Acertain string-processing language offers a primitive operation which splits astring into two pieces. since this operation involves copying the original string, ittakes n units of time for a string of length n, regardless of the location of the cut. suppose, now, that you want to break a string into many pieces. the order inwhich the breaks are made can affect the total running time. for example, if youwant to cut a 20-character string at positions 3 and 10, then making the first cutat position 3 incurs a total cost of 20 + 17 = 37, while doing position 10 first hasa better cost of 20 + 10 = 30.give a dynamic programming algorithm that, given the locations of m cuts in astring of length n, finds the minimum cost of breaking the string into m+ 1pieces.

Some of your friends have gotten into the burgeoning field of time-series data mining, in which one looks for patterns in sequences of events that occur over time. purchases at stock exchanges--what’s being bought-- are one source of data with a natural ordering in time. given a long sequence s of such events, your friends want an efficient way to detect certain "patterns" in them--for example, they may want to know if the four events buy yahoo, buy ebay, buy yahoo, buy oracle occur in this sequence s, in order but not necessarily consecutively. they begin with a collection of possible events (e.g., the possible’ transactions) and a sequence s of n of these events. a given event may occur multiple times in s (e.g., yahoo stock may be bought many times in a single sequence s). we will say that a sequence s’ is a subsequence of s if there is a way to delete certain of the events from s so that the remaining events, in order, are equal to the sequence s’. so, for example, the sequence of four events above is a subsequence of the sequence buy amazon, buy yahoo, buy ebay, buy yahoo, buy yahoo, buy oracle their goal is to be able to dream up short sequences and quickly detect whether they are subsequences of s. so this is the problem they pose to you: give an algorithm that takes two sequences of even~s--s’ of length m and s of length n, each possibly containing an event more than once--and decides in time o(m n) whether s’ is a subsequence of s