Home > Articles > Programming > Algorithms

Like this article? We recommend

Like this article? We recommend

MSD string sort 

MSD string sort

To implement a general-purpose string sort, where strings are not necessarily all the same length, we consider the characters in left-to-right order. We know that strings that start with a should appear before strings that start with b, and so forth. The natural way to implement this idea is a recursive method known as most-significant-digit-first (MSD) string sort. We use key-indexed counting to sort the strings according to their first character, then (recursively) sort the subarrays corresponding to each character (excluding the first character, which we know to be the same for each string in each subarray). Like quicksort, MSD string sort partitions the array into subarrays that can be sorted independently to complete the job, but it partitions the array into one subarray for each possible value of the first character, instead of the two or three partitions in quicksort.

Figure 5.8 Sorting a card deck with MSD string sort

End-of-string convention.

We need to pay particular attention to reaching the ends of strings in MSD string sort. For a proper sort, we need the subarray for strings whose characters have all been examined to appear as the first subarray, and we do not want to recursively sort this subarray. To facilitate these two parts of the computation we use a private two-argument toChar() method to convert from an indexed string character to an array index that returns -1 if the specified character position is past the end of the string. Then, we just add 1 to each returned value, to get a nonnegative int that we can use to index count[]. This convention means that we have R+1 different possible character values at each string position: 0 to signify end of string, 1 for the first alphabet character, 2 for the second alphabet character, and so forth. Since key-indexed counting already needs one extra position, we use the code int count[] = new int[R+2]; to create the array of frequency counts (and set all of its values to 0). Note: Some languages, notably C and C++, have a built-in end-of-string convention, so our code needs to be adjusted accordingly for such languages.

Figure 5.9 Overview of MSD string sort

With these preparations, the implementation of MSD string sort, in Algorithm 5.2, requires very little new code. We add a test to cutoff to insertion sort for small subarrays (using a specialized insertion sort that we will consider later), and we add a loop to key-indexed counting to do the recursive calls. As summarized in the table at the bottom of this page, the values in the count[] array (after serving to count the frequencies, transform counts to indices, and distribute the data) give us precisely the information that we need to (recursively) sort the subarrays corresponding to each character value.

Figure 5.10 Typical candidate for MSD string sort

Specified alphabet. 

The cost of MSD string sort depends strongly on the number of possible characters in the alphabet. It is easy to modify our sort method to take an Alphabet as argument, to allow for improved efficiency in clients involving strings taken from relatively small alphabets. The following changes will do the job:

  • Save the alphabet in an instance variable alpha in the constructor.
  • Set R to alpha.R() in the constructor.
  • Replace s.charAt(d) with alpha.toIndex(s.charAt(d)) in charAt().

In our running examples, we use strings made up of lowercase letters. It is also easy to extend LSD string sort to provide this feature, but typically with much less impact on performance than for MSD string sort.

at completion of phase for dth character

value of count[r] is

r = 0

r = 1

r between 2 and R-1

r = R

r = R+1

count frequencies

0 (not used)

number of strings of length d

number of strings whose
dth character value is r-2

transform
counts to
indices

start index of subarray for strings of length d

start index of subarray for strings whose
dth character value is r-1

not
used

distribute

start index of subarray for strings
whose dth character value is r

not
used

1 + end index of subarray for strings of length d

1 + end index of subarray for strings whose dth character value is r-1

not
used

Interpretation of count[] values during MSD string sort

The code in Algorithm 5.2 is deceptively simple, masking a rather sophisticated computation. It is definitely worth your while to study the trace of the top level at the bottom of this page and the trace of recursive calls on the next page, to be sure that you understand the intricacies of the algorithm. This trace uses a cutoff-for-small-subarrays threshold value (M) of 0, so that you can see the sort to completion for this small example. The strings in this example are taken from Alphabet.LOWERCASE, with R = 26; bear in mind that typical applications might use Alphabet.EXTENDED.ASCII, with R = 256, or Alphabet.UNICODE, with R = 65536. For large alphabets, MSD string sort is so simple as to be dangerous—improperly used, it can consume outrageous amounts of time and space. Before considering performance characteristics in detail, we shall discuss three important issues (all of which we have considered before, in Chapter 2) that must be addressed in any application.

In our running examples, we use strings made up of lowercase letters. It is also easy to extend LSD string sort to provide this feature, but typically with much less impact on performance than for MSD string sort.

