Find index with minimum sum of prefix and suffix sums in an array
In this article, we will explore the problem of finding the index with the minimum sum of prefix and suffix sums in an array. We'll describe the problem statement, provide a step-by-step explanation of the algorithm to solve it, and analyze the time complexity of the code. By the end of this article, you'll have a clear understanding of how to find the index that minimizes the sum of prefix and suffix sums in an array.
Problem Statement
Given an array of integers, we need to find an index such that the sum of the prefix elements up to that index and
the sum of the suffix elements from that index onward are minimized. Formally, we want to find an index
i
that minimizes the sum prefixSum[0..i] + suffixSum[i..n-1]
, where n
is the
size of the array.
Solution Idea
To solve this problem efficiently, we can pre-calculate prefix sums and suffix sums for each index and then iterate through the array to find the index that minimizes the sum of prefix and suffix sums.
Standard Pseudocode
Here's the pseudocode that outlines the approach to find the index with the minimum sum of prefix and suffix sums:
function minPrefixSuffix(arr):
Initialize an array prefix of size n
Initialize an array suffix of size n
Initialize a variable position to 0
Calculate prefix[0] = arr[0]
Calculate suffix[n-1] = arr[n-1]
for i from 1 to n-1:
prefix[i] = prefix[i-1] + arr[i]
for i from n-2 to 0:
suffix[i] = suffix[i+1] + arr[i]
Initialize a variable sum = prefix[0] + suffix[0]
for i from 1 to n-1:
if prefix[i] + suffix[i] < sum:
position = i
sum = prefix[i] + suffix[i]
return position
Algorithm Explanation
-
We start by initializing two arrays
prefix
andsuffix
, both of sizen
wheren
is the size of the input array. These arrays will store the prefix sums and suffix sums for each index. -
We calculate the first element of
prefix
by directly copying the first element of the input array:prefix[0] = arr[0]
. -
Similarly, we calculate the last element of
suffix
by copying the last element of the input array:suffix[n-1] = arr[n-1]
. -
We then use a loop to calculate the rest of the
prefix
array. For each indexi
from 1 ton-1
, we calculateprefix[i]
by adding the current elementarr[i]
to the previous prefix sumprefix[i-1]
. -
We use another loop to calculate the
suffix
array in reverse. For each indexi
fromn-2
to 0, we calculatesuffix[i]
by adding the current elementarr[i]
to the next suffix sumsuffix[i+1]
. -
We initialize a variable
sum
to store the minimum sum of prefix and suffix sums. We also initialize a variableposition
to store the index that corresponds to the minimum sum. -
We then iterate through the array using a loop from 1 to
n-1
. For each indexi
, we check if the sum ofprefix[i]
andsuffix[i]
is less than the current minimumsum
. If it is, we updateposition
with the current indexi
and updatesum
with the new minimum sum. -
After iterating through the array, the variable
position
will hold the index that corresponds to the minimum sum of prefix and suffix sums. -
We return the value of
position
as the result.
Code Solution
/*
C program for
Find index with minimum sum of prefix and
suffix sums in an Array
*/
#include <stdio.h>
void minPrefixSuffix(int arr[], int n)
{
// This is use collect prefix sum
int prefix[n];
// This is use collect suffix sum
int suffix[n];
int position = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
// Calculate prefix sum
for (int i = 1; i < n; ++i)
{
prefix[i] = prefix[i - 1] + arr[i];
}
// Calculate suffix sum
for (int i = n - 2; i >= 0; --i)
{
suffix[i] = suffix[i + 1] + arr[i];
}
// First pair of prefix and suffix
int sum = prefix[0] + suffix[0];
for (int i = 1; i < n; ++i)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
}
// Display calculated position
printf(" Position : %d", position);
}
int main(int argc, char
const *argv[])
{
int arr[] = {
3 , -6 , 1 , 6 , -7 , -4 , -3 , -2 , -1 , 6
};
int n = sizeof(arr) / sizeof(arr[0]);
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
minPrefixSuffix(arr, n);
return 0;
}
Output
Position : 4
/*
Java program for
Find index with minimum sum of prefix and suffix sums in an array
*/
public class PrefixSuffix
{
public void minPrefixSuffix(int[] arr, int n)
{
// This is use collect prefix sum
