Chapter 5 Optimization Algorithm and Verification Algorithm
5.3 Pigeonhole-principle-based Verification Algorithm
In our previous study [37], we have proposed a pigeonhole-principle-based fast backtracking algorithm (PPBFB) to obtain the lower bound of our problem in about 5
days using an AMD Opteron 1.6GHz PC. Here, the concept of PPBFB will be reviewed first and then, the reductions of equivalent queries (Technique 3 in Section 3.2.2.3) are also cooperated with PPBFB to accelerate the speed of the verification.
The refined version of PPBFB is called pigeonhole-principle-based verification algorithm (PPV). Finally, the lower bound is also acquired by PPV in only 12.83 minutes using an Intel Core 2 Duo 3.16 GHz PC.
The concept of PPBFB is to conduct an exhaustive worst-first search. It rates the lower bound by making use of the extended pigeonhole principle proposed by us [18]
and then backtracks as early as possible to save the search time. The refined version of PPBFB, PPV, is illustrated in Figure 20. The key idea of PPV is to consider the sizes of the two sets in the state when the search proceeds. The rectangles in Figure 20 represent the states. is the i-th possible choice among all secret codes made by the codebreaker at the p-th query. rp, max means the class which results in the most number of queries among 14 classes after the p-th query. qmax is the theoretical lower bound which means a fewest number of queries required to reach the final state, i.e.,
p
g
i,{ }
c ,φ or φ,{ }
c , from the current state and h is the lower bound we intend to verify.Theoretically, a search algorithm has to explore all valid 5040 secret codes at each query. However, in fact, PPV only need to explore 1 representative query at the first query, to expand 20 queries at the second query, and to expand 356.50 queries in average at the third query due to the equivalence property. For the codemaker, only the worst case among the 14 classes has to be expanded. The so-called “worst case”
denotes the class which will result in the most number of queries needed by the codebreaker.
Figure 20. The sketch of the PPV algorithm
The extended pigeonhole principle [18] is employed to estimate the lower bounds of the number of queries needed among 14 classes. The idea of the estimations of lower bounds is similar to that proposed in Chapter 2. In other words, the actual number of queries needed is more than or equal to the most number, qmax, of lower bounds among 14 classes. Therefore, our verification program is not necessary to search the whole game tree. It can backtrack to the parent node to expand other branches if the condition holds:
( p
+q
max)
≥h
, where we set h = 8.The main idea of the estimation of lower bounds by using the extended pigeonhole principle is that the query made by the codebreaker in each ply may divide the elements of the two sets in the current state evenly. Hence, this ideal strategy can minimize the height of subtree rooted in the current node. That is to say that there exists a “theoretical optimal” strategy for the codebreaker in the following queries such that all the elements of the two sets in each state may be divided evenly. The actual number of queries is thus more than or equal to the value of estimations. Note that we use the function, Get_lower_bound, to rate the lower bounds in Figure 20.
The detailed calculation of the lower bounds, the entire algorithms, and other improvements can be found in [37][38]. Hence, the details are omitted here.
After the careful implementation of our program based on PPV, The verification program was run on a dedicated PC equipped with an Intel Core 2 Duo 3.16 GHz CPU to verify the lower bound required for AB game with an unreliable response. If we set the value of h, which indicates the lower bound we want to verify, to 8, our program executed for about 12.83 minutes and the final output is “success!” finally.
In other words, the minimum number of queries is at least 8 in the worst case without respect to any strategies used by the codebreaker. Note that the upper bound of this problem is obtained in Lemma 7 as well. Thus, we have the following theorem which shows that the lower bound as well as the exact bound of the game is 8.
Theorem 3. For AB game with an unreliable response, 8 queries are necessary and
sufficient to identify a secret code in the worst case.5.4 Chapter Conclusion
This chapter utilizes two advanced algorithms to address AB game with an unreliable response. The first one is two-phase optimization algorithm (TPOA). With the well-designed hashing function and the simple heuristic of evaluation, the results obtained by TPOA are better than those of the previous work [37]. In other words, TPOA is more effective and efficient. Note that the upper bound of the game is declined from 9 to 8 in this refined approach.
