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CS- 514 Final Project

How circular arrays behave under successive rounds of uniform insertions and deletions Di o go Andrade G á bor Rudolf. CS- 514 Final Project. Experiments. Consider a circular array: Define the number of elements (n). Define the load factor r = n / array size.

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CS- 514 Final Project

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  1. How circular arrays behave under successive rounds of uniform insertions and deletions Diogo Andrade Gábor Rudolf CS-514 Final Project

  2. Experiments • Consider a circular array: • Define the number of elements (n). • Define the load factor r =n / array size. • Define the number of rounds of insertions & deletions (I). • Initialize the array with n elements, and then run for I iterations consisting of one insertion and deletion each. The positions for insertion and the deleted elements are selected uniformly.

  3. Data generated by experiments • Distribution, average and standard deviation of the number of shifts per iteration • Number of clusters • Distribution of cluster sizes in the “stationary” state • Measure of clustering (sum of the log of gaps)

  4. Results: Convergence • The number of clusters per iteration and the gap measure converge after some iterations, independent of the array parameters and the how the array is initialized. • 3 different initializations: • Random • One big chunk • Successive insertions

  5. Results: Convergence

  6. Histogram of Shifts and Cluster Sizes

  7. Distribution of Shifts • The probability of having to make k shifts after an insertion can be determined by the sizes of the clusters: P[k shifts] = (# clusters with size >= k) / size

  8. Approximation by Geometric & Modified Geometric Distribution

  9. Results: Dependency on load factor • The number of shifts per iteration and the average number of clusters depend only on the load factor of the array. • The gap measure depends on the load factor and on the array size.

  10. Results: Shifts - Dependency on load factor

  11. Results: gap measure - dependency on load factor and size

  12. Hashing with Linear Probing • The experiment models the behavior of a dynamic hash table with open addressing using linear probing. • The static case was studied extensively, see for example Knuth, 1963. • We compare the long-term behavior with the static case as described by Knuth’s formulas.

  13. Comparison with static case • Expected number of shifts for inserting last element in static case (Knuth’s formula) • Long-term behavior in our experiment

  14. Future Work • Proving convergence results • Derive formulas for distribution, average and deviation of shifts • Further comparison with Knuth’s results • Analyze the time it takes to reach “stationary” state from different initial arrays (most importantly for successive insertions, which correspond to a hash table)

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