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13
Succinct indexable dictionaries with applications to encoding kary trees and multisets
 In Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
"... We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,...,m − 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i) which returns the ith smallest ele ..."
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Cited by 193 (7 self)
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We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,...,m − 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i) which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n,m)+ o(n)+O(lg lg m) bits to store a set of size n, where B(n,m) = ⌈ lg ( m) ⌉ n is the minimum number of bits required to store any nelement subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: • an informationtheoretically optimal representation of a kary cardinal tree that supports standard operations in constant time, • a representation of a multiset of size n from {0,...,m − 1} in B(n,m+n) + o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time, and • a representation of a sequence of n nonnegative integers summing up to m in B(n,m + n) + o(n) bits that supports prefix sum queries in constant time. 1
Monotone Minimal Perfect Hashing: Searching a Sorted Table with O(1) Accesses
"... A minimal perfect hash function maps a set S of n keys into the set { 0, 1,..., n − 1} bijectively. Classical results state that minimal perfect hashing is possible in constant time using a structure occupying space close to the lower bound of log e bits per element. Here we consider the problem of ..."
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Cited by 20 (8 self)
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A minimal perfect hash function maps a set S of n keys into the set { 0, 1,..., n − 1} bijectively. Classical results state that minimal perfect hashing is possible in constant time using a structure occupying space close to the lower bound of log e bits per element. Here we consider the problem of monotone minimal perfect hashing, in which the bijection is required to preserve the lexicographical ordering of the keys. A monotone minimal perfect hash function can be seen as a very weak form of index that provides ranking just on the set S (and answers randomly outside of S). Our goal is to minimise the description size of the hash function: we show that, for a set S of n elements out of a universe of 2 w elements, O(n log log w) bits are sufficient to hash monotonically with evaluation time O(log w). Alternatively, we can get space O(n log w) bits with O(1) query time. Both of these data structures improve a straightforward construction with O(n log w) space and O(log w) query time. As a consequence, it is possible to search a sorted table with O(1) accesses to the table (using additional O(n log log w) bits). Our results are based on a structure (of independent interest) that represents a trie in a very compact way, but admits errors. As a further application of the same structure, we show how to compute the predecessor (in the sorted order of S) of an arbitrary element, using O(1) accesses in expectation and an index of O(n log w) bits, improving the trivial result of O(nw) bits. This implies an efficient index for searching a blocked memory.
Simple and spaceefficient minimal perfect hash functions
 In Proc. of the 10th Intl. Workshop on Data Structures and Algorithms
, 2007
"... Abstract. A perfect hash function (PHF) h: U → [0, m − 1] for a key set S is a function that maps the keys of S to unique values. The minimum amount of space to represent a PHF for a given set S is known to be approximately 1.44n 2 /m bits, where n = S. In this paper we present new algorithms for ..."
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Cited by 14 (7 self)
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Abstract. A perfect hash function (PHF) h: U → [0, m − 1] for a key set S is a function that maps the keys of S to unique values. The minimum amount of space to represent a PHF for a given set S is known to be approximately 1.44n 2 /m bits, where n = S. In this paper we present new algorithms for construction and evaluation of PHFs of a given set (for m = n and m = 1.23n), with the following properties: 1. Evaluation of a PHF requires constant time. 2. The algorithms are simple to describe and implement, and run in linear time. 3. The amount of space needed to represent the PHFs is around a factor 2 from the information theoretical minimum. No previously known algorithm has these properties. To our knowledge, any algorithm in the literature with the third property either: – Requires exponential time for construction and evaluation, or – Uses nearoptimal space only asymptotically, for extremely large n.
External perfect hashing for very large key sets
 In Proceedings of the 16th ACM Conference on Information and Knowledge Management (CIKM’07
, 2007
"... A perfect hash function (PHF) h: S → [0, m − 1] for a key set S ⊆ U of size n, where m ≥ n and U is a key universe, is an injective function that maps the keys of S to unique values. A minimal perfect hash function (MPHF) is a PHF with m = n, the smallest possible range. Minimal perfect hash functio ..."
