4.2 Sorting and Searching
This section under construction.
The sorting problem is to rearrange a set of items in ascending order. One reason that it is so useful is that it is much easier to search for something in a sorted list than an unsorted one. In this section, we will consider in detail two classical algorithms for sorting and searching, along with several applications where their efficiency plays a critical role.
Binary search.
In the game of "twenty questions", your task is to guess the value of a hidden number that is one of the N integers between 0 and N1. (For simplicity, we will assume that N is a power of two.) Each time that you make a guess, you are told whether your guess is too high or too low. An effective strategy is to maintain an interval that contains the hidden number, guess the number in the middle of the interval, and then use the answer to halve the interval size. Questions.java implements this strategy, which is an example of the general problemsolving method known as binary search. Correctness proof.
First, we have to convince ourselves that the method is correct:
that it always leads us to the hidden number. We do so by establishing
the following facts:
 The interval always contains the hidden number.
 The interval sizes are the powers of two, decreasing from N.
The first of these facts is enforced by the code; the second follows by noting that if the interval size (hilo) is a power of two, then the next interval size is (hilo)/2, which is the next smaller power of two. These facts are the basis of an induction proof that the method operates as intended. Eventually, the interval size becomes 1, so we are guaranteed to find the number.
 Running time analysis. Since the size of the interval decreases by a factor of 2 at each iteration (and the base case is reached when N = 1), the running time of binary search is lg N.
 Linearlogarithm chasm. The alternative to using binary search is to guess 0, then 1, then 2, then 3, and so forth, until hitting the hidden number. We refer to such an algorithm as a bruteforce algorithm: it seems to get the job done, but without much regard to the cost (which might prevent it from actually getting the job done for large problems). In this case, the running time of the bruteforce algorithm is sensitive to the input value, but could be as much as N and has expected value N/2 if the input value is chosen at random. Meanwhile, binary search is guaranteed to use no more than lg N steps.
 Binary representation. If you look back to Program 1.3.7, you will recognize that binary search is nearly the same computation as converting a number to binary! Each guess determines one bit of the answer. In our example, the information that the number is between 0 and 127 says that the number of bits in its binary representation is 7, the answer to the first question (is the number less than 64?) tells us the value of the leading bit, the answer to the second question tells us the value of the next bit, and so forth. For example, if the number is 77, the sequence of answers no yes yes no no yes no immediately yields 1001101, the binary representation of 77.
 Inverting a function.
As an example of the utility of binary search in scientific computing,
we revisit a problem that we consider the problem of inverting an increasing
function. To fix ideas, we refer to the Gaussian distribution Φ when describing
the method. Given a value y, our task is to find a value x such that
Φ(x) = y.
In this situation, we use real numbers as the endpoints of our
interval, not integers, but we use the same essential method as for guessing a hidden
integer: we halve the size of the interval at
each step, keeping x in the interval, until the
interval is sufficiently small that we know
the value of x to within a desired precision δ.
We start with
an interval (lo, hi) known to contain x and use the following recursive strategy:
 Compute m = lo + (hi  lo) / 2
 Base case: If (hi  lo) is less than δ, then returm m as an estimate of x
 Recursive step: otherwise, test whether Φ(m) > y. If so look for x in (lo, m); if not look for x in (m, hi)
The key to this method is the idea that the function is increasing  for any values a and b, knowing that Φ(a) < Φ(b) tells us that a < b, and vice versa. In this context, binary search is often called bisection search because we bisect the interval at each stage.
 Binary search in a sorted array. During much of the last century people would use a publication known as a phone book to look up a person's phone number. Entries appears in order, sorted by a key that identifies it (the person's name) n both cases). A bruteforce solution would be to start at the beginning, examine each entry one at a time, and continue until you find the name. No one uses that method: instead, you open the book to some interior page and look for the name on that page. If it is there, you are done; otherwise, you eliminate either the part of the book before the current page or the part of the book after the current page from consideration and repeat.
 Exception filter. We now use binary search to solve the existence problem: is a given key in a sorted database of keys? For example, when checking the spelling of a word, you need only know whether your word is in the dictionary and are not interested in the definition. In a computer search, we keep the information in an array, sorted in order of the key. The binary search code in BinarySearch.java differs from our other applications in two details. First, the file size N need not be a power of two. Second, it has to allow the possibility that the item sought is not in the array. The client program implements an exception filter: it reads a sorted list of strings from a file (which we refer to as the whitelist) and an arbitrary sequence of strings from standard input and prints those in the sequence that are not in the whitelist.
