The two key ingredients mentioned earler are present here: The following program accomplishes all the necessary tasks: Binary Search: Method A

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1 Improving Binary Search Techniques Robert Virgile, A vco Systems Textron Abstract Binary search is an efficient method for locating a specific observation in a large, sorted master file. In a binary search program, each iteration through the DATA step (or through a portion of the DATA step) eliminates half the remaining master file observations from those that might be the sought after observation. With a master file of 2**n observations, a maximum of n+l iterations are required to locate any particular observation. This paper develops two improvements to binary search techniques applicable when a number of records must be located. The examples locate 200 to 1000 target records out of a SAS data set with 100,000 obserr vations. Sorting the targets before beginning the search makes possible two improvements over a standard binary search. First, since the targets are sorted, the kth target must be positioned in the master file after the k l target. Therefore it is necessary to search the entire master file for the first target only. Second, sorting implies that it is not statistically optimum to look halfway through the remaining observations each time. This paper develops a program that selects a more efficient test point in the master file. Background Theory An information system should allow you to extract a small amount of information from a large file, without running up a tremendous bill. Binary search is one technique which allows this to happen. The object of binary search is to locate one particular observation in a large, sorted master file.* The sort variable (KEY) in the master file takes on a unique value for each observation. The following diagram portrays the structure of the master file. Given a master file with 100,000 observations, sequential sear~h using a SET statement would necessitate examining 50,000 observations on average in order to locate one particular observation. A binary search would locate an observation faster by initially examining the midpoint of the master file instead of the first observation. Each iteration through a DATA step (or through a section of a DATA step) eliminates half the remaining observations in the master file from those that might be thc sought after record. A simplified initial pass through the data might look like this: DATA HALF; LOW = 1; HIGH = TOTALOBS; SET MASTER POINT=POINTTO NOBS=TOTALOBS; IF KEY < 'Searched For Value' THEN LOW=POINTTO; ELSE HIGH = POINTTO; STOP; [Il <HIGH I 100,000. (100,000) observations <\""?;- <e:;0i NTTOI each. having i~s I a unique 1+ \I" (50001) value for,\ ' KEY <::#>"W:::J (1) LOW = the lowest observation number that could possibly correspond to the observation we are trying to locate. HIGH = the highest observation that could correspond to the observation being sought. 100,000 observations each having a unique value for KEY value Lowest value KEY I KEY::;; a variable whose value equals the searched for value when we have found the correct observation. The master file is sorted by KEY. POINTTO = the number of the observation being examined where the variable KEY will be compared to some searched for value. TOTALOBS = the total number of observations in the master file. Note: SAS is a registered trademark of SAS Institute, Inc., Cary, NC. *Binary search programs can be modified to accurately process cases where the master file does not contain the sought after record. In order to simplify the programs and analyses, and in order to emphasize the major points of this paper, we will deal with examples where the master file is guaranteed to contain the record that we are seeking. This program selects an observation halfway through the master file and examines the value of KEY on that Observation. If KEY is less than the searched for value, the program eliminates all Observations before the POINTTQ observation from further consideration. Otherwise, the program eliminates all observations after the POINTTO observation. Note that the value of TOTALOBS is detcrmined when the DATA step is compiled. The proper value is available by the time HIGH=TOTALOBS is executed. 433

2 To turn the above code into a workable program, two key ingredients must be added. First, the searching code must be repeated, each time eliminating half the remaining observations from consideration. Second, some method for ending the search must be incorporated. That is, the program must discern when the POINTTO observation's KEY matches the target value. The following program accomplishes all the necessary tasks: Binary Search: Method A SET FINDME 1* Contains the variable TARGET *1 ; LOW - I; HIGH - TOTALOBS; ITERATE - I; POINTTO - CEIL«HlGH + LOW) / 2);... DO UNTIL (COWS-'COME HOME');... SET ENTIRE.MASTER POINT-POINTTO NOBS-TOTALOBS;... IF KEY < TARGET THEN LOW-POINTTO+I;... ELSE IF KEY> TARGET THEN HlGH-POINTTO-I;... POINTTO - CEIL«HIGH + LOW) / 2); ITERATE - ITERATE + I; IF POINTTO - HIGH THEN DO; SET ENTIRE.MASTER POINT-HIGH; IF KEY - TARGET THEN DO; ITERATE - ITERATE + I; SET ENTIRE.MASTER POINT_LOW; 1* End of IF POINTTO - HIGH loop * /... 