Share my O(n) C++ DP solution with thinking process and explanation


  • 45
    K

    1. Problem


    Given a positive integer n, return the number of all possible attendance records with length n, which will be regarded as rewardable. The answer may be very large, return it after mod 10⁹ + 7.

    A student attendance record is a string that only contains the following three characters:

    1. 'A' : Absent.
    2. 'L' : Late.
    3. 'P' : Present.

    A record is regarded as rewardable if it doesn't contain more than one 'A' (absent) or more than two continuous 'L' (late).


    2. Thinking process


    2.1 Divide the whole problem into sub-problems


    Before introducing the way to calculate the number of all possible attendance records with length n, we divide the problem into 3 parts.


    As the attendance records is made by 3 characters ('P', 'L', 'A'), the total number can be divided into

    Total = ended with P + ended with L + ended with A.


    If we define following series

    T(n) is the total number of all possible attendance records with length n.

    P(n) is the total number of all possible attendance records ended with 'P' with length n.

    L(n) is the total number of all possible attendance records ended with 'L' with length n.

    A(n) is the total number of all possible attendance records ended with 'A' with length n.


    It can be inferred that

    T(n) = A(n) + P(n) + L(n), n ≥ 1.


    2.2 Solve the sub-problems by dynamic programming


    As I use dynamic programming, I need to find out the recursive relation in 3 sub-problems.


    2.2.1 Calculate P(n)

    It can be inferred that


    If we add a 'P' to an attendance records with length n - 1, we will get an attendance records ended with 'P' with length n.


    For an attendance record with length n - 1,

    • If its (n - 1)th character is 'P' ---- CAN add 'P'. ("PP")

    • If its (n - 1)th character is 'A' ---- CAN add 'P'. ("AP")

    • If its (n - 1)th character is 'L' ---- CAN add 'P'. ("LP")


    which means

    P(n) = A(n - 1) + P(n - 1) + L(n - 1), n ≥ 2.


    and we have initial value for the recursive relation

    A(1) = P(1) = L(1) = 1.


    2.2.2 Calculate L(n)

    Similarly,

    If we add a 'L' to an attendance records with length n - 1, we will get an attendance records ended with 'L' with length n.

    But the resulting attendance records must be regarded as rewardable!


    As the rule is that a record is regarded as rewardable if it doesn't contain

    more than two continuous 'L' (late).


    We need to consider the situations when we can add 'L' to an attendance record with length n - 1 and it's still regarded as rewardable.


    For an attendance record with length n - 1,

    • If its (n - 1)th character is 'P' ---- CAN add 'L'. ("PL")

    • If its (n - 1)th character is 'A' ---- CAN add 'L'. ("AL")


    • If its (n - 1)th character is 'L':

      • If its (n - 2)th character is 'A' ---- CAN add 'L'. ("ALL")

      • If its (n - 2)th character is 'P' ---- CAN add 'L'. ("PLL")

      • If its (n - 2)th character is 'L' ---- CAN NOT add 'L'. ("LLL" breaks the rule)


    which means

    L(n) = A(n - 1) + P(n - 1) + A(n - 2) + P(n - 2), n ≥ 3


    and we have initial value for the recursive relation

    A(1) = P(1) = 1.

    A(2) = 2, P(2) = 3.

    and

    L(1) = 1, L(2) = 3.


    2.2.3 Calculate A(n)

    Similarly,

    If we add a 'A' to an attendance records with length n - 1, we will get an attendance records ended with 'A' with length n.

    But the resulting attendance records must be regarded as rewardable!


    As the rule is that a record is regarded as rewardable if it doesn't contain

    more than one 'A' (absent).


    We need to consider the situations when we can add 'A' to an attendance record with length n - 1 and it's still regarded as rewardable.


    For an attendance record with length n - 1,

    • If its (n - 1)th character is 'A' ---- CAN NOT add 'A'. ("AA" breaks the rule)

    • If its (n - 1)th character is 'P' and has no 'A' ---- CAN add 'A'.

    • If its (n - 1)th character is 'L' and has no 'A' ---- CAN add 'A'.


    If we define series

    noAP(n) is the total number of all possible attendance records ended with 'P' with length n and with no 'A'.

    noAL(n) is the total number of all possible attendance records ended with 'L' with length n and with no 'A'.


    It can be inferred that

    A(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    and we have initial value for the recursive relation

    A(1) = 1.

    noAP(1) = noAL(1) = 1.


    2.2.4 Calculate noAP(n) and noAL(n)

    In 2.2.3, 2 new series noAP(n) and noAL(n) is introduced. Now, we focus on the recursive relation in noAP(n) and noAL(n).


    For noAP(n), we need to consider the situations when we can add 'P' to an attendance record with length n - 1 and no 'A' and it's still regarded as rewardable.


    Since noAP(n) has no 'A', we don't need to consider the situation when its (n - 1)th character is 'A'.


    For an attendance record with length n - 1, we can get only 2 situations

    • If its (n - 1)th character is 'P' and has no 'A' ---- CAN add 'P'.

