Sorting, Measures of disorder, and Worst-case performance

Author: Morwenn

docs/Measures-of-presortedness.md

@@ -329,7 +329,7 @@ In other domains, that value is called *F* (for *Footrule*). It is no more helpf
 
 ### *Par*
 
-*Par* was described by V. Estivill-Castro and D. Wood in *A New Measure of Presortedness* as follows:
+*Par* is described by V. Estivill-Castro and D. Wood in *A New Measure of Presortedness* as follows:
 
 > *Par(X)* = min { *p* | *X* is *p*-sorted }
 
@@ -343,6 +343,10 @@ The following definition is also given to determine whether a sequence is *p*-so
 
 In their subsequent papers, those authors consistently use *Dis* instead of *Par*, often accompanied by a link to *A New Measure of Presortedness*.
 
+### *Pos*
+
+*Pos* is described by V. Estivill-Castro and D. Wood in *Sorting, Measures of disorder, and Worst-case performance* as "the position number": the number of elements in a sequence that are not in their sorted position. This definition matches that of *Ham*, the Hamming distance between a sequence and its sorted permutation.
+
 ### *Radius*
 
 T. Altman and Y. Igarashi mention the concept of *k*-sortedness and the measure *Radius*(*X*) in *Roughly Sorting: Sequential and Parallel Approach*. However *k*-sortedness is the same as *p*-sortedness, and *Radius* is just another name for *Par* (and thus for *Dis*).

include/cpp-sort/probes/ham.h

@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 2016-2022 Morwenn
+ * Copyright (c) 2016-2025 Morwenn
  * SPDX-License-Identifier: MIT
  */
 #ifndef CPPSORT_PROBES_HAM_H_
@@ -110,7 +110,7 @@ namespace probe
             static constexpr auto max_for_size(Integer n)
                 -> Integer
             {
-                return n == 0 ? 0 : n;
+                return n;
             }
         };
     }

tests/probes/ham.cpp

@@ -1,11 +1,13 @@
 /*
- * Copyright (c) 2016-2022 Morwenn
+ * Copyright (c) 2016-2025 Morwenn
  * SPDX-License-Identifier: MIT
  */
 #include <forward_list>
 #include <iterator>
 #include <vector>
 #include <catch2/catch_test_macros.hpp>
+#include <rapidcheck.h>
+#include <rapidcheck/catch.h>
 #include <cpp-sort/probes/ham.h>
 #include <cpp-sort/utility/size.h>
 #include <testing-tools/distributions.h>
@@ -46,4 +48,8 @@ TEST_CASE( "presortedness measure: ham", "[probe][ham]" )
 
         CHECK( ham(collection) == 0 );
     }
+
+    rc::prop("Ham(X) ≠ 1", [](const std::vector<int>& sequence) {
+        return cppsort::probe::ham(sequence) != 1;
+    });
 }

tests/probes/relations.cpp

@@ -83,6 +83,13 @@ TEST_CASE( "relations between measures of presortedness", "[probe]" )
         return rem(sequence) <= 2 * exc(sequence);
     });
 
+    // Sorting, Measures of disorder, and Worst-case performance
+    // by Vladimir Estivill-Castro and Derick Wood
+
+    rc::prop("Rem(X) ≤ Ham(X)", [](const std::vector<int>& sequence) {
+        return rem(sequence) <= ham(sequence);
+    });
+
     // Encroaching lists as a measure of presortedness
     // by Steven S. Skiena