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Find the sum of contiguous subarray within a one-dimensional array of numbers which has the largest sum. State: Sum(i) = Sum(i-1) + a[i] if Sum(i-1) + a[i]>0 =0 otherwise Solution: Max ( Sum(i) ) i Є [0,n-1] Given a rod of length n inches and an array of prices that contains prices of all pieces of size smaller than n Determine the maximum value obtainable by cutting up the rod and selling the pieces. State: cutRod(n)=max(price[i] + cutRod(n-i-1)) for all i in {0,..,n-1}. Solution cutRod(L) Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack. You cannot break an item, either pick the complete item, or don’t pick it (0-1 property). State: C(i,j) = max( C(i-1,j-w[i]) + v[i] , C(i-1,j) ) (Note : Check for j-w[i]>=0) Solution: C(n,W) Find the length of the longest subsequence of a given sequence such that all elements of the subsequence are sorted in increasing order. State: S(i) = max(S(k) + 1) a[k]<a[i] Solution: Max (S(i)) where k=0,..,i-1 and Given two strings of size m, n and set of operations replace (R), insert (I) and delete (D) all at equal(or different) cost. Find minimum number of edits (operations) required to convert one string into another. State E(i, j) = min( E(i-1, j) + D, • E(i, j-1) + I, E(i-1, j-1) + R if S[i]!=S[j] E(i-1,j-1) + 0 if S[i]=S[j] ) Solution E(n,m) • Given a string composed of lowercase alphabets 'a'-'z' and '?'. replace every '?' in S with a random lowercase letter ('a' – 'z') – uniform probability Find expected number of palindromic substrings in S Test Case 1 aaa 6 Test Case 2 z?? 3.11 There are 26^2 = 676 equally likely possibilities for the letters used to replace the question marks. Here are all possible outcomes: The string "zzz" has 6 palindromic substrings. Each of the 25 strings "zaz", "zbz", ..., "zyz" has 4 palindromic substrings. Each of the 25 strings "zza", "zzb", ..., "zzy" has 4 palindromic substrings. Each of the 25 strings "zaa", "zbb", ..., "zyy" has 4 palindromic substrings. Each of the remaining 600 possible strings only has the 3 single-letter palindromic substrings. The expected number of palindromic substrings can be computed simply as the average over all 676 possible cases. Hence, the correct return value is (6 + 75*4 + 600*3) / 676.