Last time: search strategies
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- Sheryl Mills
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1 Last time: search strategies Uninfrmed: Use nly infrmatin available in the prblem frmulatin Breadth-first Unifrm-cst Depth-first Depth-limited Iterative deepening Infrmed: Use heuristics t guide the search Best first: Greedy search queue first ndes that maimize heuristic desirability based n estimated path cst frm current nde t gal; A* search queue first ndes that maimize sum f path cst s far and estimated path cst t gal. Iterative imprvement keep n memry f path; wrk n a single current state and iteratively imprve its value. Hill climbing select as new current state the successr state which maimizes value. Simulated annealing refinement n hill climbing by which bad mves are permitted, but with decreasing size and frequency. Will find glbal etremum. CS 460, Sessins 8-9 1
2 Eercise: Search Algrithms The fllwing figure shws a prtin f a partially epanded search tree. Each arc between ndes is labeled with the cst f the crrespnding peratr, and the leaves are labeled with the value f the heuristic functin, h. Which nde (use the nde s letter) will be epanded net by each f the fllwing search algrithms? (a) Depth-first search (b) Breadth-first search (c) Unifrm-cst search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 460, Sessins 8-9 2
3 Depth-first search Nde queue: initializatin # state depth path cst parent # 1 A CS 460, Sessins 8-9 3
4 Depth-first search Nde queue: add successrs t queue frnt; empty queue frm tp # state depth path cst parent # 2 B C D A CS 460, Sessins 8-9 4
5 Depth-first search Nde queue: add successrs t queue frnt; empty queue frm tp # state depth path cst parent # 5 E F G H B C D A CS 460, Sessins 8-9 5
6 Depth-first search Nde queue: add successrs t queue frnt; empty queue frm tp # state depth path cst parent # 5 E F G H B C D A CS 460, Sessins 8-9 6
7 Eercise: Search Algrithms The fllwing figure shws a prtin f a partially epanded search tree. Each arc between ndes is labeled with the cst f the crrespnding peratr, and the leaves are labeled with the value f the heuristic functin, h. Which nde (use the nde s letter) will be epanded net by each f the fllwing search algrithms? (a) Depth-first search (b) Breadth-first search (c) Unifrm-cst search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 460, Sessins 8-9 7
8 Breadth-first search Nde queue: initializatin # state depth path cst parent # 1 A CS 460, Sessins 8-9 8
9 Breadth-first search Nde queue: add successrs t queue end; empty queue frm tp # state depth path cst parent # 1 A B C D CS 460, Sessins 8-9 9
10 Breadth-first search Nde queue: add successrs t queue end; empty queue frm tp # state depth path cst parent # 1 A B C D E F G H CS 460, Sessins
11 Breadth-first search Nde queue: add successrs t queue end; empty queue frm tp # state depth path cst parent # 1 A B C D E F G H CS 460, Sessins
12 Eercise: Search Algrithms The fllwing figure shws a prtin f a partially epanded search tree. Each arc between ndes is labeled with the cst f the crrespnding peratr, and the leaves are labeled with the value f the heuristic functin, h. Which nde (use the nde s letter) will be epanded net by each f the fllwing search algrithms? (a) Depth-first search (b) Breadth-first search (c) Unifrm-cst search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 460, Sessins
13 Unifrm-cst search Nde queue: initializatin # state depth path cst parent # 1 A CS 460, Sessins
14 Unifrm-cst search Nde queue: add successrs t queue s that entire queue is srted by path cst s far; empty queue frm tp # state depth path cst parent # 1 A B D C CS 460, Sessins
15 Unifrm-cst search Nde queue: add successrs t queue s that entire queue is srted by path cst s far; empty queue frm tp # state depth path cst parent # 1 A B D E F G H C CS 460, Sessins
