# Remove entries from array to sort it and maximize sum of elements

This challenge is from an admission test to a closed number cyber security course. Anyway it doesn't have to do with cyber security, it's just to test the students logical and coding skills.

Write a program that removes entries from an array so that the remaining values are sorted in a strictly decreasing order and their sum is the maximized among all other possible decreasing sequences.

## Input and Output

Input will be an array of integer values strictly greater than 0 and all different from each other. You are free to choose whether to read input from file, command line or stdin.

Output will be a descending-sorted subarray of the input one, whose sum is greater than any other possible descending-sorted subarray.

Note: [5, 4, 3, 2] is a subarray of [5, 4, 1, 3, 2], even if 4 and 3 are not adjacent. Just because the 1 was popped.

## Bruteforce solution

The simplest solution of course would be iterate among all possible combinations of the given array and search for a sorted one with the greatest sum, that would be, in Python:

import itertools

def best_sum_desc_subarray(ary):
best_sum_so_far = 0
best_subarray_so_far = []
for k in range(1, len(ary)):
for comb in itertools.combinations(ary, k):
if sum(comb) > best_sum_so_far and all(comb[j] > comb[j+1] for j in range(len(comb)-1)):
best_subarray_so_far = list(comb)
best_sum_so_far = sum(comb)
return best_subarray_so_far


Unfortunately, since checking if the array is sorted, and calculating the sum of it's elements is $\inline&space;\Theta&space;\left(&space;k&space;\right&space;)$ and since this operation will be done $\inline&space;\binom{n}{k}&space;=&space;\frac{n!}{k!(n-k)!}$ times for $k$ from $1$ to $n$, the asymptotic time complexity will be

$\Theta\left(\sum_{k=1}^{n}&space;k&space;\frac{n!}{k!&space;(n-k)!}&space;\right)&space;=&space;\Theta(n&space;2^{n-1})&space;=&space;\Theta(n&space;2^{n})$

## Challenge

Your goal is to achieve a better time complexity than the bruteforce above. The solution with the smallest asymptotic time complexity is the winner of the challenge. If two solutions have the same asymptotic time complexity, the winner will be the one with the smallest asymptotic spatial complexity.

Note: You can consider reading, writing and comparing atomic even on large numbers.

Note: If there are two or more solutions return either of them.

## Test cases

Input:  [200, 100, 400]
Output: [400]

Input:  [4, 3, 2, 1, 5]
Output: [4, 3, 2, 1]

Input:  [50, 40, 30, 20, 10]
Output: [50, 40, 30, 20, 10]

Input:  [389, 207, 155, 300, 299, 170, 158, 65]
Output: [389, 300, 299, 170, 158, 65]

Input:  [19, 20, 2, 18, 13, 14, 8, 9, 4, 6, 16, 1, 15, 12, 3, 7, 17, 5, 10, 11]
Output: [20, 18, 16, 15, 12, 7, 5]

Input:  [14, 12, 24, 21, 6, 10, 19, 1, 5, 8, 17, 7, 9, 15, 23, 20, 25, 11, 13, 4, 3, 22, 18, 2, 16]
Output: [24, 21, 19, 17, 15, 13, 4, 3, 2]

Input:  [25, 15, 3, 6, 24, 30, 23, 7, 1, 10, 16, 29, 12, 13, 22, 8, 17, 14, 20, 11, 9, 18, 28, 21, 26, 27, 4, 2, 19, 5]
Output: [25, 24, 23, 22, 17, 14, 11, 9, 4, 2]

• Related. (I can't check right now whether the two algorithms are in fact equivalent, but I think they might be.) – Martin Ender Jan 30 '18 at 20:24
• Comments are not for extended discussion; this conversation has been moved to chat. – Martin Ender Feb 20 '18 at 8:41
• @AndersKaseorg I completely overlooked it. Thanks for pointing it out. – Marco Jan 1 at 20:30

# Haskell, $O(n \log n)$ time, $O(n)$ space

{-# LANGUAGE MultiParamTypeClasses #-}

import qualified Data.FingerTree as F

data S = S
{ sSum :: Int
, sArr :: [Int]
} deriving (Show)

instance Monoid S where
mempty = S 0 []
mappend _ s = s

instance F.Measured S S where
measure = id

bestSubarrays :: [Int] -> F.FingerTree S S
bestSubarrays [] = F.empty
bestSubarrays (x:xs) = left F.>< sNew F.<| right'
where
(left, right) = F.split (\s -> sArr s > [x]) (bestSubarrays xs)
sLeft = F.measure left
sNew = S (x + sSum sLeft) (x : sArr sLeft)
right' = F.dropUntil (\s -> sSum s > sSum sNew) right

bestSubarray :: [Int] -> [Int]
bestSubarray = sArr . F.measure . bestSubarrays


### How it works

bestSubarrays xs is the sequence of subarrays of xs that are on the efficient frontier of {largest sum, smallest first element}, ordered from left to right by increasing sum and increasing first element.

