DISTRIBUTED ALGORITHMS
AND BIOLOGICAL SYSTEMS
Nancy Lynch, Saket Navlakha
BDA-2014
October, 2014
Austin, Texas
Distributed Algorithms + Biological Systems
• Distributed algorithms researchers have been considering
biological systems, looking for:
• Biological problems and behaviors that they can model and study
using distributed algorithms methods, and
• Biological strategies that can be adapted for use in computer
networks.
• Yields interesting distributed algorithms results.
• Q: But what can distributed algorithms contribute to the
study of biological systems?
• This talk:
• Overview fundamental ideas from the
distributed algorithms research area, for
biology researchers.
• Consider how these might contribute to
biology research.
What are distributed algorithms?
• Abstract models for systems consisting of many interacting
components, working toward a common goal.
• Examples:
• Wired or wireless network of computers,
communicating or managing data.
• Robot swarm, searching an unknown
terrain, cleaning up, gathering
resources,…
What are distributed algorithms?
• Abstract models for systems consisting of many interacting
components, working toward a common goal.
• Examples:
• Wired or wireless network of computers,
communicating or managing data.
• Robot swarm, searching an unknown
terrain, cleaning up, gathering
resources,…
• Social insect colony, foraging, feeding,
finding new nests, resisting predators,…
• Components generally interact
directly with nearby components
only, using local communication.
Distributed algorithms research
• Models for distributed systems.
• Problems to be solved.
• Algorithms, analysis.
• Lower bounds, impossibility results.
• Problems: Communication, consensus,
data management, resource allocation,
synchronization, failure detection,…
• Models:
• Interacting automata.
• Local communication: individual message-
passing, local broadcast, or shared memory.
• Metrics: Time, amount of communication,
local storage.
Distributed algorithms research
• Algorithms:
• Some use simple rules, some use
complex logical constructions.
• Local communication.
• Designed to minimize costs, according to
the cost metrics.
• Often designed to tolerate limited failures.
• Analyze correctness, costs, and faulttolerance mathematically.
• Try to “optimize”, or at least obtain
algorithms that perform “very well”
according to the metrics.
Distributed algorithms research
• Lower bounds, impossibility results:
• Theorems that say that you can’t solve a
problem in a particular system model, or
you can’t solve it with a certain cost.
• Distributed computing theory includes
hundreds of impossibility results.
• Unlike theory for traditional sequential
algorithms.
• Why?
• Distributed models are hard to cope with.
• Locality of knowledge and action imposes
strong limitations.
Formal modeling and analysis
• Distributed algorithms can get
complicated:
• Many components act concurrently.
• May have different speeds, failures.
• Local knowledge and action.
• In order to reason about them
carefully, we need clear, rigorous
mathematical foundations.
• Impossibility results are
meaningless without rigorous
foundations.
• Need formal modeling frameworks.
Key ideas
• Model systems using interacting automata.
• Not just finite-state automata, but more elaborate automata that
may include complex states, timing information, probabilistic
behavior, and both discrete and continuous state changes.
• Timed I/O Automata, Probabilistic I/O Automata.
• Formal notions of composition and abstraction.
• Support rigorous correctness proofs and cost analysis.
Key ideas
• Distinguish among:
• The problems to be solved,
• The physical platforms on which the problems should be solved, and
• The algorithms that solve the problems on the platforms.
• Define cost metrics, such as time, local storage space, and
amount of communication, and use them to analyze and
compare algorithms and prove lower bounds.
• Many kinds of algorithms.
• Many kinds of analysis.
• Many lower bound methods.
• Q: How can this help biology research?
Biology research
• Model a system of insects, or cells,
using interacting automata.
• Define formally:
• The problems the systems are solving (distinguishing cells, building
structures, foraging, reaching consensus, task allocation, …),
• The physical capabilities of the systems, and
• The strategies (algorithms) that are used by the systems.
• Identify cost metrics (time, energy,…)
• Analyze, compare strategies.
• Prove lower bounds expressing inherent limitations.
• Use this to:
• Predict system behavior.
• Explain why a biological system has evolved to have the structure it
has, and has evolved or learned to behave as it does.
Example 1: Leader election
• Ring of processes.
• Computers, programs, robots, insects, cells,…
• Communicate with neighbors by sending “messages”, in
synchronous rounds.