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Cited by 14 (2 self)
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A perfect hash function (PHF) h: S → [0, m − 1] for a key set S ⊆ U of size n, where m ≥ n and U is a key universe, is an injective function that maps the keys of S to unique values. A minimal perfect hash function (MPHF) is a PHF with m = n, the smallest possible range. Minimal perfect hash functions are widely used for memory efficient storage and fast retrieval of items from static sets. In this paper we present a distributed and parallel version of a simple, highly scalable and nearspace optimal perfect hashing algorithm for very large key sets, recently presented in [4]. The sequential implementation of the algorithm constructs a MPHF for a set of 1.024 billion URLs of average length 64 bytes collected from the Web in approximately 50 minutes using a commodity PC. The parallel implementation proposed here presents the following performance using 14 commodity PCs: (i) it constructs a MPHF for the same set of 1.024 billion URLs in approximately 4 minutes; (ii) it constructs a MPHF for a set of 14.336 billion 16byte random integers in approximately 50 minutes with a performance degradation of 20%; (iii) one version of the parallel algorithm distributes the description of the MPHF among the participating machines and its evaluation is done in a distributed way, faster than the centralized function.
Succinct Data Structures for Retrieval and Approximate Membership
"... Abstract. The retrieval problem is the problem of associating data with keys in a set. Formally, the data structure must store a function f: U → {0, 1} r that has specified values on the elements of a given set S ⊆ U, S  = n, but may have any value on elements outside S. All known methods (e. g. ..."
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Cited by 13 (6 self)
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Abstract. The retrieval problem is the problem of associating data with keys in a set. Formally, the data structure must store a function f: U → {0, 1} r that has specified values on the elements of a given set S ⊆ U, S  = n, but may have any value on elements outside S. All known methods (e. g. those based on perfect hash functions), induce a space overhead of Θ(n) bits over the optimum, regardless of the evaluation time. We show that for any k, query time O(k) can be achieved using space that is within a factor 1 + e −k of optimal, asymptotically for large n. The time to construct the data structure is O(n), expected. If we allow logarithmic evaluation time, the additive overhead can be reduced to O(log log n) bits whp. A general reduction transfers the results on retrieval into analogous results on approximate membership, a problem traditionally addressed using Bloom filters. Thus we obtain space bounds arbitrarily close to the lower bound for this problem as well. The evaluation procedures of our data structures are extremely simple. For the results stated above we assume free access to fully random hash functions. This assumption can be justified using space o(n) to simulate full randomness on a RAM. 1
Theory and Practise of Monotone Minimal Perfect Hashing
"... Minimal perfect hash functions have been shown to be useful to compress data in several data management tasks. In particular, orderpreserving minimal perfect hash functions [12] have been used to retrieve the position of a key in a given list of keys: however, the ability to preserve any given orde ..."
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Cited by 13 (6 self)
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Minimal perfect hash functions have been shown to be useful to compress data in several data management tasks. In particular, orderpreserving minimal perfect hash functions [12] have been used to retrieve the position of a key in a given list of keys: however, the ability to preserve any given order leads to an unavoidable �(n log n) lower bound on the number of bits required to store the function. Recently, it was observed [1] that very frequently the keys to be hashed are sorted in their intrinsic (i.e., lexicographical) order. This is typically the case of dictionaries of search engines, list of URLs of web graphs, etc. We refer to this restricted version of the problem as monotone minimal perfect hashing. We analyse experimentally the data structures proposed in [1], and along our way we propose some new methods that, albeit asymptotically equivalent or worse, perform very well in practise, and provide a balance between access speed, ease of construction, and space usage. 1
Practical perfect hashing in nearly optimal space
 Information Systems
"... A hash function is a mapping from a key universe U to a range of integers, i.e., h: U↦→{0, 1,...,m−1}, where m is the range’s size. A perfect hash function for some set S ⊆ U is a hash function that is onetoone on S, where m≥S. A minimal perfect hash function for some set S ⊆ U is a perfect hash ..."
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Cited by 2 (1 self)
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A hash function is a mapping from a key universe U to a range of integers, i.e., h: U↦→{0, 1,...,m−1}, where m is the range’s size. A perfect hash function for some set S ⊆ U is a hash function that is onetoone on S, where m≥S. A minimal perfect hash function for some set S ⊆ U is a perfect hash function with a range of minimum size, i.e., m=S. This paper presents a construction for (minimal) perfect hash functions that combines theoretical analysis, practical performance, expected linear construction time and nearly optimal space consumption for the data structure. For n keys and m=n the space consumption ranges from 2.62n to 3.3n bits, and for m=1.23n it ranges from 1.95n to 2.7n bits. This is within a small constant factor from the theoretical lower bounds of 1.44n bits for m=n and 0.89n bits for m=1.23n. We combine several theoretical results into a practical solution that has turned perfect hashing into a very compact data structure to solve the membership problem when the key set S is static and known in advance. By taking into account the memory hierarchy we can construct (minimal) perfect hash functions for over a billion keys in 46 minutes using a commodity PC. An open source implementation of the algorithms is available
NearOptimal Space Perfect Hashing Algorithms
"... Abstract. A perfect hash function (PHF) is an injective function that maps keys from a set S to unique values. Since no collisions occur, each key can be retrieved from a hash table with a single probe. A minimal perfect hash function (MPHF) is a PHF with the smallest possible range, that is, the ha ..."
