Think Global Act Local - ETH Z 2012...Think Global – Act Local . Town Planning Patrick Geddes ....
Transcript of Think Global Act Local - ETH Z 2012...Think Global – Act Local . Town Planning Patrick Geddes ....
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ETH Zurich – Distributed Computing – www.disco.ethz.ch
Roger Wattenhofer
Think Global – Act Local
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Town Planning Patrick Geddes
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Architecture Buckminster Fuller
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Computer Architecture Caching
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Robot Gathering e.g., [Degener et al., 2011]
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Natural Algorithms
[Bernard Chazelle, 2009]
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Algorithmic Trading
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Think Global – Act Local
…is there a theory?
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Complexity Theory
Can a Computer Solve Problem P in Time t?
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Complexity Theory
Can a Computer Solve Problem P in Time t?
Network
(Think Global – Act Local)
Distributed
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Distributed (Message-Passing) Algorithms
• Nodes are agents with unique ID’s that can communicate with neighbors by sending messages. In each synchronous round, every node can send a (different) message to each neighbor.
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Distributed (Message-Passing) Algorithms
• Nodes are agents with unique ID’s that can communicate with neighbors by sending messages. In each synchronous round, every node can send a (different) message to each neighbor.
• Distributed (Time) Complexity: How many rounds does problem take?
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An Example
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How Many Nodes in Network?
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How Many Nodes in Network?
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How Many Nodes in Network?
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How Many Nodes in Network?
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How Many Nodes in Network?
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How Many Nodes in Network?
1
1
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How Many Nodes in Network?
2
1
1
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4
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1
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With a simple flooding/echo process, a network can find the number of nodes in time 𝑂(𝐷), where 𝐷 is the diameter (size) of the network.
How Many Nodes in Network?
2
1
1
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1
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10
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Diameter of Network?
• Distance between two nodes = Number of hops of shortest path
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Diameter of Network?
• Distance between two nodes = Number of hops of shortest path
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Diameter of Network?
• Distance between two nodes = Number of hops of shortest path
• Diameter of network = Maximum distance, between any two nodes
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Diameter of Network?
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Diameter of Network?
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Diameter of Network?
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Diameter of Network?
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Diameter of Network?
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Diameter of Network?
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Diameter of Network?
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(even if diameter is just a small constant)
Pair of rows connected neither left nor right? Communication complexity: Transmit Θ(𝑛2) information over O(𝑛) edges Ω(𝑛) time!
[Frischknecht, Holzer, W, 2012]
Networks Cannot Compute Their Diameter in Sublinear Time!
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What about a “local” task?
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Example: Minimum Vertex Cover (MVC)
• Given a network with n nodes, nodes have unique IDs.
• Find a Minimum Vertex Cover (MVC)
– a minimum set of nodes such that all edges are adjacent to node in MVC
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Example: Minimum Vertex Cover (MVC)
• Given a network with n nodes, nodes have unique IDs.
• Find a Minimum Vertex Cover (MVC)
– a minimum set of nodes such that all edges are adjacent to node in MVC
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Example: Minimum Vertex Cover (MVC)
• Given a network with n nodes, nodes have unique IDs.
• Find a Minimum Vertex Cover (MVC)
– a minimum set of nodes such that all edges are adjacent to node in MVC
69
17
11
10 7
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On MVC
• Find an MVC that is “close” to minimum (approximation)
• Trade-off between time complexity and approximation ratio
• MVC: Various simple (non-distributed) 2-approximations exist!
• What about distributed algorithms?!?
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𝑆1
Finding the MVC (by Distributed Algorithm)
• Given the following bipartite graph with 𝑆0 = 𝛿 𝑆1
• The MVC is just all the nodes in 𝑆1
• Distributed Algorithm…
𝑆0
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𝑆1
Finding the MVC (by Distributed Algorithm)
𝑆0
• Given the following bipartite graph with 𝑆0 = 𝛿 𝑆1
• The MVC is just all the nodes in 𝑆1
• Distributed Algorithm…
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𝑆1
Finding the MVC (by Distributed Algorithm)
𝑆0
• Given the following bipartite graph with 𝑆0 = 𝛿 𝑆1
• The MVC is just all the nodes in 𝑆1
• Distributed Algorithm…
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𝑆1
𝑆0
7 7 7 7 7 7 7 7 3 2 1 3 4 1 1 2 1 4 4 2 2 4 1 1
𝑁2(node in 𝑆0) 𝑁2(node in 𝑆1)
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𝑆1
𝑆0
7 7 7 7 7 7 7 7 3 2 1 3 4 1 1 2 1 4 4 2 2 4 1 1
𝑁2(node in 𝑆0) 𝑁2(node in 𝑆1) Graph is “symmetric”, yet highly non-regular!
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Lower Bound: Results
• We can show that for 𝜖 > 0, in 𝑡 time, the approximation ratio is at least
• Constant approximation needs at least Ω(log Δ) and Ω( log 𝑛) time.
• Polylog approximation Ω(log Δ/ log log Δ) and Ω( log 𝑛/ log log 𝑛).