Example 1: Leader election
• Ring of processes.
• Computers, programs, robots, insects, cells,…
• Communicate with neighbors by sending “messages”, in
•
•
•
•
synchronous rounds.
Problem: Exactly one process should (eventually)
announce that it is the leader.
Motivation:
A leader in a computer network or robot swarm could take
charge of a computation, telling everyone else what to do.
A leader ant could choose a new nest.
Leader election
• Problem: Exactly one process should
announce that it is the leader.
• Suppose:
• Processes start out identical.
• Behavior of each process at each round is determined by its
current state and incoming messages.
• Theorem: In this case, it’s impossible to elect a leader.
• No distributed algorithm could possibly work.
• Proof: By contradiction. Suppose we have an algorithm
that works, and reach a contradictory conclusion.
Leader election
• Problem: Exactly one process should
•
•
•
•
•
•
•
•
•
announce that it is the leader.
Initially identical, deterministic processes
Theorem: Impossible.
Proof: Suppose we have an algorithm that works.
All processes start out identical.
At round 1, they all do the same thing (send the same
messages, make the same changes to their “states”).
So they are again identical.
Same for round 2, etc.
Since the algorithm solves the problem, some process
must eventually announce that it is the leader.
But then everyone does, contradiction.
So what to do?
• Suppose the processes aren’t identical---they
have unique ID numbers.
• Algorithm:
• Send a message containing your ID clockwise.
• When you receive an ID, compare it with your own ID.
• If the incoming ID is:
• Bigger, pass it on.
• Smaller, discard it
• Equal, announce that you are the leader.
• Elects the process with the largest identifier.
• Takes 𝑂(𝑛) communication rounds, 𝑂(𝑛2 𝑏) bits of
communication, where 𝑏 bits are used to represent an
identifier.
What if we don’t have IDs?
• Suppose processes are identical, no IDs.
• Allow them to make random choices.
• Assume the processes know n, the total number of
processes in the ring.
• Algorithm:
• Toss an unfair coin, with probability 1/n of landing on heads.
• If you toss heads, become a “leader candidate”.
• It’s “pretty likely” that there is exactly one candidate.
• The processes can verify this by passing messages
around and seeing what they receive.
• If they did not succeed, try again.
Example 2: Maximal Independent Set
• Assume a general graph network,
with processes at the nodes:
• Problem: Select some of the
processes, so that they form a
Maximal Independent Set.
• Independent: No two neighbors are both in the set.
• Maximal: We can’t add any more nodes without violating
independence.
• Motivation:
• Communication networks: Selected processes can take charge of
communication, convey information to all the other processes.
• Developmental biology: Distinguish cells in fruit fly’s nervous system
to become “Sensory Organ Precursor” cells.
Maximal Independent Set
• Problem: Select some of the processes, so that they form
a Maximal Independent Set.
• Independent: No two neighbors are both in the set.
• Maximal: We can’t add any more nodes without violating
independence.
Maximal Independent Set
• Problem: Select some of the processes, so that they form
a Maximal Independent Set.
• Independent: No two neighbors are both in the set.
• Maximal: We can’t add any more nodes without violating
independence.
Distributed MIS problem
• Assume processes know a
“good” upper bound on n.
• No IDs.
• Problem: Processes should cooperate in a distributed
(synchronous, message-passing) algorithm to compute an
MIS of the graph.
• Processes in the MIS should output in and the others
should output out.
• Unsolvable by deterministic algorithms, in some graphs.
• Probabilistic algorithm:
Probabilistic MIS algorithm
• Algorithm idea:
• Each process chooses a random
ID from 1 to N.
• N should be large enough so it’s
likely that all IDs are distinct.
• Neighbors exchange IDs.
• If a process’s ID is greater than all its neighbors’ IDs, then the
process declares itself in, and notifies its neighbors.
• Anyone who hears a neighbor is in declares itself out, and
notifies its neighbors.
• Processes reconstruct the remaining graph, omitting those
who have already decided.
• Continue with the reduced graph, until no nodes remain.
Example
• All nodes start out identical.
7
Example
8
1
10
16
11
2
9
5
• Everyone chooses an ID.
13
7
Example
8
1
10
16
11
2
9
5
13
• Processes that chose 16 and 13 are in.
