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R = P id + Vkey + VG + P
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has to consider if a data or pointer value exists at the level The example contains data at only two levels {1, 5} and pointers are needed on all levels but the bottom one {1} Denoting the size of a segment of a record at level i as Ri , we nd that the number of segments per block
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Bfsi = dens
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The density for a B-tree data le implementation can again be estimated with Eq 347 as dens = 069% Large segment sizes can lower the density further if spanning of blocks by single segments is avoided Since segments sequences can span blocks, a number of block accesses may be required to nd a speci c segment at one level The expected number of blocks comprising an entire segment sequence of fanout y is bi (yi ) =
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+ yi /Bfsi
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The extra 1 block appears because a segment sequence may start at an arbitrary 2 position in the rst block 1, the fanout yi is too high at level i because the number of block If bi accessed will be large on one level There will also be fewer segments in the last block of the sequence, so that for arbitrary sequences the number of accesses can be precisely estimated using the procedure leading to Eq 4-12, assuming an equal frequency of access to all segments of the sequence The number of expected block accesses are the sum of accesses to the rst block, accesses to reach intermediate blocks, if any, and accesses to reach the remaining segments in the nal block, all divided by the number of segments in the sequence 1 yi
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bi 1 1 2 Bfsi bi 1
bG =
j Bfsi + bi yi 1 Bfsi 2
Bfsi
bi + 1 2
4-16
Bfsi so that the lesser number of segments The approximation applies when yi in the rst and last blocks does not a ect the result greatly In mumps we nd on level x the directory of le names or globals for a user At this level the name strings are limited to 8 characters and no other key is needed If the average global-variable name is 5 characters, the segment size Rx = 9 and yx = (512 4)/9 = 56 When there are many global les, including unsubscripted globals, it is likely that more than one block is required for the directory level of the user The initial segment for the drug le will be created when the rst patient is stored by a statement as SET drug(patno)= Manuel P Cair The segment in level x will consist only of the key drug and a pointer, and a block on level x 1 will be allocated to store the key value from patno and the string with Cair s name and the end-of-string mark We assume for patno a 9 digit or ve character encoding The above patient segment will require Rx 1 = 1 + 5 + 15 = 21 characters, but as soon as lower-level segments are entered, a pointer eld will be inserted, so that the level x 1 segments will actually become 23 characters in length This will make Bfsx 1 = 512 4 = 22 so that there can be this many 23 patient segments in a block For the 450 patients seen in the clinic, the segment sequence will occupy 21 blocks These have to be searched sequentially for every
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reference, requiring an average of 11 block accesses on this level An approach to avoid this high cost is given at the end of this section We will summarize the remaining levels of the example rapidly since we nd that all further segment sequences will t into a block On level x 2 the visit dates are stored as keys This means that no SET statements with only two subscripts will be executed This level will be lled indirectly when the dateno is used as a subscript to lower levels On level x 3, we nd the problem treated during the visit Again, the problem is encoded as the key, "Asthma" icd=4939 icdno = 49390, and the key is set implicitly If no drugs are prescribed for the problem the hierarchy terminates here with a segment of type 1 Level x 4 provides pointers to the drug description An alternative design could store the drug name as a data eld on this level It again contains segments having only a key and a pointer Level x 5 contains the actual drug data as a sequence of three segments This level would be loaded by statements such as SET drug(patno,dateno,icdno,dno,1)="Decadron" SET drug(patno,dateno,icdno,dno,2)=15 SET drug(patno,dateno,icdno,dno,3)=3
Naked Variables To simplify and speed execution of such sequences, an option is available that will allow reuse of a previously reached node, so that redundant searches down the tree can be eliminated The absence of a variable name implies the reuse of the previous global variable and all but the last one of the subscripts Further subscripts leading to lower levels can be appended Using the naked variable notation, the last two statements become
SET (2)=15 SET (3)=3
The number of blocks accesses to reach this level (x 5), assuming one block access for the directory level, is TF (1 + 11 + 1 + 1 + 1 + 1)(s + r + btt) = 16(s + r + btt) Some seeks may be avoided, especially on level x 1, if locality is maintained Further naked accesses to the same segment sequence are free if the bu ers are retained, at least until the end of a block is encountered
Optimal fanout would avoid sibling over ow blocks When the fanout y Bfs, the hierarchy becomes a poor match for the le, as seen on the patient level x 1 A solution is to change the hierarchy originally envisaged for the le We will apply this notion to the patient level of our example In order to obtain better performance, this level might be split into two levels, perhaps by using half of the patient number as the three-character key for each level The name will appear only on the lower level (x 2) Now Rx 1 = 1 + 3 + 2 = 6, Bfsx 1 = 84 and Rx 2 = 1 + 3 + 15 + 2 = 21, Bfsx 2 = 24 (Eq 4-14) The original fanout y = 450 is distributed over two levels, so that yx 1 , yx 2 = y 22 Both yx 1 < Bfsx 1 and yx 2 < Bfsx 2 , so that most segment sequences will t into one block The block fetches for the patient data have been reduced from 11 to slightly over 2 This le will have 6 levels, and T f 6(s + r + btt)
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