16 Unifrm-cst search Nde queue: add successrs t queue s that entire queue is srted by path cst s far; empty queue frm tp # state depth path cst parent # 1 A B D E F G H C CS 460, Sessins
17 Eercise: Search Algrithms The fllwing figure shws a prtin f a partially epanded search tree. Each arc between ndes is labeled with the cst f the crrespnding peratr, and the leaves are labeled with the value f the heuristic functin, h. Which nde (use the nde s letter) will be epanded net by each f the fllwing search algrithms? (a) Depth-first search (b) Breadth-first search (c) Unifrm-cst search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 460, Sessins
18 Greedy search Nde queue: initializatin # state depth path cst ttal parent # cst t gal cst 1 A CS 460, Sessins
19 Greedy search Nde queue: Add successrs t queue, srted by cst t gal. # state depth path cst ttal parent # cst t gal cst 1 A B D C Srt key CS 460, Sessins
20 Greedy search Nde queue: Add successrs t queue, srted by cst t gal. # state depth path cst ttal parent # cst t gal cst 1 A B G E H F D C CS 460, Sessins
21 Greedy search Nde queue: Add successrs t queue, srted by cst t gal. # state depth path cst ttal parent # cst t gal cst 1 A B G E H F D C CS 460, Sessins
22 Eercise: Search Algrithms The fllwing figure shws a prtin f a partially epanded search tree. Each arc between ndes is labeled with the cst f the crrespnding peratr, and the leaves are labeled with the value f the heuristic functin, h. Which nde (use the nde s letter) will be epanded net by each f the fllwing search algrithms? (a) Depth-first search (b) Breadth-first search (c) Unifrm-cst search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 460, Sessins
23 A* search Nde queue: initializatin # state depth path cst ttal parent # cst t gal cst 1 A CS 460, Sessins
24 A* search Nde queue: Add successrs t queue, srted by ttal cst. # state depth path cst ttal parent # cst t gal cst 1 A B D C Srt key CS 460, Sessins
25 A* search Nde queue: Add successrs t queue frnt, srted by ttal cst. # state depth path cst ttal parent # cst t gal cst 1 A B G E H D F C CS 460, Sessins
26 A* search Nde queue: Add successrs t queue frnt, srted by ttal cst. # state depth path cst ttal parent # cst t gal cst 1 A B G E H D F C CS 460, Sessins
27 Eercise: Search Algrithms The fllwing figure shws a prtin f a partially epanded search tree. Each arc between ndes is labeled with the cst f the crrespnding peratr, and the leaves are labeled with the value f the heuristic functin, h. Which nde (use the nde s letter) will be epanded net by each f the fllwing search algrithms? (a) Depth-first search (b) Breadth-first search (c) Unifrm-cst search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 460, Sessins
28 Last time: Simulated annealing algrithm Idea: Escape lcal etrema by allwing bad mves, but gradually decrease their size and frequency. - Nte: gal here is t maimize E. CS 460, Sessins
29 Last time: Simulated annealing algrithm Idea: Escape lcal etrema by allwing bad mves, but gradually decrease their size and frequency. < - - Algrithm when gal is t minimize E. CS 460, Sessins
30 This time: Outline Game playing The minima algrithm Resurce limitatins alpha-beta pruning Elements f chance CS 460, Sessins
31 What kind f games? Abstractin: T describe a game we must capture every relevant aspect f the game. Such as: Chess Tic-tac-te Accessible envirnments: Such games are characterized by perfect infrmatin Search: game-playing then cnsists f a search thrugh pssible game psitins Unpredictable ppnent: intrduces uncertainty thus game-playing must deal with cntingency prblems CS 460, Sessins
