A serial search algorithm

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Notice that a language with slightly different semantics would "solve" this problem, or at least mitigate the problem: All you need is to have range constraints on integers, thus serial and binary search in java allowing the numbers to overflow unnoticed. This is the standard out of the box behaviour of Ada, for example. Couldn't the problem have been avoided by using Java's BigInteger in the first place?

For the search algorithm there is constant time overhead in doing so, but this algorithm would only run about 30 iterations at the array size where the programming error becomes a factor. No matter which preferred remedy -- bounds checking or infinite precision[1] -- is selected, the error is ultimately Joshua Bloch's. It's amusing that he assigns blame to Programming Pearls when that book includes the passage:. How would you prove that the program is free of run-time errors such as division by zero, word overflow, variables out of declared range, [ It serial and binary search in java that Bloch thought that question was rhetorical rather than something that programmers should seriously consider.

Arrays in Java are indexed using ints, not serial and binary search in java else. So to avoid the error, you'd have to use BigIntegers for all the intermediate computations, and then convert the BigInteger to an int whenever you are about to dereference it. I guess that would be optimizing for correctness serial and binary search in java the expense of writability.

On the other hand it would be rather silly to bring in BigInteger when Java arrays have a size limit of their own. It serial and binary search in java kind of grates me that Joshua Bloch's blog carefully avoids taking any kind of responsibility for the error in the Java code. We could, for instance, begin with cleaning up our language by no longer calling a bug a bug but by calling it an error.

It is much more honest because it squarely puts the blame where it belongs, viz. The animistic metaphor of the bug that maliciously sneaked in while the programmer was not looking is intellectually dishonest as it disguises that the error is the programmer's own creation. The nice thing of this simple change of vocabulary is that it has such a profound effect: I'm sorry if I didn't make this clear in the original posting: The program contained a bug.

I take full responsibility for it in the JDK. So I was going to ask you why you didn't use the standard library binary search, until I realized you were the one writing it. Welcome to LtU, Mr. Thanks for "owning up" to the bug: I'll try not to beat you up with Dijkstra quotes: I'm actually more interested in your response on a different issue, namely this remark pointed out by "crux":.

This line also caught my eye in the original post, as it seemed to me unfairly critical of the idea of proofs about programs sorry, Frank: Perhaps you just serial and binary search in java to call attention to the difficulties in applying proof techniques to real programs, but I suspect that many people reading the Google blog wouldn't make fine distinctions in this area. These sorts of problems relating to the difference between Java ints and mathematical integers are just the sort of thing that automated proof systems are likely to catch.

I have a vague sense that Coq in particular has some "standard libraries" relating machine-style and mathematical integers which might be used to catch such a bug though perhaps not in the context of C programs. Can anyone more familiar with Coq verify this? I no longer have access to the Coq book that may left me with this impression.

Anyway, my reaction to the quoted remark was that it's unfair to criticize proofs about programs broadly on the basis of this particualar kind of failed human-readable proof. Of course, a good machine-checked proof requires a lot of foundation in specification, etc. Indeed I was trying to call attention to the difficulties in applying proof techniques to real programs, but mainly I was tipping my hat to Donald Knuth, who said in "Beware of bugs in the above code; I have only proved it correct, not tried it" [Personal communication to Peter van Emde Boas].

Like Kunth, I believe that formal proofs are not sufficient to guarantee the "correctness" of a program, because serial and binary search in java are no guarantees that the model against which the serial and binary search in java was proved mirrors reality.

Therefore, I do think it's critical that you test programs even if you've proven them correct. Also I think it's critical that you get other smart people to read them. And yes, I believe that you should use languages that eliminate entire classes of bugs. As I'm sure you're aware, Java does this to some degree, even though it doesn't eliminate the possibility of arithmetic overflow. Simply put, I believe that you should do everything in your power to ensure that your programs are correct.

There is no silver bullet. Agilists may say that testing alone is sufficient; it isn't. Formalists may say that proofs alone are sufficient; they aren't.

I'm a firm believer in the efficacy of testing, and of formal methods, but I think they work best together. You need to write your proof at the "level" of the programming language, otherwise the proof might not reflect the real semantics.

This is why the "algebra of programming" approach discussed here many times is so intersting and important. Since algebraic manipulation of code is such a desried goal, functional programming raises so much interest even though not serial and binary search in java here thinks it is the holy grail. You need proofs that are mechanically verifiable, since proofs can contain errors just as programs can.

Indeed, many programmers prefer tests over proofs because they have more faith in tests - since they know how often their "proofs" are no no good they learned how to write bad proofs in school - their grades proved it! They simply know that a proof written by them on paper can be junk and they won't know it without outside help, while a serial and binary search in java test case is easy to spot.

This is why people need more knowledge on writing mechanized proofs or why such techniques should enter mainstream compilers. Using proof assistants or theorem provers is too much of a black art. Of course, if proofs are done by program transformation see item 1 they are also much easier to verify.

As I read my messages again, I realize that they are harsher than Serial and binary search in java intended. But I do feel that the words we use to talk about programming are important. I wonder about the points made regarding defensive programming.

