instances may intersect Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29
69 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p) Want: Protocol for k i=1 BBB using protocol for BFS intersection Randomly relabel intermediate results of functions and stack them on top of each other Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29
70 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p) Want: Protocol for k i=1 BBB using protocol for BFS intersection Randomly relabel intermediate results of functions and stack them on top of each other If pair of pointer chasing instances gives the same element, BFS trees intersect Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29
71 Step 3 CC 1/20 (BFS tree intersection) CC 1/10 ( k i=1 BBB) for k = n Θ(1/p) Want: Protocol for k i=1 BBB using protocol for BFS intersection Randomly relabel intermediate results of functions and stack them on top of each other If pair of pointer chasing instances gives the same element, BFS trees intersect k p n and random scrambling = If no pair gives the same element (and no Θ(logn)-to-1 mapping), BFS trees unlikely to intersect Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 20/29
72 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 21/29 Step 2 Statement: IC µk,1/(2n 2 )( k i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n
73 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 21/29 Step 2 Statement: IC µk,1/(2n 2 )( k i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n How: Product distribution: information cost = k i=1 information cost on instance i
74 Step 2 Statement: IC µk,1/(2n 2 )( k i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n How: Product distribution: information cost = k i=1 information cost on instance i Trivial, if all instances must be solved. The problem asks only for (the instances) Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 21/29
75 Step 2 Statement: IC µk,1/(2n 2 )( k i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n How: Product distribution: information cost = k i=1 information cost on instance i Trivial, if all instances must be solved. The problem asks only for (the instances) For specific instance, (other instances) = false most of the time Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 21/29
76 Step 2 Statement: IC µk,1/(2n 2 )( k i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n How: Product distribution: information cost = k i=1 information cost on instance i Trivial, if all instances must be solved. The problem asks only for (the instances) For specific instance, (other instances) = false most of the time Fix at random other instances s.t. (other instances) = false protocol must solve the instance Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 21/29
77 Step 2 Statement: IC µk,1/(2n 2 )( k i=1 BBB) k IC µ,1/n2(bbb) Ω(kn) for k n How: Product distribution: information cost = k i=1 information cost on instance i Trivial, if all instances must be solved. The problem asks only for (the instances) For specific instance, (other instances) = false most of the time Fix at random other instances s.t. (other instances) = false protocol must solve the instance Information cost won t decrease significantly on (other instances) = true Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 21/29
78 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 22/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n)
79 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 22/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) What is known: communication complexity for pointer chasing is Ω(n) for uniform distribution [Nisan, Wigderson 1993], [Guha, McGregor 2007]
80 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 22/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) What is known: communication complexity for pointer chasing is Ω(n) for uniform distribution [Nisan, Wigderson 1993], [Guha, McGregor 2007] Obstacles: 1. Need a proof for information complexity 2. Equality of pointer chasing instances Need to account for impact of Θ(logn)-to-1 maps
81 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 23/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Obstacle 1: Need a proof for information complexity
82 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 23/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Obstacle 1: Need a proof for information complexity Use [Jain, Radhakrishnan, Sen 2003]? Π = constant-round protocol revealing information IC with error ǫ: There is a protocol Π with total communication IC/δ 2 that errs with probability ǫ+δ i.e., small information small communication
83 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 23/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Obstacle 1: Need a proof for information complexity Use [Jain, Radhakrishnan, Sen 2003]? Π = constant-round protocol revealing information IC with error ǫ: There is a protocol Π with total communication IC/δ 2 that errs with probability ǫ+δ i.e., small information small communication Won t suffice for us: δ = o(1/n)
84 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 24/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 1): Use techniques of [JRS] to produce a protocol Π Π is deterministic errs with twice the probability sends messages of length IC p O(1) with probability 1 p Ω(1)
85 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 24/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 1): Use techniques of [JRS] to produce a protocol Π Π is deterministic errs with twice the probability sends messages of length IC p O(1) with probability 1 p Ω(1) Typically concise protocol
