### Abstract

A probabilistic analysis is presented of certain pointer-based implementations of dictionaries, linear lists, and priority queues; in particular, simple list and d-heap implementations. Under the assumption of equiprobability of histories, i.e., of paths through the internal state space of the implementation, it is shown that the integrated space and time costs of a sequence of n supported operations converge as n → ∞ to Gaussian random variables. For list implementations the mean integrated spatial costs grow asymptotically as n^{2}, and the standard deviations of the costs as n^{ 3 2}. For d-heap implementations of priority queues the mean integrated space cost grows only as n^{2}√log n, i.e., more slowly than the worst-case integrated cost. The standard deviation grows as n^{ 3 2}. These asymptotic growth rates reflect the convergence as n → ∞ of the normalized structure sizes to datatype-dependent deterministic functions of time, as earlier discovered by Louchard. This phenomenon is clarified with the aid of path integrals. Path integral techniques, drawn from physics, greatly facilitate the computation of the cost asymptotics. This is their first application to the analysis of dynamic data structures.

Original language | English (US) |
---|---|

Pages (from-to) | 232-260 |

Number of pages | 29 |

Journal | Journal of Complexity |

Volume | 7 |

Issue number | 3 |

DOIs | |

State | Published - 1991 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Analysis
- Computational Mathematics

### Cite this

**A path integral approach to data structure evolution.** / Maier, Robert S.

Research output: Contribution to journal › Article

*Journal of Complexity*, vol. 7, no. 3, pp. 232-260. https://doi.org/10.1016/0885-064X(91)90035-V

}

TY - JOUR

T1 - A path integral approach to data structure evolution

AU - Maier, Robert S

PY - 1991

Y1 - 1991

N2 - A probabilistic analysis is presented of certain pointer-based implementations of dictionaries, linear lists, and priority queues; in particular, simple list and d-heap implementations. Under the assumption of equiprobability of histories, i.e., of paths through the internal state space of the implementation, it is shown that the integrated space and time costs of a sequence of n supported operations converge as n → ∞ to Gaussian random variables. For list implementations the mean integrated spatial costs grow asymptotically as n2, and the standard deviations of the costs as n 3 2. For d-heap implementations of priority queues the mean integrated space cost grows only as n2√log n, i.e., more slowly than the worst-case integrated cost. The standard deviation grows as n 3 2. These asymptotic growth rates reflect the convergence as n → ∞ of the normalized structure sizes to datatype-dependent deterministic functions of time, as earlier discovered by Louchard. This phenomenon is clarified with the aid of path integrals. Path integral techniques, drawn from physics, greatly facilitate the computation of the cost asymptotics. This is their first application to the analysis of dynamic data structures.

AB - A probabilistic analysis is presented of certain pointer-based implementations of dictionaries, linear lists, and priority queues; in particular, simple list and d-heap implementations. Under the assumption of equiprobability of histories, i.e., of paths through the internal state space of the implementation, it is shown that the integrated space and time costs of a sequence of n supported operations converge as n → ∞ to Gaussian random variables. For list implementations the mean integrated spatial costs grow asymptotically as n2, and the standard deviations of the costs as n 3 2. For d-heap implementations of priority queues the mean integrated space cost grows only as n2√log n, i.e., more slowly than the worst-case integrated cost. The standard deviation grows as n 3 2. These asymptotic growth rates reflect the convergence as n → ∞ of the normalized structure sizes to datatype-dependent deterministic functions of time, as earlier discovered by Louchard. This phenomenon is clarified with the aid of path integrals. Path integral techniques, drawn from physics, greatly facilitate the computation of the cost asymptotics. This is their first application to the analysis of dynamic data structures.

UR - http://www.scopus.com/inward/record.url?scp=33745687155&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745687155&partnerID=8YFLogxK

U2 - 10.1016/0885-064X(91)90035-V

DO - 10.1016/0885-064X(91)90035-V

M3 - Article

AN - SCOPUS:33745687155

VL - 7

SP - 232

EP - 260

JO - Journal of Complexity

JF - Journal of Complexity

SN - 0885-064X

IS - 3

ER -