32 Searching fr the net mve Cmpleity: many games have a huge search space Chess: b = 35, m=100 ndes = if each nde takes abut 1 ns t eplre then each mve will take abut millennia t calculate. Resurce (e.g., time, memry) limit: ptimal slutin nt feasible/pssible, thus must apprimate 1. Pruning: makes the search mre efficient by discarding prtins f the search tree that cannt imprve quality result. 2. Evaluatin functins: heuristics t evaluate utility f a state withut ehaustive search. CS 460, Sessins
33 Tw-player games A game frmulated as a search prblem: Initial state:? Operatrs:? Terminal state:? Utility functin:? CS 460, Sessins
34 Tw-player games A game frmulated as a search prblem: Initial state: Operatrs: Terminal state: Utility functin: bard psitin and turn definitin f legal mves cnditins fr when game is ver a numeric value that describes the utcme f the game. E.g., -1, 0, 1 fr lss, draw, win. (AKA payff functin) CS 460, Sessins
35 Game vs. search prblem CS 460, Sessins
36 Eample: Tic-Tac-Te CS 460, Sessins
37 Type f games CS 460, Sessins
38 Type f games CS 460, Sessins
39 The minima algrithm Perfect play fr deterministic envirnments with perfect infrmatin Basic idea: chse mve with highest minima value = best achievable payff against best play Algrithm: 1. Generate game tree cmpletely 2. Determine utility f each terminal state 3. Prpagate the utility values upward in the three by applying MIN and MAX peratrs n the ndes in the current level 4. At the rt nde use minima decisin t select the mve with the ma (f the min) utility value Steps 2 and 3 in the algrithm assume that the ppnent will play perfectly. CS 460, Sessins
40 Generate Game Tree CS 460, Sessins
41 Generate Game Tree CS 460, Sessins
42 Generate Game Tree CS 460, Sessins
43 Generate Game Tree 1 ply 1 mve CS 460, Sessins
44 CS 460, Sessins A subtree win lse draw
45 CS 460, Sessins What is a gd mve? win lse draw
46 Minima Minimize ppnent s chance Maimize yur chance CS 460, Sessins
47 Minima MIN Minimize ppnent s chance Maimize yur chance CS 460, Sessins
48 Minima MAX MIN Minimize ppnent s chance Maimize yur chance CS 460, Sessins
49 Minima MAX MIN Minimize ppnent s chance Maimize yur chance CS 460, Sessins
50 minima = maimum f the minimum 1 st ply 2 nd ply CS 460, Sessins
51 Minima: Recursive implementatin Cmplete:? Optimal:? Time cmpleity:? Space cmpleity:? CS 460, Sessins
52 Minima: Recursive implementatin Cmplete: Yes, fr finite state-space Optimal: Yes Time cmpleity: O(b m ) Space cmpleity: O(bm) (= DFS Des nt keep all ndes in memry.) CS 460, Sessins
53 1. Mve evaluatin withut cmplete search Cmplete search is t cmple and impractical Evaluatin functin: evaluates value f state using heuristics and cuts ff search New MINIMAX: CUTOFF-TEST: cutff test t replace the terminatin cnditin (e.g., deadline, depth-limit, etc.) EVAL: evaluatin functin t replace utility functin (e.g., number f chess pieces taken) CS 460, Sessins
54 Evaluatin functins Weighted linear evaluatin functin: t cmbine n heuristics f = w 1 f 1 + w 2 f w n f n E.g, w s culd be the values f pieces (1 fr prawn, 3 fr bishp etc.) f s culd be the number f type f pieces n the bard CS 460, Sessins
55 Nte: eact values d nt matter CS 460, Sessins
56 Minima with cutff: viable algrithm? Assume we have 100 secnds, evaluate 10 4 ndes/s; can evaluate 10 6 ndes/mve CS 460, Sessins
57 2. α-β pruning: search cutff Pruning: eliminating a branch f the search tree frm cnsideratin withut ehaustive eaminatin f each nde α-β pruning: the basic idea is t prune prtins f the search tree that cannt imprve the utility value f the ma r min nde, by just cnsidering the values f ndes seen s far. Des it wrk? Yes, in rughly cuts the branching factr frm b t b resulting in duble as far lk-ahead than pure minima CS 460, Sessins
58 α-β pruning: eample MAX 6 MIN CS 460, Sessins
59 α-β pruning: eample MAX 6 MIN CS 460, Sessins
60 α-β pruning: eample MAX 6 MIN CS 460, Sessins
