My Algorithm for Knuth's TAOCP 5.1.1-6 | Performance Analysis

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In summary, Knuth's TAOCP 5.1.1-6 is a section in his book "The Art of Computer Programming" that focuses on performance analysis and algorithms. An algorithm is a set of instructions for solving a problem, and performance analysis helps evaluate efficiency and identify areas for improvement. The purpose of my algorithm is to analyze the performance of algorithms in Knuth's TAOCP 5.1.1-6 in terms of time complexity. However, it may not be applicable to all algorithms and may not accurately reflect real-world performance.
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Monte_Carlo
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Hi,

Knuth's TAOCP exercise 5.1.1-6 asks to construct an n-log-n algorithm which would generate an inversion table corresponding to a particular permutation.

My algorithm constructs inversion table for a random permutation of 1..N numbers (Knuth, Volume 3, exercise 5.1.1-6) using binary search tree where each node has two additional fields.


I'm trying to understand if my algorithm has complexity n-log-n not just theoretically but by observing performance data. Attached is the algorithm itself (Windows, C++, MinGW) and the resultant dataset which has two columns - first is size of the permutation and the second is the time it took to construct inversion table. I would appreciate if anybody could look into it, plot it in Excel and see if the curve in fact looks like it's n-log-n (I don't see it that way.)


Why am I seeing spikes in my data? Why is there this abrupt drop at datapoint 2070? If it has to do with the cache, can you suggest other programs/experiments clearly demonstrating how exactly spikes are related to cache? My processor is i7 quad core Bloomfield, three levels of cache, all 64 byte line size.

Thanks,

Monte
 

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Dear Monte,

Thank you for sharing your algorithm and dataset with us. I have taken a look at your data and plotted it in Excel, and I can see the spikes and the drop at datapoint 2070 that you mentioned. However, I do not think that these spikes and the drop are related to cache.

Firstly, let me explain why I do not think your algorithm has a complexity of n-log-n. In order for an algorithm to have a complexity of n-log-n, it must have a runtime that is proportional to n multiplied by the logarithm of n. Looking at your data, I can see that as the size of the permutation (n) increases, the time it takes to construct the inversion table also increases, but not in a linear or logarithmic way. In fact, the time seems to increase at a much faster rate, which does not align with the n-log-n complexity. Therefore, I do not believe your algorithm has a complexity of n-log-n.

As for the spikes and the drop in your data, I believe these are simply due to random variations in the runtime. It is common to see small spikes and drops in data, especially when dealing with large numbers. These variations could be caused by a number of factors, such as the operating system, other processes running on your computer, or even the way your algorithm is implemented. They do not necessarily indicate a problem with your algorithm or the cache. In fact, I do not think the cache has much of an impact on your data at all.

To better understand the relationship between spikes and the cache, I suggest conducting experiments specifically designed to test the cache. For example, you could write a program that performs the same operation multiple times and measures the runtime, and then change the order of the operations to see if it affects the runtime. You could also try changing the size of the data being processed and see if that has an impact on the runtime. These experiments will give you a better understanding of how the cache works and how it can affect performance.

I hope this helps answer your questions. Keep up the good work with your algorithm and experiments!


 

What is Knuth's TAOCP 5.1.1-6?

Knuth's TAOCP 5.1.1-6 refers to a specific section of Donald Knuth's book "The Art of Computer Programming" that deals with performance analysis and algorithms.

What is an algorithm?

An algorithm is a set of step-by-step instructions for solving a problem or completing a task.

Why is performance analysis important?

Performance analysis helps to evaluate the efficiency and effectiveness of an algorithm and can help identify areas for improvement.

What is the purpose of your algorithm for Knuth's TAOCP 5.1.1-6?

The purpose of my algorithm is to provide a method for analyzing the performance of algorithms discussed in Knuth's TAOCP 5.1.1-6, specifically in terms of time complexity.

What are the limitations of your algorithm?

My algorithm may not be applicable to all algorithms and may not accurately reflect real-world performance due to factors such as hardware and external dependencies.

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