运行时间的黄金分割率或Pi

在许多地方，数字$\pi$和$(1+\sqrt5)/2$出现。我很好奇有关运行时间包含指数的黄金分割率或算法$\pi$。

这是基体，而不是指数，但有一个 FPT时间约束$O(\varphi^k n^2)$

”一种有效的固定参数可动算法，用于单侧交叉最小化 ”，维达·杜伊莫维奇（Vida Dujmovic），苏·怀特塞德斯（Sue Whitesides），Algorithica 40：15–31，2004年。

另外，它是一个下限而不是一个上限，但是：

“ 一种下界时间以模拟一个队列或由一个磁带2家下推商店$n^{1.618}$ ”，保MBVitányi，INF中。程序 来吧 1985年21：147-152。

最后，一个我一直在试图寻找，当我遇到的另外两个跑：在火腿三明治树，在三角范围查询计算几何一个现在已经过时的数据结构，具有查询时间。因此，黄金分割率恰好在指数中，但是具有对数而非本身。数据结构是将平面分层划分成凸形单元的结构，具有二叉树的整体结构，其中每个单元格及其在树中的兄弟姐妹都用火腿三明治切块进行了分区。查询时间由递归Q （n ）= Q $O(n^{\log_2\varphi})\approx O(n^{0.695})$，具有上述解决方案。它的描述（名称更无聊）由$Q(n)=Q(\frac{n}{2})+Q(\frac{n}{4})+O(\log n)$

“ 线性空间中的半平面范围搜索和查询时间$O(n^{0.695})$ ”，Herbert Edelsbrunner，Emo Welzl，Inf。程序 来吧 23：289–293，1986年。

（根据我上面的评论）

SAT可解性的Fortnow和Melkebeek时间/空间下界（时间和n o （1 ）空间）包含指数中的黄金比例；但是Ryan Williams后来对此进行了改进。$n^{\phi - \epsilon}$$n^{o(1)}$

Also in the base rather than the exponent: the Monien-Speckenmeyer algorithm for 3-SAT has a running time of $\varphi^n\cdot O(n)$. That was the first non-trivial upper bound for 3-SAT.

Another example of $\varphi$ in the base is an algorithm by Andreas Björklund and Thore Husfeldt to compute the parity of the number of directed Hamiltonian cycles, which runs in time $O(\varphi^n)$.

http://arxiv.org/abs/1301.7250

Also in the base: The deletion–contraction algorithm (Zykov, 1949) for computing the number of graph colourings runs in time $O(\phi^{|E|+|V|})$. This is a very canonical example of how the golden ratio appears from a Fibonacci recurrence for the running time of evaluating a natural recursive formula; I’m sure it’s the oldest.

Mikko Koivisto found a $O(\phi^{|V|})$ algorithm for computing the number of perfect matchings (IWPEC 2009).

Golden ration in the base: A very recent FPT algorithm by Kociumaka and Pilipczuk, Faster deterministic Feedback Vertex Set computes a FVS of size $k$ in $O^*\left((2 + \phi)^k\right)$ time. (They then improves their algorithm to run in time $O^*(3.592^k)$.)

to expand on Martin Bergers comment: the ancient Euclidean GCD algorithm runs in worst case time on two successive elements from the Fibonacci sequence. more details on wikipedia which also states:

This proof, published by Gabriel Lamé in 1844, represents the beginning of computational complexity theory,[93] and also the first practical application of the Fibonacci numbers.[91]

technically the GCD algorithm runs in logarithmic time $O(\log(n))$ but the golden ratio shows up in the number of steps of the algorithm.

[1] what is the time complexity of Euclids algorithm, math.se