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					  CISC 886: MultiAgent Systems
            Fall 2004




Search Algorithms for
      Agents

         Sachin Kamboj
Outline
   Introduction
   Path-Finding Problems
       Formal Definition
       Asynchronous Dynamic Programming
       Learning Real Time A*
       Moving Target Search
       Real –Time Bidirectional Search
   Constraint Satisfaction Problems
       Formal Definition
       Filtering Algorithm
       Hyper-Resolution Based Consistency Algorithm
       Asynchronous Backtracking
       Distributed Constraint Optimization Problems
           Adopt (Asynchronous Distributed Optimization)
           OptAPO (OPTimal      Asynchronous Partial Overlay)
Introduction
   Search:
       an umbrella term for various problem solving techniques
        in AI
       used when the sequence of actions required for solving a
        problem is not known a priori
           hence trial and error exploration of the alternatives is
            required
   Search algorithms are designed to solve three classes
    of problems:
       Path-finding problems
       Constraint satisfaction problems
       Competitive games
Introduction
   A whole set of search algorithms exist for single
    agents
       have known properties (like time and space complexity).
       have been used effectively to solve a large number of AI
        problems.
       Examples: BFS, DFS, Branch and Bound, A*
   So, why use multiple agents?
       Agents have limited rationality
           search is often intractable
           may not have a complete picture of the problem
           may not have the required computational capability
       Agents may be self interested
Introduction
   Approach
       If we represent the search problem as a graph, we
        can solve it by accumulating local computations
        for each node in the graph.
           Local computations can be executed asynchronously
            and concurrently




                             Agent 2
                                            Agent 3
                 Agent 1
Introduction
   Advantages of asynchronous search
    algorithms:
       Local computations needed will fit within the
        limited rationality of the agents
       Execution order of these algorithms can be highly
        flexible and arbitrary
Path Finding Problems
  Example 1: Finding a path through a Maze
              Goal




Start
Example 2: Solving the 8-puzzle problem
                     4 2
                   1 3 5
                   6 7 8



       1 4 2       1 4 2        1 2
           3 5     3 3 5      3 4 5
       6 7 8       6 7 8      6 7 8



       Initial     1 4 2       Goal
       State                   State
                   6 3 5
                     7 8
Formal Definition
   A path finding problem consists of the
    following components:
       A set of nodes, N, each representing a state
       A set of directed links, L, each representing an
        operator available to a problem solving agent
       A unique start state, S
       A set of goal states, G
       A set of weights, W, associated with each link
           represent the cost of applying the operator
           called the “distance” between the nodes
       Neighbors are nodes that have directed links
        between them
Principle of Optimality
   States that a path is optimal if and only if
    every segment of it is optimal
Asynchronous Dynamic Programming
   Let:
       h*(i) = shortest distance from node i to the goal
       k(i,j) = cost of link between i and j
       f*(j) = shortest distance from node i to goal via a
                 neighboring node j
                           f*(j) = k(i,j) +h*(j)
       By the principle of optimality:
                            h*(i) = minj f*(j)
       Asynchronous dynamic programming computes
        h* by repeating the local computations of each
        node
Asynchronous Dynamic Programming
   Assumes the following situation:
       For each node, i, there exists a process
        corresponding to i
       Each process records h(i), which is the estimated
        value of h*(i).
           The initial value of h*(i) is arbitrary (e.g., , 0) except for
            the goal nodes
       For each goal node g, h(g) is 0.
       Each process can refer to h values of neighboring
        nodes (via shared memory or message passing)
Asynchronous Dynamic Programming
   Each process updated h(i) by the following
    procedure:
       For each neighboring node j:
           Compute f(j) = k(i,j) + h(j) where
               h(j) is the current estimated distance from j to a goal node
               k(i,j) is the cost of the link from i to j
       update h(i) as follows
           h(i) ← minj f(j)
Asynchronous Dynamic Programming
   Example:

                             
                             3           
                                         1
                                     2