• Processes that chose 11, 5, 2, and 10 are out.
7
Example
4
12
• Undecided (gray) processes choose new IDs.
18
7
Example
4
12
• Processes that chose 12 and 18 are in.
• Process that chose 7 is out.
18
Example
12
• Undecided (gray) process chooses a new ID.
Example
12
• It’s in.
Properties of the algorithm
• If it ever finishes, it produces a Maximal Independent Set.
• It eventually finishes (with probability 1).
• The expected number of rounds until it finishes is
𝑂(log 𝑛).
More examples
• Building spanning trees that minimize various network
cost measures.
• Motivation:
• Communication networks: Use the tree for sending messages from
the leader to everyone else.
• Slime molds: Build a system of tubes that can convey food
efficiently from a source to all the mold cells.
• Building other network structures:
• Routes
• Clusters with leaders.
• …
More examples
• Reaching consensus, in the presence of faulty
components (stopping, Byzantine).
• Motivation:
• Agree on aircraft altimeter readings.
• Agree on processing of data transactions.
• Ants: Agree on a new nest location.
• Lower bounds:
• Byzantine-failure consensus requires at least 3𝑓 + 1 processes
(and 2𝑓 + 1 connectivity) to tolerate 𝑓 faulty processes.
• Even stopping-failure consensus requires at least 𝑓 + 1 rounds.
• Algorithms:
• Multi-round exchange of values, looking for a dominant preference
[Lamport, Pease, Shostak]; achieves optimal bounds.
Other examples
• Communication
• Resource allocation
• Task allocation
• Synchronization
• Data management
• Failure detection
•…
Recent Work: Dynamic Networks
• Most of distributed computing
theory deals with fixed, wired
networks.
• Now we study dynamic networks, which change while they
are operating.
• E.g. wireless networks.
• Participants may join, leave, fail, and recover.
• May move around (mobile systems).
Computing in Dynamic Graph Networks
• Network is a graph that changes arbitrarily from round to
round (but it’s always connected).
• At each round, each process sends a message, which is
received by all of its neighbors at that round.
• Q: What problems are solvable, and at what cost?
• Global message broadcast,
• Determining the minimum input.
• Counting the total number of nodes.
• Consensus
• Clock synchronization
•…
Robot Coordination Algorithms
• A swarm of cooperating robots, engaged in:
• Search and rescue
• Exploration
• Robots communicate, learn about their
environment, perform coordinated activities.
• We developed distributed algorithms to:
• Keep the swarm connected for communication.
• Achieve “flocking” behavior.
• Map an unknown environment.
• Determine global coordinates, working from local sensor
readings.
Biological Systems as Distributed
Algorithms
• Biological systems consist of many
components, interacting with nearby
components to achieve common goals.
• Colonies of bacteria, bugs, birds, fish,…
• Cells within a developing organism.
• Brain networks.
Biological Systems as Distributed
Algorithms
• They are special kinds of distributed
algorithms:
• Use simple chemical “messages”.
• Components have simple “state”, follow
simple rules.
• Flexible, robust, adaptive.
Problems
• Leader election: Ants choose a queen.
• Maximal Independent Set: In fruit fly
development, some cells become sensory
organs.
• Building communication structures:
• Slime molds build tubes to connect to food.
• Brain cells form circuits to represent
memories.
More problems
• Consensus: Bees agree on location of a
new hive.
• Reliable local communication: Cells use
chemical signals.
• Robot swarm coordination: Birds, fish,
bacteria travel in flocks / schools /
colonies.
Biological Systems as Distributed
Algorithms
• So, study biological systems as distributed
algorithms.
• Models, problem statements, algorithms,
impossibility results.
• Goals:
• Use distributed algorithms to understand
biological system behavior.
• Use biological systems to inspire new
distributed algorithms.
Example 3: Ant (Agent) Exploration
• [Lenzen, Lynch, Newport, Radeva, PODC 2014]
• 𝑛 ants exploring a 2-dimensional grid for food, which is
•
•
•
•
hidden at distance at most 𝐷 from the nest.
No communication.
Ants can return to the nest (origin) at any time, for free.
In terms of 𝐷 and 𝑛, we get an upper bound on the
expected time for some ant to find the food.
Similar to [Feinerman, Korman].
