SingleSellProfit.python

SingleSellProfit.python

#   Let min = arr[0]
#   For k = 1 to length(arr):
#       If arr[k] < min, set min = arr[k]
#       If profit < arr[k] - min, set profit = arr[k] - min
#
# This is short, sweet, and uses only O(n) time and O(1) memory.  The beauty of
# this solution is that we are quite naturally led there by thinking about how
# to update our answer to the problem in response to seeing some new element.
# In fact, we could consider implementing this algorithm as a streaming
# algorithm, where at each point in time we maintain the maximum possible
# profit and then update our answer every time new data becomes available.
#
# The final version of this algorithm is shown here:

def DynamicProgrammingSingleSellProfit(arr):
    # If the array is empty, we cannot make a profit.
    if len(arr) == 0:
        return 0

    # Otherwise, keep track of the best possible profit and the lowest value
    # seen so far.
    profit = 0
    cheapest = arr[0]

    # Iterate across the array, updating our answer as we go according to the
    # above pseudocode.
    for i in range(1, len(arr)):
        # Update the minimum value to be the lower of the existing minimum and
        # the new minimum.
        cheapest = min(cheapest, arr[i])

        # Update the maximum profit to be the larger of the old profit and the
        # profit made by buying at the lowest value and selling at the current
        # price.
        profit = max(profit, arr[i] - cheapest)

    return profit

# To summarize our algorithms, we have seen
#
# Naive:                        O(n ^ 2)   time, O(1)     space
# Divide-and-conquer:           O(n log n) time, O(log n) space
# Optimized divide-and-conquer: O(n)       time, O(log n) space
# Dynamic programming:          O(n)       time, O(1)     space