Kolmogorov

Complexity of a sequence = The shortest algorithm that produces it.

Martin-Löf

• A sequence is random if it passes all statistical tests.
• It cannot be produced by a program shorter than itself.

The digits of $\pi$ are not random in this sense.

Not just “difficult to compute”, there is no consistent way to define the shortest algorithm.

It’s impossible to find a way to ensure that a sequence is random. It’s similar to the halting problem in a way.

My Idea

It might be possible to define an algorithm in terms of a Turing machine. If the instructions of a Turing machine are shorter than the sequence itself, it’s not random.

Another idea

It says $P(b_n | s_{n-1}) = \frac{1}{2}$ for a binary sequence if the random process is ideal. But expecting this might be against randomness.

Pseudorandomness

Producing seemingly random sequences by algorithms, given a secret key.

The sequence is determined and can be inferred with high computing power. But for practical purposes, it’s random.

Uses of Randomness

• Randomized Algorithms: When a definite algorithm is too costly, it’s possible to run randomized algorithms for e.g. checking whether a number is prime.
• Cryptography: When a message needs to be encrypted with a secure key, the most secure way to produce a key is randomness.

Keywords:

• dominant points
• consecutive vectors
• toothbrush shape
• distance metric
• smallest perpendicular distance

Q1: How usual calculation of distance is done?

Minor DPs are deleted in approximation. A minor DP is a DP where the perpendicular distance between the point and the straight line is minimum.

a a
 b

Here b is deleted when its distance to the line a-a is minimum.

The perpendicular distance is calculated using

[ d_i = \sqrt{\frac{((x_i - x_a) (y_b - y_a) - (y_i - y_a) (x_b -x_a))^2}{(x_a - x_b)^2 + (y_a - y_b)^2}} ]

for lines between points $p_a$ and $p_b$ and the point $p_i$.

Q2: What is a toothbrush shape?

It’s something like

aaaaaa
bbbbbbbbbbbbbbbbbbbbbb

Hence the toothbrush.

Though I don’t get why is this particularly important.

Q3: Which information is included in distance metric?

Angle acuteness is added to the information described above.

Q4: How the distance metric differs from others?

It includes angle acuteness in the metric and the more acute the angle, the less likely it’s removed from DP set.

Q5: What’s baseline for performance and how does this improve it?

As the number of DPs decrease, RMSE of the new metric decreases. For large number of DPs it doesn’t matter much. (So performance penalty may not pay off)