In programmers life algorithms and data structures is most important subject if they want to go out in the programming world and make some bucks. Today, We will see what they do and where they are used with simplest examples. This list is prepared keeping in mind their use in competitive programming and current development practices.
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Python dictionaries are implemented using hash tables. It is an array whose indexes are obtained using a hash function on the keys. The goal of a hash function is to distribute the keys evenly in the array. A good hash function minimizes the number of collisions e.g. Different keys having the same hash. Python does not have this kind of hash function. Chapter 12: Dictionary (Hash Tables). By a hash algorithm is the remainder after dividing this value by the hash table size. So, for example, Amy’s hash function returns values from 0 to 25. She divided by the table. Of their numeric value, structures can use one or more fields. Hash functions are only.
1. Sort Algorithms
Sorting is the most heavily studied concept in Computer Science. Idea is to arrange the items of a list in a specific order. Though every major programming language has built-in sorting libraries, it comes in handy if you know how they work. Depending upon requirement you may want to use any of these.
More importantly one should know when and where to use them. Some examples where you can find direct application of sorting techniques include:
2. Search Algorithms
Binary Search (in linear data structures)
Binary search is used to perform a very efficient search on sorted dataset. The time complexity is O(log2N). Idea is to repeatedly divide in half the portion of the list that could contain the item, until we narrow it down to one possible item. Some applications are:
Depth/Breadth First Search (in Graph data structures)
DFS and BFS are tree/graph traversing and searching data structures. We wouldn’t go deep into how DFS/BFS work but will see how they are different through following animation.
Applications:
3. Hashing
Hash lookup is currently the most widely used technique to find appropriate data by key or ID. We access data by its index. Previously we relied on Sorting+Binary Search to look for index whereas now we use hashing.
The data structure is referred as Hash-Map or Hash-Table or Dictionary that maps keys to values, efficiently. We can perform value lookups using keys. Idea is to use an appropriate hash function which does the key -> value mapping. Choosing a good hash function depends upon the scenario.
Applications:
4. Dynamic Programming
Dynamic programming (DP) is a method for solving a complex problem by breaking it down into simpler subproblems. We solve the subproblems, remember their results and using them we make our way to solve the complex problem, quickly.
*writes down “1+1+1+1+1+1+1+1 =” on a sheet of paper* What’s that equal to?
*counting* Eight! *writes down another “1+” on the left* What about that? *quickly* Nine! How’d you know it was nine so fast? You just added one more So you didn’t need to recount because you remembered there were eight! Dynamic Programming is just a fancy way to say ‘remembering stuff to save time later’
Applications:
5. Exponentiation by squaring
Say you want to calculate 232. Normally we’d iterate 32 times and find the result. What if I told you it can be done in 5 iterations?
Exponentiation by squaring or Binary exponentiation is a general method for fast computation of large positive integer powers of a number in O(log2N). Not only this, the method is also used for computation of powers of polynomials and square matrices.
Application:
6. String Matching and Parsing
Pattern matching/searching is one of the most important problem in Computer Science. There have been a lot of research on the topic but we’ll enlist only two basic necessities for any programmer.
KMP Algorithm (String Matching)
Knuth-Morris-Pratt algorithm is used in cases where we have to match a short pattern in a long string. For instance, when we Ctrl+F a keyword in a document, we perform pattern matching in the whole document.
Regular Expression (String Parsing)
Many a times we have to validate a string by parsing over a predefined restriction. It is heavily used in web development for URL parsing and matching.
7. Primality Testing Algorithms
There are deterministic and probabilistic ways of determining whether a given number is prime or not. We’ll see both deterministic and probabilistic (nondeterministic) ways.
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Sieve of Eratosthenes (deterministic)
If we have certain limit on the range of numbers, say determine all primes within range 100 to 1000 then Sieve is a way to go. The length of range is a crucial factor, because we have to allocate certain amount of memory according to range.
For any number n, incrementally testing upto sqrt(n) (deterministic)
In case you want to check for few numbers which are sparsely spread over a long range (say 1 to 1012), Sieve won’t be able to allocate enough memory. You can check for each number n by traversing only upto sqrt(n) and perform a divisibility check on n.
Fermat primality test and Miller–Rabin primality test (both are nondeterministic)
Both of these are compositeness tests. If a number is proved to be composite, then it sure isn’t a prime number. Miller-Rabin is a more sophisticated one than Fermat’s. Infact, Miller-Rabin also has a deterministic variant, but then its a game of trade between time complexity and accuracy of the algorithm.
Application:
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We’ll discuss some advanced algorithms every competitive programmer should know in the next post. Meanwhile master the above algorithms or share in the comments about what you think every beginner-intermediate programmer should know.
