Abstract:
A data structure and related data storage and retrieval method rapidly provide a count of elements stored or referenced by a hierarchical structure of ordered elements (e.g., a tree), access to elements based on their ordinal value in the structure, and identification of the ordinality of elements. In an ordered tree implementation of the invention, a count of elements stored in each subtree is stored, i.e., the cardinality of each subtree is stored either at or associated with a higher level node pointing to that subtree or at or associated with the head node of the subtree. In addition to data structure specific requirements (e.g., creation of a new node, reassignment of pointers, balancing, etc.) data insertion and deletion includes steps of updating affected counts. Elements may be target data itself (e.g., data samples, prime numbers); keys or indices associated with target data (e.g., social security numbers of employees, product numbers and codes, etc. uses to reference associated data records, etc.); or internal memory pointer to keys or data stored external to the data structure. The invention is applicable to varied hierarchical storage structures including, for example, binary trees, AVL trees (height-balanced binary trees), b-trees, etc. (population based structures) and digital trees (i.e., tries—expanse based structures).

Description:
BACKGROUND 
     Computer processors and associated memory components continue to increase in speed. As hardware approaches physical speed limitations, however, other methods for generating appreciable decreases in data access times are required. One method is via effective data management, achieved by the appropriate choice of data structure and related storage and retrieval algorithms. For example, various prior art data structures and related storage and retrieval algorithms have been developed for data management including arrays, hashing, binary trees, AVL trees (height-balanced binary trees), b-trees, and skiplists. An enhanced storage structure is further described in pending U.S. patent application Ser. No. 09/457,164, filed Dec. 8, 1999, entitled “A Fast, Efficient, Adaptive, Hybrid Tree”, assigned in common with the instant application and incorporated herein by reference in its entirety. 
     While such data structures accommodate storage, searching and retrieval of data, they do not readily support other operations including, for example, providing a count of a number of data entries or keys (e.g., indices) falling within selected ranges or “expanses” of the structure. Nor do such constructs readily support identification of keys or indices based on their ordinal value among the stored data, or the retrieval of data based on an ordinal value or range of ordinal values of their associated keys or indices. Instead, these prior art structures require partial or complete traversal of the data to provide a count of values satisfying specified ordinal criteria. Accordingly, a need exists for a data structure which supports identification of keys and indices and data based on ordinal values within a set and further provides a count of values based on ranges of key (or index) values. 
     SUMMARY OF THE INVENTION 
     The present invention is a data structure and related data storage and retrieval method that rapidly provides a count of elements stored or referenced by a hierarchical structure of ordered elements (e.g., a tree), access to elements based on their ordinal value in the structure, and identification of the ordinality of elements. In an ordered tree implementation of the invention, a count of elements stored in each subtree is stored, i.e., the cardinality of each subtree is stored either at or associated with a higher level node pointing to that subtree or at or associated with the head node of the subtree. In addition to data structure specific requirements (e.g., creation of a new node, reassignment of pointers, balancing, etc.) data insertion and deletion includes steps of updating affected counts. Elements may be target data itself (e.g., data samples, prime numbers); keys or indices associated with target data (e.g., social security numbers of employees, product numbers and codes, etc. used to reference associated data records, etc.); or internal memory pointers to keys and/or data stored outside the data structure. While the invention is applicable to varied hierarchical storage structures including, for example, binary trees, AVL trees (height-balanced binary trees), b-trees, etc. (population based structures) and digital trees (i.e., tries—expanse based structures), a preferred embodiment of the invention incorporates a hybrid tree structure as described and set forth in above referenced U.S. Patent Application. 
     According to an aspect of the invention, a computer memory is configured to store data for access by an application program being executed on a data processing system. Stored in memory is a hierarchical data structure, the data structure storing an ordered set of keys. The structure includes a root node and a plurality of first level data structures, a subset of the ordered set of keys uniquely associated with respective ones of the first level data structures. Each of the first level data structures have associated therewith a count value representing a number of entries of an associated one of the subsets. The entries may correspond to the keys, particularly in those structures wherein keys must be unique. 
     According to an aspect of the invention, the hierarchical data structure may be a digital tree, or “trie” or similar “expanse” based data storage structure. Conversely, a feature of the invention includes “population” based structures, such as b-trees and the various types of binary trees. 
     According to another feature of the invention, the count values are stored in memory in association with the root node, the root node including addresses of each of the first level data structures. Each of the addresses may stored in memory in association with the root node as a pointer originating at the root node and terminating at a respective one of the first level data structures. 
     According to another feature of the invention, each of the first level structures further includes a plurality of directors (e.g., pointers or directed edges) to respective second level data structures and/or nodes. The first level data structures may further include interior nodes referencing other nodes and leaf nodes containing or referencing the keys. 
