Distance maximizing in array
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Distance Maximizing in Array. Yang Liu. Problem. Given an array of integers, find the maximum of j- i subject to that A[ i ]<A[j] Example Input: 5 2 3 1 7 Output: 4(=4-0). Brute Force. For(dist=n-1; dist>=0; dist--) for( i =0; i <n-dist; i ++) if(A[ i+dist ]>A[ i ]

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Distance Maximizing in Array

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Distance maximizing in array

Distance Maximizing in Array

Yang Liu


Problem

Problem

  • Given an array of integers, find the maximum of j-i subject to that A[i]<A[j]

    Example

    Input: 5 2 3 1 7

    Output: 4(=4-0)


Brute force

Brute Force

For(dist=n-1; dist>=0; dist--)

for(i=0; i<n-dist; i++)

if(A[i+dist]>A[i]

return dist;


O n algorithm

O(n) Algorithm

  • Observation

    For input “5 2 3 1 7”,

    • 3 never could be i which maximize j-I since 2 is before 3.

    • 1 never could be j since 1 is before 7

  • Idea:

    • Use lArray to store possible is.

    • Use rArray to store possible js.


Possible i s

Possible is

k=0

I[k]=A[0]

Iind[k]=0;

For(i=1;i<n;i++)

if(A[i]<I[k])

k=k+1;

I[k]=A[i]

Iind[k]=i;

A=9 7 3 1 10 2 8 6 5 4

I=9 7 3 1

Iind=0 1 2 3


Possible j s

Possible js

k=0

J[k]=A[n-1]

Jind[k]=n-1;

For(i=n-2;i>=0;i--)

if(A[i]>J[k])

k=k+1;

J[k]=A[i]

Jind[k]=i;

A=9 7 3 1 10 2 8 6 5 4

J= 10 6 5 4

Iind= 4 7 8 9


Finding maximum distance

Finding Maximum Distance

9 7 3 1

I=9 7 3 1

Iind=0 1 2 3

The furthest i for J[0]

J= 10 6 5 4

Iind= 4 7 8 9

10 6 5 4

dist: 4(=4-0)


Finding maximum distance1

Finding Maximum Distance

9 7 3 1

I=9 7 3 1

Iind=0 1 2 3

The furthest i for J[1]

J= 10 6 5 4

Iind= 4 7 8 9

10 6 5 4

dist: 5(=7-2)


Finding maximum distance2

Finding Maximum Distance

9 7 3 1

I=9 7 3 1

Iind=0 1 2 3

The furthest i for J[2]

J= 10 6 5 4

Iind= 4 7 8 9

10 6 5 4

dist: 6(=8-2)


Finding maximum distance3

Finding Maximum Distance

9 7 3 1

I=9 7 3 1

Iind=0 1 2 3

The furthest i for J[3]

J= 10 6 5 4

Iind= 4 7 8 9

10 6 5 4

dist: 7(=9-2)


Finding maximum distance4

Finding Maximum Distance

  • Queue Q to store Iind

  • Stack S to store Jind

    maxDist=0;

    While(S is not empty)

    jInd=S.pop();

    while(Q is not empty)

    iInd=Q.front();

    if(A[jInd]>A[iInd])

    maxDist=(jInd-iInd>maxDist)?jInd-iInd:maxDist;

    break;

    else

    Q.deque();


Exercise 1

Exercise 1

  • You have an array A such that A[i] is the price of a stock on day i.

    You are only permitted to buy one share of the stock and sell one share of the stock.

    Design an algorithm to find the best times to buy and sell.


Exercise 2

Exercise 2

Given an array A, find maximum A[i]-A[j] where i<j

Example

Input: A: 2 9 3 1 4

Output: 8(=9-1)


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