## 4.2 Sorting and Searching

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Join Stack Overflow to learn, share knowledge, and build your career. A linear search looks down a list, one item at a time, without jumping. In complexity terms this is an O n search - the time taken to search the list gets bigger at the same rate as the list does. A binary search is when you start with the middle of a sorted list, and see whether that's greater than or less than the value you're looking for, which determines whether the value is in the first or second half of the list.

Jump to the half way through the sublist, and compare again etc. This is pretty much how humans typically look up a word in a dictionary although we use better heuristics, obviously - if you're looking for "cat" you don't start off at "M". In complexity terms this is an O log n search - the number of search operations grows more slowly than the list does, because you're halving the "search space" with each operation. As an example, suppose you were looking for U in an A-Z list of letters index ; we're looking for the value at index Compare list[12] 'M' with 'U': Smaller, look further on.

Think of it as two different ways of finding your way in serial and binary search in data structure program phonebook. A linear search is starting at the beginning, reading every name until you find what you're looking for. A binary search, on the other hand, is when you open the book usually in the middlelook at the name on top of the page, and decide if the name you're looking for is bigger or smaller than the one you're looking for.

If the name you're serial and binary search in data structure program for is bigger, then you continue searching the upper part of the book in this very fashion. A linear search works serial and binary search in data structure program looking at each element in a list of data until it either finds the target or reaches the end. This results in O n performance on a given list. A binary search comes with the prerequisite that the data must be sorted.

We can leverage this information to decrease the number of items we need to look at to find our serial and binary search in data structure program. We know that if we look at serial and binary search in data structure program random item in the data let's say the middle item and that item is greater than our target, then all items to the right of that item will also be greater than our target.

This means that we only need to look at the left part of the data. Basically, each time we search for the target and miss, we can eliminate half of the remaining items. This gives us a nice O log n time complexity. So you should never sort data just to perform a single binary search later on. But if you will be performing many searches say at least O log n searchesit may be worthwhile to sort the data so that you can perform binary searches. You might also consider other data structures such as a hash table in such situations.

A linear search starts at the beginning of a list of values, and checks 1 by 1 in order for the result you are looking for. A binary search starts in the middle of a sorted array, and determines which side if any serial and binary search in data structure program value you are looking for is on. That "half" of the array is then searched again in the same fashion, dividing the results in half by two each time.

Make sure to deliberate about whether the win of the quicker binary search is worth the cost of keeping the list sorted to be able to use the binary search. Open the book at the half way point and look at the page.

Ask yourself, should this person be to the left or to the right. Repeat this procedure until you find the page where the entry should be and then either apply the same process to columns, or just search linearly along the names on the page as before. Linear search also referred to as sequential search looks at each element in sequence from the start to see if the desired element is present in the data structure.

When the amount of data is small, this search is fast. Its easy but work needed is in proportion to the amount of data to be searched. Doubling the number of elements will double the time to search if the desired element is not present. Binary search is efficient for larger array.

In this we check the middle element. If the value is bigger that what we are looking for, then look in the first half;otherwise,look in the second half.

Repeat this until the desired item is found. The table must be sorted for binary search. It eliminates half the data at each iteration. If we have elements to search, binary search takes about 10 steps, linear search steps. Binary Search finds the middle element of the array. Checks that middle value is greater or lower than the search value. If it is smaller, it gets the left side of the array and finds the middle element of that serial and binary search in data structure program.

If it is greater, gets the right part of the array. It loops the operation until it finds the searched value. Or if there is no value in the array finishes the search. Also you can see serial and binary search in data structure program information about Linear and Binary Search here: Thank you for your interest in this question.

Bill the Lizard k Please read the appropriate sections in your course material which, has hopefully, been selected and prepared by your instructor s. Failing that, a general wikipedia, c2 or google search can answer may of these sort of questions. A linear search would ask: The binary search would ask: Binary search requires the input data to be sorted; linear search doesn't Binary search requires an ordering comparison; linear search only requires equality comparisons Binary search has complexity O log n ; linear search has complexity O n as discussed earlier Binary search requires random access to the data; linear search only requires sequential access this can be very important - it means a linear search can stream data of arbitrary size.

Jon Skeet k A better analogy would be the "guess my number between 1 and game" with responses of "you got it", "too high", or "too low". The dictionary analogy seems fine to me, though it's a better match for interpolation search.

Dictionary analogy is better for me Apr 4 '14 at With dictionary approach, the take away is sorting. So the importantly you must make sure the data is sorted before the binary search is started.

If not you will be jumping all over the oceans without finding the value: If you do not mark the already tried ones, this can become worse. So always do the sorting. Some Java based binary search implementation is found here digizol. Yes, the requirement that the input data is sorted is my first bullet point Mia Clarke 6, 3 41 I would like to add one difference- For linear search values need not to be sorted. But for binary search the values must serial and binary search in data structure program in sorted order.

Pick a random name "Lastname, Firstname" and look it up in your phonebook. Time both methods and report back! Prabu - Incorrect - Best case would be 1, worstwith an average of May 4 '09 at Linear Search looks through items until it finds the searched value.

O n Example Python Code: O logn Example Python Code: Stack Overflow works best with JavaScript enabled.

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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:.