informed search a* algorithm - sharif
TRANSCRIPT
CE417: Introduction to Artificial IntelligenceSharif University of TechnologyFall 2019
“Artificial Intelligence: A Modern Approach”, Chapter 3Most slides have been adopted from Klein and Abdeel, CS188, UC Berkeley.
Informed SearchA* Algorithm
Soleymani
Outline} Heuristics} Greedy (best-first) search} A* search} Finding heuristics
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Uninformed Search
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Uniform Cost Search
} Strategy: expand lowest path cost
} The good: UCS is complete and optimal!
} The bad:} Explores options in every “direction”} No information about goal location Start Goal
…
c £ 3
c £ 2
c £ 1
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UCS Example
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Informed Search
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Search Heuristics§ Aheuristicis:
§ Afunctionthatestimates howcloseastateistoagoal§ Designedforaparticularsearchproblem§ Examples:Manhattandistance,Euclideandistanceforpathing
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Heuristic Function} Incorporating problem-specific knowledge in search
} Information more than problem definition} In order to come to an optimal solution as rapidly as possible
} ℎ 𝑛 : estimated cost of cheapest path from 𝑛 to a goal} Depends only on 𝑛 (not path from root to 𝑛)} If 𝑛 is a goal state then ℎ(𝑛)=0} ℎ(𝑛) ≥ 0
} Examples of heuristic functions include using a rule-of-thumb,an educated guess, or an intuitive judgment
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Example: Heuristic Function
h(x)
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Greedy Search
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Greedy search} Priority queue based on ℎ(𝑛)
} e.g., ℎ'() 𝑛 =straight-line distance from n to Bucharest
} Greedy search expands the node that appears to beclosest to goal
Greedy
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Example: Heuristic Function
h(x)
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Romania with step costs in km
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search
} Expand the node that seems closest…
} What can go wrong?
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Greedy best-first search
} Strategy: expand a node that you think is closest to agoal state} Heuristic: estimate of distance to nearest goal for each state
} A common case:} Best-first takes you straight to the (wrong) goal
} Worst-case: like a badly-guided DFS
…b
…b
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Properties of greedy best-first search} Complete? No
} Similar to DFS, only graph search version is complete in finite spaces} Infinite loops, e.g., (Iasi to Fagaras) Iasi à Neamt à Iasi à Neamt
} Time} 𝑂(𝑏𝑚), but a good heuristic can give dramatic improvement
} Space} 𝑂(𝑏𝑚): keeps all nodes in memory
} Optimal? No
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Greedy Search
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A* Search
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A* Search
UCS Greedy
A*
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A* search} Idea: minimizing the total estimated solution cost} Evaluation function for priority 𝑓 𝑛 = 𝑔 𝑛 + ℎ(𝑛)
} 𝑔 𝑛 = cost so far to reach 𝑛} ℎ 𝑛 = estimated cost of the cheapest path from 𝑛 to goal} So, 𝑓 𝑛 = estimated total cost of path through 𝑛 to goal
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start n… goal…
Actual cost 𝑔 𝑛 Estimated cost ℎ 𝑛
𝑓 𝑛 = 𝑔 𝑛 + ℎ(𝑛)
CombiningUCSandGreedy
} Uniform-cost orders by path cost, or backward cost g(n)} Greedy orders by goal proximity, or forward cost h(n)
} A* Search orders by the sum: f(n) = g(n) + h(n)
S a d
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Gh=5
h=6
h=2
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h=6 h=0
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h=7
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e h=11
Example:Teg Grenager
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dG
G
g=0h=6
g=1h=5
g=2h=6
g=3h=7
g=4h=2
g=6h=0
g=9h=1
g=10h=2
g=12h=0
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WhenshouldA*terminate?
} Should we stop when we enqueue a goal?
} No: only stop when we dequeue a goal
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G
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2h=1
h=2
h=0h=3
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IsA*Optimal?
• What went wrong?• Actual bad goal cost < estimated good goal cost• We need estimates to be less than actual costs!
