# Princeton University algorithm course algorithm part I mergesort Merge Sorting

Source: Internet
Author: User

Origin: von noriman was first implemented on edvac.

Basic Idea:

• Divide an array into (divide array into two halves)
• Recursive sorting of each part (recursively sort each half)
• Merge two parts (merge two halves)

Merging and sorting reflects the idea of division and governance (divide and conquer)

Demo:

1. Given the original array a [], the sub-arrays from Lo to mid, Mid + 1 to Hi of this array are sorted separately.

2. Copy the array to the auxiliary array (auiiary array), subscripts I and j For the first element of the two parts, respectively, and subscripts K for the first element of the original array

3. Compare the subscripts of I and J, assign a smaller value to the element at the position of K subscripts, and then increment the subscripts of K and value assignment;
In this demonstration, the element of the J subject is relatively small, so a is assigned to the K position, and then increments K and J, that is, J + 1, k + 1

4. Repeat the above process until the comparison is complete.

Implementation in Java

`Public class merge {Private Static void Merge (comparable [] A, comparable [] aux, int Lo, int mid, int hi) {assert issorted (A, lo, mid ); // check a [lo .. mid] whether to order assert issorted (A, Mid + 1, hi); // check a [Mid + 1 .. hi] whether to order for (int K = lo; k <= Hi; k ++) // copy the array aux [k] = A [k]; int I = Lo, j = Mid + 1; for (int K = lo; k <= Hi; k ++) {if (I> mid) A [k] = aux [J ++]; else if (j> Hi) A [k] = aux [I ++]; else if (less (Aux [J], aux [I]) A [k] = aux [J ++]; else a [k] = aux [I ++];} assert issorted (A, lo, hi);} Private Static void sort (comparable [] A, comparable [] aux, int low, int hi) {If (Hi <= LO) return; int mid = lo + (Hi-Lo)/2; sort (A, aux, lo, mid ); sort (A, aux, Mid + 1, hi); merge (A, aux, lo, mid, hi);} public static void sort (comparable []) {aux = new comparable [. length]; sort (A, aux, 0,. length-1 );}}`

Note: assert function: checks the value in the expression. If it is true, the program runs normally. If it is false, an exception is thrown to terminate the operation.

Performance analysis:

The algorithm complexity is N * log (n)

Optimization:

Problem: Additional memory needs to be opened based on array size N for Merge Sorting

In-Place algorithm (in-place algorithm): an algorithm that occupies less space than or equal to C log (n.

Insert sorting, select sorting, and Hill sorting are all in-situ algorithms. Merge Sorting is not an in-situ algorithm. Wiki reference

Kronrod found in-place merge in 1969, but it does not seem so useful (challenge for the bored)

Practical improvements)

Improvement 1: Use Insert sorting for small arrays

• Merging and sorting requires a lot of overhead for opening up small sub-arrays (opening up arrays not only occupies memory for elements, but also has fixed overhead for the array itself)
• When the sub-array size exceeds 7, stop (cutoff) Using Insert sort
`private static void sort(Comparable[] a, Comparable[] aux, int low, int hi){    if （hi <= lo + CUTOFF - 1)    {        Insertion.sort(a, lo, hi);        return;        }    int mid = lo + (hi - lo) / 2;    sort(a, aux, lo, mid);    sort(a, aux, mid + 1, hi);    merge(a, aux, lo, mid, hi);}`

Improvement 2: When the array is sorted properly, the computation is stopped.

• After the two parts have been sorted, if the last element in the first half is greater than the first element in the second half, the whole sequence is ordered.
`private static void sort(Comparable[] a, Comparable[] aux, int low, int hi){    if (hi <= lo) return;    int mid = lo + (hi - lo) / 2;    sort(a, aux, lo, mid);    sort(a, aux, mid + 1, hi);    if (!less(a[mid + 1], a[mid])) return;    merge(a, aux, lo, mid, hi);}`

Princeton University algorithm course algorithm part I mergesort Merge Sorting

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