1* End of DO UNTIL loop * / Where the first program tried to locate Searched for Value, this program reads the variable TARGET from the data set FINDME in order to determine the value to be located. The two key ingredients mentioned earler are present here: I. The search is repeated, each time halving the number of potentially matching observations. The six flagged statements above are the guts of this binary search, the repetitive process of (1) selecting a search point halfway through the master file, (2) reading the selected observation, and (3) eliminating half the remaining observations from consideration. 2. The program takes steps to identify a match between the value of KEY in the master file and the target value being sought (the condition which signifies the end of the search). There are two steps taken to identify this match. First, the ELSE DO statement (five lines into the DO UNTIL loop) is executed when KEY equals TARGET. Second, the loop beginning with IF POINTTO = HIGH THEN DO indicates a near match. Since the formula for POINTTO is CEIL«HlGH+LOW)/2), the condition -of POINTTO=HIGH is met whenever the binary search has eliminated all but one or two adjacent observations as potential matches. At that point, binary search becomes irrelevant, and we examine the remaining observation(s) directly. Enhancements When a number of observations are to be located in the master file, sorting the target data set permits streamlining the search process. Once a target has been matched to an observation in the master file, the following searches should look in the master file from that point onward. For example, if we locate the first of 1000 targets at observation 157 in the master file, then there is no need to examine the first 156 observations when searching for the second target. The following program incorporates this feature by adding a variable GLOBALOW, which is the observation number at which the previous target was located. The flagged statements have been added or changed compared to the previous program. This program contains three additional variables: TARGET = read from the data set FINDME, and corresponds to the searched for value. When KEY in the master file is equal to TARGET, we have located the correct observation. ITERATE = the number of SET statements executed in order to locate the correct observation. This variable measures program efficiency, and is not needed to perform the searching operation. COWS = a dummy variable never referenced outside the DO UNTIL loop. (The DO UNTIL condition' is never satisfied; the loop is exited via RETURN statements inside the loop.) 434

3 Binary Search: Method B SET FINDME 1* Contains IOOO'sorted observations */ ;... RETAIN GLOBALOW I;... LOW = GLOBALOW; HIGH = TOTALOBS; ITERATE = I; DO UNTIL (COWS = 'COME HOME'); SET ENTIRE.MASTER POINT=POINTTO NOBS=TOT ALOBS; IF KEY < TARGET THEN LOW=POINTTO+I; ELSE IF KEY> TARGET THEN HIGH=POINTTO-I;... GLOBALOW = POINTTO; IF POINTTO = HIGH THEN DO; SET ENTIRE.MASTER POINT=HIGH; IF KEY = TARGET THEN DO;... GLOBALOW = HIGH; SET ENTIRE.MASTER POINT=LOW;... GLOBALOW = LOW; END' /* End of IF POINTTO=HIGH loop * / /* End of DO UNTIL loop * / Up until this point, these programs have always selected the midpoint of the remaining observations as the ne'xt observation to be examined. However, when the targets have been sorted, this midpoint is not the most efficient record to search. In fact, the probabilities are overwhelming that the first target out of 1000 will be located within the first 1% of the master file. A further modification to the program would initially examine a closer point than the midpoint. Binary Search: Method C... IF _N_=I THEN SET FINDME POINT=GLOBALOW NOBS=NT ARGETS; SET FINDME 1* Contains 1000 sorted observations *1 ;... RETAIN GLOBALOW I NFOUND 0; LOW = GLOBALOW; HIGH = TOT ALOBS; ITERATE = I;... POINTTO = LOW + CEIL«HIGH-LOW)/(NTARGETS-NFOUND»; DO UNTIL (COWS = 'COME HOME'); SET ENTIRE.MASTER POINT=POINTTO NOBS=TOTALOBS; IF KEY < TARGET THEN LOW=POINTTO+I; ELSE IF KEY> TARGET THEN HIGH=POINTTO-I; GLOBALOW = POINTTO; IF POINTTO = HIGH THEN DO; SET ENTIRE.MASTER POINT=HIGH; IF KEY = TARGET THEN DO; GLOBALOW = HIGH; ITERATE = ITER A TE + I; SET ENTIRE.MASTER POINT=LOW; GLOBALOW = LOW; END' /* End of IF POINTTO=HIGH loop' / /* End of DO UNTIL loop' / NTARGETS ::: the total number of targets to be located FOUND"" the number of targets already located Thus NT ARGETS - NFOUND is the number of targets yet to be located. This program will replace the POINTTO formula with LOW + CEIL«HIGH-LOW)/(NTARGETS-NFOUND)) (instead of the midpoint (HIGH+LOW)/2) as an intuitive approximation of a reasonable search point. Since this new formula typically selects a point less than the midpoint, a good outcome would be KEY> TARGET. That would mean we have located a small block of observations that contains the observation we want. Notice that this program proceeds with a binary search regardless of the" outcome of the first search, This is a desirable method if the first search is "successful", that is, if the target is positioned before the searched key in the master file. What about the opposite case, when the target lies in the larger, unexplored section of the master file? In that case, the same principle applies, which is,that the target is very likely to lie in the first portion of the remaining observations. Repeated non-binary searches are called for until we locate a relatively small block thitt contains the target. The following program performs such a search: 435

