Shell Sort

Discussion

The worse-case time complexity of shell sort depends on the increment sequence. For the increments 1 4 13 40 121..., which is what is used here, the time complexity is O(n^3/2). For other increments, time complexity is known to be O(n^4/3) and even O(n·lg²(n)). Neither tight upper bounds on time complexity nor the best increment sequence are known.

Because shell sort is based on insertion sort, shell sort inherits insertion sort's adaptive properties. The adapation is not as dramatic because shell sort requires one pass through the data for each increment, but it is significant. For the increment sequence shown above, there are log₃(n) increments, so the time complexity for nearly sorted data is O(n·log₃(n)).

Because of its low overhead, relatively simple implementation, adaptive properties, and sub-quadratic time complexity, shell sort may be a viable alternative to the O(n·lg(n)) sorting algorithms for some applications when the data to be sorted is not very large.

Algorithm

h = 1
while h < n, h = 3*h + 1
while h > 0,
    h = h / 3
    for k = 1:h, insertion sort a[k:h:n]
    → invariant: each h-sub-array is sorted
end

Properties

• Not stable
• O(1) extra space
• O(n^3/2) time as shown (see below)
• Adaptive: O(n·lg(n)) time when nearly sorted

• References

  Algorithms in Java, Parts 1-4, 3rd edition by Robert Sedgewick. Addison Wesley, 2003.
  Programming Pearls by Jon Bentley. Addison Wesley, 1986.
  Quicksort is Optimal by Robert Sedgewick and Jon Bentley, Knuthfest, Stanford University, January, 2002.
  Bubble-sort with Hungarian ("Csángó") folk dance YouTube video, created at Sapientia University, Tirgu Mures (Marosvásárhely), Romania.
  Select-sort with Gypsy folk dance YouTube video, created at Sapientia University, Tirgu Mures (Marosvásárhely), Romania.
  The Beauty of Sorting YouTube video, Dynamic Graphics Project, Computer Systems Research Group, University of Toronto.