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Despite the apparent simplicity of hashing, much of the analysis is quite difficult and many questions remain unresolved Also there are many interesting ideas that generally attempt to make it unlikely that worst-case possibilities of hashing arise
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An early paper on hashing is [ I I] A wealth of information on the subject, including an analysis of hashing with linear probing, is presented in [6] Double hashing is analyzed in [5] and [7] Yet another collision resolution scheme, coalesced hashing, is described in [12] An excellent survey on the subject is [8], and [9] contains suggestions for and pitfalls in choosing hash functions Precise analytic and simulation results for all the methods described in this chapter are available in [4] Uniform hashing, in which no clustering exists, is optimal with respect to the cost of a successful search [131 If the input keys are known in advance, perfect hash functions, which do not allow collisions, exist [I] Some more complicated hashing schemes, for which the worst case depends not on the particular input but on random numbers chosen by the algorithm, appear in [2] and [3] These schemes guarantee that only a constant number of collisions occur in the worst case (although construction of a hash function can take a long time in the unlikely case of bad random numbers) They are useful for implementing tables in hardware One method of implementing Exercise 208 is described in [lo] 1 J L Carter and M N Wegman, "Universal Classes of Hash Functions," Jo~irnal of Computer and System Sciences 18 ( 19791, 143-154 2 M Dietzfelbinger, A R Karlin, K Melhorn, F Meyer auf def Heide, H Rohnert, and R E Tarjan, "Dynamic Perfect Hashing: Upper and Lower Bounds," SIAM Journal on Computing 23 (19941, 738-76 1 3 R J Enbody and H C Du, "Dynamic Hashing Schemes," Computing Surveys 20 (1988), 85-1 13 4 G H Gonnet and R Baeza-Yates, Handbook of Algorithms and Data Structures, 2d ed, Addison-Wesley, Reading, Mass, 199 1 5 L J Guibas and E Szemeredi, "The Analysis of Double Hashing," Jo~irnalof Computer and Sjstem Sciences 16 (1978), 226-274 6 D E Knuth, The Art of Comp~iter Programming, Vol3: Sorting and Searching, 2d ed, Addison-Wesley, Reading, Mass, 1998 7 G Lueker and M Molodowitch, "More Analysis of Double Hashing," Combinatorica 13 (1993), 83-96 8 W D Maurer and T G Lewis, "Hash Table Methods," Computing Surveys 7 ( 1975),5-20 9 B J McKenzie, R Harries, and T Bell, "Selecting a Hashing Algorithm," Sofnvare-Practice and Experience 20 (19901, 209-224
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10 R Morris, "Scatter Storage Techniques," Comnzutzications of the ACM 11 ( 1968), 3 8 4 4 I 1 W W Peterson, "Addressing for Random Access Storage," IBM Journal ofResearclz atzd De~lelopmeizt1 ( 1957), 130-146 12 J S Vitter, "lrnplementations for Coalesced Hashing," Ir~forn~ation Processing Letters 11 (1980), 84-86 13 A C Yao, "Uniform Hashing Is Optimal," Journal of tlze ACM 32 ( 1985), 687-693
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A Priority Queue: The Binary Heap
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The priority queue is a fundamental data structure that allows access only to the minimum item In this chapter we discuss one implementation of the priority queue data structure, the elegant b i n a q heap The binary heap supports the insertion of new items and the deletion of the minimum item in logarithmic worst-case time It uses only an array and is easy to implement In this chapter, we show: the basic properties of the binary heap, how the insert and deleteMin operations can be performed in logarithmic time, a linear-time heap construction algorithm, a C++ implementation, using both the generic and STL protocols in Section 79, an easily implemented sorting algorithm heapsort, that runs in O(N log N ) time but uses no extra memory, and the use of heaps to implement external sorting
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