search tamara berg cs 560 artificial intelligence many slides throughout the course adapted from dan...
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Search
Tamara BergCS 560 Artificial Intelligence
Many slides throughout the course adapted from Dan Klein, Stuart Russell, Andrew Moore, Svetlana Lazebnik, Percy Liang, Luke Zettlemoyer
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Course Information
• Instructor: Tamara Berg ([email protected])• Office Hours: FB 236, Mon/Wed 11:25-12:25pm• Course website: http://tamaraberg.com/teaching/Fall_15/• Course mailing list: [email protected]
• TA: Patrick (Ric) PoirsonTA office hours: SN 109, Tues/Thurs 4-5pm
• Announcements, readings, schedule, etc, will all be posted to the course webpage. Schedule may be modified as needed over the semester. Check frequently!
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Announcements for today
• HW1 will be released on the course website later today, due Sept 10, 11:59pm.– Start early!
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Recall from last class
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Search problem components• Initial state• Actions• Transition model
– What state results fromperforming a given action in a given state?
• Goal state• Path cost
– Assume that it is a sum of nonnegative step costs
• The optimal solution is the sequence of actions that gives the lowest path cost for reaching the goal
Initialstate
Goal state
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Example: Romania• On vacation in Romania; currently in Arad• Flight leaves tomorrow from Bucharest
• Initial state– Arad
• Actions– Go from one city to another
• Transition model– If you go from city A to
city B, you end up in city B
• Goal state– Bucharest
• Path cost– Sum of edge costs (total distance
traveled)
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State space• The initial state, actions, and transition model
define the state space of the problem– The set of all states reachable from initial state by any
sequence of actions– Can be represented as a directed graph where the
nodes are states and links between nodes are actions
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Vacuum world state space graph
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Search• Given:
– Initial state
– Actions
– Transition model
– Goal state
– Path cost
• How do we find the optimal solution?– How about building the state space graph and then using
Dijkstra’s shortest path algorithm?• Complexity of Dijkstra’s is O(E + V log V), where V is the size of the
state space
• The state space may be huge!
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Search: Basic idea
• Let’s begin at the start state and expand it by making a list of all possible successor states
• Maintain a frontier – the set of all leaf nodes available for expansion at any point
• At each step, pick a state from the frontier to expand
• Keep going until you reach a goal state or there are no more states to explore.
• Try to expand as few states as possible
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Tree Search Algorithm Outline
• Initialize the frontier using the start state• While the frontier is not empty– Choose a frontier node to expand according to search strategy
and take it off the frontier– If the node contains the goal state, return solution– Else expand the node and add its children to the frontier
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Tree search example
Start: AradGoal: Bucharest
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Tree search example
Start: AradGoal: Bucharest
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Tree search example
Start: AradGoal: Bucharest
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Tree search example
Start: AradGoal: Bucharest
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Tree search example
Start: AradGoal: Bucharest
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Tree search example
Start: AradGoal: Bucharest
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Tree search example
Start: AradGoal: Bucharest
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Handling repeated states• Initialize the frontier using the starting state• While the frontier is not empty
– Choose a frontier node to expand according to search strategy and take it off the frontier
– If the node contains the goal state, return solution– Else expand the node and add its children to the frontier
• To handle repeated states:– Keep an explored set; which remembers every expanded node– Every time you expand a node, add that state to the
explored set; do not put explored states on the frontier again– Every time you add a node to the frontier, check whether it already
exists in the frontier with a higher path cost, and if yes, replace that node with the new one
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Search without repeated states
Start: AradGoal: Bucharest
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Search without repeated states
Start: AradGoal: Bucharest
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Search without repeated states
Start: AradGoal: Bucharest
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Search without repeated states
Start: AradGoal: Bucharest
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Search without repeated states
Start: AradGoal: Bucharest
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Search without repeated states
Start: AradGoal: Bucharest
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Search without repeated states
Start: AradGoal: Bucharest
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Searching• Initialize the frontier using the starting state• While the frontier is not empty
– Choose a frontier node to expand according to search strategy and take it off the frontier
– If the node contains the goal state, return solution– Else expand the node and add its children to the frontier
• To handle repeated states:– Keep an explored set; which remembers every expanded node– Every time you expand a node, add that state to the
explored set; do not put explored states on the frontier again– Every time you add a node to the frontier, check whether it already exists
in the frontier with a higher path cost, and if yes, replace that node with the new oneRemaining question: What should our search strategy be,
ie how do we choose which frontier node to expand?
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Uninformed search strategies
• A search strategy is defined by picking the order of node expansion
• Uninformed search strategies use only the information available in the problem definition– Breadth-first search– Depth-first search– Iterative deepening search– Uniform-cost search
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Informed search strategies
• Idea: give the algorithm “hints” about the desirability of different states – Use an evaluation function to rank nodes and
select the most promising one for expansion
• Greedy best-first search• A* search
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Uninformed search
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Breadth-first search
• Expand shallowest node in the frontier
Example state space graph for a tiny search
problem
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)
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Breadth-first search
• Expand shallowest node in the frontier• Implementation: frontier is a FIFO queue
Example state space graph for a tiny search
problem
Example from P. Abbeel and D. Klein
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Depth-first search
• Expand deepest node in the frontier
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)
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Depth-first search
• Expansion order: (S,d,b,a,c,a,e,h,p,q,q,r,f,c,a,G)
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Depth-first search
• Expand deepest unexpanded node• Implementation: frontier is a LIFO queue
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Analysis of search strategies
• Strategies are evaluated along the following criteria:– Completeness
• does it always find a solution if one exists?
– Optimality • does it always find a least-cost solution?
– Time complexity • how long does it take to find a solution?
– Space complexity• maximum number of nodes in memory
• Time and space complexity are measured in terms of – b: maximum branching factor of the search tree– d: depth of the optimal solution– m: maximum length of any path in the state space (may be infinite)
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Properties of breadth-first search
• Complete? Yes (if branching factor b is finite)
• Optimal? Not generally – the shallowest goal node is not necessarily
the optimal oneYes – if all actions have same cost
• Time? Number of nodes in a b-ary tree of depth d: O(bd)(d is the depth of the optimal solution)
• Space? O(bd)
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BFS
Depth Nodes Time Memory
2 110 0.11 ms 107 kilobytes
4 11,110 11 ms 10.6 megabytes
6 10^6 1.1 s 1 gigabyte
8 10^8 2 min 103 gigabytes
10 10^10 3 hrs 10 terabytes
12 10^12 13 days 1 petabyte
14 10^14 3.5 years 99 petabytes
16 10^16 350 years 10 exabytes
Time and Space requirements for BFS with b=10; 1 million nodes/second; 1000 bytes/node
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Properties of depth-first search
• Complete?Fails in infinite-depth spaces, spaces with loopsModify to avoid repeated states along path
complete in finite spaces
• Optimal?No – returns the first solution it finds
• Time? May generate all of the O(bm) nodes, m=max depth of any nodeTerrible if m is much larger than d
• Space? O(bm), i.e., linear space!
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Iterative deepening search
• Use DFS as a subroutine1. Check the root2. Do a DFS with depth limit 13. If there is no path of length 1, do a DFS search
with depth limit 24. If there is no path of length 2, do a DFS with
depth limit 3.5. And so on…
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Iterative deepening search
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Iterative deepening search
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Iterative deepening search
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Iterative deepening search
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Properties of iterative deepening search
• Complete?Yes
• Optimal?Not generally – the shallowest goal node is not
necessarily the optimal oneYes – if all actions have same cost
• Time?(d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd)
• Space?O(bd)