Lecture 3 — Maximum Contiguous Subsequence Sum and The Substitution Method

1.1 Algorithm 1: Brute Force Immediate from the definition is an algorithm with O(n3) work and O(log n) span. This algorithm examines all possible combinations of subsequences and for each one of them, it computes the sum and takes the maximum. Note that every subsequence of s can be represented by a starting position i and an ending position j. We will use the shorthand si.. j to denote the subsequence 〈si , si+1, . . . , s j〉. For each subsequence i.. j, we can compute its sum by applying a plus reduce. This does O( j− i) work and O(log( j− i)) span. Furthermore, all the subsequences can be examined independently in parallel (using, e.g., tabulate). This leads the following bounds: †Lecture notes by Guy E Blelloch, Margaret Reid-Miller, and Kanat Tangwongsan.