Triplets and factors

Weekly challenge 241 — 30 October 2023

Week 241 - 30 Oct 2023

Task 1

You are given an array (3 or more members) of integers in increasing order and a positive integer. Write a script to find out the number of unique Arithmetic Triplets satisfying the following rules:

- i < j < k
- nums[j] - nums[i] == diff
- nums[k] - nums[j] == diff

Example 1Input: @nums = (0, 1, 4, 6, 7, 10) $diff = 3 Output: 2 Index (1, 2, 4) is an arithmetic triplet because both 7 - 4 == 3 and 4 - 1 == 3. Index (2, 4, 5) is an arithmetic triplet because both 10 - 7 == 3 and 7 - 4 == 3.Example 2Input: @nums = (4, 5, 6, 7, 8, 9) $diff = 2 Output: 2 (0, 2, 4) is an arithmetic triplet because both 8 - 6 == 2 and 6 - 4 == 2. (1, 3, 5) is an arithmetic triplet because both 9 - 7 == 2 and 7 - 5 == 2.

On the face of it, this is dead simple: just have three nested loops over i, j and k and check for the specified conditions:

for $i (0 .. $last - 2) { for $j (1 .. $last - 1) { for $k (2 .. $last) { $count ++ if ($nums[$j] - $nums[$i] == diff and $nums[$k] - $nums[$j] == $diff); } } }

And that works. But if the number of numbers in @nums is say 100, then the innermost loop will be executed nearly a million times, and that is going to take a while.

So we need to optimise. Firstly, let's note that the numbers are sorted. That means
that when we're in the j loop, if numbers i and j don't differ by diff then there's no
need to bother with the k loop. And if number j differs from number i by __more__ than
diff we can give up on the j loop because we know that any further numbers in that
loop will differ from the i number by more than diff.

And of course we can do much the same in the k loop, abandoning it as soon as the k number exceeds the j number by more than diff.

With these optimisations I tried 1000 random numbers in (0 .. 9999) in `@nums`

and
`$diff = 99`

and it completed in a fraction of a second, finding 6 triplets.

So I think that's good enough. I tried (for about 5 minutes) to think up some sequence where the optimisations would not help - and failed.

#!/usr/bin/perl use v5.16; # The Weekly Challenge - 2023-10-30 use utf8; # Week 241 task 1 - Arithmetic triplets use strict; # Peter Campbell Smith use warnings; # Blog: http://ccgi.campbellsmiths.force9.co.uk/challenge arithmetic_triplets ([0, 1, 4, 6, 7, 10], 3); arithmetic_triplets ([(4, 5, 6, 7, 8, 9)], 2); # generate 200 sorted unique numbers in (0 .. 1999) my ($j, @nums, $next, $count, @used); $count = 0; while ($count < 200) { $next = int(rand(2000)); next if $used[$next]; push(@nums, $next); $count ++; $used[$next] = 1; } @nums = sort { $a <=> $b } @nums; arithmetic_triplets (\@nums, 19); sub arithmetic_triplets { my (@nums, $diff, $last, $i, $j, $k, $count, $explain, $ji_diff, $kj_diff); # initialise @nums = @{$_[0]}; $diff = $_[1]; $last = @nums - 1; # loop over triplets $explain = ''; $count = 0; # loop over i any i (except the last 2) could be part of a triplet for $i (0 .. $last - 2) { # loop over j for $j (1 .. $last - 1) { $ji_diff = $nums[$j] - $nums[$i]; # if they differ by more than $diff then we can abandon this j last if $ji_diff > $diff; # unless this pair of i and j differ by $diff there's no need to check k next unless $ji_diff == $diff; # loop over k for $k (2 .. $last) { $kj_diff = $nums[$k] - $nums[$j]; # we can abandon this k if k differs from j by more than $diff last if ($kj_diff) > $diff; # and at last we've found an answer! if ($kj_diff == $diff) { $count ++; $explain .= qq{ \$nums[$i] = $nums[$i], \$nums[$j] = $nums[$j], \$nums[$k] = $nums[$k]\n}; } } } } # show results say qq[\nInput: \@nums = (] . join(q[, ], @nums) . q[)]; say qq[ \$diff = $diff]; say qq[Output: $count\n] . ($explain ? substr($explain, 0, -1) : ''); }

Input: @nums = (0, 1, 4, 6, 7, 10) $diff = 3 Output: 2 $nums[1] = 1, $nums[2] = 4, $nums[4] = 7 $nums[2] = 4, $nums[4] = 7, $nums[5] = 10 Input: @nums = (4, 5, 6, 7, 8, 9) $diff = 2 Output: 2 $nums[0] = 4, $nums[2] = 6, $nums[4] = 8 $nums[1] = 5, $nums[3] = 7, $nums[5] = 9 Input: @nums = (2, 12, 36, 46, 80, 87, 97, 100, 109, 111, 114, 122, 142, 151, 158, 172, 182, 185, 195, 197, 200, 206, 210, 219, 230, 236, 237, 256, 260, 270, 275, 276, 277, 292, 298, 301, 304, 312, 313, 319, 329, 344, 348, 350, 359, 360, 367, 371, 376, 380, 408, 431, 435, 436, 442, 444, 452, 471, 512, 534, 546, 547, 559, 567, 571, 573, 582, 584, 587, 592, 596, 615, 624, 625, 651, 660, 672, 673, 683, 690, 709, 710, 729, 748, 761, 767, 798, 800, 811, 827, 830, 834, 842, 846, 872, 879, 887, 917, 920, 925, 949, 958, 967, 988, 993, 995, 1002, 1017, 1022, 1029, 1039, 1040, 1042, 1045, 1058, 1071, 1086, 1089, 1134, 1136, 1138, 1142, 1149, 1185, 1191, 1212, 1240, 1253, 1254, 1255, 1262, 1280, 1302, 1330, 1338, 1346, 1353, 1379, 1393, 1413, 1422, 1428, 1437, 1448, 1462, 1467, 1479, 1490, 1492, 1503, 1523, 1539, 1542, 1550, 1551, 1553, 1557, 1573, 1575, 1578, 1603, 1609, 1614, 1620, 1636, 1640, 1641, 1652, 1655, 1669, 1677, 1684, 1705, 1714, 1725, 1737, 1752, 1754, 1755, 1766, 1770, 1771, 1786, 1796, 1814, 1821, 1825, 1839, 1852, 1867, 1884, 1891, 1904, 1910, 1913, 1916, 1930, 1962, 1984, 1992) $diff = 19 Output: 4 $nums[26] = 237, $nums[27] = 256, $nums[30] = 275 $nums[40] = 329, $nums[42] = 348, $nums[46] = 367 $nums[81] = 710, $nums[82] = 729, $nums[83] = 748 $nums[82] = 729, $nums[83] = 748, $nums[85] = 767

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