05-chapt4b

# 05-chapt4b - CSE630 LocalSearch(Iterative improvement Prof...

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CSE 630: Heuristic Search contd., Local Search (Iterative  improvement) Prof. Naeem Shareef

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2 Path Planning Assume we have a continuous moving robot What is the state space ?
3 Formulation #1 Cost of one horizontal/vertical step = 1 Cost of one diagonal step = 2

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4 Optimal Solution This path is the shortest in the discretized state space, but not in the original continuous space
5 Formulation #2 sweep-line

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6 Formulation #2
7 States

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8 Successor Function
9 Solution Path A path-smoothing post-processing step is usually needed to shorten the path further

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10
11 Solution Path The shortest path in this state space is also the shortest in the original continuous space

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12 (n 2 -1)-puzzle 1 2 3 4 5 6 7 8 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 ....
13 15-Puzzle Introduced (?) in 1878 by Sam Loyd, who dubbed himself “America’s greatest puzzle-expert”

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14 15-Puzzle Sam Loyd offered \$1,000 of his own money to the first person who would solve the following problem: 12 14 11 15 10 13 9 5 6 7 8 4 3 2 1 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 ?
15 But no one ever won the prize !!

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16 How big is the state space of the (n 2 -1)- puzzle? 8-puzzle ?? states
17 How big is the state space of the (n 2 -1)- puzzle? 8-puzzle 9! = 362,880 states 15-puzzle 16! ~ 2.09 x 10 13 states 24-puzzle 25! ~ 10 25 states But only half of these states are reachable from any given state (but you may not know that in advance)

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18 The second state is not reachable from the first, and Sam Loyd took no risk with his money . .. 15- Puzzle Sam Loyd of f ered \$1,000 of his own money t o t he f ir st per son who would solve t he f ollowing pr oblem: 12 14 11 15 10 13 9 5 6 7 8 4 3 2 1 ? 9 5 6 7 8 4 3 2 1
19 Let the goal be: A tile j appears after a tile i if either j appears on the same row as i to the right of i , or on another row below the row of i . For every i = 1, 2, . .., 15, let n i be the number of tiles j < i that appear after tile i N = n 2 + n 3 + + n 15 + row number of empty tile Permutation Inversions 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1

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20 Let the goal be: Permutation Inversions 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 6 9 5 7 8 4 3 2 1 n 2 = 0 n 3 n 4 n 5 n 6 n 7 = 1 n 8 n 9 n = 4 n = 0 n N = 7 + 4
21 8-puzzle 362,880 states 15-puzzle 13 24-puzzle 10 states 100 millions states/sec 0.036 sec ~ 55 hours > 10 9 years 8-, 15-, 24-Puzzles

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22 8-Queens Problem Place 8 queens in a chessboard so that no two queens are in the same row, column, or diagonal. A solution Not a solution
23 Formulation #1 States: all arrangements of 0, 1, 2, . .., 8 queens on the board Initial state: 0 queens on the board Successor function: each of the successors is obtained by adding one queen in an empty square Arc cost: irrelevant Goal test: 8 queens are on the board, with no queens attacking each other ~ 64 x 63 x ... x 57 ~ 3 x 10 14 states

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24 Formulation #2 States: all arrangements of k = 0, 1, 2, . .., 8 queens in the k leftmost columns with no two queens attacking each other Initial state: 0 queens on the board Successor function: each successor is obtained by adding one queen in any square that is
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## This note was uploaded on 04/13/2010 for the course CSE 630 taught by Professor Naeemshareef during the Spring '10 term at Ohio State.

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05-chapt4b - CSE630 LocalSearch(Iterative improvement Prof...

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