                        1                   1
                                                 0
                             1   1
                                         1

           initial       3
                                     2       3   goal
           state             
                             2           
                                         3
                                         2       state
Asynchronous Dynamic Programming
   Is the algorithm complete?
       Yes
   Is the algorithm optimal?
       Yes
   Are there any problems?
       cannot be used for reasonably large path-finding
        problems
           we cannot afford to have processes for all the nodes
Learning Real-Time A*
   Used when:
       only one agent is present
           not possible to perform local computations for all nodes
       when planning and execution needs to be
        interleaved
   In this algorithm:
       the agents selectively execute the computations
        for the current node
       each agent repeats the following procedure:
           Lookahead: calculate f(j) = k(i,j) + h(j)
           Update: the estimate of node i as h(i) ← minj f(j)
           Action Selection: Move to the neighbor j that has the
            minimum f(j) value. Ties are broken randomly
Learning Real-Time A*
   Requirement:
       the initial value of h must be optimistic, i.e.
                                 h(i)  h*(i)
   Is the algorithm complete?
       Yes, in a finite number of nodes with positive link costs, in
        which there exists a path from every node to a goal node,
        and starting with non-negative initial estimates, LRTA* will
        eventually reach a goal node
   Is the algorithm optimal?
       Requires repeated trials for optimality
       If the initial estimates are admissible, then over repeated
        problem solving trials, the values learned by LRTA* will
        eventually converge to their actual distances along every
        optimal path to the goal node
Moving Target Search
   Allows the goal state to change during the
    course of the search
   For example, a robot’s task is to reach
    another robot which is in fact moving as well
       The target robot may
           cooperatively try to reach the problem solving robot
           actively avoid the problem solving robot
           move independent of the problem solving robot
   In order to guarantee success, the problem
    solver must be able to move faster than the
    target
Moving Target Search
   Is a generalization of LRTA*
   The algorithm:
       does NOT maintain a single heuristic of the
        distance to the target goal
       instead tries to acquire heuristic information for
        each potential target location.
           Thus, MTS maintains a matrix of heuristic values,
            representing the function h(x,y) for all pairs of states x
            and y
           The matrix is updated on each move of the problem
            solver and the target.
Moving Target Search
   Let xi and xj be the current and neighboring
    positions of the problem solver and yi and yj
    be the current and neighboring positions of
    the target.
   Assume all edges in the graph have unit
    cost
   When the problem solver moves:
    1.   Calculate h(xj,yi) for each neighbor xj of xi.
    2.   Update the value of h(xi,yi) as follows:
              h(xi,yi) ← max ( h(xi,yi) , minxj{h(xj,yi) + 1} )
    3.   Move to the neighbor xj with the minimum h(xj,yi), i.e.
         assign the value of xj to xi. Ties are broken randomly.
Moving Target Search
   When the problem solver moves:
        1.   Calculate h(xi,yj) for the target’s new position yj.
        2.   Update the value of h(xi,yi) as follows:
                      h(xi,yi) ← max ( h(xi,yi) , h(xj,yj) – 1 )
        3.   Reflect the target’s new position as the new goal of the
             problem solver, i.e. assign the value of yj to yi.
   Is the algorithm complete?
       Yes, A problem solver executing MTS is
        guaranteed to eventually reach the target
   Is the algorithm optimal?
       No
Real –Time Bidirectional Search
   Two problem solvers starting from the initial and
    goal states physically move towards each other.
   Planning and execution are interleaved
   The following steps are repeatedly executed until
    the two problem solvers meet in the problem space:
    1.   Control Strategy: Select a forward (step2) or backward
         move (step3)
    2.   Forward Move: The problem solver starting from the
         initial stage (i.e. the forward problem solver) moves
         towards the problem solver starting from the goal state.
    3.   Backward Move: The problem solver starting from the
         goal stage (i.e. the backward problem solver) moves
         towards the problem solver starting from the initial state.
Real –Time Bidirectional Search
   Can be classified into two categories:
       Centralized RTBS
           The best action is selected among all possible moves of
            the two problem solvers
           The control strategy selects which of the two problem
            solvers to run depending on what the best action is
           Two centralized RTBS algorithms (based on LRTA* and
            RTA*) can be implemented
       Decoupled RTBS
           The two problem solvers independently make their own
            decisions.
           The control strategy alternatively runs the forward and
            backward problem solvers
           MTS can be used for implementing decoupled RTBS.
Constraint Satisfaction
Problems
Example 1: Scheduling a set of tasks
   A set of exams need to be scheduled during
    the last week of December. No more than 5
    exams can be scheduled on a Tuesday and no
    more than 7 exams on any other day………
Example 2: Graph-Coloring Problem
                                X1                X2
        { red, blue, yellow }                          { red, blue, yellow }



