add gap_sequence argument to Shellsort

sorts-r-us/sorts.py

@@ -19,7 +19,7 @@ def insertion_sort(arr, inplace=True):
     return a
 
 
-def sedgewick_shell_gaps(num_terms=float('inf'), upper_bound=float('inf')):
+def sedgewick_gaps(num_terms=float('inf'), upper_bound=float('inf')):
     """Generates Sedgewick's increments for Shellsort.
 
     The first 20 terms of the sequence are:
@@ -72,7 +72,38 @@ def tokuda_gap(i):
     """
     return 1 if i==0 else ((9**(i+1)>>(i<<1))-4)//5 + 1
 
-def shellsort(arr, inplace=True):
+def tokuda_gaps(num_terms=float('inf'), upper_bound=float('inf')):
+    """Generates the sequence of Tokuda gaps for Shellsort.
+
+    The first 20 terms of the sequence are:
+    [1, 4, 9, 20, 46, 103, 233, 525, 1182, 2660, 5985, 13467, 30301, 68178, 153401, 345152, 776591, 1747331, 3931496, 8845866, ...]
+
+    h_i = ceil( (9*(9/4)**i-4)/5 ) for i>=0.
+
+    If 9*(9/4)**i-4)/5 is not an integer, I believe this is the same as
+    h_i = ((9**(i+1)>>(i<<1))-4)//5 + 1,
+    and I believe the above should be non-integer valued for all i>0.
+    (We have to explicitly return 1 when i=0, as the above formula would return 2.)
+    """
+    if num_terms>=1 and upper_bound>=1:
+        yield 1 #first term
+
+    i=1 #i+1 = number of terms
+    term = 4 #2nd term
+
+    while i+1<= num_terms and term <= upper_bound:
+        yield term
+        i += 1
+        term = ((9**(i+1)>>(i<<1))-4)//5 + 1
+
+def ciura_gaps(num_terms=float('inf'), upper_bound=float('inf')):
+    """Generates the Ciura gap sequence for Shellsort
+
+    The sequence is [1, 4, 10, 23, 57, 132, 301, 701, 1750, h_k, ...]
+    """
+    yield 1
+
+def shellsort(arr, inplace=True, gap_sequence='Sedgewick'):
     """Sorts an array using Shellsort with Sedgewick's increments.
 
     With these increments, Shellsort operates in O(n^4/3) time worst case,
@@ -82,7 +113,12 @@ def shellsort(arr, inplace=True):
     a = arr if inplace else arr.copy()
     #Storing the gaps in an array requires O(log(n)) extra space.
     #Order the gaps in descending order.
-    gaps = list(sedgewick_shell_gaps(upper_bound=len(a)-1))[::-1]
+    if gap_sequence == 'Sedgewick':
+        gaps = list(sedgewick_gaps(upper_bound=len(a)-1))[::-1]
+    elif gap_sequence == 'Tokuda':
+        gaps = list(tokuda_gaps(upper_bound=len(a)-1))[::-1]
+    else:
+        raise ValueError(f'Unknown gap sequence: {gap_sequence}')
 
     #Iterate through gaps h in descending order.
     for h in gaps: