Most important sub-sections and problems in Dynamic Planning

Problem description:
Given a sequence of A1, A2,..., and an consisting of n integers (including negative integers), calculate the maximum value of the subsegment and.

When All integers are negative, the maximum sub-segment is defined as 0.
According to this definition, the optimal value is:
 
For example, when (A1, A2, A3, A4, A5, A6) = (-2, 11,-4,-5,-2,

The maximum sub-segment is:

11 + (-4) + 13 = 20
1. The maximum sub-segment and the simple problemAlgorithm:

Code:

# Include <iostream> <br/> using namespace STD; <br/> int maxsum (int A [], int N, Int & besti, Int & bestj) {<br/> int sum = 0; <br/> int I, j, k; <br/> for (I = 1; I <= N; I ++) <br/> for (j = I; j <= N; j ++) <br/>{< br/> int thissum = 0; <br/> for (k = I; k <= J; k ++) thissum + = A [k]; <br/> If (thissum> sum) {<br/> sum = thissum; <br/> besti = I; <br/> bestj = J; <br/>}< br/> return sum; <br/>}< br/> int main () {<br/> int N, A [100], M, I, j, maxsum; <br/> cout <"Enter the number of elements in the integer sequence N:" <Endl; <br/> CIN> N; <br/> cout <"Enter the value of each element in the sequence a [I] (total" <n <) "<Endl; <br/> // For (M = 1; m <= N; I ++) <br/> // CIN> A [m]; <br/> for (m = 0; m <n; m ++) <br/> CIN> A [m]; <br/> int B [100]; <br/> for (m = 0; m <n; m ++) <br/> B [M + 1] = A [m]; <br/> maxsum = maxsum (B, n, I, j); <br/> cout <"the maximum sub-segments of an integer sequence and are: "<maxsum <Endl; <br/> cout <" besti = "<I <Endl; <br/> cout <"bestj =" <j <Endl; <br/> system ("pause"); <br/>}

The time complexity of this algorithm: O (N3 ).

This algorithm can be improved to reduce the time complexity to O (n2 ).

Code:

# Include <iostream> <br/> using namespace STD; <br/> int maxsum (int A [], int N, Int & besti, Int & bestj) {<br/> int sum = 0; <br/> int I, j, k; <br/> for (I = 1; I <= N; I ++) {<br/> int thissum = 0; <br/> for (j = I; j <= N; j ++) <br/>{< br/> thissum + = A [J]; <br/> If (thissum> sum) {<br/> sum = thissum; <br/> besti = I; <br/> bestj = J; <br/>}< br/> return sum; <br/>}< br/> int main () {<br/> int n, a [100], M, I, j, maxsum; <br/> cout <"Enter the number of elements in the integer sequence N:" <Endl; <br/> CIN> N; <br/> cout <"Enter the value of each element in the sequence a [I] (total" <n <")" <Endl; <br/> // For (M = 1; m <= N; I ++) <br/>/CIN> A [m]; <br/> for (m = 0; m <n; m ++) <br/> CIN> A [m]; <br/> int B [100]; <br/> for (m = 0; m <n; m ++) <br/> B [M + 1] = A [m]; <br/> maxsum = maxsum (B, n, I, j); <br/> cout <"the maximum sub-segments of an integer sequence and are: "<maxsum <Endl; <br/> cout <" besti = "<I <Endl; <br/> cout <"bestj =" <j <Endl; <br/> system ("pause"); <br/>}< br/>

 

2. Maximum sub-section and issue division and Control Law:

Code:

// Maximum sub-segment sum and divide the algorithm. T (n) = O (nlog (n )).

# Include <iostream> <br/> using namespace STD; <br/> int maxsubsum (int A [], int left, int right) {<br/> int sum = 0; <br/> If (Left = right) sum = A [left]> 0? A [left]: 0; <br/> else {<br/> int center = (left + right)/2; <br/> int leftsum = maxsubsum (, left, center); <br/> int rightsum = maxsubsum (A, center + 1, right); <br/> int S1 = 0; <br/> int lefts = 0; <br/> for (INT I = center; I >= left; I --) {<br/> lefts + = A [I]; <br/> If (lefts> S1) S1 = lefts; <br/>}< br/> int S2 = 0; <br/> int rights = 0; <br/> for (INT I = center + 1; I <= right; I ++) {<br/> rights + = A [I]; <br/> If (rights> S2) S2 = rights; <br/>}< br/> sum = S1 + S2; <br/> If (sum <leftsum) sum = leftsum; <br/> If (sum <rightsum) sum = rightsum; <br/>}< br/> return sum; <br/>}< br/> int main () {<br/> int N, A [100], M, maxsum; <br/> cout <"Enter the number of elements in the integer sequence N:" <Endl; <br/> CIN> N; <br/> cout <"Enter the value of each element in the sequence a [I] (total" <n <")" <Endl; <br/> for (m = 0; m <n; m ++) <br/> CIN> A [m]; <br/> int B [100]; <br/> for (m = 0; m <n; m ++) <br/> B [M + 1] = A [m]; <br/> maxsum = maxsubsum (B, 1, n); <br/> cout <"maximum sub-segment of an integer sequence and" <maxsum <Endl; <br/> system ("pause"); <br/>}< br/>

 

3. Dynamic Planning Algorithm for maximum sub-segments and problems:

Code:

// Maximum child segment sum, dynamic planning, T (n) = O (n ).

# include <iostream> <br/> using namespace STD; <br/> int maxsum (int n, int A []) {<br/> int sum = 0; <br/> int B = 0; <br/> for (INT I = 1; I <= N; I ++) {<br/> If (B> 0) B + = A [I]; <br/> else B = A [I]; <br/> If (B> sum) sum = B; <br/>}< br/> return sum; <br/>}< br/> int main () {<br/> int n, a [100], M, maxsum; <br/> cout <"Enter the number of elements in the integer sequence N:" <Endl; <br/> CIN> N; <br/> cout <"Enter the value of each element in the sequence a [I] (total" <n <) "<Endl; <br/> for (m = 0; m <n; m ++) <br/> CIN> A [m]; <br/> int B [100]; <br/> for (m = 0; m <n; m ++) <br/> B [M + 1] = A [m]; <br/> maxsum = maxsum (n, B ); <br/> cout <"maximum sub-segment of the integer sequence and" <maxsum <Endl; <br/> system ("pause "); <br/>}< br/>