To go from bestSubarrays xs to bestSubarrays (x:xs), we

1. split the sequence into a left side with first elements less than x, and a right side with first elements greater than x,
2. find a new subarray by prepending x to the rightmost subarray on the left side,
3. drop the prefix of subarrays from the right side with smaller sum than the new subarray,
4. concatenate the left side, the new subarray, and the remainder of the right side.

A finger tree supports all these operations in $O(\log n)$ time.

# Perl

This should be O(n^2) in time and O(n) in space

Give numbers separated by space on one line to STDIN

#!/usr/bin/perl -a
use strict;
use warnings;

# use Data::Dumper;
use constant {
INFINITY => 9**9**9,
DEBUG    => 0,
};

# Recover sequence from the 'how' linked list
sub how {
my @z;
for (my $h = shift->{how};$h; $h =$h->[1]) {
push @z, $h->[0]; } pop @z; return join " ", reverse @z; } use constant MINIMUM => { how => [-INFINITY, [INFINITY]], sum => -INFINITY, next => undef, }; # Candidates is a linked list of subsequences under consideration # A given final element will only appear once in the list of candidates # in combination with the best sum that can be achieved with that final element # The list of candidates is reverse sorted by final element my$candidates = {
# 'how' will represent the sequence that adds up to the given sum as a
# reversed lisp style list.
# so e.g. "1, 5, 8" will be represented as [8, [5, [1, INFINITY]]]
# So the final element will be at the front of 'how'
how  => [INFINITY],
# The highest sum that can be reached with any subsequence with the same
# final element
sum  => 0,
# 'next' points to the next candidate
next => MINIMUM,   # Dummy terminator to simplify program logic
};

for my $num (@F) { # Among the candidates on which an extension with$num is valid
# find the highest sum
my $max_sum = MINIMUM; my$c = $candidates; while (num < c->{how}[0]) { if (c->{sum} > max_sum->{sum}) { max_sum = c; c =$$c->{next}; } else { # Remove pointless candidate $$c =$$c->{next}; } } my$new_sum = $max_sum->{sum} +$num;
if ($$c->{how}[0] != num) { # Insert a new candidate with a never before seen end element # Due to the unique element rule this branch will always be taken$$c = { next => $$c }; } elsif (new_sum <=$$c->{sum}) {
# An already known end element but the sum is no improvement
next;
}
$$c->{sum} = new_sum;$$c->{how} = [$num,$max_sum->{how}];
# print(Dumper($candidates)); if (DEBUG) { print "Adding$num\n";
for (my $c =$candidates; $c;$c = $c->{next}) { printf "sum(%s) = %s\n", how($c), $c->{sum}; } print "------\n"; } } # Find the sequence with the highest sum among the candidates my$max_sum = MINIMUM;
for (my $c =$candidates; $c;$c = $c->{next}) {$max_sum = $c if$c->{sum} > $max_sum->{sum}; } # And finally print the result print how($max_sum), "\n";


This answer expands on Ton Hospel's one.

The problem can be solved with dynamic programming using the recurence

$$T(i) = a_i + \max \left[ \{0\} \cup \{ T(j) | 0 \leq j < i \wedge a_i \leq a_j \} \right ]$$

where $(a_i)$ is the input sequence and $T(i)$ the maximally achievable sum of any decreasing sub-sequence ending with index $i$. The actual solution may then be retraced using $T$, as in the following rust code.

fn solve(arr: &[usize]) -> Vec<usize> {
let mut tbl = Vec::new();
// Compute table with maximum sums of any valid sequence ending
// with a given index i.
for i in 0..arr.len() {
let max = (0..i)
.filter(|&j| arr[j] >= arr[i])
.map(|j| tbl[j])
.max()
.unwrap_or(0);
tbl.push(max + arr[i]);
}
// Reconstruct an optimal sequence.
let mut sum = tbl.iter().max().unwrap_or(&0).clone();
let mut limit = 0;
let mut result = Vec::new();

for i in (0..arr.len()).rev() {
if tbl[i] == sum && arr[i] >= limit {
limit = arr[i];
sum -= arr[i];
result.push(arr[i]);
}
}
assert_eq!(sum, 0);
result.reverse();
result
}

let mut s = String::new();