Many practices of defensive programming -- e. What will it take for these PL niceties to gain broader acceptance, similar to how garbage collection is now something expected by most programmers today?

Many languages don't limit the size of their integers arbitrarily: Even in those which do, the range of an integer is usually in the same order as the size of the address space. Considering the size of an array element it's extremely rare to need a binary search among a billion of bytesthe array size will overflow sooner than integers used to count them. Java limits array sizes to 32 bits, and OCaml on bit platforms to 22 bits. These are more serious problems, and unfixable by adjusting a line of arithmetic.

No; an int on bit platforms is 31 bits in size. However, O'Caml also has int32 for when that's critical, the downside being that int32 is a boxed type. Of course, O'Caml also has arbitrarily-sized nums. But for the binary search problem, you may want to see this post on statically fixed array dimensions in O'Caml using phantom types. This is really exactly the wrong attitude to have. It puts the blame in the wrong place, and perpetuates the very thing you criticize.

After all, in ten years, maybe it won't be "integers" but rather "rationals" consider ZUIs or "XML" or "URLs" or something like that which serial and binary search in java not implemented the way you want it. No language is going to provide everything you need, with the performance profile you need, in its standard library, and what's indispensible for one programmer is irrelevant to another.

That's why we have programming languages which let you compose things rather than just serial and binary search in java. Furthermore, programming languages don't "have integers". If you need integers, it is your job to code them or pull them in from some library. Blaming the language is disingenuous, especially when it comes to C or Java, which have a trillion freely available libraries.

Even if you can blame the standard libraries for not attending to a ubiquitous need, who is the one who decides to use the standard libraries? What serial and binary search in java only provide is special syntax for arrays of a certain size.

It might be the case that there is no way to code up larger arrays without a performance hit larger than you would get by building them in, but that is a different matter and goes to expressivity. BTW, I take this attitude not because I think it's a crime to use ints where integers are required, but rather because the claim that "ints should be integers" is simply not constructive in a context where you can define whatever type you like.

If you can level that criticism at int, purely because of how it's named and how other people use it and what they may or not prescribe, then it becomes admissible to level the same sort of criticism at other elements of the language.

But that sort of criticism is not intrinsic to the language at all. The former reflects rather on particular users, and the latter goes against the basic tenet which languages like Ruby, however, flagrantly and, I think, very wrongly flout that a thing is not defined by what you name it. When language designs involve decisions that introduce discontinuity problems, that is something that can reasonably be blamed on the language. When these discontinuity problems penetrate deeply into the 'standard' libraries, that is something that can reasonably be blamed on serial and binary search in java language.

As you say, we use programming languages which let you compose things. Wherever there is a composition 'gotcha' safety issue or performance issue or security issue or arbitrary limitation against composition, you have serial and binary search in java fair target for criticism. The use of fixed-width integers called 'int', and their use throughout the standard libraries, is such a case.

Don't be so quick to blame the user when it is the language-designer who sets precedents, influences culture, and decides the initial libraries and composability profile for the language. On bit platforms where it is possible to allocate multi-gigabyte arrays, this starts becoming a serious limitation.

The fact that these data structures happen to match with types is almost a coincidence. My types are about energy management and memory deallocation. It's nice to keep in mind that arrya indexes and array sizes are not the same thing in all languages. In some languages a[10] means that the indexes are Personally I think there are two problems. One is that people treat ints as if they were integers, much in the same way that some people less nowadays, I guess treat floats and doubles as if they were reals.

The second is that I suspect the proof that Bentley serial and binary search in java was probably a proof about an imaginary program of an imaginary programming language. The truth is that you cannot strictly prove anything about C programs or Java programs because they don't have an axiomatization!

And even if they did it would be — to borrow the standards groups' jargon — "informative" not "normative".

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Searching a list of values is a common task. An application program might retrieve a student record, bank account record, credit record, or any other type of record using a search algorithm. Some of the most common search algorithms are serial search , binary search and search by hashing. The tool for comparing the performance between the different algorithms is called run-time analysis.

Here, we present search by hashing , and discuss the performance of this method. But first, we present a simple search method, the serial search and its run-time analysis. In a serial search, we step through an array or list one item at a time looking for a desired item.

The search stops when the item is found or when the search has examined each item without success. This technique is probably the easiest to implement and is applicable to many situations. The running-time of serial search is easy to analyze. We will count the number of operations required by the algorithm, rather than measuring the actual time. For searching an array, a common approach is to count one operation each time that the algorithm accesses an element of the array.

Usually, when we discuss running times, we consider the "hardest" inputs, for example, a search that requires the algorithm to access the largest number of array elements. This is called the worst-case running time. For serial search , the worst-case running time occurs when the desired item is not in the array. In this case, the algorithm accesses every element. Thus, for an array of n elements, the worst-case time for serial search requires n array accesses.

An alternative to worst-case running time, is the average-case running time, which is obtained by averaging the different running times for all inputs of a particular kind. For example, if our array contains ten elements, then if we are searching for the target that occurs at the first location, then there is just one array access. If we are searching for the target that occurs at the second location, then there are two array accesses.

