ALG0183 Algorithms & Data Structures Lecture 6 The maximum contiguous subsequence sum problem....

ALG0183 Algorithms & Data Structures by Dr Andy Brooks

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ALG0183 Algorithms & Data Structures

Lecture 6The maximum contiguous subsequence sum problem.

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Weiss Chapter 5.3

There are many algorithms to solve this problem and their performances vary dramatically.


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The maximum contiguous subsequence sum problem.

• Given (possibly negative) integers A1, A2,...,AN, find (and identify the sequence corresponding to) the maximum value of:

• The maximum contiguous subsequence sum is zero if all the integers are negative.

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j

kk i

A


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Examples

• If the input is {-2, 11, -4, 13, -5, 2}the maximum sum is 20.

• If the input is {1, -3, 4, -2, -1, 6}the maximum sum is 7.

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The case where all the numbers are positive is not interesting.


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“Always consider emptiness” Weiss

• If all the integers were negative, the algorithm could return the smallest negative integer as the maximum contiguous subsequence sum.

• This would, however, be ignoring the empty subsequence of zero integers whose sum is 0.– Remember, the empty set is a subset of any set:∀A: ∅⊆ A

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“Be aware that emptiness is always a possibility and that in many instances it is not a special case at all.” Weiss


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The obvious O(N3) algorithm.exhaustive search or brute-force

• An obvious solution is work out the sum of all possible subsequences, updating maximum sum as required.

• A loop is needed to iterate through all the possible starting subscripts of the subsequences.

• An inner loop is needed to iterate through all the possible ending subscripts of the subsequences.

• For each subsequence, a loop is needed to compute the sum of the elements.

• We have a loop inside a loop inside a loop.– Informally, Big-Oh is O(N3).

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A cubic maximum contiguous subsequence sum algorithm.


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MaxSumTest.javathe beginning

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static means “any changes to a static member are made to all objects”private means “cannot be accessed by members of another class”


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A more formal calculation of Big-Oh.

• The summation is carried out the following number of times:

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3 2

1

3 21

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jN N

i j i k i

N N N

1

( 1)

2

n

x

n nx

2

1

( 1)(2 1)

6

n

x

n n nx

1

1n

x

n

How do we get the RHS from the LHS?


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1

( 1)

2

n

x

n nx

2

1

( 1)(2 1)

6

n

x

n n nx

1 1j

k i

j i

( 1)( 2)( 1)

2

n

j i

n i n ij i

expand terms in brackets

1 is added to itself (j-i+1) times

1

1n

x

n

1 +2+3+...+(n-i+1)

A lot of work.


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An improved O(N2) algorithm.

• To improve Big-Oh from O(N3) to O(N2) requires removing a loop (if possible).

• For a value of i, the summation includes just one more element from the previous calculation. This previous calculation is thrown away by the O(N3) algorithm.

• Making use of the previous calculation allows us to remove a loop and make the algorithm O(N2).

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1j j

k j kk i k i

A A A


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A quadratic maximum contiguous subsequence sum algorithm.


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Theorem 5.2 Weiss

• Let Ai,j be any sequence with sum Si,j < 0. If q > j, then Ai,q is not the maximum subsequence.– A larger subsequence can be obtained by

excluding the negative subsequence Ai,j.

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Figure 5.6 © Addison-Wesley

A maximum subsequence cannot begin with a negative number.


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Observation

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“All contiguous subsequences that border the maximum contiguous subsequence must have negative (or 0) sums (otherwise, we would include them).”

Weiss


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Observation

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At some i, at the first value of j for which the sum of the subsequence becomes negative, we know that:

a) The maximum occured between i and j-1, orb) The maximum started before i, orc) The maximum starts after j.

Any new maximum will occur after index j, so i can be updated to j+1.

For cases a) and b) we have already met and computed the maximum.

maximum - maximum subsequence


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...,1,2,3,-7,...At some i, we begin the usual process of summing subsequences.At some j, here when -7 is read, the sum is found to be negative.There is no point in considering further sequences starting at 1, 2, or 3.

a negative subsequence

Any subsequence that begins with a negative subsequence can be improved by eliminating that negative subsequence.


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A linear maximum contiguous subsequence sum algorithm.


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Lecture activity: trace maxSubSum3

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thisSum maxSum seqStart seqEnd

j=0 2 5 -9 6 -1 3 -5 4

j=1 2 5 -9 6 -1 3 -5 4

j=2 2 5 -9 6 -1 3 -5 4

j=3 2 5 -9 6 -1 3 -5 4

j=4 2 5 -9 6 -1 3 -5 4

j=5 2 5 -9 6 -1 3 -5 4

j=6 2 5 -9 6 -1 3 -5 4

j=7 2 5 -9 6 -1 3 -5 4


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From O(N3) to O(N2) to O(N): be amazed!Of course, it is usually not possible to improve an algorithm´s performance quite so much.

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Figure 5.1 ©Addison Wesley

ALG0183 Algorithms & Data Structures Lecture 6 The maximum contiguous subsequence sum problem....

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