usaco-guide/content/4_Silver/6_Silver_Psum.md

slug	title	author	order
/silver/prefix-sums	Prefix Sums	Eric Wei	6

Given an array A_1,A_2,\ldots,A_N, answer Q queries of the following form: compute A_L+A_{L+1}+\cdots+A_R.

Standard

• CSES Range Sum Queries I
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Tutorials

This technique is also known as cumulative sum or partial sums.

• Intro to USACO 11
• CPH 9.1

USACO Silver Problems

These problems are relatively straightforward.

• USACO Breed Counting
• USACO Subsequences Summing to Seven

Now we'll look at some extensions.

Max Subarray Sum

• Maximum Subarray Sum

(Note: This problem has a solution known as Kadane's Algorithm. Please don't use that solution; try to solve it with prefix sums.)

Extension:

• CSES Maximum Subarray Sum II

2D Prefix Sums

Given a 2-dimensional array of size NxM, answer Q queries of the following form: Find the sum of all elements within the rectangle of indices (x1,y1) to (x2,y2).

• CSES Forest Queries
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• USACO Painting the Barn (Gold)
  • combine with max subarray sum!

Difference Array

Task: Given an array of size N, do the following operation Q times: add X to the values between i and j. Afterwards, print the final array.

Solution: Consider the array formed by a_i-a_{i-1}. When processing a range addition, only two values in this difference array change! At the end, we can recover the original array using prefix sums. (The math is left as an exercise to the reader.)

• USACO Haybale Stacking

Prefix Minimum, XOR, etc.

Similar to prefix sums, you can also take prefix minimum or maximum; but you cannot answer min queries over an arbitrary range with prefix minimum. (This is because minimum doesn't have an inverse operation, the way subtraction is to addition.) On the other hand, XOR is its own inverse operation, meaning that the XOR of any number with itself is zero.

• USACO My Cow Ate My Homework
• CSES Range XOR Queries

More Complex Applications

Instead of storing just the values themselves, you can also take a prefix sum over i\cdot a_i, or 10^i \cdot a_i, for instance. Some math is usually helpful for these problems; don't be afraid to get dirty with algebra!

For instance, let's see how to quickly answer the following type of query: Find

1\cdot a_l+2\cdot a_{l+1}+3\cdot a_{l+2}+\cdots+(r-l+1)\cdot a_{r}.

First, define the following:

ps[i] = a_1+a_2+a_3+a_4+\cdots+a_i

ips[i] = 1\cdot a_1+2\cdot a_2+\cdots+i\cdot a_i

Then, we have:

l\cdot a_l + (l+1) \cdot a_{l+1} + \cdots + r \cdot a_r = ips[r]-ips[l-1]

(l-1) \cdot a_l + (l-1) \cdot a_{l+1} + \cdot + (l-1) \cdot a_r = (l-1)(ps[r]-ps[l-1])

And so,

1\cdot a_l + 2 \cdot a_{l+1} + \cdots + (r-l+1) \cdot a_r = ips[r]-ips[l-1]-(l-1)(ps[r]-ps[l-1])

Which is what we were looking for!

• AtCoder Multiple of 2019 (You may want to solve the below problem "Subsequences Summing to Seven" before doing this one.)
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