The code in Algorithm 5.2 is deceptively simple, masking a rather sophisticated computation. It is definitely worth your while to study the trace of the top level at the bottom of this page and the trace of recursive calls on the next page, to be sure that you understand the intricacies of the algorithm. This trace uses a cutoff-for-small-subarrays threshold value (M) of 0, so that you can see the sort to completion for this small example. The strings in this example are taken from Alphabet.LOWERCASE, with R = 26; bear in mind that typical applications might use Alphabet.EXTENDED_ASCII, with R = 256, or Alphabet.UNICODE16, with R = 65536. For large alphabets, MSD string sort is so simple as to be dangerous—improperly used, it can consume outrageous amounts of time and space. Before considering performance characteristics in detail, we shall discuss three important issues (all of which we have considered before, in Chapter 2) that must be addressed in any application.

Figure 5.11 Trace of MSD string sort: top level of sort(a, 0, 13, 0)

Small subarrays.

The basic idea behind MSD string sort is quite effective: in typical applications, the strings will be in order after examining only a few characters in the key. Put another way, the method quickly divides the array to be sorted into small subarrays. But this is a double-edged sword: we are certain to have to handle huge numbers of tiny subarrays, so we had better be sure that we handle them efficiently. Small subarrays are of critical importance in the performance of MSD string sort. We have seen this situation for other recursive sorts (quicksort and mergesort), but it is much more dramatic for MSD string sort. For example, suppose that you are sorting millions of ASCII strings (R = 256) that are all different, with no cutoff for small subarrays. Each string eventually finds its way to its own subarray, so you will sort millions of subarrays of size 1. But each such sort involves initializing the 258 entries of the count[] array to 0 and transforming them all to indices. This cost is likely to dominate the rest of the sort. With Unicode (R = 65536) the sort might be thousands of times slower. Indeed, many unsuspecting sort clients have seen their running times explode from minutes to hours on switching from ASCII to Unicode, for precisely this reason. Accordingly, the switch to insertion sort for small subarrays is a must for MSD string sort. To avoid the cost of reexamining characters that we know to be equal, we use the version of insertion sort given at the top of the page, which takes an extra argument d and assumes that the first d characters of all the strings to be sorted are known to be equal. The efficiency of this code depends on substring() being a constant-time operation. As with quicksort and mergesort, most of the benefit of this improvement is achieved with a small value of the cutoff, but the savings here are much more dramatic. The diagram at right shows the results of experiments where using a cutoff to insertion sort for subarrays of size 10 or less decreases the running time by a factor of 10 for a typical application.

Figure 5.12 Trace of recursive calls for MSD string sort (no cutoff for small subarrays, subarrays of size 0 and 1 omitted)

public static void sort(String[] a, int lo, int hi, int d)
{  // Sort from a[lo] to a[hi], starting at the dth character.
   for (int i = lo; i <= hi; i++)
      for (int j = i; j > lo && less(a[j], a[j-1], d); j--)
         exch(a, j, j-1);
}
private static boolean less(String v, String w, int d)
{  return v.substring(d).compareTo(w.substring(d)) < 0;  }

Insertion sort for strings whose first d characters are equal

Equal keys.

A second pitfall for MSD string sort is that it can be relatively slow for subarrays containing large numbers of equal keys. If a substring occurs sufficiently often that the cutoff for small subarrays does not apply, then a recursive call is needed for every character in all of the equal keys. Moreover, key-indexed counting is an inefficient way to determine that the characters are all equal: not only does each character need to be examined and each string moved, but all the counts have to be initialized, converted to indices, and so forth. Thus, the worst case for MSD string sorting is when all keys are equal. The same problem arises when large numbers of keys have long common prefixes, a situation often found in applications.

Figure 5.13 Effect of cutoff for small subarrays in MSD string sort

Extra space.

To do the partitioning, MSD uses two auxiliary arrays: the temporary array for distributing keys (aux[]) and the array that holds the counts that are transformed into partition indices (count[]). The aux[] array is of size N and can be created outside the recursive sort() method. This extra space can be eliminated by sacrificing stability (see Exercise 5.1.17), but it is often not a major concern in practical applications of MSD string sort. Space for the count[] array, on the other hand, can be an important issue (because it cannot be created outside the recursive sort() method) as addressed in Proposition D below.

Random string model.

To study the performance of MSD string sort, we use a random string model, where each string consists of (independently) random characters, with no bound on their length. Long equal keys are essentially ignored, because they are extremely unlikely. The behavior of MSD string sort in this model is similar to its behavior in a model where we consider random fixed-length keys and also to its performance for typical real data; in all three, MSD string sort tends to examine just a few characters at the beginning of each key, as we will see.

