Stable Minimum Storage Merging by Symmetric Comparisons Pok-Son Kim1? and Arne Kutzner2

Posted 2021-06-23 rsapaper

Stable Minimum Storage Merging by Symmetric Comparisons Pok-Son Kim1? and Arne Kutzner2

http://itbe.hanyang.ac.kr/ak/papers/esa2004.pdf

// symMerge merges the two sorted subsequences data[a:m] and data[m:b] using
// the SymMerge algorithm from Pok-Son Kim and Arne Kutzner, "Stable Minimum
// Storage Merging by Symmetric Comparisons", in Susanne Albers and Tomasz
// Radzik, editors, Algorithms - ESA 2004, volume 3221 of Lecture Notes in
// Computer Science, pages 714-723. Springer, 2004.
//
// Let M = m-a and N = b-n. Wolog M < N.
// The recursion depth is bound by ceil(log(N+M)).
// The algorithm needs O(M*log(N/M + 1)) calls to data.Less.
// The algorithm needs O((M+N)*log(M)) calls to data.Swap.
//
// The paper gives O((M+N)*log(M)) as the number of assignments assuming a
// rotation algorithm which uses O(M+N+gcd(M+N)) assignments. The argumentation
// in the paper carries through for Swap operations, especially as the block
// swapping rotate uses only O(M+N) Swaps.
//
// symMerge assumes non-degenerate arguments: a < m && m < b.
// Having the caller check this condition eliminates many leaf recursion calls,
// which improves performance.
func symMerge(data Interface, a, m, b int) {
   // Avoid unnecessary recursions of symMerge
   // by direct insertion of data[a] into data[m:b]
   // if data[a:m] only contains one element.
   if m-a == 1 {
      // Use binary search to find the lowest index i
      // such that data[i] >= data[a] for m <= i < b.
      // Exit the search loop with i == b in case no such index exists.
      i := m
      j := b
      for i < j {
         h := int(uint(i+j) >> 1)
         if data.Less(h, a) {
            i = h + 1
         } else {
            j = h
         }
      }
      // Swap values until data[a] reaches the position before i.
      for k := a; k < i-1; k++ {
         data.Swap(k, k+1)
      }
      return
   }

   // Avoid unnecessary recursions of symMerge
   // by direct insertion of data[m] into data[a:m]
   // if data[m:b] only contains one element.
   if b-m == 1 {
      // Use binary search to find the lowest index i
      // such that data[i] > data[m] for a <= i < m.
      // Exit the search loop with i == m in case no such index exists.
      i := a
      j := m
      for i < j {
         h := int(uint(i+j) >> 1)
         if !data.Less(m, h) {
            i = h + 1
         } else {
            j = h
         }
      }
      // Swap values until data[m] reaches the position i.
      for k := m; k > i; k-- {
         data.Swap(k, k-1)
      }
      return
   }

   mid := int(uint(a+b) >> 1)
   n := mid + m
   var start, r int
   if m > mid {
      start = n - b
      r = mid
   } else {
      start = a
      r = m
   }
   p := n - 1

   for start < r {
      c := int(uint(start+r) >> 1)
      if !data.Less(p-c, c) {
         start = c + 1
      } else {
         r = c
      }
   }

   end := n - start
   if start < m && m < end {
      rotate(data, start, m, end)
   }
   if a < start && start < mid {
      symMerge(data, a, start, mid)
   }
   if mid < end && end < b {
      symMerge(data, mid, end, b)
   }
}