Project 4: Hannah and Otto Solution

The purpose of this assignment is to give you some practice designing and implementing an algorithm based on the dynamic programming algorithm design strategy.

The Problem Hannah and her friend Otto like palindromes.1 In fact, they like palindromes so much that, when they see a word that is not a palindrome, they each wonder about how to insert letters into that word to make it into a palindrome. They are willing to insert characters that the beginning, the end, or the middle of any word. That’s why Hannah and Otto want a program that will read a list of words and compute the shortest palindrome that can be formed by inserting letters into each word. Here are some examples.

Word
Letters to insert
Resulting palindrome
roger
E, G
rEGoger
mimi
I
Imimi
benny
E,B,Y
YbennEBy

Your task Your job is to write a program that uses dynamic programming to solve this problem. Specifically, you should write a C++ program that does precisely these things:

    1. Read a word, consisting of a sequence of lowercase letters, from standard input.

    2. Compute the length of the shortest palindrome that can be formed by inserting characters into that word. Compute the palindrome itself, using uppercase for the inserted letters. If there is more than one shortest palindrome—which happens often—compute the one that is alphabetically first.2

    3. Print a line containing the length and the palindrome itself, separated by a space.

    4. Return to Step 1 above to read and process another instance. Terminate when standard input reaches end-of-file.

Here are a few example inputs with the correct output for each one.

Sample Input 1
Sample Output 1
leonardo
13
leoDRAnardoEL
michelangelo
21
OLmicheGNAlangeHCIMlo
donatello
14
donatellETANoD
raphael
11
LEraHphaRel
splinter
15
RETNILPsplinter
april
9 LIRPapril
casey
9 YESAcasey
shredder
10
shredderHS
kraang
10
GNkraaRKng

An additional much longer sample input, along with its correct corresponding output, is available on the course website.


1You will probably recall that a palindrome is a string that reads the same both forward and backward.
    • Specifically, use the ASCII ordering, in which capital letters come before lowercase letters.


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Some hints

    1. Focus first on finding the length of the shortest palindrome, rather on computing the shortest palindrome itself. When you get this part, not only will you have already earned substantial partial credit, but you should also have a much better idea about how to compute the palindromes themselves.

    2. A correct dynamic programming solution to this problem should take (n2) time. For comparison, my solution takes about 1.5 seconds on my laptop to solve the 62,886 instances in project4-2.in. Though the specific speeds are not too important, if your program takes more than a few seconds on this input, something is wrong.

    3. Because your solution should use dynamic programming, the first step is to write a recurrence that describes the solution you want to compute, in terms of the solutions to smaller subproblems. Since this

is a string problem with a single input S[0; : : : ; n 1], one natural place to start is to think about all of the substrings of S. To define a substring, we need to know the position of its first character and its last character, which we’ll call i and j respectively. Based on that idea, we can define p(i; j) this way:


Let p(i; j) denote the length of the shortest palindrome that can be formed by inserting into S[i; : : : ; j].


Then to solve our original problem, we simply need to compute p(0; n    1).

    4. Next, we need to figure out how the p values are related to each other. That is: Given values for i and j, can we write an equation for p(i; j) in terms of other p values?

In this case we are interested in the shortest palindrome that can be formed from S[i; : : : ; j]. Let’s call that string Xi;j. We know, from the definition of the word palindrome, that the first and last letters of Xi;j are the same—otherwise, Xi;j would not be a palindrome. We also know that every character in Xi;j comes from one of two places: It could be a character originally from S (these are shown in lowercase in our examples) or it could have been inserted to complete the palindrome (which we show in uppercase). These two possibilities, applied to the first and last characters of Xi;j, give us four cases.

case
first character of Xij
last character of Xij
1
original
original
2
original
inserted
3
inserted
original
4
inserted
inserted

Let’s think about each of these cases, in order.

Case 1: This case refers to the situation in which both the first and last characters of Xij are originals. If that’s true, then we know that Xi;j looks like this:

Xi;j = S[i]Xi+1;j  1S[j]

This is useful, because it begins to connect the solution for (i; j) to the solution to a smaller subprob-lem, specifically to (i + 1; j 1). However, this case only applies when S[i] = S[j], because otherwise we would not be getting a palindrome.

Case 2: Next, suppose that the first character of Xi;j is an original, and the last one was inserted to match it. If that’s true, then we know that Xi;j looks like this:

Xi;j = S[i]Xi+1;jS[i]:

Again, this gives us a connection to a smaller subproblem.


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Case 3: The last character of Xi;j is an original, and the first one was inserted to match it. This is just like Case 1, but in reverse:

Xi;j = S[j]Xi;j  1S[j]

Case 4: Both the first and last characters of Xij were inserted. This case never happens in shortest palindromes, because if it did, we could get a shorter palindrome by deleting the first and last characters. Therefore, we can completely ignore this case.

That leaves three possibilities. How do we know which one forms the correct palindrome? Since we want the shortest palindrome, the answer is to try each one, and keep the one that leads to the shortest string.

All of this is enough to write our recurrence for p(i; j). We take the general forms for Xij from above, and add up the length of each of those strings;
p(i; j) =
(min f
p(i + 1; j) + 2; p(i; j  1) + 2

g
if S[i] = S[j]
:

min
p(i + 1; j  1) + 2; p(i + 1; j) + 2; p(i; j  1) + 2

if S[i] = S[j]


f

g

6


The +2 terms come from that fact that we are adding two additional characters—one at the front and one at the back—to the smaller string (either Xi+1;j 1, Xi+1;j, or Xi;j 1) that we expand to get Xij. It turns out, though, that if Case 1 applies, that it always will be the shortest length. That allows us to simplify the recurrence a bit, by leaving Cases 2 and 3 out of the top line:

p(i; j) =
(min p(i + 1; j) + 2; p(i; j  1) + 2

if S[i] = S[j]

p(i + 1; j  1) + 2

if S[i] = S[j]

f
g
6

This simplified version has a box around it because that formula is the most important thing in this docu-ment—a correct solution to Project 3 will come from turning that recurrence into an implemented algo-rithm.

    5. Now that we have a recurrence, there are a few issues left to resolve.

        (a) We need two loops to fill in the dynamic programming table, one for i and one for j. What are the correct upper and lower bounds for each loop? Should they be increasing or decreasing? What size should the table be?

        (b) What is the base case (or base cases) for p?

        (c) Given the completed DP table, how can you compute the correct shortest palindrome? (Alterna-tively, can you build the correct palindrome along the way?)

These details are left for you to resolve, but feel free to ask for help.

Notes    A few possibly helpful comments:

For calibration purposes, my solution has 66 lines of code. Yours may be longer, but if you find yourself writing huge amounts of code, something has probably gone wrong.

Keep in mind that you are expected to submit your own original work for this problem. Accessing reference material for C++ in general is fine and encouraged. However, you are strongly discouraged from conducting web searches for code specific to palindromes.

What to Submit You should submit, using the department’s dropbox website, a single C++ source file con-taining all of the code for your program. We will compile this program using this command line:

g++ -Wall -std=c++11 yourfile.cpp



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