int[] prefix = new int[n];
// This is use collect suffix sum
int[] suffix = new int[n];
int position = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
// Calculate prefix sum
for (int i = 1; i < n; ++i)
{
prefix[i] = prefix[i - 1] + arr[i];
}
// Calculate suffix sum
for (int i = n - 2; i >= 0; --i)
{
suffix[i] = suffix[i + 1] + arr[i];
}
// First pair of prefix and suffix
int sum = prefix[0] + suffix[0];
for (int i = 1; i < n; ++i)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
}
// Display calculated position
System.out.print(" Position : " + position);
}
public static void main(String[] args)
{
PrefixSuffix task = new PrefixSuffix();
int[] arr = {
3 , -6 , 1 , 6 , -7 , -4 , -3 , -2 , -1 , 6
};
int n = arr.length;
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task.minPrefixSuffix(arr, n);
}
}
Output
Position : 4
// Include header file
#include <iostream>
using namespace std;
/*
C++ program for
Find index with minimum sum of prefix and suffix sums in an Array
*/
class PrefixSuffix
{
public: void minPrefixSuffix(int arr[], int n)
{
// This is use collect prefix sum
int prefix[n];
// This is use collect suffix sum
int suffix[n];
int position = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
// Calculate prefix sum
for (int i = 1; i < n; ++i)
{
prefix[i] = prefix[i - 1] + arr[i];
}
// Calculate suffix sum
for (int i = n - 2; i >= 0; --i)
{
suffix[i] = suffix[i + 1] + arr[i];
}
// First pair of prefix and suffix
int sum = prefix[0] + suffix[0];
for (int i = 1; i < n; ++i)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
}
// Display calculated position
cout << " Position : " << position;
}
};
int main()
{
PrefixSuffix *task = new PrefixSuffix();
int arr[] = {
3 , -6 , 1 , 6 , -7 , -4 , -3 , -2 , -1 , 6
};
int n = sizeof(arr) / sizeof(arr[0]);
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task->minPrefixSuffix(arr, n);
return 0;
}
Output
Position : 4
// Include namespace system
using System;
/*
Csharp program for
Find index with minimum sum of prefix and
suffix sums in an Array
*/
public class PrefixSuffix
{
public void minPrefixSuffix(int[] arr, int n)
{
// This is use collect prefix sum
int[] prefix = new int[n];
// This is use collect suffix sum
int[] suffix = new int[n];
int position = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
// Calculate prefix sum
for (int i = 1; i < n; ++i)
{
prefix[i] = prefix[i - 1] + arr[i];
}
// Calculate suffix sum
for (int i = n - 2; i >= 0; --i)
{
suffix[i] = suffix[i + 1] + arr[i];
}
// First pair of prefix and suffix
int sum = prefix[0] + suffix[0];
for (int i = 1; i < n; ++i)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
}
// Display calculated position
Console.Write(" Position : " + position);
}
public static void Main(String[] args)
{
PrefixSuffix task = new PrefixSuffix();
int[] arr = {
3 , -6 , 1 , 6 , -7 , -4 , -3 , -2 , -1 , 6
};
int n = arr.Length;
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task.minPrefixSuffix(arr, n);
}
}
Output
Position : 4
package main
import "fmt"
/*
Go program for
Find index with minimum sum of prefix and suffix sums in an Array
*/
func minPrefixSuffix(arr[] int, n int) {
// This is use collect prefix sum
var prefix = make([] int, n)
// This is use collect suffix sum
var suffix = make([] int, n)
var position int = 0
prefix[0] = arr[0]
suffix[n - 1] = arr[n - 1]
// Calculate prefix sum
for i := 1 ; i < n ; i++ {
prefix[i] = prefix[i - 1] + arr[i]
}
// Calculate suffix sum
for i := n - 2 ; i >= 0 ; i-- {
suffix[i] = suffix[i + 1] + arr[i]
}
// First pair of prefix and suffix
var sum int = prefix[0] + suffix[0]
for i := 1 ; i < n ; i++ {
if prefix[i] + suffix[i] < sum {
position = i
// We get new pair
sum = prefix[i] + suffix[i]
}
}
// Display calculated position
fmt.Print(" Position : ", position)
}
func main() {
var arr = [] int {3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 }
var n int = len(arr)
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
minPrefixSuffix(arr, n)
}
Output
Position : 4
<?php
/*
Php program for
Find index with minimum sum of prefix and suffix sums in an Array
*/
class PrefixSuffix
{
public function minPrefixSuffix($arr, $n)
{
// This is use collect prefix sum
$prefix = array_fill(0, $n, 0);
// This is use collect suffix sum
$suffix = array_fill(0, $n, 0);
$position = 0;
