Elementary Sorts

Intersection of two sets

The solution to this problem is to carry out an efficient sorting, after which check the sorted arrays for matches.

Permutation

An even simpler task, here you need to sort two arrays and compare them, for sorting you can take any subquadratic algorithm.

Dutch national flag

his problem can also be viewed in terms of rearranging elements of an array. Suppose each of the possible elements could be classified into exactly one of three categories (bottom, middle, and top). For example, if all the elements are in 0 ... 1, the bottom could be defined as elements in 0 ... 0.25 (not including 0.25), the middle as 0.25 ... 0.5 (not including 0.5) and the top as 0.5 and greater. (The choice of these values illustrates that the categories need not be equal ranges). The problem is then to produce an array such that all "bottom" elements come before (have an index less than the index of) all "middle" elements, which come before all "top" elements.

One algorithm is to have the top group grow down from the top of the array, the bottom group grow up from the bottom, and keep the middle group just above the bottom. The algorithm indexes three locations, the bottom of the top group, the top of the bottom group, and the top of the middle group. Elements that are yet to be sorted fall between the middle and the top group.[4] At each step, examine the element just above the middle. If it belongs to the top group, swap it with the element just below the top. If it belongs in the bottom, swap it with the element just above the bottom. If it is in the middle, leave it. Update the appropriate index. Complexity is Θ(n) moves and examinations.

The full description is located in wikipedia.