On Solving Recurrences
博士 === 國立臺灣大學 === 資訊工程學研究所 === 89 === It is inevitable to solve a recurrence for algorithm evaluation. Knuth introduces many famous concrete methods. Among them, generating function is the most powerful, but a large portion of them are still basic tricks...
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| Format: | Others |
| Language: | en_US |
| Published: |
2001
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| Online Access: | http://ndltd.ncl.edu.tw/handle/14822225166992325004 |
| Summary: | 博士 === 國立臺灣大學 === 資訊工程學研究所 === 89 === It is inevitable to solve a recurrence for algorithm evaluation.
Knuth introduces many famous concrete methods.
Among them, generating function is the most powerful,
but a large portion of them are still basic tricks
such as summation, counting and induction.
We solve three recurrences in this work.
Basic tricks still dominate in those derivation.
Especially in chapter 4 we solve exactly a minimal equation
which generating function can not help.
It shows that basic techniques of summation still play an important role.
Chapter 2 studies the termination of Takeuchi function.
This fact is important for Takeuchi function to be an benchmark function.
We use induction to prove the fact.
Not only is our approach simpler and more intuitive but also
cant he proof be generalized to $m$-dimensional Takeuchi function.
Chapter 3 studies the average number of inversion of DHOT.
The derivation is guided by generation function but a sub-summation
which is complex and important to the final result is obtained by basic counting and summation.
Chapter 4 solves a minimal equation
which originates from a circuit problem.
Our solution is exact.
It is difficult to have an exact solution for a recurrence
with max and min operation mixed.
So our exact solution is very unique.
The method we use are induction and combinatorial tricks.
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