"RMQ" "Sparse_table algorithm"

Defined:

RMQ (range minimum/maximum query), or interval-most-valued queries , refers to an issue such as:

For sequence A of length n, answer a number of questions RMQ (A,I,J) (I,j<=n), returning the minimum/large value between the i,j of the column A.

There are a lot of solutions to this problem, and violence (and, of course, basically no questions will get you out of the way), line-segment Trees (O (Nlogn)), and a very well-known algorithm for on-line processing of RMQ problems-- sparse_table algorithm , called St algorithm, the algorithm can achieve the efficiency of Query O (1) after preprocessing O (Nlogn). The main idea is DP.

(i) pretreatment, DP

Set A[i] is the number of intervals requiring the most value of the series,Dp[i][j] represents the number of consecutive 2j from the first of the maximum value .

For example, a[10] = 3 2 4 5 6 8 1 2 9 7;

Initial:

Dp[1][0] Represents the 1th number, length is the maximum value of 20=1, that is, element 3. similarly f[1][1] = max (3,2) = 3, F[1][2]=max (3,2,4,5) = 5,f[1][3] = max (3,2,4,5,6,8,1,2) = 8;

And we can easily see dp[i][0] = A[i].

State transition equation:

We divide dp[i][j] into two segments of the same length and 2j-1, a section of dp[i][2j-1] and a dp[i+2j-1][2j-1]. Using the example above, when I=1,j=3 is the 3,2,4,5 and 6,8,1,2 the two paragraphs.

DP[I,J] is the maximum of the respective maximum values for these two paragraphs. So we got the state transition equation:

　　DP[I][J] = max (dp[i][j-1], Dp[i + 2 (j-1)][j-1];

1 /*pretreatment->o (NLOGN)*/2 voidINITRMQ (intN)3 {4      for(inti =1; I <= N; i++) dp[i][0] =A[i];5      for(intj =1; J < -; i++)6          for(inti =1; I <= N; J + +)7DP[I][J] = max (dp[i][j-1], Dp[i + (1<< (J-1))][j-1]);8}

Note the order of loops in the code, the outer layer is J, and the inner is i!!

(b) Enquiry

If the query interval is a[i, j], then we need to find the smallest power that covers this closed interval (the left boundary takes I, the right bounds fetch j) (can overlap, for example, query a[5,9], we can query a[5678] and a[6789]).

Because the length of this interval len = j-i + 1, so we can take K=log2 (len), then there are: RMQ (A, I, j) = max{Dp[i][k], Dp[j-2k+1][k]}.

For example, the maximum value of the query interval [2,8], k = log2 (8-2 + 1) = 2, which is Max (dp[2][2],dp[8-2 ^ 2 +)) = Max (F[2, 2],f[5, 2]);

intLOG2[MAXN];voidinit () {log2[0] = -1; log2[1] =0; intI, J;  for(i =2, j = 0; i < MAXN; i++)    {        if(I > (1<< (j+1)) J + +; Log2[i]=J; }}intQueryintLintR) {    if(L >r) Swap (L, R); if(L = = R)returndp[l][0]; intLen = r-l+1; intk = Log[len];//here is the log2[] array to save the log value of each number, of course, can also be evaluated with the following formula;//int k = (int) (log (DES-SRC + 1.0)/log (2.0));    returnMax (Dp[l][k], dp[r-(1&LT;&LT;K) +1][k]);}

"RMQ" "Sparse_table algorithm"