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UVa 713: Adding Reversed Numbers is a straight-forward problem that can be solved using an ad-hoc algorithm. We have discussed a similar problem in a previous blog-post. However, this problem imposes following additional constraint—numbers will be at most 200 characters long–which necessitates the use of BigInteger. Following Java code shows such an ad-hoc algorithm that solves this problem by considering the stated constraints.
Please leave a comment if you have any question. Thanks.
The UVa 10664: Luggage is a typical example of the problems that can be solved using dynamic programming (DP) technique. In fact, after further analysis, this problem can be realized as a special case of Subset-Sum problem, which we have discussed in a recent post.
The following Java code snippet outlines an algorithm using dynamic programming to solve this problem. Notice that the function solveLuggageProblem applies a bottom-up DP to construct dpTable. The boolean value of each dpTable[i][j] implies that whether it is possible to achieve weight j from the weights of 1..i suitcases. In this way, it determines whether halfWeight — the half of the total weights (of the suitcases)– can be derived from using 1..n suitcases, i.e., whether the weights of suitcases can be distributed equally into the boots of two cars.
Please leave a comment if you have any question regarding this problem or implementation. Thanks.
SPOJ 97. Party Schedule (PARTY) with F#.
SPOJ 8545. Subset Sum (Main72) with Dynamic Programming and F#.
The UVa 102: Ecological Bin Packing can be solved using brute-force search, as outlined in the following Java code snippet. Note that, the primary function getOPTConfiguration iterates over all possible bin arrangements, and determines the arrangement that requires minimum bottle moves for a given problem instance.
Please leave a comment if you have any question/comment. Happy coding!
The Subset Sum (Main72) problem, officially published in SPOJ, is about computing the sum of all integers that can be obtained from the summations over any subset of the given set (of integers). A naïve solution would be to derive all the subsets of the given set, which unfortunately would result in time complexity, given that
is the number of elements in the set.
This post outlines a more efficient (pseudo-polynomial) solution to this problem using Dynamic Programming and F#. Additionally, we post C# code of the solution.
Note that we have solved a similar problem in Party Schedule (PARTY) with F# blog-post.
This problem provides a set of integers , and specifies the following constraints–
The size of the given set, i.e., 
Given this input, we would like to find all the integers: and
is the sum of the items of any subset over
. Afterward, we sum all these integers, and return it as the result to the problem instance.
In essence, we reduce this problem as follows: Given , can we express it using any subset over
? If yes, we include it in the solution set for summation. Interestingly, we realize that the stated problem is a special case of a more general problem called Subset Sum, given that the sum is
.
What would be the maximum possible value for ? Indeed,
is not practical at all, as
can be bounded by the following upper limit:
, i.e., the summation of all the items in
. This observation effectively reduces the search space to
, for a given
.
It implies that a naïve algorithm would require to iterate all the subsets over and verify whether their sum is within
. Recall that, due to its exponential time complexity, it is quite impractical .
Using dynamic programming technique, a pseudo-polynomial algorithm can be derived, as the problem has an inherent optimal substructure property. That is, a solution of an instance of the problem can be expressed as the solutions of its subproblems, as described next.
We define as the function that determines whether the summation over any subset
can result in the integer
. So, it yields
if sum
can be derived over any subset
, otherwise,
. Also, note that,
and
.
To define the recurrence, we describe in terms of its smaller subproblems as follows.
In Eq. (1), the first case refers to the fact that is larger than
. Consequently,
can not be included in the subset to derive
. Then, the case 2 of Eq. (1) expresses the problem
into two subproblems as follows: we can either ignore
though
, or we can include it. Using any case stated in Eq. (1), if we can derive
i.e.
= true, we can include it in the solution set.
As we can see overlapping subproblems, we realize that we can effectively solve them using a bottom-up dynamic programming approach. What about the base cases?
Using a table– dp, we can implement the stated algorithm as follows.
In essence, the Nth row in the table provides the set of integers that can be derived by summing over any subset . Thereby, we compute the summation of all these integers that satisfies subsum(N,j) = true, and returns it as the result.
Full source code of the solution can be downloaded from this gist. For C# source code, please visit following gist. Please leave a comment if you have any question/suggestion regarding this post.
Enough said… now, it’s time for a 
and a new problem. See you soon; till then, happy problem-solving!
SPOJ 97. Party Schedule (PARTY) with F#.
UVa 10664. Luggage.
The Bytelandian Gold Coins problem, officially published in SPOJ, concerns computing the maximum dollars that can be exchanged for a Bytelandian gold coin. In this post, we outline a solution to this problem with memoization and F#.
The problem definition enforces following rules to perform the exchange. Consider, a Bytelandian gold coin —
It can be exchanged to three other coins, i.e., Alternatively, Our objective is to derive an algorithm that maximizes the dollars exchanged from the gold coin .
From the above interpretation, it is evident that the maximum achievable dollars, (from the exchange of coin
) can be computed as follows.
It effectively demonstrates an optimal substructure and therefore, hints to a dynamic programming (DP) technique to solve it. That is, for a coin , the optimal value of dollar is given by the following function.
We employ a top-down DP approach, as it seems more efficient than a bottom-up approach in this context. It is due to the fact that a bottom-up approach generally requires an OPT table to persist results of smaller subproblems. As in this case, the value of can be very large (i.e.,
, a bottom-up DP would require a very large array, and performs more computations. Hence, for the overlapping subproblems, we employ memoization.
The following code snippet outlines the implementation of Memo.
Full source code of the solution can be downloaded from this gist. Please leave a comment if you have any question/suggestion regarding this post.
Happy problem-solving! 