4 Binary Search: Method D IF N =1 THEN SET FINDME POINT=GLOBALOW - - NOBS=NTARGETS; SET FINDME 1* Contains 1000 sorted observations */ ~ RETAIN GLOBALOW I NFOUND 0; LOW = GLOBALOW; HIGH = TOT ALOBS; ITERATE = I;... INCRMENT CEIL«HIGH-LOW)/(NTARGETS-NFOUND));... POINTTO = LOW + INCRMENT;... DO WHILE (POINTTO < HIGH);... SET ENTIRE.MASTER POINT=POINTTO NOBS=TOTALOBS;... IF KEY < TARGET THEN LOW=POINTTO+l;... ELSE IF KEY> TARGET THEN HIGH=POINTTO-I; GLOBALOW = POINTTO; END'... POINTTO = POINTTO + INCRMENT; POINTTO - CEIL«HiGH + LOW) / 2); DO UNTIL (COWS = 'COME HOME'); SET ENTIRE.MASTER POINT=POINTTO NOBS=TOTALOBS; IF KEY < TARGET THEN LOW=POINTTO+I; ELSE IF KEY> TARGET THEN HIGH=POINTTO-I; GLOBALOW = POINTTO; NFOUND = NFOUND + I; END' IF POINTTO = HIGH THEN DO; SET ENTIRE.MASTER POINT=HIGH; IF KEY = TARGET THEN DO; GLOBALOW = HIGH; NFOUND = NFOUND + I; SET ENTIRE.MASTER POINT=LOW; GLOBALOW = LOW; NFOUND = NFOUND + I;!* End of IF POINTTO=HIGH loop' / /* End of the DO UNTIL loop * / Notice the logic of the first loop (DO WHILE) accomplishes two purposes. First, it keeps searching small block after small block until it locates a block containing the target key. Second, it automatically branches to the binary search (DO UNTIL) if the direct access formula for POINTTO suggests a search point larger than the total number of observations in the master file. Four programs have now been presented, employing four similar methods of searching through a master file. The first program, which we call Method A, performs a standard binary search for each target. The second program (Method B) first sorts the targets and then performs a standa rd binary search for each target. However, Method B searches a smaller portion of the master file for each successive target by eliminating all master file observations that lie before the previously found target. The third program (Method C) is the same as Method B, except that the first search attempt for each target is nonmbinary. The fourth program (Method D) is the same as Method B, except that all search attempts are nonmbinary until a relatively small block containing the target is located. Tests of all four programs were conducted using five sample sets of targets. The five samples contained from 200 to 1000 targets to be located in a master file of 100,000 observations. The four methods are compared on the basis of (1) number of iterations per target locateq, and (2) CPU time, breaking out separm ately the CPU time required to sort the targets. The following table presents test results. Interpretation of Results The test results show, as expected, that each enhancement to the search program reduced the number of iterations per target and the total CPU time. The best results show a reduction of over 75% of CPU time. What makes these enhancements so effective? In particular, why is the binary search ineffective compared to nonbinary searching that uses the formula for POINTTO of: LOW + CEIL«HIGH-LOW)/(NTARGETS-NFOUND)) What makes this a good point? Why are the probabilities overwhelming that the first of 1000 targets will be located within the first 1% of the master file? Two factors determine the relative effectiveness of a non-binary search point: (1) What is the probability that the TARGET value lies at or before the search point?, and (2) Compared to a binary search, how many iterations will be saved if the TARGET value does indeed lie before the search point? This section considers principles of probability theory that underlie the answers to the above questions. If we select observation n from the master file of 100,000 observations, what is the probability that the first of 1000 (sorted) targets lies at or before observation n? In general, the formula is: Probability that first Probability that all target is located BEFORE::: 1 M targets are located observation n AFTER observation n The amount on the right hand side of this equation is easy to compute. Assuming that the 1000 targets have been randomly selected: Probability that ANY given target is located AFTER observation n Probability that ALL 1000 targets are located AFTER observation n (100,000 - n) / 100,000 PI PI'*IOO,OOO 436

5 Test Program Results Number of Taraets It em M easure d Metho d A Method B Method C Method D 200 CPU Time: Seconds Percent Iterations: Average Percent CPU Time: Seconds Percent i Iterations: Average Percent I i 600 CPU Time: Seconds Percent Iterations: Average i Percent CPU Time: Seconds Percent Iterations! Average Percent f i 1000 CPU Time: Seconds ! Percent ! Iterations: Average I Percent CPU Tlme = Search Tlme + Sorting Time (using PROC SORTT) Iterations = Number of SET Statements Executed All percents based on comparison to Method A To get a feeling for the implications of these equations, let's figure out the probability that the first of 1000 targets will be located within the first 1000 observations (i.e., the first 1%) of a master file of 100,000 observations. Probability that the = I (99,000/100,000)"1000 first target is located BEFORE observation 1000 '" Similar