Advanced data structures peter brass pdf


 

Peter Brass . book Data Structures + Algorithms = Programs, and Algorithms and Data Advanced Institute of Science and Technology in July Advanced Data Structures presents a comprehensive look at the ideas, analysis PETER BRASS received a Ph.D. in mathematics at the Technical University of. Computational Geometry - Advanced Data Structures - by Peter Brass. Peter Brass, City College, City University of New York Access. PDF; Export citation.

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Advanced Data Structures Peter Brass Pdf

Advanced Data. Structure. (). By. Mahesh R Sanghavi & Deepali Pawar Peter Brass, ―Advanced Data Structures‖, Cambridge University Press. Advanced data structures by Peter Brass Cambridge University Press Full Text: PDF. Author: Richard Jankowski. Published in. Advanced Data Structures [Peter Brass] on goudzwaard.info *FREE* shipping on qualifying offers. Advanced Data Structures presents a comprehensive look at the .

Williamson and David B. Garey and David S. Johnson, W. Cormen, Charles E. Leiserson, Ronald L. Moret and Henry D. The effect of data structures on program design is investigated. The uses of data structures and algorithms in a variety of application areas are covered. Focus is on algorithmic thinking, performance guarantees and boundary cases, and efficient solutions to practical problems. Advanced topics will cover a selection of modern algorithms, many of which come from real-world applications. These policies include cheating, fabrication, falsification and forgery, multiple submission, plagiarism, complicity and computer misuse. If there is reason to believe you have been involve in academic dishonesty, you will be referred to the Office of Student Judicial Affairs. You will be given the opportunity to review the charge s. If you believe you are not responsible, you will have the opportunity for a hearing.

Thus the record and array data structures are based on computing the addresses of data items with arithmetic operations; while the linked data structures are based on storing addresses of data items within the structure itself.

Many data structures use both principles, sometimes combined in non-trivial ways as in XOR linking The implementation of a data structure usually requires writing a set of procedures that create and manipulate instances of that structure.

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The efficiency of a data structure cannot be analyzed separately from those operations. This observation motivates the theoretical concept of an abstract data type, a data structure that is defined indirectly by the operations that may be performed on it, and the mathematical properties of those operations including their space and time cost.

Many high-level programming languages and some higher-level assembly languages, such as MASM, on the other hand, have special syntax or other built-in support for certain data structures, such as vectors one-dimensional arrays in the C language or multi-dimensional arrays in Pascal. Most programming languages feature some sort of library mechanism that allows data structure implementations to be reused by different programs.

Modern languages usually come with standard libraries that implement the most common data structures. NET Framework. Modern languages also generally support modular programming, the separation between the interface of a library module and its implementation. Some provide opaque data types that allow clients to hide implementation details.

NET Framework may use classes for this purpose. Many known data structures have concurrent versions that allow multiple computing threads to access the data structure simultaneously. References [1] Paul E. Black ed. National Institute of Standards and Technology. Addison-Wesley, 3rd edition, In linked data structures, the links are usually treated as special data types that can only be dereferenced or compared for equality.

Linked data structures are thus contrasted with arrays and other data structures that require performing arithmetic operations on pointers. This distinction holds even when the nodes are actually implemented as elements of a single array, and the references are actually array indices: as long as no arithmetic is done on those indices, the data structure is essentially a linked one. Linking can be done in two ways - Using dynamic allocation and using array index linking.

Linked data structures include linked lists, search trees, expression trees, and many other widely used data structures. They are also key building blocks for many efficient algorithms, such as topological sort[1] and set union-find. It is not necessary that it should be stored in the adjacent memory locations. Every structure has a data field and an address field.

The Address field contains the address of its successor. Linked list can be singly, doubly or multiply linked and can either be linear or circular. Basic Properties Objects, called nodes, are linked in a linear sequence A reference to the first node of the list is always kept. This is called the 'head' or 'front'. Example in Java This is an example of the node class used to store integers in a Java implementation of a linked list. Linked data structure Search trees A search tree is a tree data structure in whose nodes data values can be stored from some ordered set, which is such that in an in-order traversal of the tree the nodes are visited in ascending order of the stored values.

Basic Properties Objects, called nodes, are stored in an ordered set. In-order traversal provides an ascending readout of the data in the tree Sub trees of the tree are in themselves, trees.