Insertion sort.
Insertion sort is a bruteforce sorting algorithm that is based on a simple method that people often use to arrange hands of playing cards. Consider the cards one at a time and insert each into its proper place among those already considered (keeping them sorted). The following code mimics this process in a Java method that sorts strings in an array:The outer for loop sorts the first i entries in the array; the inner for loop can complete the sort by putting a[i] into its proper position in the array.
public static void sort(String[] a) { int N = a.length; for (int i = 1; i < N; i++) for (int j = i; j > 0; j) if (a[j1].compareTo(a[j]) > 0) exch(a, j, j1); else break; }
 Mathematical analysis.
The inner loop of the insertion sort code is within
a double for loop, which suggests that the running time is quadratic, but we
cannot immediately draw this conclusion because of the break.
 Best case. When the input array is already in sorted order, the inner for loop amounts to nothing more than a comparison (to learn that a[j1] is less than a[j]) and the break, so the total running time is linear.
 Worst case. When the input is reverse sorted, the inner loop fully completes without a break, so the frequency of execution of the instructions in the inner loop is 1 + 2 + ... + N1 ~ N^2 and the running time is quadratic.
 Average case. When the input is randomly ordered To understand the performance of insertion sort for randomly ordered, we expect that each new element to be inserted is equally likely to fall into any position, so that element will move halfway to the left on average. Thus, we expect the running time to be 1/2 + 2/2 + ... + (N1)/2 ~ N^2 / 2.
 Sorting other types of data. We want to be able to sort all types of data, not just strings. For sorting objects in an array, we need only assume that we can compare two elements to see whether the first is bigger than, smaller than, or equal to the second. Java provides the Comparable interface for precisely this purpose. Simply put, a class that implements the Comparable interface promises to implement a method compareTo() for objects of its type so that that a.compareTo(b) returns a negative integer if a is less than b, a positive integer if a is greater than b, and 0 if a is equal to b. The precise meanings of less than, greater than, and equal to are up to the data type, though implementations that do not respect the natural laws of mathematics surrounding these concepts will yield unpredictable results. With this convention, Insertion.java implements insertion sort so that it sorts arrays of Comparable objects.
 Empirical analysis. Program InsertionTest.java tests our hypothesis that insertion sort is quadratic for randomly ordered files. It relies on the helper data type Stopwatch.java.
 Sensitivity to input. Note that InsertionTest.java takes a commandline parameter M and runs M experiments for each array size, not just one. One reason for doing so is that the running time of insertion sort is sensitive to its input values. It is not correct to flatly predict that the running time of insertion sort will be quadratic, because your application might involve input for which the running time is linear.
Mergesort.
To develop a faster sorting method, we use a divideandconquer approach to algorithm design that every programmer needs to understand. This nomenclature refers to the idea that one way to solve a problem is to divide it into independent parts, conquer them independently, and then use the solutions for the parts to develop a solution for the full problem. To sort an array with this strategy, we divide it into two halves, sort the two halves independently, and then merge the results to sort the full array. This method is known as mergesort. To sort a[lo, hi), we use the following recursive strategy: base case: If the subarray size is 0 or 1, it is already sorted.
 recursive step: Otherwise, compute m = lo + (hi  lo)/2, sort (recursively) the two subarrays a[lo, m) and a[m, hi), and merge them to produce a sorted result.
Merge.java is an implementation. As usual, the easiest way to understand the merge process is to study a trace of the contents of the array during the merge.
 Mathematical analysis.
The inner loop of mergesort is centered on the auxiliary array. The
two for loops involve N iterations (and creating the array takes
time proportional to N), so the frequency of execution of the instructions in
the inner
loop is proportional to the sum of the subarray sizes for all calls to the
recursive function.
The value of this quantity emerges when we arrange the calls
on levels according to their size.
On the first level, we have 1 call for size N, on the second level,
we have 2 calls for size N/2, on the third level, we have 4 calls for size N/4, and
so forth, down to the last level with N/2 calls of size 2.
There are precisely lg N levels, giving the grand total N lg N for the frequency of
execution of the instructions in the inner loop of mergesort.
This equation justifies a hypothesis that the running time of mergesort is
linearithmic.