• But now, assume bounds on:
• The size of an ant’s memory.
• The fineness of probabilities used in
an ant’s random choices.
A(ge)nt Exploration
• Algorithm (for each ant):
• Multi-phase algorithm.
• In successive phases, search to successively greater distances.
• Distances are determined by step-by-step random choices, using
successively smaller probabilities at successive phases.
• Expected time for some ant to find the food is (roughly)
𝑂
𝐷2
𝑛
+𝐷 .
• Works even in the “nonuniform” case,
where the ants don’t have a good estimate
of 𝐷.
• Analysis: Basic conditional probability
methods, Chernoff bounds.
A(ge)nt Exploration
• Lower bound: Assume 𝑛 is polynomially bounded by
𝐷. Then there is some placement for the food such that,
with high probability, no ant finds it in time much less
than Ω 𝐷2 .
• Lower bound proof ideas:
• Model the movement of an ant through its state space as a Markov
chain.
• Enhance this Markov chain with information about the ant’s
movement in the grid.
• Use properties of the enhanced Markov chain to
analyze probabilities of reaching various locations in
the grid within certain amounts of time.
Example 4: Ant Task Allocation
• [Cornejo, Dornhaus, Lynch, Nagpal 2014]
• 𝑛 ants allocate themselves among a fixed, known
•
•
•
•
set of tasks, such as foraging for food, feeding
larvae, and cleaning the nest.
No communication among ants.
Obtain information from the environment
(somehow) about tasks’ current energy
requirements.
Try to minimize sum of squares of energy deficits
for all tasks.
Special case we’ve studied so far:
• All ants are identical.
• Synchronous rounds of decision-making.
• Simple binary information (deficit/surplus for each task).
Ant Task Allocation
• Algorithm (for each ant):
• Probabilistic decisions, based on binary deficit/surplus information.
• Five states: Resting, FirstReserve, SecondReserve, TempWorker,
CoreWorker, plus “potentials” for all tasks.
• Ants in Worker states work on tasks; others are idle.
• TempWorkers are more biased towards becoming idle.
• Reserve workers are biased towards the task they last worked on.
• Detailed transition rules and task assignment rules.
• Theorem: Assume “sufficiently many” ants. In a period of
length Θ(log 𝑛) with unchanging demand, whp, the ants
converge to a fixed allocation that satisfies all demands.
• Proof: Analyzes how oscillations get dampened.
Limited ant memory size…
Low c
Efficie
Robu
Adapt
Distributed
Networks
Modularity
Stochastic
Optimizatio
Operating Principles
4 examples
Slime mold foraging & MST construction
[Afek et al. Science 2010]
[Tero et al. Science 2010]
E coli foraging & consensus navigation
[Shklarsh et al. PLoS Comput. Biol 2013]
d network design
Fly brain development & MIS
Synaptic pruning & network design
SOP selection in fruit flies
• During nervous system development,
some cells are selected as sensory
organ precursors (SOPs)
• SOPs are later attached to the fly's
sensory bristles
• Like MIS, each cell is either:
- Selected as a SOP
- Laterally inhibited by a
neighboring SOP so it cannot
become a SOP
Trans vs. cis inhibition
Notch
Delta
Trans model
• Recent findings suggest that Notch is also
suppressed in cis by delta’s from the same cell
• Only when a cell is ‘elected’ it communicates its
decision to the other cell
Cis+Trans model
Miller et al Current Biology 2009, Sprinzak et al Nature 2010, Barad et al Science Signaling 2010
MIS vs SOP
• Stochastic
• Proven for MIS
• Experimentally validated for
SOP
• Compared to previous algs:
• Unlike Luby, SOP cells do not
know its number of neighbors
(nor network topology)
• Constrained by time
• An uninhibited cell
eventually becomes a SOP
• For SOP, messages are binary
• Reduced communication
• A node (cell) only sends
messages if it joins the MIS
Can we improve MIS
algorithms by understating
how the biological process is
performed?
