The Reve’s Puzzle Revisited

2021 ◽  
Vol 58 (2) ◽  
pp. 11-18
Author(s):  
Abdullah-Al-Kafi Majumdar
Keyword(s):  
Dynamic Programming ◽  
Dynamic Programming Equation ◽  
Minimum Number

The Reve’s puzzle, introduced by the English puzzlist, H.E. Dudeney, is a mathematical puzzle with 10 discs of different sizes and four pegs, designated as S, P1, P2 and D. Initially, the n (  1) discs rest on the source peg, S, in a tower (with the largest disc at the bottom and the smallest disc at the top). The objective is to move the tower from the peg S to the destination peg D, in a minimum number of moves, under the condition that each move can transfer only one disc from one peg to another such that no disc can ever be placed on top of a smaller one. This paper considers the solution of the dynamic programming equation corresponding to the Reve’s puzzle.

scholarly journals Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

2021 ◽  
Vol 174 (1) ◽  
Author(s):  
Amirlan Seksenbayev
Keyword(s):  
Dynamic Programming ◽  
Asymptotic Expansions ◽  
Dual Problem ◽  
Infinite Length ◽  
Expected Time ◽  
Selection Strategies ◽  
Dynamic Programming Equation ◽  
Optimal Values ◽  
Design Selection ◽  
Selection Of

AbstractWe study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size $n$ n . In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length $k$ k from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with $O(1)$ O ( 1 ) remainder in the first problem and $O(k)$ O ( k ) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

Optimality conditions for a Markov decision chain with unbounded costs

1980 ◽  
Vol 17 (04) ◽  
pp. 996-1003
Author(s):  
D. R. Robinson
Keyword(s):  
Dynamic Programming ◽  
Optimality Conditions ◽  
Average Cost ◽  
Optimality Equation ◽  
Dynamic Programming Equation ◽  
Potential Condition ◽  
Markov Decision ◽  
General Conditions

It is known that when costs are unbounded satisfaction of the appropriate dynamic programming ‘optimality' equation by a policy is not sufficient to guarantee its average optimality. A ‘lowest-order potential' condition is introduced which, along with the dynamic programming equation, is sufficient to establish the optimality of the policy. Also, it is shown that under fairly general conditions, if the lowest-order potential condition is not satisfied there exists a non-memoryless policy with smaller average cost than the policy satisfying the dynamic programming equation.

On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting

1971 ◽  
Vol 8 (03) ◽  
pp. 551-560 ◽  
Author(s):  
R. Morton
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Diffusion Processes ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Multidimensional Case ◽  
Potential Cost ◽  
One Dimensional ◽  
Dynamic Programming Equation ◽  
Stationary Diffusion

Summary Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting

1971 ◽  
Vol 8 (3) ◽  
pp. 551-560 ◽  
Author(s):  
R. Morton
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Diffusion Processes ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Multidimensional Case ◽  
Potential Cost ◽  
One Dimensional ◽  
Dynamic Programming Equation ◽  
Stationary Diffusion

SummaryBecause there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

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