Advantages and disadvantages Advantages against arrays Compared to arrays, linked data structures allow more flexibility in organizing the data and in allocating space for it. In arrays, the size of the array must be specified precisely at the beginning, this can be a potential waste of memory. A linked data structure is built dynamically and never needs to be bigger than the programmer requires.

It also requires no guessing in terms of how much space you must allocate when using a linked data structure. This is a feature that is key in saving wasted memory. In array, the array elements have to be in contiguous connected and sequential portion of memory. But in linked data structure, the reference to each node gives us the information where to find out the next one. The nodes of a linked data structure can also be moved individually to different locations without affecting the logical connections between them, unlike arrays.

With due care, a process can add or delete nodes to one part of a data structure even while other processes are working on other parts. On the other hand, access to any particular node in a linked data structure requires following a chain of references that stored in it.

If the structure has n nodes, and each node contains at most b links, there will be some nodes that cannot be reached in less than logb n steps. For many structures, some nodes may require worst case up to n1 steps. In contrast, many array data structures allow access to any element with a constant number of operations, independent of the number of entries. Broadly the implementation of these linked data structure is through dynamic data structures.

It gives us the chance to use particular space again. Memory can be utilized more efficiently by using this data structures.

Memory is allocated as per the need and when memory is not further needed, deallocation is done. General disadvantages Linked data structures may also incur in substantial memory allocation overhead if nodes are allocated individually and frustrate memory paging and processor caching algorithms since they generally have poor locality of reference.

In some cases, linked data structures may also use more memory for the link fields than competing array structures. This is because linked data structures are not contiguous.

Instances of data can be found all over in memory, unlike arrays. In arrays, nth element can be accessed immediately, while in a linked data structure we have to follow multiple pointers so element access time varies according to where in the structure the element is. In some theoretical models of computation that enforce the constraints of linked structures, such as the pointer machine, many problems require more steps than in the unconstrained random access machine model.

Galler and Michael J. An improved equivalence algorithm. The paper originating disjoint-set forests. The concept was originally introduced by Jacobson [1] to encode bit vectors, unlabeled trees, and planar graphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use them in-place, without decompressing them first.

A related notion is that of a compressed data structure, in which the size of the data structure depends upon the particular data being represented. Suppose that is the information-theoretical optimal number of bits needed to store some data. A representation of bits of space, bits of space, and bits of space. It uses an idea similar to that for range-minimum queries; there are a constant number of recursions before stopping at a subproblem of a limited size. The bit array is partitioned into large blocks of size bits and small blocks of size ; each such entry takes bits.

For each large bits for a total stores the rank of block, the rank of its first bit is stored in a separate table of each of the bits of storage. Within a large block, another directory small blocks it contains.

The difference here is that it only needs bits for each entry, since only the differences from the rank of the first bit in the containing large block need to be stored. Thus, this table takes a total of lookup table ; this requires auxiliary tables take To answer a query for space. A for can then be used that stores the answer to every possible rank query on a bit string of length space, this data structure supports rank queries in time and in constant time, a constant time algorithm computes bits of storage space.

Thus, since each of these bits of Succinct data structure In practice, the lookup table can be replaced by bitwise operations and smaller tables to perform find the number of bits set in the small blocks. This is often beneficial, since succinct data structures find their uses in large data sets, in which case cache misses become much more frequent and the chances of the lookup table being evicted from closer CPU caches becomes higher.

A more complicated structure using constant time. There is a succinct static dictionary which attains this bound, namely space.

Examples When a sequence of variable-length items needs to be encoded, the items can simply be placed one after another, with no delimiters. A separate binary string consisting of 1s in the positions where an item begins, and 0s every where else is encoded along with it. Given this string, the function can quickly determine where each item begins, given its index.

The number of different binary trees on nodes is. For large , this is about ; thus we need at least about bits to encode it. A succinct binary tree therefore would occupy only bits per node.

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References [1] Jacobson, G. J Succinct static data structures. Raman, S. S Rao Grossi Grabowski, V. Mkinen, G. Navarro Experimental Algorithms: I Munro SIAM J. Succinct data structure [9] Ptracu, M. Foundations of Computer Science, Implicit data structure In computer science, an implicit data structure is a data structure that uses very little memory besides the actual data elements i.

These are storage schemes which retain no pointers and represent the file of n k-key records as a simple n by k array n thus retrieve faster. In implicit data structures the only structural information to be given is to allow the array to grow and shrink as n.