$prefix[0] = $arr[0];
$suffix[$n - 1] = $arr[$n - 1];
// Calculate prefix sum
for ($i = 1; $i < $n; ++$i)
{
$prefix[$i] = $prefix[$i - 1] + $arr[$i];
}
// Calculate suffix sum
for ($i = $n - 2; $i >= 0; --$i)
{
$suffix[$i] = $suffix[$i + 1] + $arr[$i];
}
// First pair of prefix and suffix
$sum = $prefix[0] + $suffix[0];
for ($i = 1; $i < $n; ++$i)
{
if ($prefix[$i] + $suffix[$i] < $sum)
{
$position = $i;
// We get new pair
$sum = $prefix[$i] + $suffix[$i];
}
}
// Display calculated position
echo(" Position : ".$position);
}
}
function main()
{
$task = new PrefixSuffix();
$arr = array(3, -6, 1, 6, -7, -4, -3, -2, -1, 6);
$n = count($arr);
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
$task->minPrefixSuffix($arr, $n);
}
main();
Output
Position : 4
/*
Node JS program for
Find index with minimum sum of prefix and suffix sums in an Array
*/
class PrefixSuffix
{
minPrefixSuffix(arr, n)
{
// This is use collect prefix sum
var prefix = Array(n).fill(0);
// This is use collect suffix sum
var suffix = Array(n).fill(0);
var position = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
// Calculate prefix sum
for (var i = 1; i < n; ++i)
{
prefix[i] = prefix[i - 1] + arr[i];
}
// Calculate suffix sum
for (var i = n - 2; i >= 0; --i)
{
suffix[i] = suffix[i + 1] + arr[i];
}
// First pair of prefix and suffix
var sum = prefix[0] + suffix[0];
for (var i = 1; i < n; ++i)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
}
// Display calculated position
process.stdout.write(" Position : " + position);
}
}
function main()
{
var task = new PrefixSuffix();
var arr = [3, -6, 1, 6, -7, -4, -3, -2, -1, 6];
var n = arr.length;
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task.minPrefixSuffix(arr, n);
}
main();
Output
Position : 4
# Python 3 program for
# Find index with minimum sum of prefix and suffix sums in an Array
class PrefixSuffix :
def minPrefixSuffix(self, arr, n) :
# This is use collect prefix sum
prefix = [0] * (n)
# This is use collect suffix sum
suffix = [0] * (n)
position = 0
prefix[0] = arr[0]
suffix[n - 1] = arr[n - 1]
i = 1
# Calculate prefix sum
while (i < n) :
prefix[i] = prefix[i - 1] + arr[i]
i += 1
i = n - 2
# Calculate suffix sum
while (i >= 0) :
suffix[i] = suffix[i + 1] + arr[i]
i -= 1
# First pair of prefix and suffix
sum = prefix[0] + suffix[0]
i = 1
while (i < n) :
if (prefix[i] + suffix[i] < sum) :
position = i
# We get new pair
sum = prefix[i] + suffix[i]
i += 1
# Display calculated position
print(" Position : ", position, end = "")
def main() :
task = PrefixSuffix()
arr = [3, -6, 1, 6, -7, -4, -3, -2, -1, 6]
n = len(arr)
# arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
# -----------------------------------------------
# position 0 1 2 3 4 5 6 7 8 9
# Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
# Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
# -----------------------------------------------
# position 4 contain minimum prefix and suffix
# arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
# ↑
# position
task.minPrefixSuffix(arr, n)
if __name__ == "__main__": main()
Output
Position : 4
# Ruby program for
# Find index with minimum sum of prefix and suffix sums in an Array
class PrefixSuffix
def minPrefixSuffix(arr, n)
# This is use collect prefix sum
prefix = Array.new(n) {0}
# This is use collect suffix sum
suffix = Array.new(n) {0}
position = 0
prefix[0] = arr[0]
suffix[n - 1] = arr[n - 1]
i = 1
# Calculate prefix sum
while (i < n)
prefix[i] = prefix[i - 1] + arr[i]
i += 1
end
i = n - 2
# Calculate suffix sum
while (i >= 0)
suffix[i] = suffix[i + 1] + arr[i]
i -= 1
end
# First pair of prefix and suffix
sum = prefix[0] + suffix[0]
i = 1
while (i < n)
if (prefix[i] + suffix[i] < sum)
position = i
# We get new pair
sum = prefix[i] + suffix[i]
end
i += 1
end
# Display calculated position
print(" Position : ", position)
end
end
def main()
task = PrefixSuffix.new()
arr = [3, -6, 1, 6, -7, -4, -3, -2, -1, 6]
n = arr.length
# arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
# -----------------------------------------------
# position 0 1 2 3 4 5 6 7 8 9
# Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
# Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
# -----------------------------------------------
# position 4 contain minimum prefix and suffix
# arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
# ↑
# position
task.minPrefixSuffix(arr, n)