                                     X3   { red, blue, yellow }


                                     X4   { red, blue, yellow }


   Objective:
       To paint the nodes of a graph so that any two nodes
        connected by a link do not have the same color.
            Each node has a finite number of possible colors
Formal Definition
   A constraint satisfaction problem consists of:
       A set of n variables V = {x1, x2, …, xn }
       Discrete, finite domains for each of the variables D = {
        D1, D2, …, Dn }
       A set of constraints on the value of the variables.
           The constraints are defined by predicates,
            pk(xk1, xk2, …, xkj) where each pk is the function
            pk : Dk1 x Dk2 x … x Dkj  {0 , 1}.
       The problem is to find an assignment of values to the
        variables such that all the constraints are satisfied.
   Constraint satisfaction is NP-complete in general
       A trial and error exploration of alternatives is inevitable
Relation to DAI
   We assume that the variables of the CSP are
    distributed amongst multiple agents.
   Many application problems in DAI can be
    formalized as distributed constraint satisfaction
    problems.
   For example:
       interpretation problems
       assignment problems, and
       multiagent truth maintenance problems
   For simplicity, we assume an agent for each variable
    in all the algorithms
Filtering Algorithm
   Each agent communicates its domain to its neighbor and then
    removes values that cannot satisfy constraints from its
    domain.
   More specifically, a process (agent), xi performs the
    following procedure revise(xi,xj) for each neighbor xj.
       procedure revise (xi, xj)
          for all vi  Di do
             if there is no value vj  Dj such that vj is
                consistent with vi
             then delete vi from Di; end if; end do;
   If some value of the domain is removed by performing the
    procedure revise, process xi sends the new domain to its
    neighboring processes.
   If a new domain is received from a neighbor, call procedure
    revise again.
Filtering Algorithm
   For example,
                               X1                   X2
       { red, blue, yellow }                             { red }



                                    X3   { blue }


                                    X4   { red, blue, yellow }


   As a result of the filtering algorithm, x1 will
    remove red and blue from its domain and x4
    will remove blue from its domain.
Filtering Algorithm
   If the domain of some variable becomes the empty
    set:
       the problem is over-constrained and has no solution
   If each domain has a unique value:
       the assignment of the unique values to the variables is a
        solution.
   If there exist multiple values for some variable:
       we cannot tell whether the problem has a solution or not
       further trial and error search is required to find a solution
   Filtering algorithms cannot solve CSP problems in
    general
       This algorithm is used as a preprocessing procedure
        before the application of some other method.
Hyper-Resolution Based Consistency Algorithm
   All constraints are represented as a “nogood”
       a prohibited combination of variable values.
   For example, in the figure below:
                              X1                 X2
              { red, blue }                           { red, blue }



                                   X3   { red, blue }


   A constraint between x1and x2 can be represented using two
    nogoods:
       {x1 = red, x2 = red}
       {x1 = blue, x2 = blue}
   The algorithm uses several existing nogoods and the domain
    of a variable to generate a new nogood.
Hyper-Resolution Based Consistency Algorithm
   For example, using the nogoods:
       {x1 = red, x2 = red}
       {x1 = blue, x3 = blue}
    and the domain of x1 {red, blue}, a new nogood:
       {x2 = red, x3 = blue}
    is generated
   The hyper-resolution rule is described as follows:
                        A 1 V A2 V … V Am
                         (A1  A11 … )
                         (A2  A21 … )
                        :
                        :
                         (Am  Am1 … )
                         (A11  …  A21  …  Am1 …)
Asynchronous Backtracking
   Asynchronous version of a backtracking algorithm
       standard method for solving CSPs
   Each variable/process is assigned a priority
       usually based on the alphabetical order of the variable identifiers
   Each process selects a random value from its domain
   Each process communicates its tentative variable assignments
    to its neighboring processes.
   If the current value of a process is not consistent with the
    assignment of higher priority processes, the process changes
    its value
       If no consistent value exists, generate a new nogood and send it to the
        higher priority process
       On receiving a nogood, higher priority process changes its value.
   Each process maintains the current variable assignments of
    other processes in its local_view.
       May contain obsolete information.
Asynchronous Backtracking
   Two main types of messages are
    communicated:
       ok? messages to communicate the current value
       nogood messages to communicate a new nogood
   Example:
                                (nogood {(x1, 1) })
                          X1                            X2
                               add neighbor request
               { 1, 2 }                                      { 2 } local_view {(x1, 1) }

