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Algorithms First Review

Review

Incorrect complexity of NN

Incorrect diagram of O(n log n)

NN TSP Review Assignment

  1. How long to run on set of size 4? 10? 20?
  • 4: 400ms
  • 10: ~30seconds
  • at 11 the computer balked. I presume, 20 would be close to 30,000,000,000,000 seconds given the way factorial problems scale
  1. How long to run NN on same sets?
  • 10 was usually a little faster than 4 - 0xx miliseconds - so I presume this is due to system time in getting a process stopped and started?
  • 20 didn't take much longer than 10.
  1. How close is NN's solution to the optimal?
  • the distances I was getting were usually less than 5% difference and as I have read, NN gets within 25% of the theoretical limit.
  1. How can you improve NN?
  • By starting at a pre-determined city instead of first calculating the shortest distance between any two cities
  • Minimize the possible number of cities to check prior to running the nearest neighbor algorithm, eliminating the majority of cities which would be likely to be farther than what could be considered "nearest" - though I suppose this would involve an exhaustive comparison of neighboring cities... but once that is done, stick the results in a look up table.

kNN TSP assignment

kNN

k = number of neighbors to explore

Terms

  • Computability

Computability

  • Tractability

Tractability refers to the question as to whether a problem is easily solved in practice. Algorithms that find optimal solutions become intractable quickly on large datasets. In order to find optimal solutions tractably, complicated and unintuitive algorithms have been developed. In order to find near-optimal solutions quickly, random or partial algorithms have been created.

  • Reduction of complexity of specific algorithms

FFT reduces from O(n^2) to O(n log n) via algorithm. This doesn't prove that every O(n^2) algorithm can be reduced to O(n log n).

Finding the nearest pair of points among a set of 2d points is obviously O(n^2), but using a divide and conquer algorithm reduces the complexity to O(n log n) as well.

Finding the closest pair of points: Obviously O(n^2), can be simplified to O(n log n), then again to O(n) with randomization.

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