Maximizing Stock Profit: Brute-Force and Optimized Solutions

Introduction

You are given an array prices where prices[i] represents the price of a given stock on the ith day. You aim to maximize your profit by choosing a single day to buy one stock and a different day in the future to sell that stock. Your goal is to determine the maximum profit achievable from this transaction. If no profit can be made, you should return 0.

Examples:

• Example 1:
• Input: prices = [7,1,5,3,6,4]
• Output: 5
• Explanation: Buy on day 2 (price = 1) and sell on day 5 (price = 6), profit = 6 – 1 = 5.
• Example 2:
• Input: prices = [7,6,4,3,1]
• Output: 0
• Explanation: No profitable transaction is possible.

Constraints:

• 1 <= prices.length <= 10^5
• 0 <= prices[i] <= 10^4

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Brute-Force Solution (O(n²) Time Complexity)

Approach

The brute-force solution involves checking all possible pairs of buy and sell days to calculate the maximum profit. For every day, you consider it as a potential buying day and then check all the future days as potential selling days.

Implementation

public int MaxProfitBruteForce(int[] prices) {
    int maxProfit = 0;
    int n = prices.Length;
    for (int i = 0; i < n - 1; i++) {
        for (int j = i + 1; j < n; j++) {
            int profit = prices[j] - prices[i];
            if (profit > maxProfit) {
                maxProfit = profit;
            }
        }
    }
    return maxProfit;
}

Explanation

• Outer Loop (i): Iterates through each day as a potential buying day.
• Inner Loop (j): Iterates through each day after i as a potential selling day.
• Profit Calculation: Computes the profit by subtracting the buying price (prices[i]) from the selling price (prices[j]).
• Max Profit Update: Updates maxProfit if the current profit is greater.

Time Complexity Analysis

• Time Complexity: O(n²)
• The nested loops result in O(n²) iterations.
• Space Complexity: O(1)
• Only a constant amount of additional space is used.

Drawbacks

• Inefficient for Large Inputs:
• With constraints up to 10^5 elements, the O(n²) solution becomes impractical due to time limitations.
• Redundant Calculations:
• Many unnecessary computations are performed, especially when prices are not conducive to profit.

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Optimized Solution (O(n) Time Complexity)

Why an O(n) Solution is Possible

The problem requires finding the maximum difference between two prices where the selling day comes after the buying day. By keeping track of the minimum price encountered so far and the maximum profit achievable at each step, we can solve the problem in a single pass through the array.

Approach

• Initialize Variables:
• minPrice to store the minimum price seen so far.
• maxProfit to store the maximum profit calculated so far.
• Single Pass Iteration:
• Traverse the prices array once, updating minPrice and maxProfit accordingly.

Implementation

public int MaxProfit(int[] prices) {
    if (prices == null || prices.Length < 2) return 0;

    int maxProfit = 0;
    int minPrice = prices[0];

    for (int i = 1; i < prices.Length; i++) {
        if (prices[i] < minPrice) {
            minPrice = prices[i];
        } else {
            int profit = prices[i] - minPrice;
            if (profit > maxProfit) {
                maxProfit = profit;
            }
        }
    }
    return maxProfit;
}

Explanation

• Initialization:
• Set minPrice to the first price in the array.
• Initialize maxProfit to 0.
• Iteration:
• Start from the second day (i = 1).
• Update minPrice:
  • If the current price is lower than minPrice, update minPrice.
• Calculate Profit:
  • Else, calculate the profit by subtracting minPrice from the current price.
  • If this profit is greater than maxProfit, update maxProfit.
• Return Value:
• After traversing the entire array, return maxProfit.

Time Complexity Analysis

• Time Complexity: O(n)
• The array is traversed only once.
• Space Complexity: O(1)
• Only a constant amount of additional space is used.

Why It Works

• Minimum Price Tracking:
• By always keeping track of the lowest price before the current day, we ensure that we’re buying at the lowest possible price up to that point.
• Profit Maximization:
• At each step, we check if selling on the current day yields a better profit than any we’ve seen before.
• Single Pass Efficiency:
• We avoid redundant calculations by not considering every possible pair, focusing only on the most promising candidates.

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Identifying Similar Single-Pass Problems

Characteristics of Such Problems

1. Optimal Substructure:

• The problem can be broken down into subproblems whose optimal solutions contribute to the overall optimal solution.

1. Cumulative or Running Values:

• The solution depends on cumulative values like minimums, maximums, or sums up to a certain point.

1. One-Directional Dependencies:

• Future computations depend on past or current values but not vice versa.

Common Strategies

1. Maintain Running Extremes:

• Keep track of running minimums or maximums.
• Example: Finding the maximum difference between two elements where the smaller element comes before the larger one.

1. Sliding Window Technique:

• Use a window that moves through the data to consider subsets efficiently.
• Example: Finding the maximum sum of a subarray of size k.

1. Prefix Sums and Differences:

• Precompute prefix sums to answer range sum queries in constant time.
• Example: Computing cumulative sums for quick range queries.

1. Two-Pointer Technique:

• Use two pointers to traverse the array from different ends or at different speeds.
• Example: Finding pairs in a sorted array that add up to a specific target.

1. Dynamic Programming with State Compression:

• Update states based on previous computations without storing all intermediate results.
• Example: Solving the maximum subarray problem using Kadane’s algorithm.

How to Identify Such Problems

1. Constraints Hint at Feasibility:

• Look for Problems with Nested Loops Over the Same Data:
• Large input sizes (e.g., 10^5 elements) suggest that O(n²) solutions are impractical.
• Look for solutions with linear or logarithmic time complexity.

1. Problem Requires Extremes Within Subsets:

• If you’re asked to find minimums, maximums, or optimal values within certain conditions, consider maintaining running values.

1. Repeated Calculations in Brute-Force Approach:

• If the brute-force solution involves recalculating values that could be stored, look for ways to cache or update these values on the fly.

1. Possible to Update Solution While Traversing:

• If the state of the solution can be updated incrementally as you move through the data, a single-pass solution may be possible.

Examples of Similar Problems

1. Maximum Subarray Problem (Kadane’s Algorithm):

• Problem: Find the contiguous subarray with the maximum sum.
• Approach: Keep track of the current subarray sum and the maximum sum found so far.

1. Container With Most Water:

• Problem: Find two lines that together with the x-axis form a container that holds the most water.
• Approach: Use two pointers moving towards each other to find the maximum area.

1. Longest Increasing Subsequence (Simple Version):

• Problem: Find the length of the longest increasing contiguous subsequence.
• Approach: Iterate through the array, updating the length when the sequence increases.

1. Rainwater Trapping Problem:

• Problem: Calculate how much rainwater can be trapped after raining.
• Approach: Use precomputed arrays of maximum heights to the left and right or use two pointers.

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Conclusion

By comparing both the brute-force and optimized solutions, we see that understanding the underlying patterns and relationships in the data allows us to significantly improve algorithm efficiency. The key to solving such problems lies in:

• Recognizing Patterns: Identifying that the problem can be solved by tracking minimum or maximum values.
• Optimizing Calculations: Avoiding redundant computations by updating values in a single pass.
• Analyzing Constraints: Considering input size constraints to guide the choice of algorithm.

Applying these principles not only solves the current problem efficiently but also equips you with strategies to tackle a wide range of algorithmic challenges with similar characteristics.