How can I find the maximum sum of a sub-sequence using dynamic programming?

Question

I\'m re-reading Skiena\'s Algorithm Design Manual to catch up on some stuff I\'ve forgotten since school, and I\'m a little baffled by his descriptions of Dynamic Programmin

Answer 1

My understanding of DP is about "making a table". In fact, the original meaning "programming" in DP is simply about making tables.

The key is to figure out what to put in the table, or modern terms: what state to track, or what's the vertex key/value in DAG (ignore these terms if they sound strange to you).

How about choose dp[i] table being the largest sum ending at index i of the array, for example, the array being [5, 15, -30, 10]

The second important key is "optimal substructure", that is to "assume" dp[i-1] already stores the largest sum for sub-sequences ending at index i-1, that's why the only step at i is to decide whether to include a[i] into the sub-sequence or not

dp[i] = max(dp[i-1], dp[i-1] + a[i])

The first term in max is to "not include a[i]", the second term is to "include a[i]". Notice, if we don't include a[i], the largest sum so far remains dp[i-1], which comes from the "optimal substructure" argument.

So the whole program looks like this (in Python):

a = [5,15,-30,10]

dp = [0]*len(a)
dp[0] = max(0,a[0]) # include a[0] or not

for i in range(1,len(a)):
    dp[i] = max(dp[i-1], dp[i-1]+a[i]) # for sub-sequence, choose to add or not     


 print(dp, max(dp)) 

The result: largest sum of sub-sequence should be the largest item in dp table, after i iterate through the array a. But take a close look at dp, it holds all the information.

Since it only goes through items in array a once, it's a O(n) algorithm.

This problem seems silly, because as long as a[i] is positive, we should always include it in the sub-sequence, because it will only increase the sum. This intuition matches the code

dp[i] = max(dp[i-1], dp[i-1] + a[i])

So the max. sum of sub-sequence problem is easy, and doesn't need DP at all. Simply,

sum = 0
for v in a:
     if  v >0
         sum += v

However, what about largest sum of "continuous sub-array" problem. All we need to change is just a single line of code

dp[i] = max(dp[i-1]+a[i], a[i])    

The first term is to "include a[i] in the continuous sub-array", the second term is to decide to start a new sub-array, starting a[i].

In this case, dp[i] is the max. sum continuous sub-array ending with index-i.

This is certainly better than a naive approach O(n^2)*O(n), to for j in range(0,i): inside the i-loop and sum all the possible sub-arrays.

One small caveat, because the way dp[0] is set, if all items in a are negative, we won't select any. So for the max sum continuous sub-array, we change that to

dp[0] = a[0]