The document provides a detailed explanation of the merge sort algorithm as described in 'Cracking the Coding Interview' by Gayle Laakmann McDowell. It illustrates the merge process, including the preparation of the helper array, the comparison of elements, and the copying of remaining elements, through a step-by-step walkthrough with code examples. The document emphasizes how the merge function operates within the context of the overall merge sort algorithm.

1. Merge Sort algorithm
Illustrated walkthrough

2. Reference
“Cracking the coding interview” Fifth edition.
Gayle Laakmann McDowell, 2008 - 2013

3. Merge function
This function does the most of the heavy lifting,
so we look at it first, then see it in the context of
Merge Sort algorithm

4. Merge Function
for (int i = begin; i <= last; i++) {
helper[i] = array[i];
}
Part #1: prepare helper
int left = begin;
int right = middle + 1;
int storeIndex = begin;
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
&& right <= last) {
helper[right]) {
= helper[left];
Part #2: pick smaller and
copy to the target
= helper[right];
storeIndex++;
}
int remainder = middle - left;
for (int i = 0; i <= remainder; i++) {
array[storeIndex + i] = helper[left + i];
}
Part #3: copy any
remainder from left
(right not necessary)

5. begin
middle
last
[0]
array
[1]
[2]
[3]
10
30
20
40
left sub-array
already sorted within
the sub-array
left sub-array
{begin..middle}
right sub-array
already sorted within
the sub-array
right sub-array
{middle+1..last}

6. [0]
array
[1]
[2]
[3]
10
30
20
40
helper
for (int i = begin; i <= last; i++) {
helper[i] = array[i];
}

7. [0]
[1]
[2]
[3]
array
10
30
20
40
helper
10
30
20
40
for (int i = begin; i <= last; i++) {
helper[i] = array[i];
}

8. begin
middle
last
[0]
[2]
[3]
10
array
[1]
30
20
40
10
30
20
40
left
helper
int left = begin;
int right = middle + 1;
int storeIndex = begin;

9. [0]
[2]
[3]
10
array
[1]
30
20
40
20
40
right
left
helper
10
30
int left = begin;
int right = middle + 1;
int storeIndex = begin;

10. store
Index
[0]
[2]
[3]
10
array
[1]
30
20
40
20
40
right
left
helper
10
30
int left = begin;
int right = middle + 1;
int storeIndex = begin;

11. 0 <= 1
is true
store
Index
[0]
[2]
[3]
10
array
[1]
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];
2 <= 3
is true

12. 10 <= 20
is true
store
Index
[0]
[2]
[3]
10
array
[1]
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

13. store
Index
[0]
[2]
[3]
10
array
[1]
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

14. store
Index
[0]
array
[1]
[2]
[3]
10
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

15. store
Index
[0]
array
[1]
[2]
[3]
10
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

16. 1 <= 1
is true
store
Index
[0]
array
[1]
[2]
[3]
10
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];
2 <= 3
is true

17. 30 <= 20
is false
store
Index
[0]
array
[1]
[2]
[3]
10
30
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

18. store
Index
[0]
array
[1]
[2]
[3]
10
20
20
40
20
40
right
left
helper
10
30
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

19. store
Index
[0]
array
[1]
[2]
[3]
10
20
20
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

20. store
Index
[0]
array
[1]
[2]
[3]
10
20
20
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

21. 1 <= 1
is true
store
Index
[0]
array
[1]
[2]
[3]
10
20
20
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];
3 <= 3
is true

22. 30 <= 40
is true
store
Index
[0]
array
[1]
[2]
[3]
10
20
20
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

23. store
Index
[0]
array
[1]
[2]
[3]
10
20
30
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

24. store
Index
[0]
array
[1]
[2]
[3]
10
20
30
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

25. store
Index
[0]
array
[1]
[2]
[3]
10
20
30
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

26. 2 <= 1
is false
3 <= 3
is true
store
Index
[0]
array
[1]
[2]
[3]
10
20
30
40
right
left
helper
10
30
20
while (left <= middle
if (helper[left] <=
array[storeIndex]
left++;
} else {
array[storeIndex]
right++;
}
Exit while loop
storeIndex++;
}
40
&& right <= last) {
helper[right]) {
= helper[left];
= helper[right];

27. store
Index
[0]
[2]
[3]
10
array
[1]
20
30
40
right
left
helper
1 - 2 = -1
remainder
-1
10
30
20
40
int remainder = middle - left;
for (int i = 0; i <= remainder; i++) {
array[storeIndex + i] = helper[left + i];
}

28. store
Index
[0]
[2]
[3]
10
array
[1]
20
30
40
right
left
helper
0 <= -1
is false
remainder
-1
10
30
20
40
int remainder = middle - left;
for (int i = 0; i <= remainder; i++) {
array[storeIndex + i] = helper[left + i];
}
Skip over for loop

29. store
Index
[0]
array
[1]
[2]
[3]
10
20
30
40
right
left
helper
10
30
Already there because
right sub-array already
occupies tail end of the
target array
20
40
int remainder = middle - right;
for (int i = 0; i <= remainder ; i++) {
array[storeIndex + i] = helper[ right + i];
}
Not needed

30. Merge Sort Algorithm
Now we use Merge function

31. begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

32. 0 >= 3
is false
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

33. begin
last
middle
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

34. Merge & Sort the
left sub-array first
begin
Call Stack #0
last
middle
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

35. Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

36. 0 >= 1
is false
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

37. Call Stack #1
Call Stack #0
middle
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

38. Call Stack #1
Call Stack #0
middle
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

39. Call Stack #3
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

40. 0 >=0
is true
Call Stack #3
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

41. 0 >=0
is true
Call Stack #3
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
Base condition is met!
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

42. Call Stack #1
Call Stack #0
middle
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

43. 1 >=1
is true
Call Stack #3
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

44. Call Stack #3
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
Base condition is met!
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

45. Call Stack #1
Call Stack #0
middle
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
Merge
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

46. Already sorted
(unchanged but sorted)
Call Stack #1
Call Stack #0
middle
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);
(implicit return at the end of function)

47. Merge & Sort the
right sub-array next
Call Stack #0
begin
last
middle
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);

48. Walkthrough ends here.
The right sub-array is processed the
same way as the left sub-array.
After that, Merge is called with two
already-sorted sub-arrays
Call Stack #1
Call Stack #0
begin
last
[0]
[1]
[2]
[3]
5
7
3
2
if (begin >= last) {
return;
}
int middle = (begin + last) / 2;
MergeSort(array, helper, begin, middle);
MergeSort(array, helper, middle + 1, last);
Merge(array, helper, begin, middle, last);