Figure 5.14 Characters examined by MSD string sort

Performance.

The running time of MSD string sort depends on the data. For compare-based methods, we were primarily concerned with the order of the keys; for MSD string sort, the order of the keys is immaterial, but we are concerned with the values of the keys.

  • For random inputs, MSD string sort examines just enough characters to distinguish among the keys, and the running time is sublinear in the number of characters in the data (it examines a small fraction of the input characters).
  • For nonrandom inputs, MSD string sort still could be sublinear but might need to examine more characters than in the random case, depending on the data. In particular, it has to examine all the characters in equla keys, so the running time is nearly linear in the number of characters in the data when significant numbers of equal keys are present.
  • In the worst case, MSD string sort examines all the characters in the keys, so the running time is linear in the number of characters in the data (like LSD string sort). A worst-case input is one with all strings equal.

Some applications involve distinct keys that are well-modeled by the random string model; others have significant numbers of equal keys or long common prefixes, so the sort time is closer to the worst case. Our license-plate-processing application, for example, can fall anywhere between these extremes: if our engineer takes an hour of data from a busy interstate, there will not be many duplicates and the random model will apply; for a week’s worth of data on a local road, there will be numerous duplicates and performance will be closer to the worst case.

Proposition C. To sort N random strings from an R-character alphabet, MSD string sort examines about N log R N characters, on average.

Proof sketch: We expect the subarrays to be all about the same size, so the recurrence CN = RCN/R + N approximately describes the performance, which leads to the stated result, generalizing our argument for quicksort in Chapter 2. Again, this description of the situation is not entirely accurate, because N/R is not necessarily an integer, and the subarrays are the same size only on the average (and because the number of characters in real keys is finite). These effects turn out to be less significant for MSD string sort than for standard quicksort, so the leading term of the running time is the solution to this recurrence. The detailed analysis that proves this fact is a classical example in the analysis of algorithms, first done by Knuth in the early 1970s.

As food for thought and to indicate why the proof is beyond the scope of this book, note that key length does not play a role. Indeed, the random-string model allows key length to approach infinity. There is a nonzero probability that two keys will match for any specified number of characters, but this probability is so small as to not play a role in our performance estimates.

As we have discussed, the number of characters examined is not the full story for MSD string sort. We also have to take into account the time and space required to count frequencies and turn the counts into indices.

Proposition D. MSD string sort uses between 8N 1 3R and ~7wN 1 3WR array accesses to sort N strings taken from an R-character alphabet, where w is the average string length.

Proof: Immediate from the code, Proposition A, and Proposition B. In the best case MSD sort uses just one pass; in the worst case, it performs like LSD string sort.

When N is small, the factor of R dominates. Though precise analysis of the total cost becomes difficult and complicated, you can estimate the effect of this cost just by considering small subarrays when keys are distinct. With no cutoff for small subarrays, each key appears in its own subarray, so NR array accesses are needed for just these subarrays. If we cut off to small subarrays of size M, we have about N/M subarrays of size M, so we are trading off NR/M array accesses with NM/4 compares, which tells us that we should choose M to be proportional to the square root of R.

Proposition D (continued). To sort N strings taken from an R-character alphabet, the amount of space needed by MSD string sort is proportional to R times the length of the longest string (plus N ), in the worst case.

Proof: The count[] array must be created within sort(), so the total amount of space needed is proportional to R times the depth of recursion (plus N for the auxiliary array). Precisely, the depth of the recursion is the length of the longest string that is a prefix of two or more of the strings to be sorted.

As just discussed, equal keys cause the depth of the recursion to be proportional to the length of the keys. The immediate practical lesson to be drawn from Proposition D is that it is quite possible for MSD string sort to run out of time or space when sorting long strings taken from large alphabets, particularly if long equal keys are to be expected. For example, with Alphabet.UNICODE and more than M equal 1,000-character strings, MSD.sort() would require space for over 65 million counters!

The main challenge in getting maximum efficiency from MSD string sort on keys that are long strings is to deal with lack of randomness in the data. Typically, keys may have long stretches of equal data, or parts of them might fall in only a narrow range. For example, an information-processing application for student data might have keys that include graduation year (4 bytes, but one of four different values), state names (perhaps 10 bytes, but one of 50 different values), and gender (1 byte with one of two given values), as well as a person’s name (more similar to random strings, but probably not short, with nonuniform letter distributions, and with trailing blanks in a fixed-length field). Restrictions like these lead to large numbers of empty subarrays during the MSD string sort. Next, we consider a graceful way to adapt to such situations.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020