61 α-β pruning: eample MAX 6 Selected mve MIN CS 460, Sessins
62 α-β pruning: general principle Player Oppnent m α If α > v then MAX will chse m s prune tree under n Player Similar fr β fr MIN Oppnent n v CS 460, Sessins
63 Prperties f α-β CS 460, Sessins
64 The α-β algrithm: CS 460, Sessins
65 Mre n the α-β algrithm Same basic idea as minima, but prune (cut away) branches f the tree that we knw will nt cntain the slutin. CS 460, Sessins
66 Mre n the α-β algrithm: start frm Minima CS 460, Sessins
67 Remember: Minima: Recursive implementatin Cmplete: Yes, fr finite state-space Optimal: Yes Time cmpleity: O(b m ) Space cmpleity: O(bm) (= DFS Des nt keep all ndes in memry.) CS 460, Sessins
68 Mre n the α-β algrithm Same basic idea as minima, but prune (cut away) branches f the tree that we knw will nt cntain the slutin. Because minima is depth-first, let s cnsider ndes alng a given path in the tree. Then, as we g alng this path, we keep track f: α : Best chice s far fr MAX β : Best chice s far fr MIN CS 460, Sessins
69 Mre n the α-β algrithm: start frm Minima Nte: These are bth Lcal variables. At the Start f the algrithm, We initialize them t α = - and β = + CS 460, Sessins
70 Mre n the α-β algrithm In Min-Value: MAX α = - β = + MIN Ma-Value lps ver these MAX Min-Value lps ver these α = - β = 5 α = - β = 5 α = - β = 5 CS 460, Sessins
71 Mre n the α-β algrithm In Ma-Value: MAX α = - β = + MIN Ma-Value lps ver these α = 5 β = + MAX α = - β = 5 α = - β = 5 α = - β = 5 CS 460, Sessins
72 Mre n the α-β algrithm In Min-Value: MAX MIN α = 5 β = + α = - β = + MAX Min-Value lps ver these α = - β = 5 α = - β = 5 α = - β = 5 α = 5 β = 2 End lp and return 5 CS 460, Sessins
73 Mre n the α-β algrithm In Ma-Value: MAX α = - β = + MIN Ma-Value lps ver these α = 5 β = + α = 5 β = + MAX α = - β = 5 α = - β = 5 α = - β = 5 α = 5 β = 2 End lp and return 5 CS 460, Sessins
74 Anther way t understand the algrithm Frm: Fr a given nde N, α is the value f N t MAX β is the value f N t MIN CS 460, Sessins
75 Eample CS 460, Sessins
76 α-β algrithm: CS 460, Sessins
77 Slutin NODE TYPE ALPHA BETA SCORE A Ma -I +I B Min -I +I C Ma -I +I D Min -I +I E Ma D Min -I 10 F Ma D Min -I C Ma 10 +I G Min 10 +I H Ma G Min C Ma 10 +I 10 B Min -I 10 J Ma -I 10 K Min -I 10 L Ma K Min -I NODE TYPE ALPHA BETA SCORE J Ma B Min -I A Ma 10 +I Q Min 10 +I R Ma 10 +I S Min 10 +I T Ma S Min R Ma 10 +I V Min 10 +I W Ma V Min R Ma 10 +I 10 Q Min A Ma CS 460, Sessins
78 State-f-the-art fr deterministic games CS 460, Sessins
79 Nndeterministic games CS 460, Sessins
80 Algrithm fr nndeterministic games CS 460, Sessins
81 Remember: Minima algrithm CS 460, Sessins
82 Nndeterministic games: the element f chance epectima and epectimin, epected values ver all pssible utcmes CHANCE? ?? CS 460, Sessins
83 Nndeterministic games: the element f chance Epectima CHANCE 4 = 0.5* * Epectimin CS 460, Sessins
84 Evaluatin functins: Eact values DO matter Order-preserving transfrmatin d nt necessarily behave the same! CS 460, Sessins
85 State-f-the-art fr nndeterministic games CS 460, Sessins
86 Summary CS 460, Sessins
87 Eercise: Game Playing Cnsider the fllwing game tree in which the evaluatin functin values are shwn belw each leaf nde. Assume that the rt nde crrespnds t the maimizing player. Assume the search always visits children left-t-right. (a) Cmpute the backed-up values cmputed by the minima algrithm. Shw yur answer by writing values at the apprpriate ndes in the abve tree. (b) Cmpute the backed-up values cmputed by the alpha-beta algrithm. What ndes will nt be eamined by the alpha-beta pruning algrithm? (c) What mve shuld Ma chse nce the values have been backed-up all the way? B E F G H I J K C L M N O P Q R S T U V W X Y A D Ma Min Ma Min CS 460, Sessins
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