Algorithm lecture Dynamic programming

Algorithms
Dynamic Programming

Dynamic Programming
• A strategy for designing algorithms is
dynamic programming
– A technique, not an algorithm
(like divide & conquer)
– The word “programming” is historical and
predates computer programming
• Use when problem breaks down into
recurring small subproblems

Dynamic Programming Example:
Longest Common Subsequence
• Longest common subsequence (LCS)
problem:
– Given two sequences x[1..m] and y[1..n], find
the longest subsequence which occurs in both
– Ex: x = {A B C B D A B }, y = {B D C A B A}
– {B C} and {A A} are both subsequences of both
– Brute-force algorithm: For every subsequence
of x, check if it’s a subsequence of y

LCS Algorithm
• Brute-force algorithm: 2m subsequences of
x to check against n elements of y:O(n 2m)
• Another example is finding nth Fibonacci
value.
• It can be solved three ways-
– Reursion
– Memoized
– DP

Fibonacci(Recursion)
int fib(int n)
{
if(memo[n] != null){
return memo[n]
}
if (n <= 1)
return n;
return fib(n-1) + fib(n-2);
}

Fibonacci(Memoized)
int fib(int n)
{
if(memo[n] != null){
return memo[n]
}
if (n <= 1)
result = 1
else{
result = fib(n-1) + fib(n-2);
}
fibo[n] = result;
return result;
}

Fibonacci(DP)
function fib(n) {
res[0] = 0;
res[1] = 1;
for(let i = 2; i <= n; i ++) {
res[i] = fibo[i - 1] + fibo[i - 2]
}
return return
}

Algorithm lecture Dynamic programming

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