I recently encountered a problem which states : There is an array A which contains n integers. In one operation we can swap Ai with one of its neighbours i.e., swap(Ai,Ai+1) or swap(Ai,Ai-1) provided any element in the array can be swapped atmax once. After doing some optimal number of operations, we have to find the max value of summation of i*Ai for i = 1 to n.
For ex — A : [2,1,4,3] n = 4
Ans : 30 Swap elements at index 1,2 and elements at 3,4 that gives us the final array as [1,2,3,4]. The sum is 1*1 + 2*2 + 3*3 + 4*4 = 1+4+9+16 = 30
Can someone help me with the ideas to solve this problem ?
is there any link of this problem?
nope
if i swap Ai with Ai+1 does that mean i cant swap Ai+2 with Ai+1 ?
since Ai+1 swapped.
yes
Can solve using dp.
$$$\text{dp}[i]$$$ denotes max value for prefix upto $$$i$$$. Transitions are simple, if you swap you get $$$\text{dp}[i - 2] + (i - 1)\cdot\text{a}[i] + i\cdot\text{a}[i - 1]$$$, if you don't swap you get $$$\text{dp}[i - 1] + i\cdot\text{a}[i]$$$.
$$$\text{dp}[i]$$$ is the maximum of these two values. Answer is $$$\text{dp}[n]$$$. Note that I've 1-indexed everything for brevity.
swap and dont swap cases: dp[i]=max(dp[i−1] + arr[i] × i, dp[i−2] + arr[i]×(i−1) + arr[i−1]×i)
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