No extra information is required. It is called "implicit" because most of the structure of the elements is expressed implicitly by their order. Another term used interchangeably is space efficient. Definitions of very little are vague and can mean from O 1 to O log n extra space. Everything is accessed in-place, by reading bits at various positions in the data. To achieve optimal coding, we use bits instead of bytes. Implicit data structures are frequently also succinct data structures.

Although one may argue that disk space is no longer a problem and we should not concern ourselves with improving space utilization, the issue that implicit data structures are designed to improve is main memory utilization. Hence, if a larger chunk of an implicit data structure fits in main memory the operations performed on it can be faster even if the asymptotic running time is not as good as its space-oblivious counterpart.

Furthermore, since the CPU-cache is usually much smaller than main-memory, implicit data structures can improve cache-efficiency and thus running speed, especially if the method used improves locality. Implicit data structure for weighted element For presentation of elements with different weight several data structures are required.

The structure uses one more location besides required for values of elements. The first structure supports worst case search time in terms of rank of weight of elements w. If the elements are drawn from uniform distribution, then variation of this structure takes average time.

The same result obtain for the data structures in which the intervals between consecutive values have access probabilities. It refers to a data structure whose operations are roughly as fast as those of a conventional data structure for the problem, but whose size can be substantially smaller. The size of the compressed data structure is typically highly dependent upon the entropy of the data being represented.

Important examples of compressed data structures include the compressed suffix array[1][2] and the FM-index,[3] both of which can represent an arbitrary text of characters T for pattern matching.

Given any input pattern P, they support the operation of finding if and where P appears in T. The search time is proportional to the sum of the length of pattern P, a very slow-growing function of the length of the text T, and the number of reported matches. The space they occupy is roughly equal to the size of the text T in entropy-compressed form, such as that obtained by Prediction by Partial Matching or gzip.

Moreover, both data structures are self-indexing, in that they can reconstruct the text T in a random access manner, and thus the underlying text T can be discarded. In other words, they simultaneously provide a compressed and quickly searchable representation of the text T. They represent a substantial space improvement over the conventional suffix tree and suffix array, which occupy many times more space than the size of T.

They also support searching for arbitrary patterns, as opposed to the inverted index, which can support only word-based searches. In addition, inverted indexes do not have the self-indexing feature. An important related notion is that of a succinct data structure, which uses space roughly equal to the information-theoretic minimum, which is a worst-case notion of the space needed to represent the data. In contrast, the size of a compressed data structure depends upon the particular data being represented.

When the data are compressible, as is often the case in practice for natural language text, the compressed data structure can occupy substantially less space than the information-theoretic minimum.

References [1] R. Grossi and J. Grossi, A. Gupta, and J. Ferragina and G.

Search data structure 10 Search data structure In computer science, a search data structure is any data structure that allows the efficient retrieval of specific items from a set of items, such as a specific record from a database. The simplest, most general, and least efficient search structure is merely an unordered sequential list of all the items. Locating the desired item in such a list, by the linear search method, inevitably requires a number of operations proportional to the number n of items, in the worst case as well as in the average case.

Useful search data structures allow faster retrieval; however, they are limited to queries of some specific kind. Moreover, since the cost of building such structures is at least proportional to n, they only pay off if several queries are to be performed on the same database or on a database that changes little between queries. Static search structures are designed for answering many queries on a fixed database; dynamic structures also allow insertion, deletion, or modification of items between successive queries.

In the dynamic case, one must also consider the cost of fixing the search structure to account for the changes in the database. Classification The simplest kind of query is to locate a record that has a specific field the key equal to a specified value v. Other common kinds of query are "find the item with smallest or largest key value", "find the item with largest key value not exceeding v", "find all items with key values between specified bounds vmin and vmax".

In certain databases the key values may be points in some multi-dimensional space. For example, the key may be a geographic position latitude and longitude on the Earth. In that case, common kinds of queries are find the record with a key closest to a given point v", or "find all items whose key lies at a given distance from v", or "find all items within a specified region R of the space".

A common special case of the latter are simultaneous range queries on two or more simple keys, such as "find all employee records with salary between 50, and , and hired between and ". Single ordered keys Array if the key values span a moderately compact interval. Priority-sorted list; see linear search Key-sorted array; see binary search Self-balancing binary search tree Hash table Finding the smallest element Heap Asymptotic amortized worst-case analysis In this table, the asymptotic notation O f n means "not exceeding some fixed multiple of f n in the worst case.

