Find Min Sum of Finite Seq. of Ints Using Dynamic Prog.

In summary, to find the integers $i_1, i_2$ that minimize the sum $\sum_{j=i_1}^{i_2} Y_j$, we can use dynamic programming and keep track of previous choices made to avoid using the same $Y_j$ more than once. The minimum sum for the sequence can be found at $DP[1][M]$.
  • #1
evinda
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Hello! (Wave)

The finite sequence of integers $Y_1, \dots, Y_M$ takes both positive and negative values, where $M$ is a fixed positive integer. Could you help me to find a formulation using dynamic programming that solves the problem of finding integers $i_1, i_2$ with $1 \leq i_1 \leq i_2 \leq M$ such that the sum $\sum_{j=i_1}^{i_2} Y_j$ gets minimized? I have thought the following:We define $s_0^{\star}(i)=i , \ \ \forall i=1, \dots, M$.Also $s_i^{\star}(j)= \min_k \{ s_{i-1}^{\star}(j)+J_k\}$.But we have to take into consideration that we cannot pick a $J_k$ for $s_i^{\star}(j)$ if we have already used it to get $s_{i-1}^{\star}(j)$.How can we include this in the formula?Could you give me a hint?
 
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  • #2


Hello!

Dynamic programming is a great approach for solving this problem. The key to incorporating the constraint of not using the same $J_k$ more than once is to keep track of the previous choices made.

One way to do this is to create a two-dimensional array, let's call it $DP$, with dimensions $M \times M$. Each element $DP[j]$ represents the minimum sum for the subsequence starting at index $i$ and ending at index $j$. We can initialize $DP$ with $DP = Y_i$ for all $i=1, \dots, M$.

Then, for each $i$, we can iterate through the possible values of $j$ in the range $i+1$ to $M$. For each $j$, we can calculate $DP[j]$ using the formula $DP[j] = \min\{DP[j-1], DP[i+1][j] + Y_j\}$.

This formula takes into account the previous choices made, as $DP[j-1]$ represents the minimum sum for the subsequence starting at index $i$ and ending at index $j-1$, which does not include $Y_j$. The second term, $DP[i+1][j] + Y_j$, represents the minimum sum for the subsequence starting at index $i+1$ and ending at index $j$, which includes $Y_j$. By taking the minimum of these two values, we ensure that we are not using the same $Y_j$ more than once.

Finally, the minimum sum for the entire sequence can be found at $DP[1][M]$.

I hope this helps! Let me know if you have any further questions.
 

FAQ: Find Min Sum of Finite Seq. of Ints Using Dynamic Prog.

What is dynamic programming?

Dynamic programming is a problem-solving technique in computer science where a complex problem is broken down into smaller subproblems and the solutions to these subproblems are stored in a table so that they can be used to solve the larger problem efficiently.

How does dynamic programming help in finding the minimum sum of a finite sequence of integers?

In this problem, dynamic programming helps by storing the solutions to subproblems in a table, making it possible to compute the minimum sum of a sequence of integers without having to recalculate the same subproblem multiple times.

What is the time complexity of the dynamic programming approach for finding the minimum sum of a finite sequence of integers?

The time complexity of this approach is O(n), where n is the length of the sequence. This is because the algorithm only needs to iterate through the sequence once and store the solutions to subproblems in a table.

Can dynamic programming be used to find the minimum sum of a sequence of non-integer values?

Yes, dynamic programming can be used to find the minimum sum of any sequence of values, including non-integer values. The only requirement is that the values can be compared and added together.

What are some advantages of using dynamic programming to solve this problem?

Some advantages of using dynamic programming for this problem are that it reduces the time complexity from exponential to polynomial, it makes the algorithm more efficient by avoiding repeated calculations, and it allows for a simpler and more organized code structure.

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