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Cited by 2 (0 self)
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Abstract. A perfect hash function (PHF) is an injective function that maps keys from a set S to unique values. Since no collisions occur, each key can be retrieved from a hash table with a single probe. A minimal perfect hash function (MPHF) is a PHF with the smallest possible range, that is, the hash table size is exactly the number of keys in S. Differently from other hashing schemes, MPHFs completely avoid the problem of wasted space and wasted time to deal with collisions. The study of perfect hash functions started in the early 80s, when it was proved that the theoretic information lower bound to describe a minimal perfect hash function was approximately 1.44 bits per key. Although the proof indicates that it would be possible to build an algorithm capable of generating optimal functions, no one was able to obtain a practical algorithm that could be used in real applications. Thus, there was a gap between theory and practice. The main result of the thesis filled this gap, lowering the space complexity to represent MPHFs that are useful in practice from O(n log n) to O(n) bits. This allows the use of perfect hashing in applications to which it was not considered a good option. This explicit construction of PHFs is something that the data structures and algorithms community has been looking for since the 1980s. 1.
The Context of Coordinating Groups in Dynamic Mobile Networks
"... Abstract. Contextawareness in dynamic and unpredictable environments is a wellstudied problem, and many approaches handle sensing, understanding, and acting upon context information. Entities in these environments are not in isolation, and oftentimes the manner in which entities coordinate depends ..."
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Cited by 2 (2 self)
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Abstract. Contextawareness in dynamic and unpredictable environments is a wellstudied problem, and many approaches handle sensing, understanding, and acting upon context information. Entities in these environments are not in isolation, and oftentimes the manner in which entities coordinate depends on some (implicit) notion of their shared context. In this paper, we are motivated by the need to explicitly construct notions of the context of a group that can support better coordination within the group. First we identify an efficient representation of context (both of an individual and of a group) that can be shared across wireless connections without incurring a significant communication overhead. Second we provide precise semantics for different types of groups, each with compelling use cases in these dynamic computing environments. Finally, we define and demonstrate protocols for efficiently computing groups and their context in a distributed manner. 1
Perfect hashing for data management applications
, 2007
"... Perfect hash functions can potentially be used to compress data in connection with a variety of data management tasks. Though there has been considerable work on how to construct good perfect hash functions, there is a gap between theory and practice among all previous methods on minimal perfect has ..."
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Cited by 1 (0 self)
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Perfect hash functions can potentially be used to compress data in connection with a variety of data management tasks. Though there has been considerable work on how to construct good perfect hash functions, there is a gap between theory and practice among all previous methods on minimal perfect hashing. On one side, there are good theoretical results without experimentally proven practicality for large key sets. On the other side, there are the theoretically analyzed time and space usage algorithms that assume that truly random hash functions are available for free, which is an unrealistic assumption. In this paper we attempt to bridge this gap between theory and practice, using a number of techniques from the literature to obtain a novel scheme that is theoretically wellunderstood and at the same time achieves an orderofmagnitude increase in performance compared to previous “practical ” methods. This improvement comes from a combination of a novel, theoretically optimal perfect hashing scheme that greatly simplifies previous methods, and the fact that our algorithm is designed to make good use of the memory hierarchy. We demonstrate the scalability of our algorithm by considering a set of over one billion URLs from the World Wide Web of average length 64, for which we construct a minimal perfect hash function on a commodity PC in a little more than 1 hour. Our scheme produces minimal perfect hash functions using slightly more than 3 bits per key. For perfect hash functions in the range {0,..., 2n −1} the space usage drops to just over 2 bits per key (i.e., one bit more than optimal for representing the key). This is significantly below of what has been achieved previously for very large values of n. 1.