    • If its (n - 1)th character is 'L' and has no 'A' ---- CAN add 'P'.


    which means

    noAP(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    and we have initial value for the recursive relation

    noAP(1) = noAL(1) = 1.


    For noAL(n), we need to consider the situations when we can add 'L' to an attendance record with length n - 1 and no 'A' and it's still regarded as rewardable.


    Since noAL(n) has no 'A', we don't need to consider the situation when its (n - 1)th character is 'A'.


    For an attendance record with length n - 1, we can get

    • If its (n - 1)th character is 'P' and has no 'A' ---- CAN add 'L'.("PL")

    • If its (n - 1)th character is 'L' and has no 'A'.

      • If its (n - 2)th character is 'P' and has no 'A' ---- CAN add 'L'.("PLL")

      • If its (n - 2)th character is 'L' and has no 'A' ---- CAN NOT add 'L'.("LLL" breaks the rule.)


    which means

    noAL(n) = noAP(n - 1) + noAP(n - 2), n ≥ 3.

    and we have initial value for the recursive relation

    noAP(1) = noAL(1) = 1.

    and

    noAL(2) = 2.


    2.3 Recursive relationship summarization


    The answer to the whole problem is T(n), and

    T(n) = A(n) + P(n) + L(n), n ≥ 1.


    Recursive formula:

    P(n) = A(n - 1) + P(n - 1) + L(n - 1), n ≥ 2.

    A(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    noAP(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    L(n) = A(n - 1) + P(n - 1) + A(n - 2) + P(n - 2), n ≥ 3.

    noAL(n) = noAP(n - 1) + noAP(n - 2), n ≥ 3.

    with Initial value

    A(1) = P(1) = L(1) = 1.

    noAP(1) = noAL(1) = 1.

    L(2) = 3.

    noAL(2) = 2.


    2.4 Simplifying


    When n ≥ 4, the 3 formulas

    A(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    noAP(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    noAL(n) = noAP(n - 1) + noAP(n - 2), n ≥ 3.

    can be simplified to

    A(n) = A(n - 1) + A(n - 2) + A(n - 3), n ≥ 4.


    Finally, the recursive formula group becomes

    P(n) = A(n - 1) + P(n - 1) + L(n - 1), n ≥ 2.

    L(n) = A(n - 1) + P(n - 1) + A(n - 2) + P(n - 2), n ≥ 3.

    A(n) = A(n - 1) + A(n - 2) + A(n - 3), n ≥ 4.

    Here, noAP(n) and noAL(n) disappeared.


    with Initial value

    P(1) = 1.

    L(1) = 1, L(2) = 3.

    A(1) = 1, A(2) = 2, A(3) = 4.


    2.5 Do modulus


    The result need to be returned after mod 10⁹ + 7.

    Since the result is generated by adding a lot of middle results together, in order to make sure that every middle result and the final result won't exceed INT_MAX, we need to do mod for every middle result, and for every 2-middle-result-addition.


    3. Complexity analysis


    3.1 Time complexity


    Since the algorithm is one-pass from 1 to n.

    The time complexity is O(n).


    3.2 Space complexity


    Since 3 arrays are used to save P(n), L(n), A(n), the total size is 3n.

    The space complexity is O(n).


    4. Code


    class Solution {
    public:
        int checkRecord(int n) {
            int m = 1000000007;
            int *A = new int [n];
            int *P = new int [n];
            int *L = new int [n];
            
            P[0] = 1;
            L[0] = 1;
            L[1] = 3;
            A[0] = 1;
            A[1] = 2;
            A[2] = 4;
            
            if(n == 1) return 3;
            
            for(int i = 1; i < n; i++)
            {
                A[i - 1] %= m;
                P[i - 1] %= m;
                L[i - 1] %= m;
                
                P[i] = ((A[i - 1] + P[i - 1]) % m + L[i - 1]) % m;
                
                if(i > 1) L[i] = ((A[i - 1] + P[i - 1]) % m + (A[i - 2] + P[i - 2]) % m) % m;
                
                if(i > 2) A[i] = ((A[i - 1] + A[i - 2]) % m + A[i - 3]) % m;
            }
            
            return ((A[n - 1] % m + P[n - 1] % m) % m + L[n - 1] % m) % m;
        }
    };
    


  • 0
    S

    @KJer said in Share my O(n) C++ DP solution with thinking process and explanation:

    A(n) = A(n - 1) + A(n - 2) + A(n - 3), n ≥ 4.

    for A case why it can be simplified to the above?


  • 3
    K

    @sse.michael

    A(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    noAP(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    which means

    A(n) = noAP(n), n ≥ 2.

    As

    noAL(n) = noAP(n - 1) + noAP(n - 2), n ≥ 3.

    Then

    noAL(n) = A(n - 1) + A(n - 2), n ≥ 3.

    As

    A(n) = noAP(n - 1) + noAL(n - 1), n ≥ 2.

    Then I get

    A(n) = A(n - 1) + A(n - 2) + A(n - 3), n ≥ 4.


  • 0
    S

    @KJer said in Share my O(n) C++ DP solution with thinking process and explanation:

    A(n) = noAP(n)

    yea~, i got it. thanks


  • 0
    C

    Very helpful! Thank you


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