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Given :

  • Profit[n]: a n item array indicating the profit on a day (may be positive or negative)
  • Cost[n]: a n item array indicating the cost of holding a stock on a day (always positive)

Task: Find $1 \leq i \leq j \leq n$ that maximize Total Profit divided by $\sqrt{\text{Total Cost}}$ from day $i$ to day $j$.

Extensions:

  • Does knowing that the total profit over all days = 0 help improve the algorithm?
  • Rather than maximizing the ratio, is it possible to find the maximum profit given a "Budget" (Maximum Cost)
  • Would supposing that the cost remains constant help improve the algorithm?

Attempts to solve:

  • Naive: simply loop through all pairs of $i, j$: ($O(N^2)$)
  • Dynamic programming? - don't know where to start
  • Divide and conquer? - seems applicable but hard to implement
  • Is there an $O(N)$ solution?

Intended Application:

  • I have a sequence of data points and a trendline that estimates the true mean and variance
  • I want to find stretches where the data is on a "hot" or "cold" streak (i.e. where the cumulative excess over the mean is most unexpected compared to the variance.) That's where the square root comes from
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