Divide and Conquer Algo to find maximum difference between two ordered elements

问题

Given an array arr[] of integers, find out the difference between any two elements such that larger element appears after the smaller number in arr[].

Max Difference = Max { arr[x] - arr[y] | x > y }

Examples:

If array is [2, 3, 10, 6, 4, 8, 1, 7] then returned value should be 8 (Diff between 10 and 2).
If array is [ 7, 9, 5, 6, 3, 2 ] then returned value should be 2 (Diff between 7 and 9)

My Algorithm:

I thought of using D&C algorithm. Explanation

2, 3, 10, 6, 4, 8, 1, 7

then

2,3,10,6      and     4,8,1,7

then

2,3  and 10,6  and  4,8 and 1,7

then

2 and 3   10 and 6   4 and 8    1 and 7

Here as these elements will remain in same order, i will get the maximum difference, here it's 6.

Now i will move back to merege these arrays and again find the difference between minimum of first block and maximum of second block and keep doing this till end.

I am not able to implement this in my code. can anyone please provide a pseudo code for this?

回答1:

We have max { A[i] - A[j] | i > j } = max { A[i] - min { A[j] | j < i } | i }, which yields a straightforward O(n) algorithm:

prefix_min = A[0]
result = -infinity
for i := 1 to n - 1:
    # invariant: We have prefix_min = min { A[j] | j < i }
    result = max(result, A[i] - prefix_min)
    prefix_min = min(prefix_min, A[i])

Divide & Conquer is conceptionally more complicated, but also leads to a linear time solution (with a higher constant factor).

回答2:

Say you want to find the largest Difference LD(A[])

Complete Psuedocode as desired:

Divide the array in two parts A1[] and A2[].

Find minimum & maximum element in A1[] and LD(A1).
Find minimum & maximum element in A2[] and LD(A2).

LD(A) = max( LD(A1), LD(A2), MAX(A2) - MIN(A1) )
MAX(A) = max( MAX(A1), MAX(A2) )
MIN(A) = min( MIN(A1), MIN(A2) )

Base Case (length(A) == 2):

If A[1] > A[0], 
  LD(A) = A[1] - A[0].
  MAX(A) = A[1]
  MIN(A) = A[0]
else
  LD(A) = 0.
  MAX(A) = A[0]
  MIN(A) = A[1]

Note:

If (length(A) == 1)
    LD(A) = 0
    MIN(A) = MAX(A) = A[0]

Similarly you can calculate the min and max elements in each subarray.

回答3:

Below is the code in C using Divide and Conquer.
Plz. do feel free to comment.

#include <stdio.h>

struct data{
    int min_;
    int max_;
};

struct data crossminmax(int *p,int lo,int mid,int hi){
    int i,min,max;
    struct data temp;

    min = p[mid];
    for(i=mid;i>=lo;i--){
        if(p[i] < min){
            min = p[i];
        }
    }

    max = p[mid+1];
    for(i=mid+1;i<=hi;i++){
        if(p[i] > max){
            max = p[i];
        }
    }
    temp.min_ = min;
    temp.max_ = max;
    return temp;
}

/* MinMax calculates the difference between Biggest and Smallest element  *
 * of an array using divide and conquer principles such that the position * 
 * of Biggest element is always greater than the Samllest element         */

struct data minmax(int *p,int lo,int hi){
    int mid,leftdiff,rightdiff,crossdiff;
    struct data left,right,cross,temp;

    if(lo == hi){
        temp.min_ = p[lo];
        temp.max_ = p[hi];
        return temp;
    }

    mid = (lo+hi)/2;
    left = minmax(p,lo,mid);
    right = minmax(p,mid+1,hi);

    cross = crossminmax(p,lo,mid,hi);
    leftdiff = left.max_ - left.min_;
    rightdiff = right.max_ - right.min_;
    crossdiff = cross.max_ - cross.min_;

    if(leftdiff > rightdiff && leftdiff > crossdiff){
        return left;
    }else if(rightdiff > crossdiff){
        return right;
    }else{
        return cross;
    }
}

int main(){
    int arr[] = {5,2,3,10,1,3,16,4,3};
    struct data dt;
    dt = minmax(arr,0,8);
    printf("Max difference = %d, Max Element=%d, Min Element = %d  \n",dt.max_ - dt.min_,dt.max_,dt.min_);
    return 0;
}

回答4:

largest differance in array

condition : - larger number should appear after smaller number
{ 10, 3, 6, 4, 8, 1, 7 }     6
{ 2, 3, 10, 6, 4, 8, 1 }     8
{ 7, 9, 5, 6, 3, 2 }         1
{ 1, 2, 3, 4 }               3
{ 4, 3, 2, 1  }              0



#include<stdio.h>
int main(){
  int n = 7;
  int arr[7] = { 10, 3, 6, 14, 8, 1, 7 };

  int i;
  int max_diff = 0;
  int min = arr[0];

  for( i=0; i<n; i++){
    if( (arr[i] - min) > max_diff ){
      max_diff = arr[i] - min;
    }

    if(arr[i] < min){
      min = arr[i];
    }
  }
  printf("max diff = %d", max_diff);

  return 0;
}

//time complexity O(n)

//space complexity O(1)

for better understanding please visit https://www.youtube.com/watch?v=thPG6eTPf68&t=130s

来源：https://stackoverflow.com/questions/24055608/divide-and-conquer-algo-to-find-maximum-difference-between-two-ordered-elements

标签

algorithm

divide-and-conquer