Movie
Simulations
• 2 by 6 grid
• Each cell touches all adjacent and
diagonal neighbors
Simulations
- All models assume a cell becomes a SOP
by accumulating the protein Delta until it
passes some threshold
Four different models:
1. Accumulation
- Accumulating Delta based on a Gaussian distribution
2. Fixed Accumulation
- Randomly select an accumulation rate only once
3. Rate Change
- Increase accumulation probability as time goes by
using feedback loop
4. Fixed rate
- Fix accumulation probability, use the same
probability in all rounds
Observation: Comparing the time of
experimental and simulated selection
New MIS Algorithm + Demo
MIS Algorithm (n,D) // n – upper bound on number of nodes
D - upper bound on number of neighbors
- p = 1/D
- round = round +1
- W.h.p., the algorithm computes
a MIS in O(log2n) rounds
- if round > log(n)
- All msgs are 1 bit
p = p * 2 ; round = 0 // we start a new phase
- Each processor flips a coin with probability p
- If result is 0, do nothing
- If result is 1, send to all other processors
- If no collisions, Leader; all processes exit
- Otherwise
Afek et al Science 2011, Afek et al DISC 2011
Low c
Efficie
Robu
Adapt
Distributed
Networks
Modularity
Stochastic
Optimizatio
Operating Principles
4 examples
Slime mold foraging & MST construction
[Afek et al. Science 2010]
[Tero et al. Science 2010]
E coli foraging & consensus navigation
[Shklarsh et al. PLoS Comput. Biol 2013]
d network design
Fly brain development & MIS
Synaptic pruning & network design
Slime mold network
Tokyo rail network
Very similar transport efficiency and resilience
Slime-mold model
• Slime mold operates without
centralized control or global
information processing
• Simple feedback loop between the
thickness of tube (i.e. edge weight)
and the internal protoplasmic flow
(i.e. the amount it’s used)
• Idea: reinforce preferred routes;
removed unused or overly
redundant connections
• Edges = tubes
• Nodes = junctions between tubes
Slime-mold model
Start with a randomly meshed
lattice with fine spacing (t = 0)
The flux through a tube
(edge) is calculated as:
Conductance
of the tube
At each time step, choose
random source (node1) and
sink (node2)
Source:
Sink:
Otherwise:
Pressure difference
between ends of
tube
Length of tube
Think “network flow”: the source pumps flow out,
the sink consumes it; everyone else just passes the
flow along (conservation: what enters must exit)
Updated tube/edge weights
The first term: the expansion of
tubes in response to the flux
The second term: the rate of tube
constriction, so that if there is no
flow, the tube gradually
disappears
f(|Q|) = sigmoidal curve
Measuring network quality
= Final slime
mold network
• TL = wiring length
• MD = avg minimum distance
between any pair of food sources
• FT = tolerance to disconnection after
single link failure
• Each normalized to baseline MST
= Tokyo railway
network
= minimum
spanning tree
= minimum
spanning tree
+ added links
= actual railway network
= slime mold designed network (with light)
= slime mold designed network (no light)
= simulated networks
Same perform,
lower cost for SM
Overall better FT
for real network
Cost: TLMST(
) = 1.80 and TLMST(
Efficiency: MDMST(
) = 1.75 +/- 0.30  SIMILAR
) = 0.85 and MDMST(
) = 0.85 +/- 0.04  SIMILAR
Fault tolerance: 4% of links cause real network disconnection, but 14-20% for SM
Low c
Efficie
Robu
Adapt
Distributed
Networks
Modularity
Stochastic
Optimizatio
Operating Principles
4 examples
Slime mold foraging & MST construction
[Afek et al. Science 2010]
[Tero et al. Science 2010]
E coli foraging & consensus navigation
[Shklarsh et al. PLoS Comput. Biol 2013]
d network design
Fly brain development & MIS
Synaptic pruning & network design
Bacteria foraging
• Imagine a complicated terrain with food in some local
minima, and a collection of bacterium that want to find the
food. How can they quickly succeed?
• Need to agree on a movement trajectory to collectively
find food sources based on:
• Individual knowledge / sensory information
• Knowledge from others
• Need to account for dynamic and varied environments
with little or no central coordination
• May also be helpful for robot coordination problems
Bacterial chemotaxis
• Bacteria navigate via chemotaxis, i.e. they move
according to gradients in the chemical concentration
(food)
• In areas of low concentration, bacteria tumble more (i.e.
they move randomly)
Bacteria acquire cues from
neighbors:
- Repulsion to avoid collision
- Orientation
- Attraction to avoid
fragmentation
Bacteria as automata
• Treat bacteria has individual automata with two sources of
information:
• Individual belief in the food source
• Interaction with neighbors’ beliefs about where the food source is
• Parameter w(i)t controls how much bacterium i listens to
its neighbors at time t
Independent vs fixed weights
• [[maybe include videos?]]