This table is only an approximate summary; for each data structure there are special situations and variants that may lead to different costs. Also two or more data structures can be combined to obtain lower costs. Footnotes Persistent data structure In computing, a persistent data structure is a data structure that always preserves the previous version of itself when it is modified. Such data structures are effectively immutable, as their operations do not visibly update the structure in-place, but instead always yield a new updated structure.

A persistent data structure is not a data structure committed to persistent storage, such as a disk; this is a different and unrelated sense of the word "persistent. The data structure is fully persistent if every version can be both accessed and modified.

If there is also a meld or merge operation that can create a new version from two previous versions, the data structure is called confluently persistent.

Structures that are not persistent are called ephemeral. While persistence can be achieved by simple copying, this is inefficient in CPU and RAM usage, because most operations make only small changes to a data structure.

A better method is to exploit the similarity between the new and old versions to share structure between them, such as using the same subtree in a number of tree structures. However, because it rapidly becomes infeasible to determine how many previous versions share which parts of the structure, and because it is often desirable to discard old versions, this necessitates an environment with garbage collection.

Persistent data structure 12 Partially persistent In the partial persistence model, we may query any previous version of the data structure, but we may only update the latest version.

This implies a linear ordering among the versions. Three methods on balanced binary search tree: Fat Node Fat node method is to record all changes made to node fields in the nodes themselves, without erasing old values of the fields. This requires that we allow nodes to become arbitrarily fat. In other words, each fat node contains the same information and pointer fields as an ephemeral node, along with space for an arbitrary number of extra field values.

Each extra field value has an associated field name and a version stamp which indicates the version in which the named field was changed to have the specified value. Besides, each fat node has its own version stamp, indicating the version in which the node was created.

The only purpose of nodes having version stamps is to make sure that each node only contains one value per field name per version. In order to navigate through the structure, each original field value in a node has a version stamp of zero. Complexity of Fat Node With using fat node method, it requires O 1 space for every modification: just store the new data.

Each modification takes O 1 additional time to store the modification at the end of the modification history. This is an amortized time bound, assuming we store the modification history in a growable array. For access time, we must find the right version at each node as we traverse the structure. If we made m modifications, then each access operation has O logm slowdown resulting from the cost of finding the nearest modification in the array.

Path Copying Path copy is to make a copy of all nodes on the path which contains the node we are about to insert or delete. Then you have to cascade the change back through the data structure: all nodes that pointed to the old node must be modified to point to the new node instead.

These modifications cause more cascading changes, and so on, until we reach to the root. Maintain an array of roots indexed by timestamp. The data structure pointed to by time ts root is exactly time ts date structure. Complexity of Path Copying With m modifications, this costs O logm additive lookup time. Modification time and space are bounded by the size of the structure, since a single modification may cause the entire structure to be copied.

Peter Brass

Thats O m for one update, and thus O n2 preprocessing time. A combination Sleator, Tarjan et al. Many data structures use both principles, sometimes combined in non-trivial ways as in XOR linking.

The efficiency of a data structure cannot be analyzed separately from those operations. This observation motivates the theoretical concept of an abstract data type , a data structure that is defined indirectly by the operations that may be performed on it, and the mathematical properties of those operations including their space and time cost. Elements are accessed using an integer index to specify which element is required. Typical implementations allocate contiguous memory words for the elements of arrays but this is not always a necessity.

Arrays may be fixed-length or resizable. A linked list also just called list is a linear collection of data elements of any type, called nodes, where each node has itself a value, and points to the next node in the linked list. The principal advantage of a linked list over an array, is that values can always be efficiently inserted and removed without relocating the rest of the list. Certain other operations, such as random access to a certain element, are however slower on lists than on arrays.

A record also called tuple or struct is an aggregate data structure. A record is a value that contains other values, typically in fixed number and sequence and typically indexed by names.

The elements of records are usually called fields or members. A union is a data structure that specifies which of a number of permitted primitive types may be stored in its instances, e.

Contrast with a record , which could be defined to contain a float and an integer; whereas in a union, there is only one value at a time. Enough space is allocated to contain the widest member datatype. A tagged union also called variant , variant record, discriminated union, or disjoint union contains an additional field indicating its current type, for enhanced type safety.

An object is a data structure that contains data fields, like a record does, as well as various methods which operate on the data contents.

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