Algorithms

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Algorithms

Sorting

Given: compare(a,b) to compare 2 numbers. Goal: Sort a list of numbers from smallest to largest.

\(\displaystyle O(n^2)\) Algorithms

Bubble Sort

Insertion Sort

Basic Idea:

  • Maintain a subarray which is always sorted

Algorithm:

  • Grab a number from the unsorted portion and insert it into the sorted portion
  • Repeat until no more numbers are in the unsorted subarray

Selection Sort

Basic Idea:

  • Maintain a subarray which is always sorted.
  • Every number in the sorted subarray will be smaller than the smallest number in the unsorted subarray.

Algorithm:

  • Find the smallest number in the unsorted portion and insert it into the sorted portion
  • Repeat until no more numbers are in the unsorted subarray.

Quicksort

Basic Idea:

  • Divide an conquer.
  • Very fast. Average case O(nlogn).
  • Good for multithreaded systems.

Algorithm:

\(\displaystyle O(n \log n)\) Algorithms

Merge Sort

Heap Sort

Linear Algorithms

Counting Sort

Radix Sort

Selection

Given: compare(a,b) to compare 2 numbers. A number k. Goal: Return the k'th largest number.

Median finding

Reference

  • See CLRS
  • Idea: Reinterpret your data as a 2D array of size 5 x (n/5)
  • Find the median of each column of 5 elements
  • Sort the columns by their medians
  • Now you can eliminate the upper left (1/4) of elements and the lower right (1/4) of elements
  • Recursively iterate on the remaining (1/2) of elements
  • Each iteration takes O(n). Consecutive iterations are on n/2 data so we have \(\displaystyle O(n) + O(n/2) + ... = O(n)\)
  • Worst Case \(\displaystyle O(n)\)

Quickselect

  • Worst Case \(\displaystyle O(n^2)\)
  • Average Case \(\displaystyle O(n)\)
Notes
  • Using a good \(\displaystyle O(n)\) pivot finding algorithm (See finding the median in O(n)) will reduce the worst case to \(\displaystyle O(n)\)

Graph Algorithms

Greedy

Dynamic Programming

Search

Binary Search