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A small phone book as a hash tableIn, a hash table ( hash map) is a that implements an, a structure that can map to. A hash table uses a to compute an index, also called a hash code, into an array of buckets or slots, from which the desired value can be found.Ideally, the hash function will assign each key to a unique bucket, but most hash table designs employ an imperfect hash function, which might cause hash where the hash function generates the same index for more than one key. Such collisions are always accommodated in some way.In a well-dimensioned hash table, the average cost (number of ) for each lookup is independent of the number of elements stored in the table. Many hash table designs also allow arbitrary insertions and deletions of key-value pairs, at ( ) constant average cost per operation.In many situations, hash tables turn out to be on average more efficient than or any other lookup structure. For this reason, they are widely used in many kinds of computer, particularly for,. Hash collision resolved by separate chaining.In the method known as separate chaining, each bucket is independent, and has some sort of of entries with the same index. The time for hash table operations is the time to find the bucket (which is constant) plus the time for the list operation.In a good hash table, each bucket has zero or one entries, and sometimes two or three, but rarely more than that.
Therefore, structures that are efficient in time and space for these cases are preferred. Structures that are efficient for a fairly large number of entries per bucket are not needed or desirable. If these cases happen often, the hashing function needs to be fixed. Separate chaining with linked lists Chained hash tables with are popular because they require only basic data structures with simple algorithms, and can use simple hash functions that are unsuitable for other methods. The cost of a table operation is that of scanning the entries of the selected bucket for the desired key.
If the distribution of keys is, the average cost of a lookup depends only on the average number of keys per bucket—that is, it is roughly proportional to the load factor.For this reason, chained hash tables remain effective even when the number of table entries n is much higher than the number of slots. For example, a chained hash table with 1000 slots and 10,000 stored keys (load factor 10) is five to ten times slower than a 10,000-slot table (load factor 1); but still 1000 times faster than a plain sequential list.For separate-chaining, the worst-case scenario is when all entries are inserted into the same bucket, in which case the hash table is ineffective and the cost is that of searching the bucket data structure. If the latter is a linear list, the lookup procedure may have to scan all its entries, so the worst-case cost is proportional to the number n of entries in the table.The bucket chains are often searched sequentially using the order the entries were added to the bucket. If the load factor is large and some keys are more likely to come up than others, then rearranging the chain with a may be effective.
More sophisticated data structures, such as balanced search trees, are worth considering only if the load factor is large (about 10 or more), or if the hash distribution is likely to be very non-uniform, or if one must guarantee good performance even in a worst-case scenario. However, using a larger table and/or a better hash function may be even more effective in those cases. Chained hash tables also inherit the disadvantages of linked lists. When storing small keys and values, the space overhead of the next pointer in each entry record can be significant. An additional disadvantage is that traversing a linked list has poor, making the processor cache ineffective.Separate chaining with list head cells. Hash collision by separate chaining with head records in the bucket array.Some chaining implementations store the first record of each chain in the slot array itself.The number of pointer traversals is decreased by one for most cases.
The purpose is to increase cache efficiency of hash table access.The disadvantage is that an empty bucket takes the same space as a bucket with one entry. To save space, such hash tables often have about as many slots as stored entries, meaning that many slots have two or more entries.
Separate chaining with other structures Instead of a list, one can use any other data structure that supports the required operations. For example, by using a, the theoretical worst-case time of common hash table operations (insertion, deletion, lookup) can be brought down to rather than O( n).
However, this introduces extra complexity into the implementation, and may cause even worse performance for smaller hash tables, where the time spent inserting into and balancing the tree is greater than the time needed to perform a on all of the elements of a list. A real world example of a hash table that uses a self-balancing binary search tree for buckets is the HashMap class in.The variant called uses a to store all the entries that hash to the same slot. Each newly inserted entry gets appended to the end of the dynamic array that is assigned to the slot. The dynamic array is resized in an exact-fit manner, meaning it is grown only by as many bytes as needed.
Alternative techniques such as growing the array by block sizes or pages were found to improve insertion performance, but at a cost in space. This variation makes more efficient use of and the (TLB), because slot entries are stored in sequential memory positions. It also dispenses with the next pointers that are required by linked lists, which saves space. Despite frequent array resizing, space overheads incurred by the operating system such as memory fragmentation were found to be small. An elaboration on this approach is the so-called, where a bucket that contains k entries is organized as a perfect hash table with k 2 slots.