 Quadraticlinearithmic chasm. The difference between N^2 and N lg N makes a huge difference in practical applications. Understanding the enormousness of this difference is another critical step to understanding the importance of the design and analysis of algorithms. For a great many important computational problems, a speedup from quadratic to linearithmic makes the difference between being able to solve a problem involving a huge amount of data and not being able to effectively address it at all.
 Divideandconquer algorithms. The same basic approach is effective for many important problems, as you will learn if you take a course on algorithm design.
 Reduction to sorting. A problem A reduces to a problem B if we can use a solution to B to solve A. Designing a new divideandconquer algorithm from scratch is sometimes akin to solving a puzzle that requires some experience and ingenuity, so you may not feel confident that you can do so at first. But it is often the case that a simpler approach is effective: given a new problem that lends itself to a quadratic bruteforce solution, ask yourself how you would solve it if the data were sorted in some way. For example, consider the problem of determining whether the elements in an array are all different. This problem reduces to sorting because we can sort the array, the make a linear pass through the sorted array to check whether any entry is equal to the next (if not, the elements are all different.)
Frequency counts.
FrequencyCount.java reads a sequence of strings from standard input and then prints a table of the distinct values found and the number of times each was found, in decreasing order of the frequencies. We accomplish this by two sorts. Computing the frequencies.
Our first step is to sort the strings on standard
input. In this case, we are not so much interested in the fact that the strings are
put into sorted order, but in the fact that sorting brings equal strings together.
If the input is
to be or not to be to
be be not or to to to
 Sorting the frequencies. Next, we sort the Counter objects. We can do so in client code without any special arrangements because Counter implements the Comparable interface.
 Zipf's law. The application highlighted in FrequencyCount is elementary linguistic analysis: which words appear most frequently in a text? A phenomenon known as Zipf's law says that the frequency of the ith most frequent word in a text of M distinct words is proportional to 1/i.
Longest repeated substring.
Another important computational task that reduces to sorting is the problem of finding the longest repeated substring in a given string. This problem is simple to state and has many important applications, including computerassisted music analysis, cryptography, and data compression. Think briefly about how you might solve it. Could you find the longest repeated substring in a string that has millions of characters? Program LRS.java is a clever solution that uses suffix sorting.Quicksort.
Quicksort is a divideandconquer method for sorting. It works by partitioning an array of elements into two parts, then sorting the parts independently. As we shall see, the precise position of the partition depends on the initial order of the elements in the input file. The crux of the method is the partitioning process, which rearranges the array to make the following three conditions hold: The element a[i] is in its final place in the array for i.
 None of the elements a[left], ..., a[i1] is greater than a[i].
 None of the elements in a[i+1], ..., a[right] is less than a[i].
We achieve a complete sort by partitioning, then recursively applying the method to the subfiles.
We use the following general strategy to implement partitioning. First, we arbitrarily choose a[right] to be the partitioning element  the one that will go into its final position. Next, we scan from the left end of the array until we find an element greater than the partitioning element, and we scan from the right end of the array until we find an element less than the partitioning element. The two elements that stopped the scans are obviously out of place in the final partitioned array, so we exchange them. Continuing in this way, we ensure that no array elements to the left of the left index are greater than the partitioning element, and no array elements to the right of the right index are less than the partitioning element, as depicted in the following diagram.
When the scan indices cross, all that we need to do to complete the partitioning process is to exchange a[right] with the leftmost element of the right subfile (the element pointed to by the left index i).
Program QuickSort.java implements
this algorithm.
Q + A
Q. Why do we need to go to such lengths to prove a program correct?
A. To spare ourselves considerable pain. Binary search is a notable example. For example, you now understand binary search; a classic programming exercise is to write a version that uses a while loop instead of recursion. Try solving Exercise 4.2.2 without looking back at the code in the book. In a famous experiment, J. Bentley once asked several professional programmers to do so, and most of their solutions were not correct. According to Knuth, the first binary search algorithm was published in 1946, but the first published binary search without bugs did not appear until 1962.
Q. Are there implementations for sorting and searching in the Java libarary?
A. Yes. The Java library java.util.Arrays contains the methods Arrays.sort() and Arrays.binarySearch() that implement mergesort and binary search for Comparable types and a sorting implementation for primitive types based on a version of the quicksort algorithm, which is faster than mergesort and also sorts an array in place (without using any extra space). SystemSort.java illustrates how to use Arrays.sort().
