text: # Codility - Aluminium 2014 - Part 1
  
  ## Introduction
  
  This is an analysis of Codility - Aluminium 2014 Challenge, Part 1.
  
  • The problem is, finding the maximum sum of a compact subsequence of array elements after performing a single swap operation. It’s a little tricky maximal subarray problem.
  • For example, given an array {3, -10, 4, 5}, we can swap 3 and -10 to get a compact subsequence {3, 4, 5}, which has the maximum sum 12
  • Time bound and space bound are both O(n)
  • There exists many algorithms to solve maximal subarray problems, but they cannot directly applied to this problem.
  
  ## Analysis
  
  • First, we will take a look at the solution of a CLASSIC maximal subarray problem, which means swap is not allowed.
  
  For a given array $a_0, a_1, \cdots, a_n-1$,
  
  The maximum sum of a compact subsequence can be expressed by the following formula.
  
  Beca\$use we want the maximal part from $i$ to $j$, by thinking reversely, we can calculate the minimal partial sum of sub array $a_0, a_1, \cdots, a_x-1$ as $f(x)$, minimal partial sum of sub array $a_x+1, a_x+2, \cdots, a_n-1$ as $g(x)$. Then use the whole sum to subtract them if them are minus values.
  
  let $S$ to be the accumulation of the given array.
  
  let
  
  to be the minimal partial sum of sub array $a_0, a_1, \cdots, a_x-1$.
  
  let
  
  to be the minimal partial sum of sub array $a_x+1, a_x+2, \cdots, a_n-w$.
  
  Then we calculate the following formula.
  
  • OK, then write above thought into Java, we get the following codes
explanation: The document provides an analysis of the Codility - Aluminium 2014 Challenge, Part 1. It discusses the problem of finding the maximum sum of a compact subsequence of array elements after performing a single swap operation. The text includes an example and a detailed explanation of the approach to solving the problem, including the mathematical formulas and a brief mention of the Java implementation. However, it does not contain a specific question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # Codility - Aluminium 2014 - Part 1
  
  ## Introduction
  
  This is an analysis of Codility - Aluminium 2014 Challenge, Part 1.
  
  • The problem is, finding the maximum sum of a compact subsequence of array elements after performing a single swap operation. It’s a little tricky maximal subarray problem.
  • For example, given an array {3, -10, 4, 5}, we can swap 3 and -10 to get a compact subsequence {3, 4, 5}, which has the maximum sum 12
  • Time bound and space bound are both O(n)
  • There exists many algorithms to solve maximal subarray problems, but they cannot directly applied to this problem.
  
  ## Analysis
  
  • First, we will take a look at the solution of a CLASSIC maximal subarray problem, which means swap is not allowed.
  
  For a given array $a_0, a_1, \cdots, a_n-1$,
  
  The maximum sum of a compact subsequence can be expressed by the following formula.
  
  Beca\$use we want the maximal part from $i$ to $j$, by thinking reversely, we can calculate the minimal partial sum of sub array $a_0, a_1, \cdots, a_x-1$ as $f(x)$, minimal partial sum of sub array $a_x+1, a_x+2, \cdots, a_n-1$ as $g(x)$. Then use the whole sum to subtract them if them are minus values.
  
  let $S$ to be the accumulation of the given array.
  
  let
  
  to be the minimal partial sum of sub array $a_0, a_1, \cdots, a_x-1$.
  
  let
  
  to be the minimal partial sum of sub array $a_x+1, a_x+2, \cdots, a_n-w$.
  
  Then we calculate the following formula.
  
  • OK, then write above thought into Java, we get the following codes
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides an analysis of the Codility - Aluminium 2014 Challenge, Part 1. It discusses the problem of finding the maximum sum of a compact subsequence of array elements after performing a single swap operation. The text includes an example and a detailed explanation of the approach to solving the problem, including the mathematical formulas and a brief mention of the Java implementation. However, it does not contain a specific question or a question and answer.
  
  Label: no-question