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s02lec21 - 15.053 Thursday May 2 Match game example Suppose...

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1 15.053 Thursday, May 2 z Dynamic Programming Review More examples Handouts: Lecture Notes 2 Match game example z Suppose that there are 50 matches on a table, and the person who picks up the last match wins. At each alternating turn, my opponent or I can pick up 1, 2 or 6 matches. Assuming that I go first, how can I be sure of winning the game? 3 Determining the strategy using DP z n = number of matches left (n is the state/stage ) z g(n) = 1 if you can force a win at n matches. g(n) = 0 otherwise g(n) = optimal value function. At each state/stage you can make one of three decisions: take 1, 2 or 6 matches. z g(1) = g(2) = g(6) = 1 (boundary conditions) z g(3) = 0; g(4) = g(5) = 1. The recursion: z g(n) = 1 if g(n-1) = 0 or g(n-2) = 0 or g(n-6) = 0; g(n) = 0 otherwise. z Equivalently, g(n) = 1 – min (g(n-1), g(n-2), g(n-6)). 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 The same table 5 Principle of Optimality z Any optimal policy has the property that whatever the current state and decision, the remaining decisions must constitute an optimal policy with regard to the state resulting from the current decision. z Whatever node j is selected, the remaining path from j to the end is the shortest path starting at j. 6 Finding shortest paths in graphs with no directed circuits. 1 2 5 4 3 6 7 8 5 1 3 2 1 1 2 4 7 3 6 If a network has no directed cycles, then the nodes can be labeled so that for each arc (i,j), i < j. Such a node labeling is called a topological order .
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7 Finding a topological order 1 2 5 4 3 6 7 8 5 1 3 2 1 1 2 4 7 3 6 Find a node with no incoming arc. Label it node 1. For i = 2 to n, find a node with no incoming arc from an unlabeled node. Label it node i.
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s02lec21 - 15.053 Thursday May 2 Match game example Suppose...

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