Subsequence Sum Threshold — Problem Statement & Solution Guide
Quick Answer & Algorithm Key Takeaway
Master coding challenges related to Arrays and solve the Subsequence Sum Threshold problem optimally.
O(n log n)O(n)Problem Description
Given an integer array readings, determine the greatest possible sum of a subsequence (any subset of elements, order irrelevant) that satisfies two conditions: (1) the subsequence contains at most 50 elements, and (2) it includes at least 10 elements whose value is strictly greater than 200. If no subsequence meets both criteria, output -1. The input consists of the length of the array followed by the array values. The output is a single integer representing the maximum achievable sum.
DSA Pattern Breakdown
DSA Pattern Breakdown
"Subsequence Sum Threshold"
WHY DOES IT MATTER?
The pattern combines "selection with cardinality constraints" and "mandatory element class count", a frequent motif in resource allocation, budgeting, and recommendation systems where you must guarantee a minimum representation of a premium class while maximizing overall value.
OPTIMIZATION CHALLENGE
Recognizing that only the top 10 high‑value items are mandatory lets us avoid exploring exponential subsets; the rest of the decision reduces to picking the largest remaining numbers, which is solved by a single sort or linear selection, collapsing the problem to O(n log n) or O(n).
REAL-WORLD CONNECTION
Think of a cloud‑resource scheduler that must allocate at most 50 VMs to a job, with at least 10 high‑performance instances (CPU >200). The scheduler picks the strongest high‑performance VMs first, then fills the remaining slots with the best available machines to maximize throughput, mirroring the greedy selection here.
In an interview, sort once, slice the first 10 high‑value elements, then greedily extend with the next best values. Keep the code simple: separate high/low, sort descending, and use array slicing – no DP, no bitmasking.
COMPLEXITY AT A GLANCE
O(n log n)O(n)Core Theory — Why This Approach?
The problem is a variant of the classic knapsack where the weight constraint is replaced by a cardinality limit (≤50) and a mandatory count of high‑value items (≥10 elements >200). A naïve exhaustive search would enumerate all 2^n subsets, which is infeasible even for moderate n (n can be up to 10^5 in typical interview settings). The optimal paradigm leverages greedy selection after sorting because the objective is a simple sum and the constraints are monotonic: picking larger numbers never harms feasibility, and the only non‑trivial requirement is to guarantee enough >200 elements. By first securing the ten largest >200 values and then filling the remaining slots with the next biggest numbers regardless of their value, we achieve the maximal possible sum while respecting both limits.
Interview Questions on This Problem
Q1How would you modify the solution if the threshold value (200) and the required count (10) were given as input parameters?
Treat the threshold and required count as variables; filter the array into high and low groups based on the threshold, verify high.size()≥required, then sum the top ‘required’ high elements and fill the remaining slots up to the size limit with the next largest values from the combined remainder. The algorithmic steps stay identical, only the constants change.
Q2Can this problem be solved in O(n) time without full sorting? If so, outline the approach used in large‑scale systems like ad‑ranking pipelines.
Yes. Use a linear‑time selection algorithm (quickselect) to find the 10‑th largest element >threshold and the (k‑th) largest overall where k = min(50, n). Partition the array around these pivots to extract the needed top‑10 high values and the top‑(50‑10) remaining values. This yields O(n) average time and O(1) extra space, which is crucial for streaming or distributed pipelines where full sorting is too costly.
Q3Why is it safe to apply a greedy strategy here, whereas many subset‑sum problems require DP?
Because the objective function is linear (sum) and the constraints are only about counts, not about specific sums or capacities. Adding a larger element can never reduce feasibility, so the optimal set is always composed of the largest permissible elements, eliminating the need for dynamic programming.
Examples
Input
12 250 180 210 90 300 50 400 220 130 260 80 190
Output
-1
Explanation: Only six elements exceed 200 (250,210,300,400,220,260), which is fewer than the required ten, so no valid subsequence exists.
Input
15 210 215 220 225 230 235 240 245 250 255 260 100 90 80 70
Output
2585
Explanation: Eleven elements are greater than 200. Selecting all of them yields the maximum sum 210+215+...+260=2585, which respects the length limit (11 ≤ 50) and the required count (≥10). Adding any of the remaining numbers would only increase the sum, so the optimal sum is 2585.
Input
20 300 -50 210 -20 220 -10 230 -5 240 -1 250 -2 260 -3 270 -4 280 -5 290 -6
Output
2550
Explanation: Exactly ten elements are greater than 200. Including any negative or non‑qualifying element would lower the total, so the optimal subsequence consists of those ten positive numbers. Their sum is 300+210+...+290=2550.
Input
13 210 215 220 225 230 235 240 245 250 255 260 100 150
Output
2835
Explanation: Eleven elements exceed 200; the requirement is satisfied. Adding the two additional positive numbers (100 and 150) still keeps the total length under 50 and raises the sum to 2585+250=2835, which is maximal.
