### Abstract

A simple variant of a priority queue, called a soft heap, is introduced. The data structure supports the usual operations: insert, delete, meld, and findmin. Its novelty is to beat the logarithmic bound on the complexity of a heap in a comparison-based model. To break this information-theoretic barrier, the entropy of the data structure is reduced by artificially raising the values of certain keys. Given any mixed sequence of n operations, a soft heap with error rate ε (for any 0 < ε ≤ 1/2) ensures that, at any time, at most εn of its items have their keys raised. The amortized complexity of each operation is constant, except for insert, which takes O(log 1/ε) time. The soft heap is optimal for any value of ε in a comparison-based model. The data structure is purely pointer-based. No arrays are used and no numeric assumptions are made on the keys. The main idea behind the soft heap is to move items across the data structure not individually, as is customary, but in groups, in a data-structuring equivalent of "car pooling." Keys must be raised as a result, in order to preserve the heap ordering of the data structure. The soft heap can be used to compute exact or approximate medians and percentiles optimally. It is also useful for approximate sorting and for computing minimum spanning trees of general graphs.

Original language | English (US) |
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Pages (from-to) | 1012-1027 |

Number of pages | 16 |

Journal | Journal of the ACM |

Volume | 47 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2000 |

### All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence

### Keywords

- Amortization
- Heap
- Priority queue
- Soft heap