### Abstract

Given a multiset X = {x_{1},⋯, x_{n}} of real numbers, the floating-point set summation (FPS) problem asks for S_{n} = x_{1} + ··· + x_{n}, and the floating point prefix set summation problem (FPPS) asks for S_{k} = x_{1} + ··· + X_{k} for all k = 1,⋯, n. Let E^{*} _{k} denote the minimum worst-case error over all possible orderings of evaluating Sk-We prove that if X has both positive and negative numbers, it is NP-hard to compute S_{n} with the worst-case error equal to E_{n}. We then give the first known polynomial-time approximation algorithm for computing S_{n} that has a provably small error for arbitrary X. Our algorithm incurs a worstcase error at most 2([log(n - 1)] + 1)E^{*} _{n.} ^{1} After X is sorted, it runs in O(n) time, yielding an O(n^{2})-time approximation algorithm for computing S_{k} for all k = 1,⋯, n such that the worst-case error for each Sk is less than 2[log(k - 1)1 + 1)E^{*} _{k}. For the case where X is either all positive or all negative, we give another approximation algorithm for computing S_{n} with a worst-case error at most [log log n]E^{*} _{n}. Even for unsorted X, this algorithm runs in 0(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most flog n] E_{n}, while E _{n} was known to be attainable in O(n log n) time using Huffman coding. Consequently, FPPS is solvable in O(n^{2}) time such that the worst-case error for each S_{k} is the minimum. To improve this quadratic time bound in practice, we design two on-line algorithms that calculate the next S_{k} by taking advantage of the current S _{k} and thus reduce redundant computation.

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - 25th International Colloquium, ICALP 1998, Proceedings |

Pages | 375-386 |

Number of pages | 12 |

State | Published - Dec 1 1998 |

Event | 25th International Colloquium on Automata, Languages and Programming, ICALP 1998 - Aalborg, Denmark Duration: Jul 13 1998 → Jul 17 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1443 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 25th International Colloquium on Automata, Languages and Programming, ICALP 1998 |
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Country | Denmark |

City | Aalborg |

Period | 7/13/98 → 7/17/98 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automata, Languages and Programming - 25th International Colloquium, ICALP 1998, Proceedings*(pp. 375-386). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1443 LNCS).

}

*Automata, Languages and Programming - 25th International Colloquium, ICALP 1998, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1443 LNCS, pp. 375-386, 25th International Colloquium on Automata, Languages and Programming, ICALP 1998, Aalborg, Denmark, 7/13/98.

**Efficient minimization of numerical summation errors.** / Kao, Ming Yang; Wang, Jie.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Efficient minimization of numerical summation errors

AU - Kao, Ming Yang

AU - Wang, Jie

PY - 1998/12/1

Y1 - 1998/12/1

N2 - Given a multiset X = {x1,⋯, xn} of real numbers, the floating-point set summation (FPS) problem asks for Sn = x1 + ··· + xn, and the floating point prefix set summation problem (FPPS) asks for Sk = x1 + ··· + Xk for all k = 1,⋯, n. Let E* k denote the minimum worst-case error over all possible orderings of evaluating Sk-We prove that if X has both positive and negative numbers, it is NP-hard to compute Sn with the worst-case error equal to En. We then give the first known polynomial-time approximation algorithm for computing Sn that has a provably small error for arbitrary X. Our algorithm incurs a worstcase error at most 2([log(n - 1)] + 1)E* n. 1 After X is sorted, it runs in O(n) time, yielding an O(n2)-time approximation algorithm for computing Sk for all k = 1,⋯, n such that the worst-case error for each Sk is less than 2[log(k - 1)1 + 1)E* k. For the case where X is either all positive or all negative, we give another approximation algorithm for computing Sn with a worst-case error at most [log log n]E* n. Even for unsorted X, this algorithm runs in 0(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most flog n] En, while E n was known to be attainable in O(n log n) time using Huffman coding. Consequently, FPPS is solvable in O(n2) time such that the worst-case error for each Sk is the minimum. To improve this quadratic time bound in practice, we design two on-line algorithms that calculate the next Sk by taking advantage of the current S k and thus reduce redundant computation.

AB - Given a multiset X = {x1,⋯, xn} of real numbers, the floating-point set summation (FPS) problem asks for Sn = x1 + ··· + xn, and the floating point prefix set summation problem (FPPS) asks for Sk = x1 + ··· + Xk for all k = 1,⋯, n. Let E* k denote the minimum worst-case error over all possible orderings of evaluating Sk-We prove that if X has both positive and negative numbers, it is NP-hard to compute Sn with the worst-case error equal to En. We then give the first known polynomial-time approximation algorithm for computing Sn that has a provably small error for arbitrary X. Our algorithm incurs a worstcase error at most 2([log(n - 1)] + 1)E* n. 1 After X is sorted, it runs in O(n) time, yielding an O(n2)-time approximation algorithm for computing Sk for all k = 1,⋯, n such that the worst-case error for each Sk is less than 2[log(k - 1)1 + 1)E* k. For the case where X is either all positive or all negative, we give another approximation algorithm for computing Sn with a worst-case error at most [log log n]E* n. Even for unsorted X, this algorithm runs in 0(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most flog n] En, while E n was known to be attainable in O(n log n) time using Huffman coding. Consequently, FPPS is solvable in O(n2) time such that the worst-case error for each Sk is the minimum. To improve this quadratic time bound in practice, we design two on-line algorithms that calculate the next Sk by taking advantage of the current S k and thus reduce redundant computation.

UR - http://www.scopus.com/inward/record.url?scp=0004461454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0004461454&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0004461454

SN - 3540647813

SN - 9783540647812

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 375

EP - 386

BT - Automata, Languages and Programming - 25th International Colloquium, ICALP 1998, Proceedings

ER -