Q. So why not just use them?
A. Feel free to do so. As with many topics we have studied, you will be able to use such tools more effectively if you understand the background behind them.
Q. Why doesn't the Java library use a randomized version of quicksort?
A. Good question. At the very least, the library should cutoff to some guaranteed N log N algorithm if it "realizes" it is in trouble. Perhaps to avoid side effects. Programmers may want their libraries to be deterministic for debugging. But the library only uses quicksort for primitive types when stability is not an issue, so the programmer probably wouldn't notice the randomness, except in running time.
Exercises
 Develop an implementation of Questions.java that takes the maximum number N as commandline input. Prove that your implementation is correct.
 Add code to Insertion to produce the trace given in the text.
 Add code to Merge to produce the trace given in the text.

Give traces of insertion sort and mergesort in the style of the traces in the text,
for the input
it was the best of times it was
 Describe why it is desirable to use immutable keys with binary search.

Explain why we use lo + (hi  lo) / 2 to compute the index midway
between lo and hi instead of using
(lo + hi) / 2. This can happen if the array contains
around a billion elements.
Solution. The latter fails when lo + hi overflows an int. Joshua Bloch reports that Sun's Java 1.5 implementation contained this bug. Here is another correct version using bitshifting.
int mid = lo + (hi  lo) / 2; int mid = (lo + hi) >>> 1;
 Modify BinarySearch.java so that if the search key is in the array, it returns the smallest index i for which a[i] is equal to key, and otherwise, it returns i, where i is the smallest index such that a[i] is greater than key.
 Describe what happens if you apply binary search to an unorderd array. Why shouldn't you check whether the array is sorted before each call to binary search? Could you check that the elements binary search examines are in ascending order?
 Write a program Dedup.java that reads strings from standard input and prints them on standard output with all duplicates removed (in sorted order).
 Find the frequency distribution of words in your favorite book. Does it obey Zipf's law?
 Find the longest repeated substring in your favorite book.
 Add code to LRS.java to make it print indices in the original string where the longest repeated substring occurs.
 Find a pathological input for which LRS.java runs in quadratic time (or worse).
 Show that binary search in a sorted array is logarithmic as long as it eliminates at least a constant fraction of the array at each step.
 Modify BinarySearch.java so that if the search key is not in the array, it returns the largest index i for which a[i] is smaller than key (or 1 if no such index exists).
 Analyze mergesort mathematically when N is a power of 2, as we did for binary search.
 Analyze mergesort for the case when N is not a power of two.
Creative Exercises
This list of exercises is intended to give you experience in developing fast solutions to typical problems. Think about using binary search, mergesort, or devising your own divideandconquer algorithm. Implement and test your algorithm.
 Median.
Add to StdStats.java a method
median() that computes in linearithmic time the median of a sequence
of N integers.
Hint: reduce to sorting.
 Mode.
Add to StdStats.java a method
mode() that computes in linearithmic time the mode (value that
occurs most frequently) of a sequence of N integers.
Hint: reduce to sorting.
 Integer sort.
Write a lineartime filter
IntegerSort.java that takes
from from standard input a sequence of integers that are between 0 and 99
and prints the same integers in sorted order on standard output.
For example, presented with the input sequence
98 2 3 1 0 0 0 3 98 98 2 2 2 0 0 0 2
0 0 0 0 0 0 1 2 2 2 2 2 3 3 98 98 98
 Floor and ceiling. Given a sorted array of Comparable items, write methods floor() and ceil() that returns the index of the largest (or smallest) item not larger (or smaller) than an argument item in logarithmic time.
 Closest pair. Given an array of N real numbers, write a static method to find in linearithmic time the pair of integers that are closest in value.
 Farthest pair. Given an array of N real numbers, write a static method to find in linear time the pair of integers that are farthest apart in value.
 Two sum. Write a static method that takes as argument an array of N int values and determines in linearithmic time whether any two of them sum to 0.
 Three sum. Write a static method that takes as argument an array of N int values and determines whether any three of them sum to 0. Your program should run in time proportional to N^2 log N. Extra credit: develop a program that solves the problem in quadratic time.
 Majority. An element is a majority if it appears more than N/2 times. Write a static method that takes an array of N strings as argument and identifies a majority (if it exists) in linear time.
