1/34 Informed search algorithms Chapter 4 Modified by Vali Derhami.
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Transcript of 1/34 Informed search algorithms Chapter 4 Modified by Vali Derhami.
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Informed search algorithms
Chapter 4
Modified by Vali Derhami
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Material
• Chapter 4
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Outline
• Best-first search• Greedy best-first search• A* search• Heuristics• Local search algorithms• Hill-climbing search• Simulated annealing search• Local beam search• Genetic algorithms
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Review: Tree search
• A search strategy is defined by picking the order of node expansion
•
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Best-first searchبهترين اول جستجوي
• Idea: use an evaluation function f(n) for each node– estimate of "desirability"Expand most desirable unexpanded node
• Implementation:Order the nodes in fringe in decreasing order of desirability
• Special cases:– greedy best-first search– A* search
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Romania with step costs in km
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Greedy best-first search• Evaluation function f(n) = h(n) (heuristic)
= estimate of cost from n to goal
• e.g., hSLD(n) = straight-line distance from n to Bucharest
Greedy best-first search expands the node that appears to be closest to goal
•
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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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Properties of greedy best-first search
• Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi 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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A* search
• Idea: avoid expanding paths that are already expensive– h(n) = estimated cost from n to goal– f(n) = estimated total cost of path through n to
goal–
• Evaluation function f(n) = g(n) + h(n)
• g(n) = cost so far to reach n
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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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A* search example
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Admissible heuristics
• A heuristic h(n) is admissible if for every node n,
h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.
• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal
• An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic
• Example: hSLD(n) (never overestimates the actual road distance)
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Optimality of A* (proof)• Suppose some suboptimal goal G2 has been generated and is in the
fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
• f(n)= g(n)+h(n)
• f(G2) = g(G2) since h(G2) = 0
• g(G2) > g(G) since G2 is suboptimal
• f(G) = g(G) since h(G) = 0
• f(G2) > f(G)=C* from above
•
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Optimality of A* (proof)• Suppose some suboptimal goal G2 has been generated and is in the fringe.
Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
• f(n)= g(n)+h(n)• f(G2) > C*=f(G) (1) from above • h(n) ≤ h*(n) since h is admissible• g(n) + h(n) ≤ g(n) + h*(n) =C*• f(n) ≤ C*=f(G),
With respect to (1) and (2) , f(G2) > f(n), and A* will never select G2 for expansion
•• (2)
•
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گراف جستجوي در بهينگي عدم
روش • كه شود بهينه *Aتوجه گراف جستجوي دريك است ممكن روش اين در كه چرا نيست
گذاشته كنار تكراري حالت يك به بهينه مسيرشود.
شناسايي: • تكراري حالت مذكور روش در ياداوري . گردد مي حذف شده كشف جديد مسير شود
• : حل راه•. شود گذاشته كنار دارد تر هزينه پر مسير كه گرهيبه • بهينه مسير كند تضمين كه باشد گونه بدان عملكرد نحوه
شده دنبال كه است مسيري اولين هميشه تكراري حالتيكنواخت. هزينه جستجوي همانند است
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Consistent heuristicsسازگار هيوريستيكهاي
• A heuristic is consistent if for every node n, every successor n' of n generated by any action a, h(n) ≤ c(n,a,n') + h(n')
هزينه و پسين گره هيوريستيك تابع حاصلجمع هميشه يعنيبه .رفتن است مساوي يا بيشتر والد هيوريستيك تابع از ان
». هست هم قبول قابل سازگار هيوريستيك هر• If h is consistent, we havef(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') ≥ g(n) + h(n) = f(n)
مقدار • .f(n)يعني است نزولي غير مسيري هر طول در• Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal
•i.e., f(n) is non-decreasing along any path.
•
•
•
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Optimality of A*
• A* expands nodes in order of increasing f value
• Contour i has all nodes with f=fi, where fi < fi+1
•
• Gradually adds "f-contours" of nodes
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Properties of A*• Complete? Yes (unless there are infinitely many nodes
with f ≤ f(G) )• Optimal? Yes
•
• ريشه از را جستجو مسيرهاي كه الگوريتمهايي ميان درتواند نمي ديگري بهينه الگوريتم هيچ دهند مي توسعه
از ميدهد توسع كه هايي گره تعداد كه كند كمتر *A تضمينباشد
• Time? Exponential• Space? Keeps all nodes in memory• مياورد كم حافظه بياورد كم وقت آنكه از بيشتر .الگوريتم
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حافظه با هیوریستیک جستجویمحدود
•. دارند حافظه مشکل شده مطرح الگوریتمهای
مصرفی • کاهشحافظه برای الگوریتم دو–Iterative Deepening A* (IDA*)–SMA* مشابهA* کمتر صف اندازه با
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Iterative Deepening A* (IDA*)
تکراری – شونده روشعمیق با تفاوت
هزینه • استفاده مورد . g+hیعنی fبرش عمق نه است
تا • که است عمقی جستجوی یک الگوریتم از تکرار هر
هزینه .fمحدوده رود پیشمی
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Simplified-Memory-Bounded A* (SMA*)
•. کند می استفاده موجود حافظه همه ازگسترش • را ها گره دهد می اجازه حافظه که جایی تا
میدهد.• ( ) آنکه ) لبه برگی گره بدترین شد پر حافظه که هنگامی
( ) fمقدار پاک حافظه از کند می حذف را دارد باالتری . ) میگرداند بر والدش به را گره این مقدار و کند می
داری: • برگی های گره تمام اگر چه fسوال باشد یکسانافتد؟ می اتفاقی
•SMA * یافتنی دست حل راه اگر است بهینه و کامل. باشد یافتنی دست بهینه حل راه و موجود
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SMA* Exampleاندازه به گره 3حافظه
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SMA* Code
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Admissible heuristics
E.g., for the 8-puzzle:• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance موقعيتهاي از كاشيها فاصله مجموع هدفشان
(i.e., no. of squares from desired location of each tile)
• h1(S) = ? • h2(S) = ?
••
•
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Admissible heuristics
E.g., for the 8-puzzle:• h2(n) = total Manhattan distance(i.e., no. of squares from desired location of each tile)
• h1(S) = ? 8• h2(S) = ? 3+1+2+2+2+3+3+2 = 18
•
• h1(n) = number of misplaced tiles
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Dominance• effective branching factor b*.
N + 1 = 1 + b* + b*2 + • • • + (b*)d . N=total number of nodes generated by A*, d= solution depth
, b* is the branching factor that a uniform tree of depth d would have to have in order to contain N+ 1 nodes.
• If h2(n) ≥ h1(n) for all n (both admissible)• then h2 dominates h1 , and h2 is better for search
• Typical search costs (average number of nodes expanded):• d=12 IDS = 3,644,035 nodes
A*(h1) = 227 nodes A*(h2) = 73 nodes b*=1.24
• d=24 IDS = too many nodesA*(h1) = 39,135 nodes A*(h2) = 1,641 nodes b*=1.26
••