And so on through the final target, which requires ten accesses. The average of all these searches is:. Both worst-case time and average-case time are O n , but nevertheless, the average case is about half the time of the worst-case.

A third way to measure running time is called best-case , and as the name suggests, it takes the most optimistic view. The best-case running time is defined as the smallest of all the running times on inputs of a particular size. For serial search, the best-case occurs when the target is found at the front of the array, requiring only one array access. Thus, for an array of n elements, the best-case time for serial search requires just 1 array access. Unless the best-case behavior occurs with high probability, the best-case running time is generally not used during analysis.

Hashing has a worst-case behavior that is linear for finding a target, but with some care, hashing can be dramatically fast in the average-case. Hashing also makes it easy to add and delete elements from the collection that is being searched. To be specific, suppose the information about each student is an object of the following form, with the student ID stored in the key field:.

We call each of these objects a record. Of course, there might be other information in each student record. If student IDs are all in the range The record for student ID k can be retrieved immediately since we know it is in data[k].

What, however, if the student IDs do not form a neat range like Suppose that we only know that there will be a hundred or fewer and that they will be distributed in the range We could then use an array with 10, components, but that seems wasteful since only a small fraction of the array will be used.

It appears that we have to store the records in an array with elements and to use a serial search through this array whenever we wish to find a particular student ID. If we are clever, we can store the records in a relatively small array and still retrieve students by ID much faster than we could by serial search. In this case, we can store the records in an array called data with only components.

We'll store the record with student ID k at location:. The record for student ID is stored in array component data[7]. This general technique is called hashing. Each record requires a unique value called its key. In our example the student ID is the key, but other, more complex keys are sometimes used.

A function called the hash function , maps keys to array indices. Suppose we name our hash function hash. If a record has a key of k , then we will try to store that record at location data[hash k ]. Using the hash function to compute the correct array index is called hashing the key to an array index. The hash function must be chosen so that its return value is always a valid index for the array.

Given this hash function and keys that are multiples of , every key produces a different index when it was hashed. Thus, hash is a perfect hash function. Unfortunately, a perfect hash function cannot always be found. Suppose we no longer have a student ID , but we have instead. The record with student ID will be stored in data[3] as before, but where will student ID be placed?

So there are now two different records that belong in data[3]. This situation is known as a collision. In this case, we could redefine our hash function to avoid the collision, but in practice you do not know the exact numbers that will occur as keys, and therefore, you cannot design a hash function that is guaranteed to be free of collisions. Typically, though, you do know an upper bound on how many keys there will be.

The usual approach is to use an array size that is larger than needed. The extra array positions make the collisions less likely. A good hash function will distribute the keys uniformly throughout the locations of the array.

If the array indices range from 0 to 99, then you might use the following hash function to produce an array index for a record with a given key:. One way to resolve collisions is to place the colliding record in another location that is still open. This storage algorithm is called open-addressing. Open addressing requires that the array be initialized so that the program can test if an array position already contains a record.

With this method of resolving collisions, we still must decide how to choose the locations to search for an open position when a collision occurs There are 2 main ways to do so. There is a problem with linear probing. When several different keys hash to the same location, the result is a cluster of elements, one after another. As the table approaches its capacity, these clusters tend to merge into larger and larger clusters.

This is the problem of clustering. Clustering makes insertions take longer because the insert function must step all the way through a cluster to find a vacant location. Searches require more time for the same reason. The most common technique to avoid clustering is called double hashing. With double hashing, we could return to our starting position before we have examined every available location.

An easy way to avoid this problem is to make sure that the array size is relatively prime with respect to the value returned by hash2 in other words, these two numbers must not have any common factor, apart from 1.

Two possible implementations are:. In open addressing, each array element can hold just one entry. When the array is full, no more records can be added to the table. One possible solution is to resize the array and rehash all the entries. This would require a careful choice of new size and probably require each entry to have a new hash value computed. A better approach is to use a different collision resolution method called chained hashing , or simply chaining , in which each component of the hash table's array can hold more than one entry.

We still hash the key of each entry, but upon collision, we simply place the new entry in its proper array component along with other entries that happened to hash to the same array index. The most common way to implement chaining is to have each array element be a linked list. The nodes in a particular linked list will each have a key that hashes to the same value. The worst-case for hashing occurs when every key hashes to the same array index. In this case, we may end up searching through all the records to find the target just as in serial search.

The average-case performance of hashing is complex, particularly if deletions are allowed. We will give three different formulas for the three versions of hashing: The three formulas depend on how many records are in the table.

When the table has many records, there are many collisions and the average time for a search is longer. We define the load factor alpha as follows:. For open address hashing, each array element holds at most one item, so the load factor can never exceed 1.

But with chaining, each array position can hold many records, and the load factor might be higher than 1. In the following table, we give formulas for the average-case performance of the three hashing schemes along with numerical examples.

You are given a template implementation of a hash table using open addressing with linear probing. Here is the source code:.