     According to another aspect of the invention, a computer memory for storing data for access by an application program being executed on a data processing system includes a hierarchical data structure stored in memory. The data structure stores an ordered set of keys and includes a head node addressing each of a first plurality of first level data structures. Each of the first level data structures, in turn, address respective second level data structures. First level nodes of the ordered set of keys are uniquely associated with respective ones of the first level data structures while second level nodes are uniquely associated with respective ones of the second level data structures. Each of the first and second level data structures have associated therewith a count representing a number of the keys stored in respective ones of the structures. 
     According to a feature of the invention, each of the first level nodes includes references to at least two of the second level nodes. Further, the counts may be associated with a number of the keys referenced by respective ones of the references. 
     According to another feature of the invention, the references include addresses of the second level nodes in the memory. For example, the references may be in the form of pointers to the second level nodes. 
     According to another aspect of the invention, a method of storing data in a computer memory includes storing ordered sets of keys into a plurality of data structures. Addresses of the data structures are stored in a root node and counts of one of the keys in each of the data structures are stored in association with each of the addresses. 
     According to a feature of a method according to the invention, a step of determining an ordinality of one of the keys includes adding at least one of the counts to an ordinality of the key with respect to others of the keys commonly stored in one of the data structures. 
     According to another aspect of the invention wherein the data structures include at least one first level data structure referencing a plurality of second level data structures, the method further includes a step of distributing the keys among the plurality of second data structures and storing in the first level data structure counts of the keys in each of the second level data structures. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention will be better understood from a reading of the following detailed description taken in conjunction with the drawing in which like reference designators are used to designate like elements, and in which: 
     FIGS. 1-1,  1 - 2 , and  1   a  are diagrams of a b-tree including key counts stored in respective subtree root nodes, each indicator associated with a number of keys present each of the “pointed-to”, lower level sub-subtrees; 
     FIGS. 2-1,  2 - 2 ,  2 - 3 , and  2   a  are diagrams of a b-tree including key counts stored in respective subtree root nodes, each indicator associated with a number of keys present in the subtree; 
     FIGS. 3-1,  3 - 2 ,  3 - 3 , and  3   a  are diagrams of a b-tree including key counts stored in respective subtree root nodes, a top indicator providing a total count of keys referenced by the subtree, individual indicators providing counts of sub-subtrees pointed to by the root node; 
     FIG. 4 is a flow chart of a recursive method of obtaining the ordinality of a key stored in a data structure according to the invention; 
     FIG. 5 is a flow chart of a recursive method of retrieving a key based on its ordinality as stored in a data structure according to the invention; 
     FIGS. 6-1 and  6 - 2  are a diagram of a hybrid “Judy” tree structure including key counts stored in respective subtree root nodes, each indicator associated with a number of keys present in each of the “pointed-to”, lower level sub-subtrees; 
     FIG. 7 is a diagram of a binary tree including key counts stored in respective subtree root nodes, each indicator associated with a number of keys present in respective left and right sub-subtrees; 
     FIG. 8 is a diagram of a binary tree including key counts stored in respective subtree root nodes, each indicator associated with a number of keys present in the subtree; and 
     FIG. 9 is a block diagram of a computer system which is adapted to use the present invention. 
    
    
     DETAILED DESCRIPTION 
     For ease of explanation and understanding, an index counting structure according to the invention is described with reference to several tree structures used to index and possibly store data in a memory. The memory can be any type of device and can be accessed by any type of computer or processor. However, it is understood that the scope of the invention is not limited to a particular data structure or type, or to any particular memory or processor type, except to the extent literally required by the appended claims. Thus, all embodiments of the present invention are given by way of example only and are not to be considered by way of limitation. 
     FIGS. 1-1 and  1 - 2  is a diagram of a “b-tree” including index counts stored in respective subtree root nodes, each count associated with a number of keys or entries present in each of the “pointed-to”, lower level sub-subtrees. (Unless otherwise indicated, “root” node may refer to any interior node to the extent that such node is an initial or head node of a corresponding subtree.) Thus, associated with each pointer to a subtree is a count of the number of entries contained in that subtree or, if the pointer is to a leaf, then the number of entries in that leaf. In this example, the leaf nodes of the b-tree are populated with the 168 prime numbers that are less than 1000. Prime numbers are a particularly useful example of the invention to illustrate searching to determine the number of prime numbers within representative ranges and, conversely identifying ranges of prime numbers based on their ordinal values. Pointer  101  is associated with root node  100  used to access first level node  110 . Associated with the originating “tail end” of pointer  101  at root node  100  is a count, in this case having a value of 168, indicating that pointer  101  is directed to a tree having 168 entries, i.e., the number of prime numbers less than 1000. 