A
GS
1 3h=6
h=0
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h =7
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Admissible Heuristics
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Idea: Admissibility
Inadmissible(pessimistic)heuristicsbreakoptimalitybytrappinggoodplansonthefrontier
Admissible(optimistic)heuristicsslowdownbadplansbutneveroutweightruecosts
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Admissible Heuristics
} A heuristic h is admissible (optimistic) if:
where is the true cost to a nearest goal
} Examples:
} Coming up with admissible heuristics is most of what’sinvolved in using A* in practice.
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Optimality of A* Tree Search
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OptimalityofA*TreeSearch
Assume:} A is an optimal goal node} B is a suboptimal goal node} h is admissible
Claim:
} A will exit the frontier before B
…
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OptimalityofA*TreeSearch:Blocking
Proof:} Imagine B is on the frontier} Some ancestor n of A is on the
frontier, too (maybe A!)} Claim: n will be expanded before B
1. f(n) is less or equal to f(A)
Definitionoff-costAdmissibilityofh
…
h=0atagoal
𝑓 𝑛 ≤ 𝑔 𝑛 + ℎ∗(𝑛)𝑔 𝐴 = 𝑔 𝑛 + ℎ∗(𝑛)
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OptimalityofA*TreeSearch:Blocking
Proof:} Imagine B is on the frontier} Some ancestor n of A is on the
frontier, too (maybe A!)} Claim: n will be expanded before B
1. f(n) is less or equal to f(A)2. f(A) is less than f(B)
Bissuboptimalh=0atagoal
…
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OptimalityofA*TreeSearch:Blocking
Proof:} Imagine B is on the frontier} Some ancestor n of A is on the
frontier, too (maybe A!)} Claim: n will be expanded before B
1. f(n) is less or equal to f(A)2. f(A) is less than f(B)3. n expands before B
} All ancestors of A expand before B} A expands before B} A* search is optimal
…
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A* search
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} Combines advantages of uniform-cost and greedysearches
} A* can be complete and optimal when ℎ(𝑛) has someproperties
A* search: example
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A* search: example
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A* search: example
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A* search: example
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A* search: example
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A* search: example
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Graph Search
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} Failure to detect repeated states can cause exponentially more work.
SearchTreeStateGraph
Tree Search: Extra Work!
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GraphSearch} In BFS, for example, we shouldn’t bother expanding the circled nodes (why?)
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Recall: Graph Search} Idea: never expand a state twice
} How to implement:
} Tree search + set of expanded states (“closed set”)
} Expand the search tree node-by-node, but…
} Before expanding a node, check to make sure its state has never been expanded before
} If not new, skip it, if new add to closed set
} Important: store the closed set as a set, not a list
} Can graph search wreck completeness? Why/why not?
} How about optimality?
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Optimality of A* Graph Search
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A*GraphSearchGoneWrong?
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h=2
h=1
h=4
h=1
h=0
S(0+2)
A(1+4) B(1+1)
C(2+1)
G(5+0)
C(3+1)
G(6+0)
Statespacegraph Searchtree
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Conditions for optimality of A*
} Admissibility: ℎ(𝑛) be a lower bound on the cost to reach goal} Condition for optimality of TREE-SEARCH version of A*
} Consistency (monotonicity): ℎ 𝑛 ≤ 𝑐 𝑛, 𝑎, 𝑛8 + ℎ 𝑛8} Condition for optimality of GRAPH-SEARCH version of A*
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Consistent heuristics
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} Triangle inequality
𝑛
𝑛′
𝐺ℎ(𝑛′)
ℎ(𝑛)
𝑐(𝑛, 𝑎, 𝑛′)
𝑐 𝑛, 𝑎, 𝑛8 : cost of generating 𝑛′ by applying action to 𝑛
for every node 𝑛 and every successor𝑛8 generated by any action 𝑎
ℎ 𝑛 ≤ 𝑐 𝑛, 𝑎, 𝑛8 + ℎ 𝑛8
ConsistencyofHeuristics} Main idea: estimated heuristic costs ≤ actual costs
} Admissibility: heuristic cost ≤ actual cost to goal
h(A) ≤ actual cost from A to G
} Consistency: heuristic “arc” cost ≤ actual cost for each arc
h(A) – h(C) ≤ cost(A to C)
} Consequences of consistency:} The f value along a path never decreases
h(A) ≤ cost(A to C) + h(C)
} A* graph search is optimal
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A
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G
h=4 h=11
h=2
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Admissible but not consistent: Example
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} 𝑓 (for admissible heuristic) may decrease along a path} Is there any way to make ℎ consistent?
