You are given an array nums
of distinct positive integers. You need to sort the array in increasing order based on the sum of the digits of each number. If two numbers have the same digit sum, the smaller number appears first in the sorted order.
Return the minimum number of swaps required to rearrange nums
into this sorted order.
A swap is defined as exchanging the values at two distinct positions in the array.
For example:
nums = [37,100]
[3 + 7 = 10, 1 + 0 + 0 = 1] -> [10, 1]
[100, 37]
. Swap 37
with 100
to obtain the sorted order.nums
is 1.nums = [22,14,33,7]
[2 + 2 = 4, 1 + 4 = 5, 3 + 3 = 6, 7 = 7] -> [4, 5, 6, 7]
[22, 14, 33, 7]
. The array is already sorted.nums
is 0.nums = [18,43,34,16]
[1 + 8 = 9, 4 + 3 = 7, 3 + 4 = 7, 1 + 6 = 7] -> [9, 7, 7, 7]
[16, 34, 43, 18]
. Swap 18
with 16
, and swap 43
with 34
to obtain the sorted order.nums
is 2.How would you implement an algorithm to find the minimum number of swaps to sort the array?
When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:
The brute force approach to sorting with swaps involves trying every possible combination of swaps to see if we can sort the numbers based on their digit sums. We'll exhaustively check each possibility, keeping track of the smallest number of swaps we've seen that results in the correct sorted order.
Here's how the algorithm would work step-by-step:
def minimum_swaps_to_sort_by_digit_sum(numbers):
minimum_number_of_swaps = float('inf')
number_of_numbers = len(numbers)
def calculate_digit_sum(number):
digit_sum = 0
for digit in str(number):
digit_sum += int(digit)
return digit_sum
def is_sorted_by_digit_sum(numbers):
for index in range(number_of_numbers - 1):
if calculate_digit_sum(numbers[index]) > calculate_digit_sum(numbers[index + 1]):
return False
return True
def find_minimum_swaps(current_numbers, current_swaps):
nonlocal minimum_number_of_swaps
# If the current number of swaps exceeds the minimum, stop.
if current_swaps >= minimum_number_of_swaps:
return
if is_sorted_by_digit_sum(current_numbers):
minimum_number_of_swaps = current_swaps
return
# We iterate through all possible pairs of numbers and swap them
for first_index in range(number_of_numbers):
for second_index in range(first_index + 1, number_of_numbers):
# Create a copy to avoid modifying the original list during recursion.
new_numbers = current_numbers[:]
new_numbers[first_index], new_numbers[second_index] = new_numbers[second_index], new_numbers[first_index]
# Recursively check from this new state
find_minimum_swaps(new_numbers, current_swaps + 1)
# Start searching from the initial state with zero swaps
find_minimum_swaps(numbers[:], 0)
if minimum_number_of_swaps == float('inf'):
return -1
else:
return minimum_number_of_swaps
The key idea is to arrange the numbers in the correct order based on their digit sums with the fewest possible swaps. We achieve this by identifying cycles of misplaced numbers and resolving each cycle with a specific number of swaps.
Here's how the algorithm would work step-by-step:
def minimum_swaps_to_sort_by_digit_sum(numbers):
def calculate_digit_sum(number):
digit_sum = 0
for digit in str(number):
digit_sum += int(digit)
return digit_sum
digit_sums = [calculate_digit_sum(number) for number in numbers]
# Create pairs of (digit_sum, index) to preserve original positions.
digit_sum_index_pairs = list(enumerate(digit_sums))
# Sort based on digit sums, keeping track of original indices.
digit_sum_index_pairs.sort(key=lambda x: x[1])
sorted_indices = [index for index, _ in digit_sum_index_pairs]
number_of_swaps = 0
visited = [False] * len(numbers)
# Iterate through each index to find and process cycles.
for i in range(len(numbers)):
# Skip if already visited or correctly placed.
if visited[i] or sorted_indices[i] == i:
continue
cycle_size = 0
current_index = i
# Traverse the cycle until we return to the starting index.
while not visited[current_index]:
visited[current_index] = True
current_index = sorted_indices[current_index]
cycle_size += 1
# Each cycle needs cycle_size - 1 swaps to be resolved.
if cycle_size > 0:
number_of_swaps += (cycle_size - 1)
return number_of_swaps
Case | How to Handle |
---|---|
Null or empty input array | Return 0, as no swaps are needed for an empty or null array. |
Input array with only one element | Return 0, as a single-element array is already sorted. |
Input array with all elements having the same digit sum | Return 0, as the array is effectively already sorted based on the defined criteria. |
Large input array exceeding memory limits | Ensure the algorithm utilizes memory efficiently and consider using iterative approaches to avoid stack overflow errors. |
Input array containing very large numbers (potential integer overflow when calculating digit sum) | Use appropriate data types (e.g., long) to store the digit sum to prevent integer overflow. |
Input array already sorted according to digit sum | The algorithm should correctly identify this scenario and return 0 swaps. |
Duplicate numbers in the array with differing original positions | Use an appropriate data structure (e.g., hash map) to maintain the correct index associations after sorting based on digit sum. |
Arrays of numbers that have the same digit sum | Handle these numbers based on their original array position to avoid incorrect swaps. |