 Common element.
Write a static method that takes as argument three arrays of strings,
determines whether there is any string common to all three arrays, and if so,
returns one such string. The running time of your method should be
linearithmic in the total number of strings.
Hint: sort each of the three lists, then describe how to do a "3way" merge.
 Prefixfree codes. In data compression, a set of strings is prefix free if no string is a prefix of another. For example, the set of strings 01, 10, 0010, 1010 is not prefix free because 10 is a prefix of 1010. Write a program that reads in a set of strings from standard input and determines whether the set is prefix free.
 Longest common substring.
Write a static method that finds the longest common substring of two given
strings s and t.
Hint: Suffix sort each string. Then merge the two sorted suffixes together.
 Longest repeated, nonoverlapping string. Modify LRS.java to find the longest repeated substring that does not overlap.
 Partitioning.
Write a static method that sorts a Comparable
array that is known to have at most two different values.
Hint: Maintain two pointers, one starting at the left end and moving right, the other starting at the right end and moving left. Maintain the invariant that all elements to the left of the left pointer are equal to the smaller of the two values and all elements to the right of the right pointer are equal to the larger of the two values.
 Dutch national flag problem. Write a static method that sorts a Comparable array that is known to have at most three different values. (Edsger Dijkstra named this the Dutchnationalflag problem because the result is three "stripes" of values like the three stripes in the flag.)
 Quicksort.
Write a recursive program
QuickSort.java
that sorts an array of randomly ordered distinct Comparable
elements.
Hint: Use a method like the one described in the previous exercise. First, partition the array into a left part with all elements less than v, followed by v, followed by a right part with all elements greater than v. Then, recursively sort the two parts.
Extra credit: Modify your method (if necessary) to work properly when the elements are not necessarily distinct.
 Reverse domain. Write a program to read in a list of domain names from standard input, and print the reverse domain names in sorted order. For example, the reverse domain of cs.princeton.edu is edu.princeton.cs. This computation is useful for web log analysis. To do so, create a data type Domain.java that implements the Comparable interface, using reverse domain name order.
 Local minimum in an array.
Given an array of N real numbers, write a static method to find in logarithmic
time a local minimum (an index i such that a[i1] < a[i] < a[i+1]).
Solution. Query middle value a[n/2], and two neighbors a[n/2  1] and a[n/2 + 1]. If a[n/2] is local minimum, stop; otherwise search in half with smaller neighbor.
 Discrete distribution. Design a fast algorithm to repeatedly generate numbers from the discrete distribution: Given an array a[] of nonnegative real numbers that sum to 1, the goal is to return index i with probability a[i]. Form an array s[] of cumulated sums such that s[i] is the sum of the first i elements of a[]. Now, generate a random real number r between 0 and 1, and use binary search to return the index i for which s[i] ≤ s[i+1].
 Rhyming words.
Write a program Rhymer.java that
tabulates a list that you can use to find words that rhyme. Use the following approach:
 Read in a dictionary of words into an array of strings.
 Reverse the letters in each word (confound becomes dnuofnoc, for example).
 Sort the resulting array.
 Reverse the letters in each word back to their original order.
For example, confound is adjacent to words such as astound and surround in the resulting list.
Web Exercises
 Union of intervals. Given N intervals on the real line, determine the length of their union in O(N log N) time. For example the union of the four intervals [1, 3], [2, 4.5], [6, 9], and [7, 8] is 6.5.
 Coffee can problem. (David Gries). Suppose you have a coffee can which contains an unknown number of black beans and an unknown number of white beans. Repeat the following process until exactly one bean remains: Select two beans from the can at random. If they are both the same color, throw them both out, but insert another black bean. If they are different colors, throw the black one away, but return the white one. Prove that this process terminates with exactly one bean left. What can you deduce about the color of the last bean as a function of the initial number of black and white beans? Hint: find a useful invariant maintained by the process.
 Spam campaign. To initiate an illegal spam campaign, you have a list of email addresses from various domains (the part of the email address that follows the @ symbol). To better forge the return addresses, you want to send the email from another user at the same domain. For example, you might want to forge an email from nobody@princeton.edu to somebody@princeton.edu. How would you process the email list to make this an efficient task?
 Order statistics. Given an array of N elements, not necessarily in ascending order, devised an algorithm to find the kth largest one. It should run in O(N) time on random inputs.
