You are given an integer array prices where prices[i] is the price of a given stock on the i-th day.

On each day, you may decide to buy and/or sell the stock. You can only hold at most one share of the stock at any time. However, you can buy it then immediately sell it on the same day.

Find and return the maximum profit you can achieve.

Examples

Example 1:

Input: prices = [7,1,5,3,6,4]
Output: 7
Explanation: Buy on day 2 (price = 1) and sell on day 3 (price = 5), profit = 5-1 = 4.
Then buy on day 4 (price = 3) and sell on day 5 (price = 6), profit = 6-3 = 3.
Total profit is 4 + 3 = 7.

Example 2:

Input: prices = [1,2,3,4,5]
Output: 4
Explanation: Buy on day 1 (price = 1) and sell on day 5 (price = 5), profit = 5-1 = 4.
Total profit is 4.

Example 3:

Input: prices = [7,6,4,3,1]
Output: 0
Explanation: There is no way to make a positive profit, so we never buy the stock to achieve the maximum profit of 0.

Constraints

  • 1 <= prices.length <= 3 * 10^4
  • 0 <= prices[i] <= 10^4

Thinking Process

You are given an integer array prices where prices[i] is the price of a given stock on the i-th day.

On each day, you may decide to buy and/or sell the stock. You can only hold at most one share of the stock at any time. However, you can buy it then immediately sell it on the same day.

  • Define state: what subproblem does dp[i] (or dp[i][j]) represent?
  • Recurrence: how does the answer build from smaller indices?
  • Base cases first; optimize space if only prior row/layer is needed.
1D DP recurrence dp[i] 0 1 2 ? dp[i] from smaller indices / subproblems

Common Approaches

Typical techniques for this pattern:

Approach Time Space Notes
1D DP (this problem) O(n) O(n) or O(1) Linear recurrence
2D DP O(nm) O(nm) or O(n) Grid or two-sequence problems
State machine DP O(n) O(1) Buy/sell, hold/not-hold states
Memoization (top-down) Same as DP O(n) Recursive + cache

Solution

Solution: Greedy - Capture All Gains

class Solution:
    def maxProfit(self, prices):
        if len(prices) == 0:
            return 0

        maxProfit = 0
        for i in range(1, len(prices)):
            if prices[i] > prices[i - 1]:
                maxProfit += prices[i] - prices[i - 1]

        return maxProfit

Solution Explanation

Approach: 1D DP (this problem)

Key idea: You are given an integer array prices where prices[i] is the price of a given stock on the i-th day.

How the code works:

  • Define state: what subproblem does dp[i] (or dp[i][j]) represent?
  • Recurrence: how does the answer build from smaller indices?
  • Base cases first; optimize space if only prior row/layer is needed.

Walkthrough — input prices = [7,1,5,3,6,4], expected output 7:

Buy on day 2 (price = 1) and sell on day 3 (price = 5), profit = 5-1 = 4. Then buy on day 4 (price = 3) and sell on day 5 (price = 6), profit = 6-3 = 3. Total profit is 4 + 3 = 7.

Time: O(n) where n is the length of prices array · Space: O(1)

Algorithm Explanation:

  1. Edge Case (Line 4):
    • If prices array is empty, return 0 (no profit possible)
  2. Initialize (Line 5):
    • maxProfit: Accumulator for total profit (starts at 0)
    • lastPrice: Previous day’s price (initialized but not used in this version)
  3. Calculate Profit (Lines 6-8):
    • For each day starting from index 1:
      • Calculate difference: prices[i] - prices[i - 1]
      • Add positive gains: max(0, prices[i] - prices[i - 1])
        • If price increased, add the gain to profit
        • If price decreased or stayed same, add 0
      • Accumulate: Add to maxProfit
  4. Return (Line 9):
    • Return total accumulated profit

Example Walkthrough:

Example 1: prices = [7,1,5,3,6,4]

Initial: maxProfit = 0

i=1: prices[1]=1, prices[0]=7
  Difference: 1 - 7 = -6
  Gain: max(0, -6) = 0
  maxProfit = 0 + 0 = 0

i=2: prices[2]=5, prices[1]=1
  Difference: 5 - 1 = 4
  Gain: max(0, 4) = 4
  maxProfit = 0 + 4 = 4

i=3: prices[3]=3, prices[2]=5
  Difference: 3 - 5 = -2
  Gain: max(0, -2) = 0
  maxProfit = 4 + 0 = 4

i=4: prices[4]=6, prices[3]=3
  Difference: 6 - 3 = 3
  Gain: max(0, 3) = 3
  maxProfit = 4 + 3 = 7

i=5: prices[5]=4, prices[4]=6
  Difference: 4 - 6 = -2
  Gain: max(0, -2) = 0
  maxProfit = 7 + 0 = 7