                                                 
              (ok? (x1, 1))                                     (nogood {(x
                                                                (ok? (x2, 2)) 1, 1), (x2, 2) })
                                     X3      { 1, 2 }

                               local_view {(x1, 1), (x2, 2) }
Distributed Constraint Optimization Problems
   Are a generalization of constraint satisfaction problems
   Like DCSP, DCOP includes a set of variables:
       each variable is assigned to an agent that has control over its value
   In DCSP
       the agents assign values to variables so as to satisfy the constraints on
        them
   In DCOP
       the agents must coordinate their choice of values so that a global
        objective function is optimized.
   Applications of DCOP:
       Multiagent Teamwork
       Distributed Scheduling
       Distributed Sensor Networks
Distributed Constraint Optimization Problems

   Formal Definition
       A constraint satisfaction problem consists of:
           A set of n variables V = {x1, x2, …, xn }
           Discrete, finite domains for each of the variables D = { D1, D2,
            …, Dn }
           A set of cost functions f = {f1, …, fm} .
               where each fi is a function
                fi : Di1 x Di2 x … x Dij  N U .
           The problem is to find an assignment A* = {d1, …, dn | di  Di}
            such that the global cost called F, is minimized.
               F is defined as follows:
                             m
                   F ( A)   f i ( A)
                            i 1
Distributed Constraint Optimization Problems
   Design Criteria for DCOP algorithms:
       Agents should be able to optimize a global
        function in a distributed fashion using only local
        communication
       The agents should operate asynchronously
           agents should not sit idle waiting for a particular
            message from a particular agent
       The algorithm should provide provable quality
        guarantees on system performance
Adopt (Asynchronous Distributed Optimization)
   Generalization of Asynchronous Backtracking
       with a bunch of performance tweaks.
   Starts by assigning a priority to the agents based on a
    depth-first search tree
       each node has a single parent and multiple children
       parents have higher priority than the children
       hence, does not require a linear priority ordering on the
        agents
   Constraints are only allowed between a node and any
    of its ancestors and descendants
       there can be no constraints between different subtrees of
        the DFS tree
           not a restriction of the constraint network itself
Adopt (Asynchronous Distributed Optimization)
   Example:

              x1                  x1




              x2                  x2



      x3              x4   x3              x4


       Constraint Graph         DFS Tree
Adopt (Asynchronous Distributed Optimization)
   Algorithm begins by all agents choosing their values
    concurrently
   The algorithm uses three types of messages:
       VALUE Messages:
           used to send the current selected value of the variable to the
            descendants below the node in the DFS tree
           similar to ok? messages in ABT
       THRESHOLD Messages:
           are only sent by a parent to its immediate children
           contain a single number which represents the backtrack threshold
       COST Messages:
           are a generalization of nogood messages in ABT
           contain the current context (same as in ABT) and the lb and the
            ub.
Adopt (Asynchronous Distributed Optimization)
   The algorithm calculates the local cost using the
    formula:
     (di )  ( x ,d                        f ij (di , d j )
                  j     j )CurrentContext


    where δ(di) is the local cost at xi when xi chooses d.
       This formula is used to calculate the cost of a node only
        on the basis of the constraints that the node shares with its
        ancestors (NOT its children)
            This is because the current context is built from the VALUE
             messages received by a node
       The node (xi) also calculates LB and UB
            The idea is that LB and UB are the lower and upper bounds on
             the cost seen so far for a subtrees rooted at xi.
Adopt (Asynchronous Distributed Optimization)
   For a leaf node,
       lb(di) = ub(di) = δ(di)
   For any other node,
    d  Di , lb(d )   (d )  x Children lb(d , xl )
                                     l

   For all nodes:
     LB  min dDi lb(d )
   Similar for UB
   By keeping a track of LB and UB, the agent knows
    the current lower bound and upper bound on cost in
    the subtrees
   The algorithm uses a threshold values to decide
    when to backtrack
OptAPO
   OPTimal Asynchronous Partial Overlay
       used to increase the efficiency of previous DCOP
        algorithms (eg adopt)
           previous DCOP algorithms were based on a total
            separation of the agents knowledge during the
            problem solving process
       is based on a partial centralization technique
        called cooperative mediation
           allows the agents to extend and overlap the context
            that they use for making their local decisions
OptAPO
   When an agent acts as a mediator, it
       computes a solution to the overall problem
       recommends value changes to the agents involved
        in the mediation session
Questions?

				
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