From daba8035ff9a31c70a87bc1f09fdb630ff48a387 Mon Sep 17 00:00:00 2001
From: Jörg Sommrey <28217714+jo-37@users.noreply.github.com>
Date: Mon, 27 Feb 2023 15:55:15 +0100
Subject: Solution to task 2

---
 challenge-206/jo-37/perl/ch-2.pl | 91 ++++++++++++++++++++++++++++++++++++++++
 1 file changed, 91 insertions(+)
 create mode 100755 challenge-206/jo-37/perl/ch-2.pl

diff --git a/challenge-206/jo-37/perl/ch-2.pl b/challenge-206/jo-37/perl/ch-2.pl
new file mode 100755
index 0000000000..79c301738a
--- /dev/null
+++ b/challenge-206/jo-37/perl/ch-2.pl
@@ -0,0 +1,91 @@
+#!/usr/bin/perl -s
+
+use v5.16;
+use Test2::V0 '!float';
+use PDL;
+use PDL::NiceSlice;
+
+our ($tests, $examples);
+
+run_tests() if $tests || $examples;	# does not return
+
+die <<EOS unless @ARGV && !(@ARGV % 2);
+usage: $0 [-examples] [-tests] [N1 N2...]
+
+-examples
+    run the examples from the challenge
+ 
+-tests
+    run some tests
+
+N1 N2...
+    even sized list of numbers
+
+EOS
+
+
+### Input and Output
+
+say max_pairsum(@ARGV);
+
+
+### Implementation
+
+# We are given an even sized list of numbers (N₁,...,N₂ₙ).  The list
+# shall be partitioned into pairs.  There is a target function defined
+# as the sum over the minimum of all the pairs.  This target function
+# shall be maximized, i.e. the task is to find a partitioning into pairs
+# where the target is maximal and to provide this maximum value.
+#
+# For the sake of convenience we assume all numbers being distinct.  We
+# partition the list into pairs
+# ((P¹₁, P¹₂),...,(Pⁿ₁, Pⁿ₂))
+# Neither the order of pairs, nor the order of the paired numbers are
+# relevant for the target function.  Thus we can chose an arbitrary pair
+# P = (a, b), Q = (c, d) and arrange it such that
+# a < b, c < d and a < c
+# by exchanging the pairs and/or the numbers within the pairs without
+# altering the value of the target function.  The contributions to the
+# target function are min(P) = a  and min(Q) = c.
+#
+# Now suppose b > c, i.e. the two pairs "overlap".
+# We build a different pairing where b and c are
+# exchanged and all other pairs remain unaltered, i.e.
+# P' = (a, c) and Q' = (b, d)
+# The new contributions to the target function are
+# min(P') = a = min(P) (from the precondition a < c) and
+# min(Q') = min(b, d) > min(c, d) = min(Q) (from our assumption b > c
+# and the precondition c < d)
+# All summands in the target function remain unchanged with the
+# exception of Q', which increased from Q.
+#
+# There is only one partitioning into pairs where none of the pairs
+# overlap:  Taking the pairs as adjacent elements from the sorted list.
+# From the above considerations, this partitioning maximizes the target
+# function.
+#
+# A long introduction to a short solution:
+sub max_pairsum {
+    # Sum over the first elements of successive pairs taken from the
+    # sorted list.
+    sum pdl(@_)->qsort->(0:-2:2);
+}
+
+
+### Examples and tests
+
+sub run_tests {
+    SKIP: {
+        skip "examples" unless $examples;
+
+        is max_pairsum(1,2,3,4), 4, 'example 1';
+        is max_pairsum(0,2,1,3), 2, 'example 2';
+    }
+
+    SKIP: {
+        skip "tests" unless $tests;
+	}
+
+    done_testing;
+    exit;
+}
-- 
cgit