𝑡 𝑡
[Kuhn, Moscibroda, W, journal version in submission]
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Lower Bound: Results
• We can show that for 𝜖 > 0, in 𝑡 time, the approximation ratio is at least
• Constant approximation needs at least Ω(log Δ) and Ω( log 𝑛) time.
• Polylog approximation Ω(log Δ/ log log Δ) and Ω( log 𝑛/ log log 𝑛).
𝑡 𝑡
tight for MVC
[Kuhn, Moscibroda, W, journal version in submission]
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Lower Bound: Reductions
• Many “local looking” problems need non-trivial t, in other words, the
bounds Ω(log Δ) and Ω( log 𝑛) hold for a variety of classic problems.
[Kuhn, Moscibroda, W, journal version in submission]
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Lower Bound: Reductions
• Many “local looking” problems need non-trivial t, in other words, the
bounds Ω(log Δ) and Ω( log 𝑛) hold for a variety of classic problems.
line graph
cloning
MVC through MM
line graph
[Kuhn, Moscibroda, W, journal version in submission]
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Olympics!
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e.g., dominating set approximation in planar graphs
Distributed Complexity Classification
1 log∗ 𝑛 polylog 𝑛 𝐷 poly 𝑛
various problems in growth-bounded
graphs
MIS, approx. of dominating set, vertex cover, ...
count, sum, spanning tree,
...
diameter, MST, verification of e.g. spanning tree, …
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e.g., dominating set approximation in planar graphs
Distributed Complexity Classification
1 log∗ 𝑛 polylog 𝑛 𝐷 poly 𝑛
various problems in growth-bounded
graphs
MIS, approx. of dominating set, vertex cover, ...
count, sum, spanning tree,
...
diameter, MST, verification of e.g. spanning tree, …
„easy“ „hard“
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Locality
Sublinear Algorithms
Local Algorithms
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Locality is Everywhere!
Sublinear Algorithms
Self-Stabilization
Self- Assembly
Applications e.g. Multi-Core
Dynamic Networks
Local Algorithms
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Locality is Everywhere!
Sublinear Algorithms
Self-Stabilization
Self- Assembly
Applications e.g. Multi-Core
Dynamic Networks
Local Algorithms
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[Afek, Alon, Barad, et al., 2011]
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[Afek, Alon, Barad, et al., 2011]
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Maximal Independent Set (MIS)
• Given a network with n nodes, nodes have unique IDs.
• Find a Maximal Independent Set (MIS)
– a non-extendable set of pair-wise non-adjacent nodes
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Maximal Independent Set (MIS)
• Given a network with n nodes, nodes have unique IDs.
• Find a Maximal Independent Set (MIS)
– a non-extendable set of pair-wise non-adjacent nodes
69
17
11
10 7
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Maximal Independent Set (MIS)
• Given a network with n nodes, nodes have unique IDs.
• Find a Maximal Independent Set (MIS)
– a non-extendable set of pair-wise non-adjacent nodes
69
17
11
10 7
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given: id, degree synchronized while (true) { p = 1 /(2*degree); if (random value between 0 and 1 < p) { transmit “(degree, id)”; …
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given: id, degree synchronized while (true) { p = 1 /(2*degree); if (random value between 0 and 1 < p) { transmit “(degree, id)”; …
?!
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≪
Distributed Computing Without Computing!
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Nano-Magnetic Computing
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Stone Age Distributed Computing
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nFSM: networked Finite State Machine
• Every node is the same finite state machine, e.g. no IDs
• Apart from their state, nodes cannot store anything
• Nodes know nothing about the network, including e.g. their degree
• Nodes cannot explicitly send messages to selected neighbors, i.e. nodes can only implicitly communicate by changing their state
• Operation is asynchronous
• Randomized next state okay, as long as constant number
• Nodes cannot compute, e.g. cannot count
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One, Two, Many Principle Piraha Walpiri
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One, Two, Many Principle
• Not okay
– while (k < log n) {
– At least half of neighbors in state s?
– More neighbors in state s than in state t?
• Okay
– No neighbor in state s?
– Some neighbor in state s?
– At most two neighbors in state s?
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𝑢2
𝑢1
𝑢0
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MIS
𝑢2
𝑢1
𝑢0
alone
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MIS
𝑢2
𝑢1
𝑑2 𝑢0
alone
not alone
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MIS
𝑢2
𝑢1
𝑑1 𝑑2 𝑢0
alone
not alone
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MIS
𝑢2
𝑢1
𝑑1 𝑑2 not 𝑢0
alone
not alone
MIS neighbor
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nFSM solves MIS whp in time O(log2 𝑛)
MIS
𝑢2
𝑢1
𝑑1 𝑑2 not 𝑢0
alone
not alone
MIS neighbor
[Emek, Smula, W, in submission]
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Overview General Graph
Growth-Bounded Graph
Bounded Degree Graph D
iameter
MV
C
MIS
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Overview General Graph
Growth-Bounded Graph
Bounded Degree Graph
Mes
sage
Pas
sin
g
Wir
ele
ss
Sto
ne
Age
D
iameter
MV
C
MIS
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Summary
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Thank You! Questions & Comments?
Thanks to my co-authors Yuval Emek Silvio Frischknecht Stephan Holzer Fabian Kuhn Thomas Moscibroda Jasmin Smula www.disco.ethz.ch