On the other hand, another improvement, pigeonhole-principle-based verification
algorithm (PPV), is modified from pigeonhole-principle-based fast backtracking
algorithm (PPBFB). PPV uses equivalent properties to reduce the branching factors at the first three queries. Although the final outcome is the same as that in [37], the speed of PPV is faster than PPBFB due to the reductions of equivalent queries.Moreover, the lower bound provided by PPV is 8 as well.
Fortunately, we have proved that the upper bound of the game matches the lower bound while its value is 8. Hence, the minimum number of queries for AB game with an unreliable response is 8. Furthermore, it may be interesting to deal with AB game with e unreliable responses, where e ≥ 2.
Chapter 6
Conclusion and Future Work
In this dissertation, some optimization approaches for deductive games and their variants are taken into account. Section 6.1 concludes with the proposed optimization algorithms and our contributions. Some future work is mentioned in Section 6.2.
6.1 Concluding Remarks
Two advanced algorithms and a reduction technique for deductive games are demonstrated in this study. Moreover, two promising algorithms, which are proposed before, with some modifications are introduced to solve our addressed problem as well. We summarize our main novel contributions:
(1) A more efficient complete algorithm, which is called depth-first
backtracking algorithm with branch-and-bound pruning (DBB) for
Mastermind in the expected case, is introduced to take the place of traditional approaches and meanwhile, an admissible heuristic, which can be applied to various deductive games, is presented as well. From the experiments, DBB is significantly superior to the traditional algorithms and an alternative optimal strategy is also obtained finally.(2) To date, there have been no optimal expected-case strategies for AB game in formal literature since its appearance. Thus, a refined branch-and-bound
algorithm with speed-up techniques (RBB) is demonstrated to deal with this
problem. A tactic for playing AB game optimally in the expected case is eventually attained by utilizing RBB and in addition, the corresponding expected number of queries, 26274/5040 ≈ 5.213, is derived.(3) A sophisticated method, called structural-reduction approach (SR), which aims at explaining the pessimistic situation in this game, is presented to investigate 3×n AB games. After careful theoretical analyses, optimal strategies for the codebreaker in the pessimistic situation are discovered.
Furthermore, a worthwhile formula for calculating the optimal numbers of queries required for arbitrary values of n is derived and proven successfully.
(4) Two algorithms, which are named as two-phase optimization algorithm (TPOA) and pigeonhole-principle-based verification algorithm (PPV), are surveyed for solving AB game with an unreliable response. The purpose of TPOA is to discover an upper bound of the required number of queries in this game while PPV aims at identifying a lower bound of it. Fortunately, experimental results show that the upper bound equals the lower bound and then, the exact bound of the number of queries needed, whose value is 8, is achieved.
From the survey of related papers, it reveals that the search space of many games and optimization problems are often so huge that traditional search algorithms are not able to explore it efficiently. Of course, there were plenty of pruning techniques, which were proposed before. However, slight inaccuracy of the measures of these pruning techniques may usually lead to the poor results that are far from the optimum.
In this study, our proposed search algorithms, which are replied upon the admissible heuristics, have contributed success to various deductive games. Note that
in general, the admissible heuristics can be regarded as a kind of theoretical pruning techniques since the pruning occurs but does not affect the correctness of search algorithms. In other words, the results of the search algorithms are accurate if the pruning techniques are based on theoretical analyses. Hence, it may be a trend to combine search algorithms with theoretical pruning for solving those complicated problems.
On the other hand, other optimization problems such as coding theory, circuit testing, differential cryptanalysis, and additive search problem may also be solved by taking advantage of our demonstrated methods with modifications in the future. We hope that the research results may assist other scientists with the development of their concerned issues.
