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# Program to find closest subsequence sum in Python

Suppose we have an array nums and another value goal. We want to select a subsequence of nums such that the sum of its elements is the nearest to goal. So in other words, if the sum of the subsequence's elements is s, then we want to minimize the absolute difference |s - goal|.

So we have to find the minimum possible value of |s - goal|. So, if the input is like nums = [8,-8,16,-1] goal = -3, then the output will be 2 because select the subsequence [8,-8,-1], with a sum of -1. The absolute difference is |-1 - (-3)| = abs(2) = 2, this is the minimum.

To solve this, we will follow these steps −

n := size of nums

sort nums based on -ve value after getting absolute values of x

neg := make an array of size n+1 and fill with 0

pos := make an array of size n+1 and fill with 0

for i in range n-1 to 0, decrease by 1, do

if nums[i] < 0, then

neg[i] := neg[i+1] + nums[i]

pos[i] := pos[i+1]

otherwise,

pos[i] := pos[i+1] + nums[i]

neg[i] := neg[i+1]

ans := |goal|

s := a new set ans insert 0 into it

Define a function check() . This will take a, b

if b < goal - ans or goal + ans < a, then

return False

return True

From the main method, do the following

for i in range 0 to n - 1, do

sl := a list of x for all x in s when check(x+neg[i], x+pos[i] is true]

if size of sl is same as 0, then

come out from loop

s := a new set from sl

for each x in sl, do

y := x + nums[i]

if |y - goal| < ans, then

ans := |y - goal|

if ans is same as 0, then

return 0

insert y into s

return ans

## Example

Let us see the following implementation to get better understanding

from collections import Counter def solve(nums, goal): n = len(nums) nums.sort(key=lambda x: -abs(x)) neg = [0 for _ in range(n+1)] pos = [0 for _ in range(n+1)] for i in range(n-1, -1, -1): if nums[i] < 0: neg[i] = neg[i+1] + nums[i] pos[i] = pos[i+1] else: pos[i] = pos[i+1] + nums[i] neg[i] = neg[i+1] ans = abs(goal) s = set([0]) def check(a, b): if b < goal - ans or goal + ans < a: return False return True for i in range(n): sl = [x for x in s if check(x+neg[i], x+pos[i])] if len(sl) == 0: break s = set(sl) for x in sl: y = x + nums[i] if abs(y - goal) < ans: ans = abs(y - goal) if ans == 0: return 0 s.add(y) return ans nums = [8,-8,16,-1] goal = -3 print(solve(nums, goal))

## Input

[0,1,2,3,4], [[3,1],[1,3],[5,6]]

## Output

2

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