computations for other sample values of n produce the following results: Observation Number (n) in the Master File Probability that the First Key is Located Before n (= ) In round numbers, it would take 7 binary search iterations to narrow down the location of the first target to the first 800 records (800 is approximately 100,000 divided by 2**7). More than 99% of the time, we would be saving 6 iteratipns by beginning to search at the 800th record instead of the 50,000th. The other less than 1 % of the time, we would be wasting 1 iteration. On average, the net gain is approximately: ' iterations saved We would average a similar net savings by selecting a cutoff point of 400: ' ' iteration's saved These numbers appear to be somewhat "robust." That is, the expected savings won't vary considerably from one good search formula to the next. Thus our nonbinary search formula can attempt to select a "good" search point, but need not select an "optimum" one. That ~s a fortunate result, since selection of an optimum search point may require some exceedingly complex mathematics. One "good" search point was located at: LOW + CEIL«HIGH LOW)/(NT ARGETS NFOUND)) Experiments were also run using variations on this formula: LOW + FACTOR CEIL«HIGH.LOW)/(NT ARGETS-NFOUND)) where FACTOR ranged in value from 0.9 to 1.1. These experiments showed negligible differences from the original formula (FACTOR=I), with variations typically running in the hundredths of an iteration per target. One further question remains. Why is the fall in CPU time steeper than the fall in the average number of iterations? The answer lies in the fact that some searches require more CPU time than others. In general terms, each direct access search spins the disk to the location of the desired observation. An observation that lies 50,000 records distant takes more spinning (=more computer resources, =more CPU time) than an observation that is 1000 records distant. The binary search examines more distant records than the non-binary search, hence CPU time per search is greater. 437

6 Finally. tests were run on a small data set consisting of only 10 target keys. The results are listed below. While there is some improveme,nt in the number of iterations required, the CPU time actually rises. This rise means that the reduction in iterations required is not sufficient to offset the increased overhead incurred by the combination of sorting and increased complexity of the s'earch program. Also contributing to this result is the fact that a smaller number of targets causes each search to cover a larger block of records in the maser file. When implementing these techniques, programmers should consider the relative sizes of the data sets. Given the size of the master file, different search techniques may be appropriate for different sizes of target data sets. TEST RESULTS FOR 10 TARGETS Method CPU Time Iterations A % 100% B % 96.8% C % 91.0% D ; % 87.8% The last comparison that needs to be made is comparing the direct access searches (binary or non-binary) with a sequential search. This progra,m performs a sequential search and requires the target keys to be sorted: SET FINDME; DO UNTIL (KEY=TARGET); IF KEY = TARGET THEN SET ENTIRE.MASTER; The RETURN statement is necessary when two targets have the same value. Test programs produced results as follows: Back to Reality Having completed all this work, the final step would be to examine how it can be put to use. Certainly I was able to say to my boss, "You've got to send me to SUGI 11. I'm presenting a paper!" But are there other useful applications? How often will the necessary criteria be met: 1. A large, sorted master file contains one observation per value of a KEY variable. 2. The objective is to extract a significant fraction of the observations. 3. The criteria for selecting observations are supplied from information outside the master file. To satisfy myself as to the usefulness of these search programs, I listed half a dozen potential applications: l. Master file"" Customers Target file = With overdue accounts KEY:: Customer ID 2. Master file"" Credit cards issued Target file:: Stolen credit cards KEY"", Credit card number 3. Master file = Tapes in a large tape library Target file = Containing unneeded data KEY = Tape VOLSER 4. Master file = Checking accounts Target file = Bounced checks KEY = Checking account number 5. Master file"" Taxpayers Target file = To be audited KEY = Social Security Number 6. Master file = Drivers Target file = Involved in an accident KEY = License number In addition,.inverted lists based on a single key would be candidates for a binary search. See the reference articles for details on the structure of a SAS database using inverted lists. Number of Target Keys CPU Time (Not Including Sorting) Anyone wishing to apply these programs is welcome to contact the author by writing to: Robert Virgile Avco Systems Textron 201 Lowell Street, Room 4159 Wilmington, MA References Howard, Neil, and Linda Williams Pickle, "Efficient Data Retrieval: Direct Access Using the Point Option," Proceedings.of the Ninth Annu_a! SAS Users Group International Conference. 1984, PP Ramsay, Alec, "Keyed Access to SAS Data Sets," Proceedings of the Ninth Annual SAS Users Group International Conference, 1984, pp