Route/Path Planning Using A Star and UCS

  Route/Path Planning Referensi 

  Materi kuliah IF3170 Inteligensi Buatan Teknik Course Website:

  Informatika ITB,  STEI  Teknik Informatika  IF3170

   Stuart J Russell & Peter Norvig, Artifcial Intelligence: A Modern Approach, 3rd Edition, Prentice-Hall International, Inc, 2010, Textbook Site:(2nd edition)

   Free online course materials | MIT OpenCourseWare Website: Site:

   Lecture Notes in Informed Heuristic Search,

  ICS 271 Fall 2008, Route Planning

  Source: Russell’s book Search

  O

  71 F 151

  99 S Z 211

  75

  140 R B

  97 P 120

  101 118

  D

  75 146 138 M T

  70 111

  L C

  S: set of cities i.s: A (Arad) g.s: B (Bucharest) Goal test: s = B ? Path cost: time ~ distance Uninformed Search

  Breadth-First Search (BFS)

  Treat agenda as a queue (FIFO) O

  71 F S 151

  Z

  99 211

  75 A

  80

  140 R B

  97 P 101

  120

  Simpul-E Simpul Hidup

  118 D

  A Z ,S ,T _{A A A}

  75 138 146

  M

  Z S ,T , O

  70 ^{A A A AZ} T 111

  L

  S T ,O , O ,F ,R

  C A A AZ AS AS AS

  T O ,O ,F ,R , L _{A AZ AS AS AS AT} O O ,F ,R ,L Path: A S  F  B, ^{AZ AS AS AS AT} Path-cost = 450

  O F ,R ,L _{AS AS AS AT} F R ,L , B _{AS AS AT ASF} R L ,B , D ,C ,P _{AS AT ASF ASR ASR AS R} Depth-First Search (DFS)

  Treat agenda as a stack (LIFO) O

  71 F S 151

  Z

  99 211

  75 A

  80

  140 R B

  97 P 120

  101 118

  D

  75 146 138 M

  70 T 111

  L

  Simpul-E Simpul Hidup

  C

  A Z ,S ,T _{A A A} Z O , S ,T _{A AZ A A} Path: A Z  O  S  F  B

  O S ,S ,T _{AZ AZO A A} Path-cost = 607

  S F , R ,S ,T _{AZO AZOS AZOS A A} F B , _{AZOS AZOSF} R ,S ,T _{AZOS A A} IDS

  71 O F S 151

  Z

  99 211

  75 A

  80

  140 R B

  97 P 120

  101 118

  D

  75 146 138 M

  70 T 111

  L C

  Depth=0: A: cutoff Depth=1: A  Z ,S ,T  Z : cutoff, S : cutoff, T : cutoff _{A A A A A A} Depth=2: A  Z ,S ,T O , S ,T O : cutoff  F , R ,T _{A A A AZ A A AZ AS AS A}    F : cutoff  R : cutoff  L  L : cutoff _{AS AS AT AT}

Depth=3: A  Z ,S ,T  O , S ,T  S ,S ,T  S : cutoff 

_{A A A AZ A A AZO A A AZO}

  F , R ,T B , R ,T  B _{AS AS A ASF AS A ASF} 

  Simpul- Simpul Hidup E

  A Z , T S _{A-75 A-118, A-140} Uniform Cost Search (UCS)

  Z T , S , O _{A-75 A-118 A-140 AZ-146} T S ,O , L  _{A-118 A-140 AZ-146 AT-229}

  BFS & IDS fnd path with fewest steps S O , R , L ,F _{A-140 AZ-146 AS-220 AT-229 AS-}

   If steps ≠ cost, this is not relevant (to _{239 AS-291} ,O optimal solution)

  O R , L , F ,O _{AZ-146 AS-220 AT-229 AS-239 AS-} 

  How can we fnd the shortest path ₂₉₁ (measured by sum of distances along R L , F ,O , P _{AS-220 AT-229 AS-239 AS-291 ASR-}

  ,D ,C path)? _{71 O} ^{317 ASR-340 ASR-366}

  F ₁₅₁ S Z _{99 211} L F ,O , M , P _{AT-229 AS-239 AS-291 ATL-299 ASR- 75 317 ASR-340 ASR-366} ,D ,C A _{140 80} F O ,M , P _{AS-239 AS-291 ATL-299 ASR-} R ₉₇ B P _{118 120 101 O M , P ,D AS-291 ATL-299 ASR-317 ASR- 317 ASR-340 ASR-366, ASF-450} ,D ,C B D _{75 146 138 340 ASR-366, ASF-450} ,C B ₇₀ M T ₁₁₁ L