6.2 Future Work
There are still some open issues regarding our problem domain. The optimal strategies of deductive games with much higher dimensions, which are called m×n AB games while m ≥ 4, are still unknown. It is interesting to investigate them because they may become NP-complete problems or harder problems if the value of m is getting larger constantly. Then, the boundary value of m is significant as well. Besides the original versions of much higher dimensions, other variants of deductive games are also worth studying such as static deductive games or deductive games with multiple unreliable responses. From the progress of research, 3×n deductive games in the expected case and 4×n deductive games in the worst case may be solved completely in the near future.
There are other important problems such as the Renyi-Ulam game and the counterfeit coin problem, whose styles are similar to deductive games. In fact, the Renyi-Ulam game has been widely surveyed in the fault-tolerance area and
meanwhile, the counterfeit coin problem has been discussed constantly in the information-theory area as well. However, there are still a lot of open issues about the two significant problems. These open questions are likewise worth studying in further detail for discovering their solutions.
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Appendix A. Equivalence Transformations for AB Game at the Second Query
The following equivalence transformations for the second query of AB game transform the 209 codes into their corresponding representatives.
Table 14. Equivalence transformations
Order Representative Each
query Equivalence transformations
1 0123 - -
2301 ⎟⎟⎠
3456 ⎟⎟⎠
4103 ⎟⎟⎠
4253 ⎟⎟⎠
3405 ⎟⎟⎠
3415 ⎟⎟⎠
0314 ⎟⎟⎠
4130 ⎟⎟⎠
2034 ⎟⎟⎠
2014 ⎟⎟⎠
Appendix B. Optimal Strategy for AB Game in the Expected Case
Prior to introducing the optimal strategy of AB game, its representation will be illustrated first. The lower-case alphabets, a, b, c, …, m, n, represent the 14 responses (hints), as shown in Table 15.
Table 15. The mapping between responses and representative letters
Response Representative letter Response Representative letter
[4, 0] a [1, 1] h
[3, 0] b [1, 0] i
[2, 2] c [0, 4] j
[2, 1] d [0, 3] k
[2, 0] e [0, 2] l
[1, 3] f [0, 1] m
[1, 2] g [0, 0] n
Three kinds of tokens will appear in the strategy. The first kind is four-digit Arabic numerals, which means the query made by the codebreaker. The second one is lower-case letters mentioned above, which indicate the responses. The last kind is parentheses. The tokens in parentheses refer to the optimal tactic of the state. In other words, it is an optimal game tree of that state. The tactic is constructed with a recursive form and can be treated as a game tree. For example, suppose that a game tree depicted in Figure 21 is given. Then its corresponding representation will be
“4872 ( j 7248 ( a ) f 4287 ( f 8274 ( a ) a ) a )”. Furthermore, it is easy to reconstruct the game tree from its representation with depth-first ordering.
j f a 4872
7248 4287
8274 a
7248
a
a f {7248} {4287, 8274}
{8274}
4872 ( j 7248 ( a ) f 4287 ( f 8274 ( a ) a ) a ) {4287, 4872, 7248, 8274}
4287
8274
4872
Figure 21. The transformation between the game tree and its corresponding representation
The derived optimal strategy of AB game in the average case is shown partially as follows due to space restrictions. In order to clarify the levels, we use an indent structure. We have established a website (http://www.csie.ntnu.edu.tw/~linss/
ABgame/optimal_strategy.html) that includes the full text of the optimal strategy.
0123 ( n 4567 ( l 5689 ( l 7498 ( j 8974 ( c 9874 ( a ) a )
f 8794 ( f 9748 ( a ) a )
c 7894 ( f 7948 ( a ) j 9478 ( a )
a ) a )
k 6948 ( l 8795 ( c 7895 ( a ) f 9875 ( a ) a )
k 8495 ( l 9876 ( a ) j 9854 ( a ) h 7896 ( a ) e 8796 ( a ) a )
j 8496 ( a )
h 8975 ( j 9758 ( a ) a )
g 8954 ( j 9845 ( a ) f 9458 ( a ) e 8976 ( a ) a )
f 6894 ( f 9846 ( a ) a )
e 7958 ( a )
d 6798 ( l 8945 ( a )
a )
c 6498 ( j 8946 ( a ) a )
b 6978 ( a ) a )
// The full text of the optimal strategy is included at http://www.csie.ntnu.edu.tw/
~linss/ABgame/optimal_strategy.html.
