Python Program - Maximum Subarray Problem


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Kadane's algorithm is used to find the maximum sum of a contiguous subarray. Kadane's algorithm is based on the idea of looking for all positive contiguous subarray and find the maximum sum of a contiguous subarray.

In this algorithm, a variable called max_sum is created to store maximum sum of the positive contiguous subarray till current iterated element and a variable called current_sum is created to store sum of the positive subarray which ends at current iterated element. In each iteration, current_sum is compared with max_sum, to update max_sum if it is greater than max_sum.

Example:

To understand the kadane's algorithm, lets consider an array $$Array = [-3, 1, -8, 12, 0, -3, 5, -9, 4]$$ and discuss each step taken to find the maximum sum of all positive contiguous subarray.

Insertion Sort

  max_sum = current_sum = 0

  Step 1: i = 0, Array[0] =  -3
  current_sum = current_sum + (-3) = -3
  Set current_sum = 0 because current_sum < 0

  Step 2: i = 1, Array[0] =  1
  current_sum = current_sum + 1 = 1
  update max_sum = 1 because current_sum > max_sum

  Step 3: i = 2, Array[0] =  -8
  current_sum = current_sum + (-8) = -7
  Set current_sum = 0 because current_sum < 0

  Step 4: i = 3, Array[0] =  12
  current_sum = current_sum + 12 = 12
  update max_sum = 12 because current_sum > max_sum

  Step 5: i = 4, Array[0] =  0
  current_sum = current_sum + 0 = 12

  Step 6: i = 5, Array[0] =  -3
  current_sum = current_sum + (-3) = 9

  Step 7: i = 6, Array[0] =  5
  current_sum = current_sum + 5 = 14
  update max_sum = 14 because current_sum > max_sum

  Step 8: i = 7, Array[0] =  -9
  current_sum = current_sum + (-9) = 5

  Step 9: i = 8, Array[0] =  4
  current_sum = current_sum + 4 = 9

Hence, after all iterations, the value of max_sum is 14. The stating index point and end index point of this subarray are 3 and 6 respectively.

# function for kadane's algorithm
def kadane(MyList):
  max_sum = 0
  current_sum = 0
  for i in MyList: 
    current_sum = current_sum + i
    if current_sum < 0:
      current_sum = 0
    if max_sum < current_sum:
      max_sum = current_sum
  return max_sum
  
# test kadane's algorithm code                 
MyList = [-3, 1, -8, 12, 0, -3, 5, -9, 4]
print("Maximum SubArray is:",kadane(MyList))

Output

Maximum SubArray is: 14

To get the location of maximum subarray, variables max_start and max_end are maintained with the help of variables current_start and current_end.

# function for kadane's algorithm
def kadane(MyList):
  max_sum = 0
  current_sum = 0

  max_start = 0
  max_end = 0
  current_start = 0
  current_end = 0

  for i in range(len(MyList)): 
    current_sum = current_sum + MyList[i]
    current_end = i
    if current_sum < 0:
      current_sum = 0
      # Start a new sequence from next element
      current_start = current_end + 1

    if max_sum < current_sum:
      max_sum = current_sum
      max_start = current_start
      max_end = current_end
      
  print("Maximum SubArray is:", max_sum)
  print("Start index of max_Sum:", max_start)
  print("End index of max_Sum:", max_end)
  
# test kadane's algorithm code                 
MyList = [-3, 1, -8, 12, 0, -3, 5, -9, 4]
kadane(MyList)

Output

Maximum SubArray is: 14
Start index of max_Sum: 3
End index of max_Sum: 6

Time Complexity:

The time complexity of Kadane's algorithm is $$\mathcal{O}(N)$$.


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