Maximal Subarray - Linear Algorithm

Overview

• This algorithm illustrates using a solution to a smaller problem in a solution to a larger problem

First Example: Maximal Subarray

• Problem: Find maximal subarray in array X(1 .. $n$)
• Given integers $x_1, x_2, ..., x_n$, find $a ≤ b$ that give $\displaystyle \text{max} \left(\sum_{i=a}^b x_i\right)$
• That is: find largest sum of a subsequence


• Solutions:
• Naive algorithms: $O(n^3)$ and $O(n^2)$ (ie triple and double loops)
• Divide and conquer algorithm: $O(n\lg n)$
• Dynamic Programming (DP) algorithm: $O(n)$


• Let's work though developing the DP algorithm

DP Maximal Subarray Algorithm: Assumptions

• Assume the following:
• Your algorithm examines each position 1 .. $n$


• You are currently examining position $p$
• In other words, you are working on subproblem $p$


• The maximal subarray in 1 .. $p-1$ is at $a .. b$


• How could you use the maximal subarray in $1 .. p-1$ to find the maximal subarray in $1..p$?

Using the Subproblem to Solve the Problem

• Consider finding the maximal subarray in elements $x_1 .. x_p$ for $ 2 ≤ p ≤ n$


• Subproblem solution: assume the maximal subarray in elements 1 .. $p-1$ is at $a .. b$


• There are two cases for $a .. b$?


1. $a .. b$ reaches $p-1$ (ie $b=p-1$)


2. $a .. b$ does not reach $p-1$ (ie $b < p-1$)

Finding the Maximal Subarray: Case 1

• This case ($b = p-1$, ie $a .. b$ is the solution to $1 .. p-1$) is easy:


1. If $x_p > 0$, then set $b$ to $p$


• Thus, $a\dots p$ is the solution to $1 .. p$


2. If $x_p ≤ 0, a .. b$ does not change


• Thus, $a\dots b$, the solution to $1 .. p-1$ is also the solution to $1 .. p$

Maximal Subarray: Case 2

• This case ($b < p-1$) is harder:


• What is the problem?


• What information would make things easier?
• Think about it some.

Maximal Subarray: Extra Information

• Some additional information makes the problem solvable


• Assume that $c .. d$ is the range of the best solution that ends at $d$


• When $p=d+1$ is considered, what can happen to the $c .. d$
• Consider the values of the sums $S_{ab}$ and $S_{cd}$

Maximal Subarray: Diagram

• Diagram showing ab and cd and p

Maximal Subarray: Solution

• An Ada solution: maxsub.adb (and prettified)


• A simpler solution, that only finds the max, but not where it is:

    function maxsub(A) return int -- return the sum of the maximal subarray
        bestsum: array(0..A'last) of int  -- Holds solution to each subproblem 1 .. n

        bestsum(0) := 0   -- a initial value so we can access bestsum(1-1)

        for i in A'range loop
            bestsum(i) := A(i) + max(0, bestsum(i-1))
        end loop

        return max(bestsum)