Constraints
- 1 <= readings.length <= 100000
- -10^9 <= readings[i] <= 10^9
- max_length = 50
- threshold = 200
- min_length = 10
Optimal Approach & Strategy
Sort the array descending, take the ten largest >200 values, then add the next largest values until reaching 50 elements total; this runs in O(n log n) time.
Brute Force Approach
Enumerate every possible subset, check the two constraints, and keep the maximum sum; this requires O(2^n) time and is impossible for large n.
Code Solutions
// Returns the maximum sum of a subsequence that uses at most 50 elements
// and contains at least 10 elements greater than 200. Returns -1 if impossible.
function subsequenceSumThreshold(readings) {
const largeCount = readings.filter(v => v > 200).length;
if (largeCount < 10) return -1;
const sorted = readings.slice().sort((a, b) => b - a);
const take = Math.min(50, sorted.length);
let sum = 0;
for (let i = 0; i < take; ++i) sum += sorted[i];
return sum;
}
const readline = require('readline');
const rl = readline.createInterface({ input: process.stdin, output: process.stdout });
let data = [];
rl.on('line', line => data.push(line.trim()))
.on('close', () => {
const n = parseInt(data[0], 10);
const readings = data.slice(1).join(' ').trim().split(/\s+/).map(Number);
console.log(subsequenceSumThreshold(readings));
});
#include <iostream>
#include <vector>
#include <algorithm>
// Returns the maximum sum of a subsequence that uses at most 50 elements
// and contains at least 10 elements greater than 200. Returns -1 if impossible.
long long subsequenceSumThreshold(const std::vector<int>& readings) {
int n = readings.size();
int largeCount = 0;
for (int v : readings) if (v > 200) ++largeCount;
if (largeCount < 10) return -1; // not enough large elements
// Sort descending to pick the biggest elements first.
std::vector<int> sorted = readings;
std::sort(sorted.begin(), sorted.end(), std::greater<int>());
int take = std::min(50, n);
long long sum = 0;
for (int i = 0; i < take; ++i) sum += sorted[i];
return sum;
}
int main() {
int n;
std::cin >> n;
std::vector<int> readings(n);
for (int i = 0; i < n; ++i) std::cin >> readings[i];
std::cout << subsequenceSumThreshold(readings) << std::endl;
return 0;
}
import java.util.*;
import java.util.stream.*;
public class SubsequenceSumThreshold {
// Returns the maximum sum of a subsequence that uses at most 50 elements
// and contains at least 10 elements greater than 200. Returns -1 if impossible.
public static long subsequenceSumThreshold(int[] readings) {
long largeCount = Arrays.stream(readings).filter(v -> v > 200).count();
if (largeCount < 10) return -1;
// Sort descending
Integer[] arr = Arrays.stream(readings).boxed().toArray(Integer[]::new);
Arrays.sort(arr, Collections.reverseOrder());
int take = Math.min(50, arr.length);
long sum = 0;
for (int i = 0; i < take; ++i) sum += arr[i];
return sum;
}
public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
int n = sc.nextInt();
int[] readings = new int[n];
for (int i = 0; i < n; ++i) readings[i] = sc.nextInt();
System.out.println(subsequenceSumThreshold(readings));
sc.close();
}
}
def subsequence_sum_threshold(readings):
"""Return the maximum sum of a subsequence that uses at most 50 elements
and contains at least 10 elements greater than 200. Return -1 if impossible.
"""
large_count = sum(1 for v in readings if v > 200)
if large_count < 10:
return -1
sorted_readings = sorted(readings, reverse=True)
take = min(50, len(sorted_readings))
return sum(sorted_readings[:take])
if __name__ == "__main__":
import sys
data = sys.stdin.read().strip().split()
if not data:
sys.exit(0)
n = int(data[0])
readings = list(map(int, data[1:1 + n]))
print(subsequence_sum_threshold(readings))
// Returns the maximum sum of a subsequence that uses at most 50 elements
// and contains at least 10 elements greater than 200. Returns -1 if impossible.
function subsequenceSumThreshold(readings) {
const largeCount = readings.filter(v => v > 200).length;
if (largeCount < 10) return -1;
const sorted = readings.slice().sort((a, b) => b - a);
const take = Math.min(50, sorted.length);
let sum = 0;
for (let i = 0; i < take; ++i) sum += sorted[i];
return sum;
}
const readline = require('readline');
const rl = readline.createInterface({ input: process.stdin, output: process.stdout });
let data = [];
rl.on('line', line => data.push(line.trim()))
.on('close', () => {
const n = parseInt(data[0], 10);
const readings = data.slice(1).join(' ').trim().split(/\s+/).map(Number);
console.log(subsequenceSumThreshold(readings));
});
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