𝑛
𝑛′
𝑐(𝑛, 𝑎, 𝑛’) = 1ℎ(𝑛) = 9ℎ(𝑛’) = 6⟹ ℎ 𝑛 ≰ ℎ 𝑛’ + 𝑐(𝑛, 𝑎, 𝑛’)
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𝑔 𝑛 = 5ℎ 𝑛 = 9𝑓(𝑛) = 14
𝑔 𝑛′ = 6ℎ 𝑛8 = 6𝑓(𝑛′) = 12G
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ℎD 𝑛′ = max(ℎ 𝑛8 , ℎD 𝑛 − 𝑐(𝑛, 𝑎, 𝑛′))
Consistency implies admissibility
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} Consistency ⇒Admissblity} All consistent heuristic functions are admissible} Nonetheless, most admissible heuristics are also consistent
ℎ 𝑛J ≤ 𝑐 𝑛J, 𝑎J, 𝑛K + ℎ(𝑛K)≤ 𝑐 𝑛J, 𝑎J, 𝑛K + 𝑐 𝑛K, 𝑎K, 𝑛L + ℎ(𝑛L)
…≤ ∑ 𝑐 𝑛N, 𝑎N, 𝑛NOJP
NQJ + ℎ(G)
𝑛J 𝑛K 𝑛L 𝑛P 𝐺…𝑐(𝑛J, 𝑎J, 𝑛K) 𝑐(𝑛P, 𝑎P, 𝐺)𝑐(𝑛K, 𝑎K, 𝑛L)
0 ⇒ ℎ 𝑛J ≤ cost of (every) path from 𝑛J to goal≤ cost of optimal path from 𝑛J to goal
Optimality} Tree search:
} A* is optimal if heuristic is admissible} UCS is a special case (h = 0)
} Graph search:} A* optimal if heuristic is consistent} UCS optimal (h = 0 is consistent)
} Consistency implies admissibility
} In general, most natural admissible heuristics tend to beconsistent, especially if from relaxed problems
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OptimalityofA*GraphSearch} Sketch: consider what A* does with a consistent heuristic:
} Fact 1: In graph search, A* expands nodes in increasing total f value (f-contours)
} Fact 2: For every state s, nodes that reach s optimally are expanded before nodesthat reach s suboptimally
} Result: A* graph search is optimal…
f£ 3
f£ 2
f£ 1
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Optimality of A* (consistent heuristics)Theorem: If ℎ(𝑛) is consistent, A* using GRAPH-SEARCH isoptimal
Lemma1: if ℎ(𝑛) is consistent then 𝑓(𝑛) values are non-decreasing along any path
Proof: Let 𝑛′ be a successor of 𝑛
I. 𝑓(𝑛′) = 𝑔(𝑛′) + ℎ(𝑛′)II. 𝑔(𝑛′) = 𝑔(𝑛) + 𝑐(𝑛, 𝑎, 𝑛′)III. 𝐼, 𝐼𝐼 ⇒ 𝑓 𝑛8 = 𝑔 𝑛 + 𝑐 𝑛, 𝑎, 𝑛8 + ℎ 𝑛8
IV. ℎ 𝑛 isconsistent ⇒ ℎ 𝑛 ≤ 𝑐 𝑛, 𝑎, 𝑛8 + ℎ 𝑛8 V. 𝐼𝐼𝐼, 𝐼𝑉 ⇒ 𝑓(𝑛′) ≥ 𝑔(𝑛) + ℎ(𝑛) = 𝑓(𝑛)
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Optimality of A* (consistent heuristics)Lemma 2: For every state 𝑛, nodes that reach 𝑛 optimally areexpanded before nodes that reach 𝑛 sub-optimally
Proof by contradiction:Assume that 𝑛 has been selected for expansion. Another frontier node 𝑛8must exist on the optimal path from initial node to 𝑛 (using graph separationproperty). Moreover, based on Lemma 1, 𝑓 𝑛8 ≤ 𝑓 𝑛 and thus𝑛8wouldhave been selected first.
