Dynamic Programming is a powerful technique used to solve complex problems by breaking them down into smaller sub-problems. It is a bottom-up approach in which a problem is solved by solving its sub-problems first and then combining their solutions. The concept of dynamic programming was developed by Richard Bellman in the 1950s.
Dynamic programming is a useful tool for solving problems with overlapping sub-problems. It works by breaking down a problem into smaller sub-problems and then solving them one by one.
This technique is more efficient than other methods, such as brute force or divide-and-conquer, because it only needs to solve each sub-problem once. The solutions to the sub-problems are stored in a table, and the final solution is obtained by combining the solutions to the sub-problems.
Dynamic programming can be used to solve a wide variety of problems, such as finding the shortest path between two points, determining the optimum way to cut wood, or scheduling jobs on a computer. It is also used in bioinformatics, graphics, economics, and other fields.
def fibonacci_dp(n):
mem = [0] * (n + 1)
mem[1] = 1
mem[2] = 1
for i in range(3, n+1):
mem[i] = mem[i-1] + mem[i-2]
return mem[n]
print(fibonacci_dp(8)) # prints 13
This approach is much faster than the previous.
one, as it only requires a single loop and has a linear time complexity.
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