  M P ,D ,D _{ATL-299 ASR-317 ASR-340 ATLM-} C _{364 ASR-366, ASF-450} ,C B P D ,D ,C _{ASR-317 ASR-340 ATLM-364 ASR-366,}

  Path: A S  R  P  B B , C B _{ASRP-418 ASRP-455, ASF-450}

  Path-cost = 418

  Informed Search Greedy Best-First Search 

  

Idea: use an evaluation function f(n) for each node

 f(n) = h(n)  estimates of cost from n to goal

  

e.g., h (n) = straight-line distance from n to Bucharest

_SLD

   Greedy best-frst search expands the node that appears to be closest to goal

   Romania with step costs in km

  Greedy best-frst search example

  Greedy best-frst search example

  Greedy best-frst search example

  Greedy best-frst search example

  

Problems with Greedy Best First Search



  Not complete 

  Get Stuck with Local Minima/ Plateu 

  Irrevocable (not able to be reversed/ changed) 

  

Can we incorporate heuristics in systematic search? Heuristic Search 

  Heuristic estimates value of a node  promise of a node

   difculty of solving the subproblem

   quality of solution represented by node

   the amount of information gained

   f(n)- heuristic evaluation function.

   depends on n, goal, search so far, domain A* Search 

  Idea: avoid expanding paths that are already expensive 

  Evaluation function f(n) = g(n) + h(n)  g(n) = cost so far to reach n

   h(n) = estimated cost from n to goal

   f(n) = estimated total cost of path through

  A

• search example

A

• search example

A

• search example

A

• search example

A

• search example

A

• search example

A* Special 

  Goal: fnd a minimum sum-cost path 

  Notation:  c(n,n’) - cost of arc (n,n’)

  

g(n) = cost of current path from start to node n in the

search tree.

   h(n) = estimate of the cheapest cost of a path from n to a goal.

   Special evaluation function: f = g+h

   f(n) estimates the cheapest cost solution path that goes through n.

   h*(n) is the true cheapest cost from n to a goal.  g*(n) is the true shortest path from the start s, to n.

  

If the heuristic function, h always underestimate

  Properties of A* 

  Complete? Yes, unless there are infnitely many nodes with f ≤ f(G) 

  Time? Exponential: O(b m

  ) 

  

Space? Keep all the nodes in memory: O(b

m

  ) 

  Optimal? Yes Branch-and-Bound vs A* 

  As in A*, look for a bound which is guaranteed lower than the true cost 

  

Search the branching tree in any way you like

 e.g. depth frst (no guarantee), best frst

   Cut of search if cost + bound > best solution found

   If heuristic is cost + bound, search = best frst  then BnB = A*

   Bounds often much more sophisticated  e.g. using mathematical programming optimisations

  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.

   An admissible heuristic never

overestimates the cost to reach the goal,

i.e., it is optimistic

   Example: h (n) (never overestimates the

  SLD actual road distance)

  Theorem : If h(n) is admissible, A using

• TREE-SEARCH is optimal

Admissibility

  2

  73  x y

  What must be true about h for _{h=100 h=1}

  1 

  1 A* to fnd optimal path? _{h=0 h=0}

  A* fnds optimal path if h is admissable; h is admissible when it never overestimates. g(X)+h(X)=2+100=102

   In this example, h is not G(Y)+h(Y)=73+1=74 admissible.

  Optimal path is not 

  In route fnding problems, found! straight-line distance to goal is

admissible heuristic. Because we choose Y,

rather than X which is in the optimal path.

Contoh Soal UAS Sem 2 2014/2015

  Dalam permainan video game, adakalanya entitas bergerak dalam video game perlu berpindah dari satu posisi ke posisi lain. Seringkali proses perpindahan perlu mengutamakan jalur terdekat atau biaya minimal karena berhubungan dengan poin yang diperoleh. Gambar di bawah ini menunjukkan contoh jalur yang mungkin dilewati oleh entitas bergerak dalam suatu video game. Suatu entitas akan berpindah

dari posisi titik A menuju ke posisi titik F. Jika

diperlukan informasi heuristik, nilai heuristik

  

Contoh Soal UAS Sem 2 2014/2015 (2)



  Pencarian solusi dengan: a.

  UCS b. Greedy Best First c. A Star Untuk masing-masing pendekatan tuliskan: 

  Formula 

  Iterasi 

  Simpul ekspan 

  Simpul hidup & nilai f(n) Urut abjad, simpul ekspan tidak mengulang, tidak membentuk

  Selamat Belajar