Lecture 05 Ticket

Complete before the beginning of class on Monday, 09/12. (Will be graded for completion not correctness.)

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\[\def\compare{ {\mathrm{compare}} } \def\swap{ {\mathrm{swap}} } \def\sort{ {\mathrm{sort}} } \def\true{ {\mathrm{true}} } \def\false{ {\mathrm{false}} }\]

Consider the following method that takes two arrays of numerical values, \(a\) and \(b\) as input and returns another array \(c\):

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Merge(a, b):
  i, j, k <- 1
  c <- new array of size size(a) + size(b)
  while i <= size(a) and j <= size(b) do
    if a[i] <= b[j] then
      min <- a[i]
      i <- i + 1
    else
      min <- b[j]
      j <- j + 1
    endif
    c[k] <- min
    k <- k + 1
  endwhile
  
  while i <= size(a) do
    c[k] <- a[i]
    i <- i + 1
    k <- k + 1
  endwhile
  
  while j <= size(b) do
    c[k] <- b[j]
    j <- j + 1
    k <- k + 1
  endwhile

  return c

Argue that if \(a\) and \(b\) are sorted, then the array \(c\) returned by \(\mathrm{Merge}(a, b)\) is sorted and contains every element from both \(a\) and \(b\). Your argument does not have to be too formal, but you should describe a loop invariant for the first while loop that explains how the procedure creates a sorted array.