Problem: erroneous
positive feedback leads
bacteria astray: a
subgroup gets “bad”
information and convinces
the others along an
incorrect trajectory
Solution: adaptive weights
• Bacteria adjust their communication weights based on
their own internal confidence:
• When a bacteria finds a beneficial path (strong gradient), it
downweights, listens less to its neighbors
• When it is unsure, it increases its interaction with its neighbors
• This simple idea requires short-term memory and a very
simple rule
Low c
Efficie
Robu
Adapt
Distributed
Networks
Modularity
Stochastic
Optimizatio
Operating Principles
4 examples
Slime mold foraging & MST construction
[Afek et al. Science 2010]
[Tero et al. Science 2010]
E coli foraging & consensus navigation
[Shklarsh et al. PLoS Comput. Biol 2013]
d network design
Fly brain development & MIS
Synaptic pruning & network design
Synaptic pruning & brain development
Thousands of neurons &
synapses generated per
minute
≤ Human birth
Density of synapses
decreases by 50-60%
Age 2
Adolescence
Pruning occurs in almost every brain region and organism studied
But it is unlike previous models of network
development… so why does it happen?
• Idea:
• Source-target pairs are drawn from a distribution D that represents
a prior/likelihood on signals.
• Dtrain is fixed beforehand (but unknown)
• Dtest derived from D, but different s-t pairs
• Constraints:
• Distributed no centralized controller
• Streaming process s-t pairs online
Synaptic pruning as an algorithm
1. Start with all equivalent nodes
and
blanket the space with
connections
Synaptic pruning as an algorithm
2. Receive source-target request
Target
Source
Edge
Usage
1-10
1
10-15
1
*
0
“use it or lose it”
3. Route request and keep track of
usage
Target
15
10
Source
1
Edge
Usage
1-10
1
10-15
1
6-14
1
14-30
1
*
0
“use it or lose it”
4. Process next pair…
Target
30
15
14
10
6
1
Source
Edge
Usage
1-10
2
10-15
1
6-14
2
14-30
1
1-3
1
10-19
1
6-13
1
“use it or lose it”
5. Repeat
30
15
14
10
6
19
1
3
13
Edge
Usage
1-10
14
10-15
6
6-14
12
14-30
3
1-3
7
10-19
2
6-13
1
“use it or lose it”
6. Prune at some rate:
High usage edges are important
[keep]
Low usage edges are not [prune]
30
15
14
10
6
19
1
3
13
Pruning rates have been ignored in the literature
Human frontal cortex
[Huttenlocher 1979]
Mouse somatosensory cortex
[White et al. 1997]
# of synapses / image
Decreasing pruning rates in the cortex
Rapid elimination
16 time-points
early then taper-off
41 animals
9754 images
42709 synapses
Postnatal day
Efficiency
(avg. routing distance)
Robustness
(# of alternate paths)
Decreasing rates optimize network function
Cost (# of edges)
Decreasing rates > 20% more efficient
than other rates
Cost (# of edges)
Decreasing rates have slightly
better fault tolerance
Take-aways [clean up / split]
• Biology-inspired algorithms
• Evaluated using standard cost metrics, based on formal models
• Sometimes worse performance in some aspects, but often simpler algorithms: e.g. the fly MIS
algorithm had higher run-time, but required fewer assumptions; synaptic pruning is initially
wasteful, but is more adaptive than growth-based models.
• Often no UIDs
• Dynamic and variable network topologies, similar to new wireless devices & networks
• Formal models can be used to evaluate performance and predict future system behavior
• Contribution to biology:
• Global understanding of local processes (Notch-Delta signaling, slime mold optimization criteria)
• Raised new testable hypotheses (e.g. the role of pruning rates on learning)
• [add more]
• Common algorithmic strategies we’ve seen:
• Stochasticity, to overcome noise and adversaries
• Feedback processes, to reinforce good solutions/edges/paths
• Rates of communication/contact vital to inform decision-making (MIS, brain, ants, etc.)
Conclusion