Appendix C. Publication List
(a) Referred Papers:
As a PhD student
[a1] Huang, L. T., and Lin, S. S. (2009), “Optimal analyses for 3×n AB games in the worst case,” to appear in Lecture Notes in Computer Science series.
[a2] Huang, L. T., Chen, S. T., Huang, S. J., and Lin, S. S. (2007), “An efficient approach to solve Mastermind optimally,” ICGA Journal, Vol.
30, No. 3, pp. 143–149.
[a3] Chen, S. T., Lin, S. S., and Huang, L. T. (2007), “A two-phase optimization algorithm for Mastermind,” Computer Journal, Vol. 50, No.
4, pp. 435–443.
[a4] Chen, S. T., Lin, S. S., Huang, L. T., and Hsu, S. H. (2007), “Strategy optimization for deductive games,” European Journal of Operational
Research, Vol. 183, No. 2, pp. 757–766.
As a master’s student
[a5] Huang, L. T., Chen, S. T., and Lin, S. S. (2006), “Exact-bound analyses and optimal strategies for Mastermind with a lie,” Lecture Notes in
Computer Science, Advances in Computer Games 11, Vol. 4250, pp.
195–209.
[a6] Chen, S. T., Lin, S. S., Huang, L. T., and Wei, C. J. (2004), “Towards the Exact Minimization of BDDs — An Elitism-Based Distributed Evolutionary Algorithm,” Journal of Heuristics: Special Issue on New
Advance on Parallel Meta-Heuristics for Complex Problems, Vol. 10, No.
3, pp. 337–355.
(b) Submitted Paper:
[b1] Huang, L. T., Chen, S. T., and Lin, S. S. (2009), “Optimal Algorithm for AB Game in the Expected Case,” submitted to IEEE Transactions on
Computational Intelligence and AI in Games.
(c) Conference Papers:
As a PhD student
[c1] Huang, L. T., and Lin, S. S. (2009), “Optimal analyses for 3×n AB games in the worst case,” The 12th
conference on Advances in Computer Games
(ACG12), Pamplona, Spain.[c2] Huang, L. T., Chen, S. T., Huang, S. J., and Lin, S. S. (2007), “An efficient approach to solve Mastermind optimally,” COMPUTER GAMES
WORKSHOP 2007, Amsterdam, The Netherlands.
[c3] Chen, S. T., Lin, S. S., Chang, S. W., and Huang, L. T. (2006), “A two-phase search algorithm for the set covering problem”，第十一屆人工 智慧與應用研討會，國立高雄應用科技大學，台灣，中華民國。
As a master’s student
[c4] Huang, L. T., Chen, S. T., and Lin, S. S. (2005), “Exact-bound analyses and optimal strategies for Mastermind with a lie,” The 11th
Advances in Computer Games Conference (ACG11), Taipei, Taiwan.
(d) Technical Reports:
[d1] 陳善泰、黃立德、張書維、劉耀才、江漢昇、胡淑瓊，2005，國科會 研究報告：演繹競局問題最佳化策略及其應用於容錯系統之研究 (2/2)，NSC93-2213-E-003-001。
[d2] 陳善泰、黃立德、張書維、劉耀才、江漢昇、胡淑瓊，2004，國科會 研究報告：演繹競局問題最佳化策略及其應用於容錯系統之研究
[d2] 陳善泰、黃立德、張書維、劉耀才、江漢昇、胡淑瓊，2004，國科會 研究報告：演繹競局問題最佳化策略及其應用於容錯系統之研究