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The Algorithm Design Manual: Chapter 3

3-1. A common problem for compilers and text editors is determining whether the parentheses in a string are balanced and properly nested. For example, the string ((())())() contains properly nested pairs of parentheses, which the strings )()( and ()) do not. Give an algorithm that returns true if a string contains properly nested and balanced parentheses, and false if otherwise. For full credit, identify the position of the first offending parenthesis if the string is not properly nested and balanced.

Solution:

3-3. We have seen how dynamic arrays enable arrays to grow while still achieving constant-time amortized performance. This problem concerns extending dynamic arrays to let them both grow and shrink on demand.
(a) Consider an underflow strategy that cuts the array size in half whenever the array falls below half full. Give an example sequence of insertions and deletions where this strategy gives a bad amortized cost.
(b) Then, give a better underflow strategy than that suggested above, one that achieves constant amortized cost per deletion.

Solution: 3-3. (a) Let’s assume that the array is current fulled at half. If we delete one element it will cut in half. If we now add a new element it have to expand.

(b) Instead of shrinking it by half its size I would shrink it only by a forth its size. So we have a buffer which avoids shrinking and expanding in too short periods.

3-4. Design a dictionary data structure in which search, insertion, and deletion can all be processed in O(1) time in the worst case. You may assume the set elements are integers drawn from a finite set 1, 2, .., n, and initialization can take O(n) time.

Solution: Luckily these are sets so there are no duplicates. Therefore we can use an array with size n. It takes O(n) to fill the array. Search (A[i]) can be done in O(1), also insertion (A[i] = v). To remove data we can just use a specific value like NUL which works also in O(1).

3-6. Describe how to modify any balanced tree data structure such that search, insert, delete, minimum, and maximum still take O(\log n) time each, but successor and predecessor now take O(1) time each. Which operations have to be modified to support this?

Solution: My idea is to take a search tree and use a double linked list for the predecessors and successors. You can see how this works in this picture:

Each node gets a pointer to its entry on the double linked list. Let’s see if it still works in O(\log n). Search, minimum and maximum haven’t really changed, so this works still fine. What about insert and delete? Assume that 3 isn’t inserted yet. We transverse the list up to node 2. We know that the node of 3 will be placed on the right. Now we have to check what happens with the double linked list.
Node 2 refers to list entry 2 which has a successor list entry 4. So we have to point 2 to 3 and 3 to 2 and 4 and we’re done. The same works for deletion. Successor and predecessor are available through the double linked list with O(1). And we’re done.

3-8. Design a data structure to support the following operations:
• insert(x,T) – Insert item x into the set T .
• delete(k,T) – Delete the kth smallest element from T.
• member(x,T) – Return true iff x ∈ T .
All operations must take O(\log n) time on an n-element set.

Solution: We use a basic binary search tree for this and add two counters which count the number of children nodes.

Here you can see how the counters in pink. Let’s look at 4. The pink 2 indicates that there are 2 nodes on the left, i.e. smaller than 4. And the pink 3 indicates that there are 3 nodes on the right, i.e. bigger than 4. What happens if we insert an item?
We do the standard insert traversal of a binary search tree with the difference that in each node we add a one to our counter because there will be one more node if we added our new item.

You can see what happened when we added 1. Besides adding the node we increased the left counter of 2 by one and the left counter by 4 by 1 because we traveled this way. So there’s no problem on inserting new nodes with O(\log n) complexity.
The next method is member which is basically search in a binary search tree which also works with a complexity of O(\log n).
The last one is delete(k, T) which removes the kth smallest item. Finally we can use our counters. The first counter indicates if our kth smallest item is on the left, the item or on the right.
Example 1: We want the 3th smallest item. We start at 4 and see that there are 3 items on the left, i.e. the 3th smallest item in its left children. Next we are at 2 and see that there are 1 on the left and 1 on the right, therefore we have to go right (1 left + item itself + 1 right item = 3). Now we have arrived 3 and there are no other child items therefore 3 is our 3th smallest item.

Example 2: We want the 4th smallest item. We start at 4 and see that there are 3 items to the left. Therefore the 4th smallest item is 4 itself.

This is basically some kind of binary search so it also works with O(\log n) complexity.

3-12. Suppose you are given an input set S of n numbers, and a black box that if given any sequence of real numbers and an integer k instantly and correctly answers whether there is a subset of input sequence whose sum is exactly k. Show how to use the black box O(n) times to find a subset of S that adds up to k.

Solution: The first time we enter our set S. If it returns yes we can continue otherwise it isn’t possible to form the sequence which sums up to k.
The next step is to test our Set without the first element. If the black box returns yes we can delete it from our set otherwise we know that it is needed. We do this for each element and our S shrinks to a set which sums up to k.

3-13. Let A[1..n] be an array of real numbers. Design an algorithm to perform any sequence of the following operations:
• Add(i,y) – Add the value y to the ith number.
• Partial-sum(i) – Return the sum of the first i numbers, i.e. \sum_{j=1}^{i} A[j].
There are no insertions or deletions; the only change is to the values of the numbers. Each operation should take O(\log n) steps. You may use one additional array of size n as a work space.

Solution: The general idea is to use the work space for the sums. Instead of using each i, I will only use log n of them to guarantee O(\log n) for each operation.

This function takes 1 + \log n = O(\log n) time.

Here you can see how this works on an example:

3-17. A Caesar shift (see Section 18.6 (page 641)) is a very simple class of ciphers for secret messages. Unfortunately, they can be broken using statistical properties of English. Develop a program capable of decrypting Caesar shifts of sufficiently long texts.

Solution:

3-24. What is the best data structure for maintaining URLs that have been visited by a Web crawler? Give an algorithm to test whether a given URL has already been visited, optimizing both space and time.

Solution: An easy way is to use a hash table for the domains and a hash table for paths.

3-28. You have an unordered array X of n integers. Find the array M containing n elements where Mi is the product of all integers in X except for Xi. You may not use division. You can use extra memory. (Hint: There are solutions faster than O(n^2).)

Solution: Here’s a algorithm which works in O(n \log n):
First we use our extra memory to store multiplications. We multiply \log n numbers together. This takes O(n \log n). Now we just need to find our number by using this pre-multiplications and one element of the original array. This takes for all M about O(n \log n) time. Therefore we have O(n \log n) total run time.
Example:

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  1. in solution (3-13), what if we have (2^8) elements, with your solution there will be log(2^8) workspace, every one has length (n/log(n)) = 256/8 = 32 element.

    Partial-sum(15), nearest workspace is the first one = workspace[0] – Sum(elements from index 16 till index 31), 16 steps > 8 steps(log n).

    correct me If I don’t understand your answer.

    • Yeah this solution doesn’t work. It gets worse as the array size gets bigger. I suggest you use a tree with all elements as leaf nodes, this way you are using ~2n nodes, which is under the constraint.

  2. Hello there.

    Great solutions – been very useful for me to compare while towing the same path.

    The solution to 3-8 appears to not work in certain scenarios. For example in degenerate trees of the worst case scenario. Insert/delete will be O(h).

    Could you please clarify if I am missing something?

    Thanks a lot for the great work too!