While it uses more memory ( n 2 slots for n entries, in the worst case and n × k slots in the average case), this variant has guaranteed constant worst-case lookup time, and low amortized time for insertion.It is also possible to use a for each bucket, achieving constant time for all operations with high probability. Open addressing. Hash collision resolved by open addressing with linear probing (interval=1). Note that 'Ted Baker' has a unique hash, but nevertheless collided with 'Sandra Dee', that had previously collided with 'John Smith'.In another strategy, called open addressing, all entry records are stored in the bucket array itself. When a new entry has to be inserted, the buckets are examined, starting with the hashed-to slot and proceeding in some probe sequence, until an unoccupied slot is found. When searching for an entry, the buckets are scanned in the same sequence, until either the target record is found, or an unused array slot is found, which indicates that there is no such key in the table. The name 'open addressing' refers to the fact that the location ('address') of the item is not determined by its hash value.
Main article:Hash tables are commonly used to implement many types of in-memory tables. They are used to implement (arrays whose indices are arbitrary or other complicated objects), especially in like, and.When storing a new item into a and a hash collision occurs, the multimap unconditionally stores both items.When storing a new item into a typical associative array and a hash collision occurs, but the actual keys themselves are different, the associative array likewise stores both items.
However, if the key of the new item exactly matches the key of an old item, the associative array typically erases the old item and overwrites it with the new item, so every item in the table has a unique key.Database indexing Hash tables may also be used as -based data structures and (such as in ) although are more popular in these applications. In multi-node database systems, hash tables are commonly used to distribute rows amongst nodes, reducing network traffic for hash joins.Caches. Main article:Hash tables can be used to implement, auxiliary data tables that are used to speed up the access to data that is primarily stored in slower media.
In this application, hash collisions can be handled by discarding one of the two colliding entries—usually erasing the old item that is currently stored in the table and overwriting it with the new item, so every item in the table has a unique hash value.Sets Besides recovering the entry that has a given key, many hash table implementations can also tell whether such an entry exists or not.Those structures can therefore be used to implement a, which merely records whether a given key belongs to a specified set of keys. In this case, the structure can be simplified by eliminating all parts that have to do with the entry values. Hashing can be used to implement both static and dynamic sets.Object representation Several dynamic languages, such as, and, use hash tables to implement objects. In this representation, the keys are the names of the members and methods of the object, and the values are pointers to the corresponding member or method.Unique data representation. Main article:Hash tables can be used by some programs to avoid creating multiple character strings with the same contents.
For that purpose, all strings in use by the program are stored in a single string pool implemented as a hash table, which is checked whenever a new string has to be created. This technique was introduced in interpreters under the name, and can be used with many other kinds of data ( in a symbolic algebra system, records in a database, files in a file system, binary decision diagrams, etc.).Transposition table. Main article: Implementations In programming languages Many programming languages provide hash table functionality, either as built-in associative arrays or as standard modules.
In, for example, the class provides hash tables for keys and values of arbitrary type.The programming language (including the variant which is used on ) includes the HashSet, HashMap, LinkedHashSet, and LinkedHashMap collections.In 5 and 7, the Zend 2 engine and the Zend 3 engine (respectively) use one of the hash functions from to generate the hash values used in managing the mappings of data pointers stored in a hash table. In the PHP source code, it is labelled as DJBX33A (Daniel J. Bernstein, Times 33 with Addition).' S built-in hash table implementation, in the form of the dict type, as well as 's hash type (%) are used internally to implement namespaces and therefore need to pay more attention to security, i.e., collision attacks. Python also use hashes internally, for fast lookup (though they store only keys, not values).In the, support for hash tables is provided via the non-generic Hashtable and generic Dictionary classes, which store key-value pairs, and the generic HashSet class, which stores only values.In the hash table uses the open addressing model from Ruby 2.4 onwards.In 's standard library, the generic HashMap and HashSet structs use linear probing with Robin Hood bucket stealing.defines the classes Set / IdentitySet and Dictionary / IdentityDictionary.
All Smalltalk implementations provide additional (not yet standardized) versions of WeakSet, WeakKeyDictionary and WeakValueDictionary.array variables are hash tables, and Tcl dictionaries are immutable values based on hashes. The functionality is also available as C library functions (for generic hash tables) and (for dictionary values). The performance has been independently benchmarked as extremely competitive. History The idea of hashing arose independently in different places.
In January 1953, wrote an internal IBM memorandum that used hashing with chaining., and implemented a program using hashing at about the same time. Open addressing with linear probing (relatively prime stepping) is credited to Amdahl, but (in Russia) had the same idea. See also.Related data structures There are several data structures that use hash functions but cannot be considered special cases of hash tables:., memory efficient data-structure designed for constant-time approximate lookups; uses hash function(s) and can be seen as an approximate hash table. (DHT), a resilient dynamic table spread over several nodes of a network., a structure, similar to the, but where each key is hashed first.References.
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