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Distributed Software Development
Problem Solving II
Chris Brooks
Department of Computer Science
University of San Francisco
Department of Computer Science — University of San Francisco – p. 1/
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Distributed Problem Solving
•^
Last week, we talked about
loosely coupled
distributed
problems
◦^
Primarily distributed search
•^
A large data set is divided across clients.
-^
Clients interact only with a central server; no client-clientcommunication
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“Medium-coupled” problems
•^
In medium-coupled problems, each node can do a substantialamount of computing on its own
-^
A final solution will require communication between nodes.
-^
Actions taken by one node can affect other nodes.
Department of Computer Science — University of San Fra
Example: scheduling classes
•^
Before the semester starts, Benson sends every teacher anemail: When and where do you want your classes?
-^
Each professor has their own constraints, and can come upwith choices locally:^ ◦
Brooks: no classes before 10 am, in Kudlick ◦ Wolber: No classes on Friday ◦ Galles: Mornings, nothing on Lone Mountain.
-^
Brooks is able to eliminate some possible schedules withoutconsulting with anyone else.
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Example: scheduling classes
•^
Some choices require communication with a center (Benson)
◦^
Only one class in Kudlick at a time. ◦^
Some classes may be constrained by other departments’choices^ •^
CS 110 and Calculus I should be at different times.
•^
Nodes start with a large set of constraints; some are eliminatedby the center.
-^
Hopefully at least one viable schedule remains.
◦^
If not, someone is “encouraged” to relax their constraints.
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Constraint Satisfaction
•^
A constraint satisfaction problem is one of assigning values tovariables so as to satisfy a set of constraints.
-^
Typically, any solution that satisfies the constraints is equallyacceptable.
-^
Toy problems:^ ◦
N-queens ◦^
map coloring
•^
Real problems:^ ◦
Scheduling CS classes ◦^
Building a car ◦^
Register allocation
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Constraint Satisfaction
•^
More formally, a CSP consists of:^ ◦
a set of variables
x
, x 1
, ..., x 2
}n
◦^
Each variable as a domain of possible values
D
, D 1
, ..., D 2
n
◦^
and a set of constraints
C
, C 1
•^
Unary constraints:
x <
, ymod
, etc
•^
Binary constraints:
x < y, x
y <
,^ etc
•^
N-ary constraints:
x
1
x
2
x
n^
•^
(a problem with N-ary constraints can be transformed intoa binary constraint problem, with an increase in thenumber of variables)
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Formalizing a CSP
•^
An assignment of values to variables that satisfies allconstraints is called a
consistent
solution.
•^
We might might also have an objective function
y
f
(x
, ..., x 1
)n
that lets us compare solutions.
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Approaches
•^
If the domain of all variables is continuous (i.e. real numbers)and constraints are all linear functions, we can use
linear
programming
to solve the problem.
•^
This is actually the easy version (in P).^ ◦
Express the problem as a system of equations
•^
If variables are discrete, the problem is harder.
-^
We can use
dynamic programming
•^
Often, variables have complex or symbolic values.
-^
In the most general case, we can express a CSP as a searchproblem.
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Solving CSPs with search
•^
We can use depth-first search to solve a CSP^ ◦
Begin with an initial state: no values assigned to
x
, ..., x 1
n
◦^
Sequentially assign values to variables. ◦^
We are done if we find an assignment to each variable suchthat all constraints are met.
•^
Since CSPs are commutative (for a solution, it doesn’t matterwhich order values are assigned) we can consider one variableat a time.
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Backtracking
•^
With CSPs, the challenge is deciding how to proceed when aconstraint is violated.
◦^
Some assignment of values to variables must be undone,but which? ◦^
This decision is called
backtracking
◦^
Standard DFS undoes the most recently assigned value. ◦^
This is called
chronological backtracking
◦^
Easy to implement ◦^
Problem: an early assignment may have doomed us to aninconsistent solution.
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Example
WesternAustralia
NorthernTerritory
SouthAustralia
Queensland
NewSouthWales Victoria
Tasmania
•^
Three-coloring the map ofAustralia
-^
Assigning Q = red, NSW =green, V = blue, T = redcannot lead to a solution.
-^
Different values for T willnot change this.
-^
V needs a different value.
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Asynchronous Backtracking Example
•^
x1 returns
x
, which x2 adds to its local view.
