The Drawback of Insertion Sort

Insertion Sort (Part 96) is very fast for nearly sorted data, but slow for completely unsorted data because it can only move an element one position at a time. An element near the end must be swapped many times to reach the beginning.

The Solution: Shell Sort Shell Sort (invented by Donald Shell) is an enhancement of Insertion Sort. It allows for large moves by sorting elements that are far apart, drastically reducing the number of total comparisons needed.

The Principle: The algorithm sorts elements separated by a large gap ($k$). It repeatedly reduces the gap until the final gap is $1$ (at which point it becomes a simple Insertion Sort, which is fast because the list is nearly sorted).

The Shell Sort Algorithm

The algorithm uses a sequence of decreasing gaps (e.g., $5, 2, 1$).

The Steps (Simplified):

  1. Choose a Gap ($k$): Start with a large gap (e.g., $k = N / 2$).
  2. Gapped Insertion Sort: Perform an Insertion Sort on all elements separated by that gap (e.g., compare $A_i$ with $A_{i+k}$, $A_{i+2k}$, etc.).
  3. Shrink Gap: Reduce the gap ($k \leftarrow k / 2$) and repeat the process.
  4. Final Pass: When $k=1$, the list is almost sorted, and the final Insertion Sort pass is very fast.

Z80 Implementation Focus

Shell Sort requires robust 16-bit math to calculate the memory address of the gapped elements ($A_{i+k}$), which means leveraging the multi-byte arithmetic routines (Part 8, 16) and complex pointer management.

Pointer Management: The inner loop requires two main pointers: the current position $HL$ and the gapped position $DE$. $DE$ is calculated by adding the 16-bit $k$ (the gap distance) to $HL$.

GAPPED_SORT_LOOP:
    ; Assume HL = Current Address, IX = Gap distance (16-bit)
    
    LD   DE, HL             ; DE ← Start of gapped comparison
    ADD  DE, IX             ; DE ← HL + Gap (Points to the element k distance away)
    
    ; Compare (HL) with (DE) and swap if out of order (similar to Insertion Sort)
    CALL COMPARE_HL_DE      ; Compare the two elements
    JP   C, CONTINUE_LOOP
    
    CALL SWAP_HL_DE         ; Swap the elements across the gap
    
CONTINUE_LOOP:
    ; ... (Advance HL and repeat)
    RET

Efficiency Trade-off: While much faster than $O(n^2)$ algorithms, Shell Sort is complex to code. However, its improved $O(n (\log n)^2)$ (for certain gap sequences) makes the code overhead worthwhile for sorting large arrays (over 100 elements) efficiently on the Z80.