You can add multiple items of the same type to the knapsack.

Computing each value will require time, and we need to sequentially compute such values.

Therefore, total running time is . Total space is .

The value of will contain the value of the optimal knapsack packing.

We can reconstruct the list of items in the optimal solution by maintaining and following “backpointers”

**2. Maximum Value Contiguous Subsequence.**

Given a sequence of real numbers ,

determine a contiguous subsequence for which the sum of elements in the subsequence is maximized.

**Solution: **Let** **denote the max sum over all windows ending at

It only takes linear time to run the program since it only need to solve sub-problems and each takes constnt time.

**3. Making Change**: You are given types of coin denominations of values (all integers).

Assume , so you can always make change for any amount of money C.

Give an algorithm which makes change for an amount of money C with as few coins as possible.

**Solution: **Let** **denote the min number of coins required to make changes for amount of money

It takes time because we are solving subproblems,

each of which requires minimization different terms. (Similar to integer knapsack problem)

**4. Longest Increasing Subsequence.** Given a sequence of n real numbers ,

determine a subsequence (not necessarily contiguous) of maximum length in which the values in the subsequence form a strictly increasing sequence.

**Solution: **Let** **denote the longest strictly increasing subsequence ending at position , therefore,

to find the solution overall, it has to find the maximum over all potential ending points of :

since the longest increasing subsequence could end anywhere.

The algorithm takes time to run because I have to sort subproblems and each of them takes time.

**5.Box Stacking:**

You are given a set of types of rectangular 3-D boxes, where the box has height , width and depth (all real numbers).

You want to create a stack of boxes which is as tall as possible,

but you can only stack a box on top of another box if the dimensions of the 2-D base of the lower box are each strictly larger than those of the 2-D base of the higher box.

Of course, you can rotate a box so that any side functions as its base.

It is also allowable to use multiple instances of the same type of box.

**Solution: **constraint : can only stack box on box if and

(without loss of generality, assume ).

First sort the base area by decreasing order:

then let denote the tallest box stack I could form with box on top, so

The algorithm takes time to run because I have to sort subproblems and each of them takes time.

At the end, the solution is found by taking the max of all potential top boxes of .

(Since we are not sure in the optimal solution, which box is on top)

**6. Building Bridges**.

Consider a 2-D map with a horizontal river passing through its center.

There are cities on the southern bank with x-coordinates and cities on the northern bank with x-coordinates .

You want to connect as many north-south pairs of cities as possible with bridges such that no two bridges cross.

When connecting cities, you are only allowed to connect the the ith city on the northern ban to the ith city on the southern bank.

**Solution: **consider the south bank city (below the river) ,

let denote the index of corresponding city on the north bank ( in the figure).

So we could compute the by sorting in time.

Now we want to find the longest increasing subsequence through , so it is identical to **problem 4.**

**7. Balanced Partition**:You have a set of n integers each in the range 0 . . .K.

Partition these integers into two subsets such that you minimize ,

where and denote the sums of the elements in each of the two subsets.

**Solution: **Use where and

therefore, if or

dynamic programming formula is:

this procedure takes time to compute.

To solve the original problem, let , and the purpose is to make close to 0 as possible.

what we want to find is:

**8. Edit Distance.**

Given two text strings A of length n and B of length m,

you want to transform A into B with a minimum number of operations of the following types:

delete a character from A, insert a character into A,

or change some character in A into a new character.

The minimal number of such operations required to transform A into B is called the edit distance between A and B.

**Solution: ** Use to denote the cost for inserting, deleting or replacing a letter.

The goal is to compute the minimum cost of translating A to B.

Let denote the minimum cost of translating into , then

It takes time to compute and final solution is stored in