Result: 7

Visual Representation:

Prices: [7, 1, 5, 3, 6, 4]
         ↓  ↓  ↓  ↓  ↓  ↓
Day:     0  1  2  3  4  5

Gains:
  Day 0→1: 1-7 = -6 (loss, ignore)
  Day 1→2: 5-1 = +4 (gain, capture) ✓
  Day 2→3: 3-5 = -2 (loss, ignore)
  Day 3→4: 6-3 = +3 (gain, capture) ✓
  Day 4→5: 4-6 = -2 (loss, ignore)

Total: 4 + 3 = 7

Example 2: prices = [1,2,3,4,5]

Initial: maxProfit = 0

i=1: 2-1 = 1 → maxProfit = 1
i=2: 3-2 = 1 → maxProfit = 2
i=3: 4-3 = 1 → maxProfit = 3
i=4: 5-4 = 1 → maxProfit = 4

Result: 4

Visual Representation:

Prices: [1, 2, 3, 4, 5]
Gains:  +1 +1 +1 +1
Total:  1 + 1 + 1 + 1 = 4

Example 3: prices = [7,6,4,3,1]

Initial: maxProfit = 0

i=1: 6-7 = -1 → maxProfit = 0
i=2: 4-6 = -2 → maxProfit = 0
i=3: 3-4 = -1 → maxProfit = 0
i=4: 1-3 = -2 → maxProfit = 0

Result: 0 (no positive gains)

Algorithm Breakdown

Why Greedy Works

The greedy strategy is optimal because:

  1. No Future Information: We don’t need to know future prices
  2. Local Optimal: Capturing every price increase is always beneficial
  3. No Regret: If we skip a gain today, we can’t capture it later (each day is independent)
  4. Optimal Substructure: Maximum profit = sum of all positive day-to-day gains

Key Insight: Capture All Gains

Instead of trying to find the best buy/sell pairs, we can:

  • Buy and sell on consecutive days whenever price increases
  • Sum all positive differences to get maximum profit
  • This is equivalent to buying at local minima and selling at local maxima

Example:

Prices: [1, 5, 3, 6]

Strategy 1: Buy at 1, sell at 5, buy at 3, sell at 6
  Profit: (5-1) + (6-3) = 4 + 3 = 7

Strategy 2: Buy at 1, sell at 6 (skip intermediate)
  Profit: 6-1 = 5

Strategy 3: Capture all gains (greedy)
  Day 0→1: 5-1 = 4
  Day 1→2: 3-5 = -2 (ignore)
  Day 2→3: 6-3 = 3
  Total: 4 + 3 = 7 ✓

Mathematical Proof

For any price sequence, the maximum profit is:

maxProfit = Σ max(0, prices[i] - prices[i-1]) for i from 1 to n-1

This captures all positive price movements, which is optimal.

Time & Space Complexity

  • Time Complexity: O(n) where n is the length of prices array
    • Single pass through the array
    • Constant time operations for each element
  • Space Complexity: O(1)
    • Only using a few variables
    • No additional data structures

Key Points

  1. Greedy Strategy: Capture every price increase
  2. Simple Logic: Sum all positive day-to-day differences
  3. Optimal: Greedy approach finds maximum profit
  4. No DP Needed: Simple greedy is sufficient
  5. Efficient: O(n) time, O(1) space

Common Mistakes

  1. Empty array: [] → return 0
  2. Single price: [5] → return 0 (no transaction possible)
  3. All increasing: [1,2,3,4,5] → return 4
  4. All decreasing: [5,4,3,2,1] → return 0
  5. Constant prices: [5,5,5,5] → return 0

  6. Mixed: [7,1,5,3,6,4] → return 7

  7. Wrong logic: Trying to find single best buy/sell pair
  8. Missing edge case: Not handling empty array
  9. Wrong index: Off-by-one errors in loop
  10. Negative profit: Not using max(0, ...) to ignore losses
  11. Complex solution: Using DP when greedy is sufficient

Tags

Array, Greedy, Dynamic Programming, Medium

Key Takeaways

  • Define state: what subproblem does dp[i] (or dp[i][j]) represent?
  • Recurrence: how does the answer build from smaller indices?
  • Base cases first; optimize space if only prior row/layer is needed.

References

Template Reference