 Kendall's tau distance. Given two permutations, Kendall's tau distance is the number of pairs out of position. "Bubblesort metric." Give an O(N log N) algorithm to compute the Kendall tau distance between two permutations of size N. Useful in topk lists, social choice and voting theory, comparing genes using expression profiles, and ranking search engine results.
 Antipodal points. Given N points on a circle, centered at the origin, design an algorithm that determines whether there are two points that are antipodal, i.e., the line connecting the two points goes through the origin. Your algorithm should run in time proportional to N log N.
 Antipodal points. Repeat the previous question, but assume the points are given in clockwise order. Your algorithm should run in time proportional to N.
 Identity. Given an array a[] of N distinct integers (positive or negative) in ascending order. Devise an algorithm to find an index i such that a[i] = i if such an index exists. Hint: binary search.
 L1 norm. There are N circuit elements in the plane. You need to run a special wire (parallel to the xaxis) across the circuit. Each circuit element must be connected to the special wire. Where should you put the special wire? Hint: median minimizes L1 norm.
 Finding common elements. Given two arrays of N 64bit integers, design an algorithm to print out all elements that appear in both lists. The output should be in sorted order. Your algorithm should run in N log N. Hint: mergesort, mergesort, merge. Remark: not possible to do better than N log N in comparison based model.
 Finding common elements. Repeat the above exercise but assume the first array has M integers and the second has N integers where M is much less than N. Give an algorithm that runs in N log M time. Hint: sort and binary search.
 Anagrams. Design a O(N log N) algorithm to read in a list of words and print out all anagrams. For example, the strings "comedian" and "demoniac" are anagrams of each other. Assume there are N words and each word contains at most 20 letters. Designing a O(N^2) algorithms should not be too difficult, but getting it down to O(N log N) requires some cleverness.
 Pattern recognition. Given a list of N points in the plane, find all subset of 3 or more points that are collinear.
 Pattern recognition. Given a list of N points in the plane in general position (no three are collinear), find a new point p that is not collinear with any pair of the N original points.
 Search in a sorted, rotated list. Given a sorted list of N integers that has been rotated an unknown number of positions, e.g., 15 36 1 7 12 13 14, design an O(log N) algorithm to determine if a given integer is in the list.
 Counting inversions. Each user ranks N songs in order of preference. Given a preference list, find the user with the closest preferences. Measure "closest" according to the number of inversions. Devise an N log N algorithm for the problem.
 Throwing cats from an Nstory building. Suppose that you have an N story building and a bunch of cats. Suppose also that a cat dies if it is thrown off floor F or higher, and lives otherwise. Devise a strategy to determine the floor F, while killing O(log N) cats.
 Throwing cats from a building. Repeat the previous exercise, but devise a strategy that kills O(log F) cats. Hint: repeated doubling and binary search.
 Throwing two cats from an Nstory building. Repeat the previous question, but now assume you only have two cats. Now your goal is to minimize the number of throws. Devise a strategy to determine F that involves throwing cats O(√N) times (before killing them both). This application might occur in practice if search hits (cat surviving fall) are much cheaper than misses (cat dying).
 Throwing two cats from a building. Repeat the previous question, but only throw O(√F) cats. Reference: ???.
 Nearly sorted.
Given an array of N elements, each which is at most
k positions from its target position, devise an algorithm that
sorts in O(N log k) time.
Solution 1: divide the file into N/k pieces of size k, and sort each piece in O(k log k) time, say using mergesort. Note that this preserves the property that no element is more than k elements out of position. Now, merge each blocks of k elements with the block to its left.
Solution 2: insert the first k elements into a binary heap. Insert the next element from the array into the heap, and delete the minimum element from the heap. Repeat.
 Merging k sorted lists. Suppose you have k sorted lists with a total of N elements. Give an O(N log k) algorithm to produce a sorted list of all N elements.
 Longest common reverse complemented substring. Given two DNA strings, find the longest substring that appears in one, and whose reverse WatsonCrick complement appears in the other. Two strings s and t are reverse complements if t is the reverse of s except with the following substitutions A<>T, C<>G. For example ATTTCGG and CCGAAAT are reverse complements of each other. Hint: suffix sort.