86 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 24/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 1): Use techniques of [JRS] to produce a protocol Π Π is deterministic errs with twice the probability sends messages of length IC p O(1) with probability 1 p Ω(1) Typically concise protocol Note: prob. of long message prob. of answer YES
87 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 25/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Modify [NW] for typically concise protocols and equality
88 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 25/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Modify [NW] for typically concise protocols and equality Original argument for protocol tree:
89 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 25/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Modify [NW] for typically concise protocols and equality Original argument for protocol tree: simulate in parallel trivial algorithm: make step forward when possible
90 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 25/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Modify [NW] for typically concise protocols and equality Original argument for protocol tree: simulate in parallel trivial algorithm: make step forward when possible x current = current value in this algorithm
91 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 25/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Modify [NW] for typically concise protocols and equality Original argument for protocol tree: simulate in parallel trivial algorithm: make step forward when possible x current = current value in this algorithm by induction, H(f next (x current )) = logn o(1) for 1 o(1) fraction of internal nodes for 1 o(1) fraction of leaves with o(n) communication
92 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 25/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Modify [NW] for typically concise protocols and equality Original argument for protocol tree: simulate in parallel trivial algorithm: make step forward when possible x current = current value in this algorithm by induction, H(f next (x current )) = logn o(1) for 1 o(1) fraction of internal nodes for 1 o(1) fraction of leaves with o(n) communication with this entropy, prob. of correct solution is o(1)
93 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 26/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): simulate in parallel trivial algorithm for both instances: make step forward when possible
94 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 26/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): simulate in parallel trivial algorithm for both instances: make step forward when possible (x current,y current ) = current values in this algorithm
95 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 26/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): simulate in parallel trivial algorithm for both instances: make step forward when possible (x current,y current ) = current values in this algorithm by induction, H(f next (x current )) = logn o(1) and H(g next (y current )) = logn o(1) for 1 o(1) fraction of internal nodes for 1 o(1) fraction of leaves with o(n) communication
96 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 26/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): simulate in parallel trivial algorithm for both instances: make step forward when possible (x current,y current ) = current values in this algorithm by induction, H(f next (x current )) = logn o(1) and H(g next (y current )) = logn o(1) for 1 o(1) fraction of internal nodes for 1 o(1) fraction of leaves with o(n) communication (impact of rare long messages small)
97 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 27/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Entropy of solution to each pointer chasing logn o(1) Probability Ω(1/n) for 3 4 n elements
98 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 27/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Entropy of solution to each pointer chasing logn o(1) Probability Ω(1/n) for 3 4 n elements Distributions independent: deterministic protocol pointer chasing instances held by different players product input distribution
99 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 27/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Entropy of solution to each pointer chasing logn o(1) Probability Ω(1/n) for 3 4 n elements Distributions independent: deterministic protocol pointer chasing instances held by different players product input distribution must collide with probability n/4 Ω(1/n) 2 = Ω(1/n)
100 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 27/29 Step 1 Statement: IC µ,1/n 2(BBB) Ω(n) Our solution (part 2): Entropy of solution to each pointer chasing logn o(1) Probability Ω(1/n) for 3 4 n elements Distributions independent: deterministic protocol pointer chasing instances held by different players product input distribution must collide with probability n/4 Ω(1/n) 2 = Ω(1/n) protocol errs with probability Ω(1/n)
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103 Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 28/29 Main result: Shortest Path, Perfect Matching, and Directed Connectivity require n 1+Ω(1/p) space in p passes Open Questions: Simpler proof? Improve lower bounds from Ω(n 1+1/(2p) ) to Ω(n 1+1/p )? Better bounds for maximum matching? Is looking for a few augmenting paths harder? Can the techniques be used for approximate matchings?
104 Questions? Krzysztof Onak Superlinear lower bounds for multipass graph processing p. 29/29
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