end
main()
Output
Position : 4
/*
Scala program for
Find index with minimum sum of prefix and suffix sums in an Array
*/
class PrefixSuffix()
{
def minPrefixSuffix(arr: Array[Int], n: Int): Unit = {
// This is use collect prefix sum
var prefix: Array[Int] = Array.fill[Int](n)(0);
// This is use collect suffix sum
var suffix: Array[Int] = Array.fill[Int](n)(0);
var position: Int = 0;
prefix(0) = arr(0);
suffix(n - 1) = arr(n - 1);
var i: Int = 1;
// Calculate prefix sum
while (i < n)
{
prefix(i) = prefix(i - 1) + arr(i);
i += 1;
}
i = n - 2;
// Calculate suffix sum
while (i >= 0)
{
suffix(i) = suffix(i + 1) + arr(i);
i -= 1;
}
// First pair of prefix and suffix
var sum: Int = prefix(0) + suffix(0);
i = 1;
while (i < n)
{
if (prefix(i) + suffix(i) < sum)
{
position = i;
// We get new pair
sum = prefix(i) + suffix(i);
}
i += 1;
}
// Display calculated position
print(" Position : " + position);
}
}
object Main
{
def main(args: Array[String]): Unit = {
var task: PrefixSuffix = new PrefixSuffix();
var arr: Array[Int] = Array(3, -6, 1, 6, -7, -4, -3, -2, -1, 6);
var n: Int = arr.length;
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task.minPrefixSuffix(arr, n);
}
}
Output
Position : 4
import Foundation;
/*
Swift 4 program for
Find index with minimum sum of prefix and suffix sums in an Array
*/
class PrefixSuffix
{
func minPrefixSuffix(_ arr: [Int], _ n: Int)
{
// This is use collect prefix sum
var prefix: [Int] = Array(repeating: 0, count: n);
// This is use collect suffix sum
var suffix: [Int] = Array(repeating: 0, count: n);
var position: Int = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
var i: Int = 1;
// Calculate prefix sum
while (i < n)
{
prefix[i] = prefix[i - 1] + arr[i];
i += 1;
}
i = n - 2;
// Calculate suffix sum
while (i >= 0)
{
suffix[i] = suffix[i + 1] + arr[i];
i -= 1;
}
// First pair of prefix and suffix
var sum: Int = prefix[0] + suffix[0];
i = 1;
while (i < n)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
i += 1;
}
// Display calculated position
print(" Position : ", position, terminator: "");
}
}
func main()
{
let task: PrefixSuffix = PrefixSuffix();
let arr: [Int] = [3, -6, 1, 6, -7, -4, -3, -2, -1, 6];
let n: Int = arr.count;
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task.minPrefixSuffix(arr, n);
}
main();
Output
Position : 4
/*
Kotlin program for
Find index with minimum sum of prefix and
suffix sums in an array
*/
class PrefixSuffix
{
fun minPrefixSuffix(arr: Array < Int > , n: Int): Unit
{
// This is use collect prefix sum
var prefix: Array < Int > = Array(n)
{
0
};
// This is use collect suffix sum
var suffix: Array < Int > = Array(n)
{
0
};
var position: Int = 0;
prefix[0] = arr[0];
suffix[n - 1] = arr[n - 1];
var i: Int = 1;
// Calculate prefix sum
while (i < n)
{
prefix[i] = prefix[i - 1] + arr[i];
i += 1;
}
i = n - 2;
// Calculate suffix sum
while (i >= 0)
{
suffix[i] = suffix[i + 1] + arr[i];
i -= 1;
}
// First pair of prefix and suffix
var sum: Int = prefix[0] + suffix[0];
i = 1;
while (i < n)
{
if (prefix[i] + suffix[i] < sum)
{
position = i;
// We get new pair
sum = prefix[i] + suffix[i];
}
i += 1;
}
// Display calculated position
print(" Position : " + position);
}
}
fun main(args: Array < String > ): Unit
{
val task: PrefixSuffix = PrefixSuffix();
val arr: Array < Int > = arrayOf(3, -6, 1, 6, -7, -4, -3, -2, -1, 6);
val n: Int = arr.count();
/*
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
-----------------------------------------------
position 0 1 2 3 4 5 6 7 8 9
Prefix sum [3 -3 -2 4 -3 -7 -10 -12 -13 -7]
Suffix sum [-7 -10 -4 -5 -11 -4 0 3 5 6]
-----------------------------------------------
position 4 contain minimum prefix and suffix
arr [3, -6, 1, 6, -7, -4, -3, -2, -1 , 6 ]
↑
position
*/
task.minPrefixSuffix(arr, n);
}
Output
Position : 4
Time Complexity Analysis
The time complexity of this algorithm is O(n), where n is the size of the input array. This is because we iterate through the array three times: once to calculate the prefix sums, once to calculate the suffix sums, and once to find the index with the minimum sum. The individual calculations and comparisons are done in constant time.
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