 Circular string linearization. Plasmids contain DNA in a circular molecule instead of a linear one. To facilitate search in a database of DNA strings, we need a place to break it up to form a linear string. A natural choice is the place that leaves the lexicographically smallest string. Devise an algorithm to compute this canonical representation of the circular string Hint: suffix sort.
 Find all matches. Given a text string, find all matches of the query string. Hint: combine suffix sorting and binary search.
 Longest repeated substring with less memory. Instead of using an array of substrings where suffixes[i] refers to the ith sorted suffix, maintain an array of integers so that index[i] refers to the offset of the ith sorted suffix. To compare the substrings represented by a = index[i] and b = index[j], compare the character s.charAt(a) against s.charAt(b), s.charAt(a+1) against s.charAt(b+1), and so forth. How much memory do you save? Is your program faster?
 Idle time. Suppose that a parallel machine processes n jobs. Job j is processed from s_{j} to t_{j}. Given the list of start and finish times, find the largest interval where the machine is idle. Find the largest interval where the machine is nonidle.
 Local minimum of a matrix. Given an NbyN array a of N^{2} distinct integers, design an O(N) algorithm to find a local minimum: an pair of indices i and j such that a[i][j] < a[i+1][j], a[i][j] < a[i][j+1], a[i][j] < a[i1][j], and a[i][j] < a[i][j1].
 Monotone 2d array. Give an nbyn array of elements such that each row is in ascending order and each column is in ascending order, devise an O(n) algorithm to determine if a given element x in the array. You may assume all elements in the nbyn array are distinct.
 2D maxima. Given a set of n points in the plane, point (xi, yi) dominates (xj, yj) if xi > xj and yi > yj. A maxima is a point that is not dominated by any other point in the set. Devise an O(n log n) algorithm to find all maxima. Application: on xaxis is space efficiency, on yaxis is time efficiency. Maxima are useful algorithms. Hint: sort in ascending order according to xcoordinate; scan from right to left, recording the highest yvalue seen so far, and mark these as maxima.
 Compound words. Read in a list of words from standard input, and print out all twoword compound words. If after, thought, and afterthought are in the list, then afterthought is a compound word. Note: the components in the compound word need not have the same length.
 Smith's rule. The following problem arises in supply chain management. You have a bunch of jobs to schedule on a single machine. (Give example.) Job j requires p[j] units of processing time. Job j has a positive weight w[j] which represents its relative importance  think of it as the inventory cost of storing the raw materials for job j for 1 unit of time. If job j finishes being processed at time t, then it costs t * w[j] dollars. The goal is to sequence the jobs so as to minimize the sum of the weighted completion times of each job. Write a program SmithsRule.java that reads in a command line parameter N and a list of N jobs specified by their processing time p[j] and their weight w[j], and output an optimal sequence in which to process their jobs. Hint: Use Smith's rule: schedule the jobs in order of their ratio of processing time to weight. This greedy rule turns out to be optimal.
 Sum of four primes.
The Goldbach conjecture says that all positive even integers
greater than 2 can be expressed as the sum of two primes.
Given an input parameter N (odd or even),
express N as the sum of four primes (not necessarily distinct)
or report that it is impossible to do so. To make your algorithm
fast for large N, do the following steps:
 Compute all primes less than N using the Sieve of Eratosthenes.
 Tabulate a list of sums of two primes.
 Sort the list.
 Check if there are two numbers in the list that sum to N. If so, print out the corresponding four primes.
 Typing monkeys and power laws. (Micahel Mitzenmacher) Suppose that a typing monkey creates random words by appending each of 26 possible lettter with probability p to the current word, and finishes the word with probability 1  26p. Write a program to estimate the frequency spectrum of the words produced.
 Typing monkeys and power laws.
Repeat the previous exercise, but assume that the letters az occur
proportional to the following probabilities, which are typical of
English text.
CHAR FREQ CHAR FREQ CHAR FREQ CHAR FREQ CHAR FREQ A 8.04 G 1.96 L 4.14 Q 0.11 V 0.99 B 1.54 H 5.49 M 2.53 R 6.12 W 1.92 C 3.06 I 7.26 N 7.09 S 6.54 X 0.19 D 3.99 J 0.16 O 7.60 T 9.25 Y 1.73 E 12.51 K 0.67 P 2.00 U 2.71 Z 0.09 F 2.30
 Binary search.
