Download Maximum subarray sum Example Solution 1: MaxsubSlow

Maximum subarray sum
CSC 212: Data Structures and Abstractions
02: Maximum Subarray Problem
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Given an array of integers:
Marco Alvarez
A = [a1 , a2 , . . . , an ]
sj,k = aj + aj+1 + · · · + ak =
Department of Computer Science and Statistics
University of Rhode Island
Spring 2017
k
X
ai
i=j
Definition. The maximum subarray sum of array A is the sequence A[j : k], 0 
j  k  n that maximizes sj,k , the sum of its values.
Remark. By convention, the special array element A[0] = 0 is defined.
If all elements are negative, then the solution is an empty
subarray of zero sum.
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Example
Solution 1: MaxsubSlow
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Apply Brute Force
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calculate the partial sums of every possible subarray
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for every sum, compare it to a running maximum and
update it if necessary
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Example
A
0
Analysis
3
-4
1
-4
3
7
1
-2
O(n3 )
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Solution 2: MaxsubFaster
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Example
Wasting time calculating all partial sums
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consider using prefix sums
Prefix sums
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A
0
3
-4
1
-4
3
7
1
-2
S
0
3
-1
0
-4
-1
6
7
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sum of the first t integers in A is denoted by:
t
X
St = a 1 + a 2 + · · · + a t =
ai
i=1
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Any subarray summation can be calculated by:
Sj,k = Sk
Sj
1
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Analysis
Solution 3: MaxsubFastest
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Apply same idea …
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consider using maximum suffix sums
Maximum suffix sums
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Mt is the summation value for a maximum subarray that
ends at t, denoted by:
Mt = max{0, max {sj,t }}
j=1,...,t
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O(n )
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Any maximum suffix summation can be calculated
by:
Mt = max{0, Mt
1
+ A[t]}
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Example
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Analysis
A
0
3
-4
1
-4
3
7
1
-2
M
0
3
0
1
0
3
10
11
9
O(n)
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Processing Power
Computational Cost
Size of
Input
MaxSubSlow
3
O(n )
MaxSubFaster
2
O(n )
MaxSubFastest
O(n)
n=1
1
1
1
n = 10
1,000
100
10
n = 100
1,000,000
10,000
100
n = 1000
1,000,000,000
1,000,000
1,000
n = 10000
1,000,000,000,000
100,000,000
10,000
n = 100000
1,000,000,000,000,000
10,000,000,000
100,000
n = 1000000
1,000,000,000,000,000,000
1,000,000,000,000
1,000,000
n = 10000000
1,000,000,000,000,000,000,000
100,000,000,000,000
10,000,000
http://pages.experts-exchange.com/processing-power-compared/
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References
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Algorithm Design and Applications, Goodrich &
Tamassia
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Section 1.3
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