•^
x2 checks its current assignment and its local view against thelist of nogoods.^ ◦
x
,^ (
x
is a nogood.
•^
Therefore, x2, can’t find a consistent value, and so it sends (nogood,
(x
back to x1.
•^
x1 receives the nogood. Since there are no other processesincluded, x1 knows that it must change its value.
-^
It chooses
x^1
,^ 2)
and resends
ok
(x
to x2 and x3.
•^
x2 can safely also choose 2. When x3 gets the message, itsends
ok
(x
,^ (
x
to x2.
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Asynchronous backtracking
•^
Guaranteed to terminate and find a solution if one exists(complete).
-^
Same process as single-processor backtracking
-^
If an variable can’t be assigned a value, undo thenext-most-recent variable.
-^
More communication, due to asynchronicity.
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Asynchronous Weak-commitment Search
•^
A problem with asynchronous backtracking is the staticallydefined priorities.^ ◦
In the example, a poor choice by x1 influenced everything. ◦^
A bad choice by a high-priority agent might cause a greatdeal of needless resetting of variables.
•^
Rather than just sending a nogood to the lowest-priorityprocess, we use a heuristic.
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Asynchronous Weak-commitment Search
•^
Each process gets a priority value assigned dynamically.
-^
When an agent can’t find a value that satisfies all constraints, it:^ ◦
chooses a value that minimizes constraint violations ◦ increases its priority to
k
, where
k
is the highest nogood
received. ◦^
Sends nogoods to all lower-priority processes.
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Example
a1(0) a2(0)a3(0) a4(0)
•^
One agent per row
-^
Initially all priorities are 0.
-^
All agents send ok?messages to each other.
-^
a4 is unable to find an as-signment consistent with itslocal view.
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Example
a1(0)a2(0)a3(0)a4(1)
•^
a4 sends (nogood
(a
,^ (
a
,^ (
a^3
,^ 2)
,^ (
a
messages to all other agents.
-^
a4 increments its priority.
-^
Chooses a new value thatminimizes constraints, given itslocal view.^ ◦
a
◦^
a4 sends ok? messages to allother processes.
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Example
a1(0)a2(0) a3(2)a4(1)
•^
a3 receives a4’s ok? andtries to find a new value.
-^
No consistent value^ ◦
Selects 1 to minimizeconflicts ◦^
Increments priority to 2(since a4 was 1). ◦^
Sends ok? to all lowerpriorities. ◦^
a2 and a4 are fine withthis choice.
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Example
a1(0)a2(0)a3(2)a4(1)
•^
a1 has a conflict with a3, andmoves to 2.
-^
Consistent, so no need to sendnogoods.
-^
The solution satisfies allconstraints.
-^
Notice that the original problemstemmed from a bad choice by a1.^ ◦
Unlike asynchronousbacktracking, this algorithmdoesn’t exhaustively searcha2’s choices before changinga1.
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Breakout
•^
Backtracking and weak-commitment are examples ofalgorithms that construct consistent partial solutions.^ ◦
Each node tries to select values that are consistent withsome other subset of nodes.
•^
We can also use hill-climbing to solve this problem^ ◦
Sometimes called iterative improvement.
•^
Premise start with a complete but flawed solution and try toapply improvements.
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Breakout
•^
We can use the min-conflict heuristic to guide hill-climbing.
-^
Start with a random assignment of values to variables.
-^
foreach variable:^ ◦
Choose the value that minimizes the number of conflicts
-^
Repeat until:^ ◦
A solution is found ◦ A local optimum is reached.
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Breakout
•^
Breakout
helps us jump out of local optima.
•^
Each constraint starts with a weight of 1.
-^
When a local optimum is found, all currently-violatedconstraints have their weights increased by 1.
◦^
This makes nearby states more appealing, and “pushes” thehillclimber out of the optimum.
•^
Like all hill-climbing algorithms, this is not complete, but can betuned to work well in practice.
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Distributed Breakout
•^
We can also construct a distributed version of breakout.
-^
Neighbors exchange possible assignments, and the agentwhose assignment produces maximal improvement is chosen.
-^
Agents detect that subgroups are trapped in a quasi-localminimum. (no need to solve consensus to detect a localminimum).^ ◦
xi
is in a quasi-local minimum if it is violating a constraint, and the improvement of all of its neighbors is 0. ◦^
In other words, no hill can be climbed in the agent’simmediate vicinity.
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