Summary of "Buy and Sell Stock Problem and Pow(X,N) Power exponential Problem - Leetcode | DSA Series"

Summary of the Video:

Title: Buy and Sell Stock Problem and Pow(X,N) Power Exponential Problem - Leetcode | DSA Series

Main Concepts Covered:

  1. Binary Exponentiation (Power Calculation: xn)
    • Problem: Calculate x raised to the power n efficiently, where n can be a large integer (including negative values).
    • Naive Approach: Multiply x by itself n times → O(n) time complexity → inefficient for large n (up to ±2³¹).
    • Optimized Approach: Binary Exponentiation (Exponentiation by Squaring):
      • Convert the exponent n to its binary form.
      • Use the binary digits to determine which powers of x to multiply.
      • Square x repeatedly to get powers of 2 (x¹, x², x⁴, x⁸, ...).
      • Multiply only those powers corresponding to the set bits (1s) in the binary representation of n.
      • Time complexity reduces to O(log n).
    • Handling Negative Powers:
      • If n is negative, convert the problem to (1/x)|n|.
    • Corner Cases:
      • x0 = 1 (any x except 0).
      • 0n = 0 (for n > 0).
      • 1n = 1 (any n).
      • (-1)n = 1 if n even, -1 if n odd.
    • Pseudocode Outline:
      • Initialize answer = 1, binary_form = abs(n), and base = x (or 1/x if n < 0).
      • While binary_form > 0:
        • If the last bit of binary_form is 1, multiply answer by base.
        • Square base.
        • Right shift binary_form by 1 (divide by 2).
      • Return answer.
  2. Stock Buy and Sell Problem (Leetcode: Best Time to Buy and Sell Stock)
    • Problem: Given an array of stock prices where prices[i] is the price on day i, find the maximum profit by buying on one day and selling on a later day. If no profit is possible, return 0.
    • Key Constraints:
      • You must buy before you sell (no selling before buying).
      • Only one transaction allowed (one buy and one sell).
    • Naive Approach:
      • Find minimum and maximum prices → Not valid if max price comes before min price.
    • Optimized Approach:
      • Iterate through prices imagining each day as a potential selling day.
      • Keep track of the lowest price seen so far (best_buy).
      • For each day (selling day), calculate potential profit = current price - best_buy.
      • Update maximum profit if this potential profit is higher.
      • Update best_buy if current price is lower than the recorded best_buy.
    • Time Complexity: O(n), where n is the number of days/prices.
    • Example:
      • Prices = [7, 1, 5, 3, 6, 4]
      • Best buy = 7 initially
      • Update best_buy to 1 on day 1
      • Max profit updated to 4 on day 2 (5 - 1)
      • Max profit updated to 5 on day 4 (6 - 1)
      • Final max profit = 5
    • Edge Case: If prices always decrease, max profit = 0.

Summary of Methodologies:

Binary Exponentiation Algorithm (xn):

Stock Buy and Sell Algorithm:

Category ?

Educational

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