Justify why the following modified version of binarySearch()
works. Prove that if the key is in the array, it correctly
returns the smallest index i such that a[i] = key;
if the key is not in the array, it returns i where
i is the smallest index such that a[i] > key.
// precondition array a in ascending order public static int binarySearch(long[] a, long key) { int bot = 1; int top = a.length; while (top  bot > 1) { int mid = bot + (top  bot) / 2; if (key > a[mid]) bot = mid; else top = mid; } if (a[top] == key) return top; else return top  1; }
 Range search.
Given a database of all tolls collected in NJ road system in 2006,
devise a scheme to answer queries of the form:
extract sum of all tolls collected in a given time interval.
Use a Toll data type that implements
the Comparable interface, where the key is
the time that the toll was collected.
Hint: sort by time, compute a cumulative sum of the first i tolls, then use binary search to find the desired interval.
 Longest repeated substrings. Modify LRS.java to find all longest repeated substrings.
 Nonrecursive binary search.
Write a nonrecurisve version of binary search.
public static int binarySearch(long[] a, long key) { int bot = 0; int top = a.length  1; while (bot <= top) { int mid = bot + (top  bot) / 2; if (key < a[mid]) top = mid  1; else if (key > a[mid]) bot = mid + 1; else return mid; } return 1; }
 Two sum to x.
Given a sorted list of N integers and a target integer x,
determine in O(N) time whether there
are any two that sum to exactly x.
Hint: maintain an index lo = 0 and hi = N1 and compute a[lo] + a[hi]. If the sum equals x, you are done; if the sum is less than x, decrement hi; if the sum is greater than x, increment lo. Be careful if one (or more) of the integers are 0.
 Zero of a monotonic function.
Let f be a monotonically increasing function with f(0) < 0
and f(N) > 0. Find the smallest integer i such that f(i) > 0.
Devise an algorithm that makes O(log N) calls to f().
Hint: assuming we know N, maintaing an interval [lo, hi] such that f[lo] < 0 and f[hi] > 0 and apply binary search. If we don't know N, repeatedly compute f(1), f(2), f(4), f(8), f(16), and so on until you find a value of N such that f(N) > 0.
 Bitonic max. Let a[] be an array that starts out increasing, reaches a maximum, and then decreases. Design an O(log N) algorithm to find the index of the maximum value.
 Bitonic search. An array is bitonic if it is comprised of an increasing sequence of integers followed immediately by a decreasing sequence of integers. Given a bitonic array a of N distinct integers, describe how to determine whether a given integer is in the array in O(log N) steps. Hint: find the maximum, then binary search in each piece.
 Median in two sorted arrays. Given two sorted arrays of size N_{1} and N_{2}, find the median of all elements in O(log N) time where N = N_{1} + N_{2}. Hint: design a more general algorithm that finds the kth largest element for any k. Compute the median element in the large of the two lists and; throw away at least 1/4 of the elements and recur.
 Element distinctness. Give an array of N long integers, devise an O(N log N) algorithm to determine if any two are equal. Hint: sorting brings equal values together.
 Duplicate count. Give a sorted array of N elements, possibly with duplicates, find the index of the first and last occurrence of k in O(log N) time. Give a sorted array of N elements, possibly with duplicates, find the number of occurrences of element k in O(log N) time. Hint: modify binary search.
Scientific example of sorting.
Google display search results in descending order of "importance",
a spreadsheet displays columns sorted by a particular field,
Matlab sorts the real eigenvalues of a symmetric matrix in
descending order.
Sorting also arises as a critical subroutine in many applications
that appear to have nothing to do with sorting at all including:
data compression (see the BurrowsWheeler programming assignment),
computer graphics (convex hull, closest pair),
computational biology (longest common substring discussed
below), supply chain management (schedule jobs to minimize
weighted sum of completion times), combinatorial optimization (Kruskal's
algorithm), social choice and voting (Kendall's tau distance),
Historically, sorting was most important for commercial applications,
but sorting also plays a major role in the scientific computing
infrastructure.
NASA
and the fluids mechanics community use sorting to
study problems in rarefied flow; these collision detection
problems are especially
challenging since they involve ten of billions of particles and
can only be solved on supercomputers in parallel.
Similar sorting techniques are used in some fast Nbody simulation codes.
Another important scientific application of sorting is for load balancing
the processors of a parallel supercomputers.
Scientists rely on clever sorting algorithm to perform loadbalancing
on such systems.