Algorithm lecture Dynamic programming

Algorithm lecture Dynamic programming

• 1.
  Algorithms Dynamic Programming
• 2.
  Dynamic Programming • Astrategy 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
• 3.
  Dynamic Programming Example: LongestCommon 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
• 4.
  LCS Algorithm • Brute-forcealgorithm: 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
• 5.
  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); }
• 6.
  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; }
• 7.
  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 }
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  The End