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Solve in cpp

Created on: July 4, 2025

Answered using O3 by Chat01

Question

1751659636521526385411074233014.jpg1751659636521526385411074233014.jpg
1751659649565804054801295627726.jpg1751659649565804054801295627726.jpg

Solve in cpp

Python

python
from PIL import Image
import matplotlib.pyplot as plt

img = Image.open('/mnt/data/1751659636521526385411074233014.jpg')
plt.imshow(img)
plt.axis('off')

(-0.5, 3999.5, 2999.5, -0.5)

Python

python
plt.figure(figsize=(6,10))
plt.imshow(img.crop((0,150,600,1100)))
plt.axis('off')

(-0.5, 599.5, 949.5, -0.5)

Python

python
plt.figure(figsize=(6,10))
plt.imshow(img.crop((0,400,600,1500)))
plt.axis('off')

(-0.5, 599.5, 1099.5, -0.5)

Python

python
plt.figure(figsize=(6,10))
plt.imshow(img.crop((0,1000,600,1900)))
plt.axis('off')

(-0.5, 599.5, 899.5, -0.5)

Python

python
plt.figure(figsize=(6,10))
plt.imshow(img.crop((0,1400,600,2500)))
plt.axis('off')

(-0.5, 599.5, 1099.5, -0.5)

Thought

Clarifying the problem strategy

The problem seems to involve making an array non-decreasing by adding units to contiguous subarrays. The example provided shows servers with powers [3, 4, 1, 6, 2], and the task is to adjust these such that each element is less than or equal to the next. I deduce that we can only increase values in the array and must add the least possible total amount to achieve a non-decreasing order. I’ll zoom in on the image more for a clearer understanding of the function description.

Reconsidering the incremental problem

The task is about making an array non-decreasing by adding units to contiguous subarrays. The goal is to minimize the total increment, but their example result adds more units than what I expected. My approach of scanning the array and choosing b[i] as max(a[i], b[i-1]) gives me fewer increments, which is supposed to be the optimal solution. However, the increment behavior in the example suggests they performed different operations, increasing some values more than my method would. I'll recheck to ensure everything aligns.

Recalculating the cost

Okay, it seems the problem is about minimizing the sum of the values z added across multiple operations, not minimizing the total units added to the array. Each operation adds z to a contiguous segment, and our goal is to minimize the sum of z for all operations. I need to break down the transformation to a nondecreasing array into smaller additive operations, where we can simplify it to accumulate differences and count how many operations are necessary to cover the changes across the array.

Analyzing the cost function

I think I've settled on the cost structure now: the cost is related to the positive differences between consecutive values of the array c, which represents increments. So, if c increases at each step, we record the cost as the difference; decreases don't add to the cost. The goal is to minimize this cost, which seems to just involve calculating the necessary increment at each position to keep the values nondecreasing. The sample cases confirm our approach. Now I just need to iterate and ensure everything works as expected.

Thought for 1m 24s

Answer

Idea

Think of building the final array b that must be non-decreasing and never smaller than the given power array a.

text
a[i]      … original power
inc[i]    … how much we add to a[i]
b[i] = a[i] + inc[i]   … final value

An operation “add z to a contiguous sub-array” can be split into z unit-operations that each cost 1, so the total cost is

text
cost = Σ inc[i] started for the *first* time
      = inc[0] + Σ max(0, inc[i] − inc[i-1])

Hence we only pay when the running “height” inc climbs above its previous level.

To spend the least possible, at every position we push the current element up just enough to keep the whole prefix sorted, and nothing more.

Algorithm (O (n) time, O (1) memory)

text
cur_inc = 0          // inc[i-1]
prev_b  = a[0]       // b[i-1]
ans     = 0

for i = 0 … n-1
    need_b  = max(prev_b, a[i])        // smallest allowed b[i]
    inc_i   = need_b - a[i]            // how much we must add here

    if inc_i > cur_inc                 // we raise the “water level”
        ans += inc_i - cur_inc         // pay only the extra height

    cur_inc = inc_i
    prev_b  = need_b

ans is the minimum possible sum of z-values.

All numbers fit in 64 bit (n ≤ 1e5, power[i] ≤ 1e9), so use long long.

Implementation

cpp
#include <bits/stdc++.h>
using namespace std;

/*
 * Complete the 'findMinimumSum' function below.
 * The function is expected to return a LONG_INTEGER.
 * The function accepts INTEGER_ARRAY power as parameter.
 */
long long findMinimumSum(vector<int> power) {
    const int n = power.size();
    long long ans = 0;                  // minimum sum of z's
    long long cur_inc = 0;              // inc at position i-1
    long long prev_b  = power[0];       // b at position i-1

    for (int i = 0; i < n; ++i) {
        long long need_b = max(prev_b, (long long)power[i]);
        long long inc_i  = need_b - power[i];

        if (inc_i > cur_inc)            // pay only for extra raise
            ans += inc_i - cur_inc;

        cur_inc = inc_i;
        prev_b  = need_b;
    }
    return ans;
}

Why it works

  • need_b is the smallest value that keeps the prefix sorted.
  • inc_i is therefore the minimum increment at i.
  • Paying only when inc_i exceeds the previous increment counts each new “layer” exactly once, which is the optimal cost proven by the unit-operation view.

The solution runs in linear time and constant extra memory, easily fitting the constraints.

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