Lemma 1 & 2⇒ The sequence of nodes expanded by A* (using GRAPH-SEARCH) is in non-decreasing order of 𝑓(𝑛)
Since ℎ = 0 for goal nodes, the first selected goal node for expansion is anoptimal solution (𝑓 is the true cost for goal nodes)
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Admissible vs. consistent (tree vs. graph search)
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} Consistent heuristic: When selecting a node for expansion, thepath with the lowest cost to that node has been found
} When an admissible heuristic is not consistent, a node willneed repeated expansion, every time a new best (so-far) costis achieved for it.
Contours in the state space} A* (using GRAPH-SEARCH) expands nodes in order of
increasing 𝑓 value} Gradually adds "f-contours" of nodes
} Contour 𝑖 has all nodes with 𝑓 = 𝑓N where𝑓N < 𝑓N+1
A* expands all nodes with f(n) < C*A* expands some nodes with f(n) = C* (nodes on the goal contour)A* expands no nodes with f(n) > C*⟹pruning
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PropertiesofA*
…b
…b
Uniform-Cost A*
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A* Example
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Example
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UCS Greedy A*
UCSvs A*Contours
} Uniform-cost (A* using ℎ(𝑛)=0) expandsequally in all “directions”
} A* expands mainly toward the goal, butdoes hedge its bets to ensure optimality} More accurate heuristics stretched toward the goal
(more narrowly focused around the optimal path)
Start Goal
Start Goal
States are points in 2-D Euclidean space.g(n)=distance from starth(n)=estimate of distance from goal
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Comparison
GreedyUniformCost A*
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Properties of A*} Complete?
} Yes if nodes with 𝑓 ≤ 𝑓 𝐺 = 𝐶∗ are finite} Step cost≥ 𝜀 > 0 and 𝑏 is finite
} Time?} Exponential
} But, with a smaller branching factor
¨ 𝑏d∗ed or when equal step costs 𝑏f×h∗ihh∗
} Polynomial when |ℎ(𝑥) − ℎ∗(𝑥)| = 𝑂(𝑙𝑜𝑔ℎ∗(𝑥))} However,A* is optimally efficient for any given consistent heuristic
} No optimal algorithm of this type is guaranteed to expand fewer nodes than A* (exceptto node with 𝑓 = 𝐶∗)
} Space?} Keeps all leaf and/or explored nodes in memory
} Optimal?} Yes (expanding node in non-decreasing order of 𝑓)
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Robot navigation example
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} Initial state? Red cell
} States? Cells on rectangular grid (except to obstacle)
} Actions? Move to one of 8 neighbors (if it is not obstacle)
} Goal test? Green cell
} Path cost? Action cost is the Euclidean length of movement
A* vs. UCS: Robot navigation example
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} Heuristic: Euclidean distance to goal
} Expanded nodes: filled circles in red & green} Color indicating 𝑔 value (red: lower, green: higher)
} Frontier: empty nodes with blue boundary
} Nodes falling inside the obstacle are discarded
Adopted from: http://en.wikipedia.org/wiki/Talk%3AA*_search_algorithm
Robot navigation: Admissible heuristic
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} Is Manhattan 𝑑o 𝑥, 𝑦 = 𝑥J − 𝑦J + 𝑥K − 𝑦K distancean admissible heuristic for previous example?
A*: inadmissible heuristic
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ℎ = ℎ_𝑆𝐿𝐷ℎ = 5 ∗ ℎ_𝑆𝐿𝐷
Adopted from: http://en.wikipedia.org/wiki/Talk%3AA*_search_algorithm
A*: Summary
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A*: Summary} A* uses both backward costs and (estimates of) forward
costs
} A* is optimal with admissible / consistent heuristics
} Heuristic design is key: often use relaxed problems
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