     As is conventional for a b-tree, each interior node includes an array of pointers to other nodes and a list of separators or identifiers, each representing a dividing point between values stored in corresponding subtrees or leaf nodes of the tree. Often, as in the present example, the separators or identifiers are actually the keys representing the greatest entry within an associated node pointed to by an associated pointer. The present invention further includes, at the root and each interior node (i.e., at each node pointing to another node), a count of the total number of entries contained in the corresponding nodes (i.e., subtree) pointed to by the parent node. Thus, it is possible to determine the ordinality of an entry by keeping a running count of entry totals for subtrees bypassed during tree traversal. In the present example, it would only be necessary to keep track of the total number of entries stored in subtrees and leaf nodes corresponding to key ranges less than the target key. Similarly, it is possible to traverse the tree based on key ordinality (i.e, the ordered position of the key) by identifying a key at each node associated with a desired accumulated count. 
     For example, in the b-tree of FIGS. 1-1 and  1 - 2 , root node  100  includes a pointer to a first level node  110  and a count of the number of entries in the tree. First level node  110  and each interior node  140 ,  150 ,  160 ,  180  and  190  includes an ordered array of up to eight pointers providing paths to children nodes, i.e., respective subtrees or terminal leaf nodes. Separator values corresponding to respective maximum key values contained in all but the last of the children nodes pointed to by a parent node are stored at the parent node as an ordered list. Also stored at each node, associated with respective pointers to children nodes, are count values representing the number of data entries or keys associated with the pointed-to nodes, i.e., within the referenced subtrees. Thus, node  110  accommodates a maximum of eight pointers to respective children nodes, i.e., subtrees or leaf nodes. As depicted, first level node  110  includes pointers  131 - 135  providing paths to respective second level interior nodes  140 ,  150 ,  160 ,  180  and  190 . Each of these second level interior nodes is, in turn, the root node of a subtree. Pointers not being used are given a null value indicating such. 
     Associated with each assigned pointer  131 - 135  is a respective count value  121 - 125  representing the number of entries (in this case, prime numbers) contained within or pointed to by the subtree (or leaf node) addressed by the pointer. In the example, first level node  110  includes a first pointer  131  addressing interior child node  140 . Separator  113  (FIG. 1 a ) indicates the maximum key in or referenced by child node  140  has value of 101, meaning that keys less than or equal to 101 are stored in (or referenced by) the subtree having child node  140  as its root node. Associated with pointer  131  is count  121  having a value of 26 representing the total number of entries referenced by pointer  131 . Thus, without traversing the subtree associated with pointer  131 , its can be determined at parent node  110  that the subtree not only stores keys having values of 101 and less, but that there are a total of 26 entries in that subtree (i.e., a population count of 26). 
     Pointer  132 , also originating at first level node  110 , is associated with count  122  having a value of 35 indicating that the corresponding subtree stores or references a total of 35 entries. Since pointer  132  is bounded by separators  113  and  114 , it can be further determined that the subtree includes entries having key values greater than 101 but less than or equal to 283, i.e., has an expanse of [101, 283) where “[” indicates a lower, not included limit and “)” indicates an upper included limit. Thus, in the present example, it can be determined at and from first level node  110  that there are a total of 35 prime numbers that are greater than 101 but less than or equal to 283. Similarly, pointers  133 ,  134  and  135  are associated with respective counts  123 ,  124  and  125  indicating that the pointers reference  48 ,  34  and  25  entries, respectively, and corresponding to primes greater than 283 (i.e., 293) through 599, primes greater than 599 (i.e., 601) through 823, and primes greater than 823 (i.e., 827) through the maximum value stored, in this case  997  stored in leaf node  196 . Note that, as is conventional for b-trees, there is no indication of the minimum or maximum value stored in the tree, although this value might be stored in or associated with, for example, the root node or pointer to the root node. Having the minimum and maximum key value stored there would avoid tree traversal to determine these values. 
     Each second level node  140 ,  150 ,  160 ,  180  and  190  also includes up to eight pointers to lower level nodes (in this case, terminal leaf nodes); counts, associated with each assigned, non-nil pointer representing the total number of entries referenced by the pointer; and up to seven (i.e., N−1 where N is the maximum number of pointers per node) separator values indicating the maximum entry value stored in the respective “left” subtree. Thus, second level node  140  includes an array of pointers  142  and associated counts referencing terminal leaf nodes  143 - 147 . Similarly, second level node  150  is the root node of a subtree including terminal leaf nodes  153 - 158 ; second level node  160  points to and maintains a count of the number of entries in terminal leaf nodes  163 - 170 ; node  180  partitions a total of 34 entries into six terminal leaf nodes  183 - 188 , respectively storing six, eight, four, five, four and seven entries therein; and second level node  190  includes the final 25 primes less than 1000, i.e.,  827 - 997 ; stored in terminal leaf nodes  193 - 196 . Each of the second level nodes include counts of the number of entries contained in the pointed-to terminal leaf nodes so that indications to (i.e., traversal of) leaf nodes not containing the target key are avoided. 
     Operationally, conventional b-tree building, insertion, deletion and rebalancing must be augmented to include maintenance of the count values. Thus, counts must be increment at all nodes referencing a new entry as part of node (or leaf) creation. Similarly, deletion of a leaf (or key) requires a corresponding adjustment of the affected count values. 
     FIGS. 2-1,  2 - 2 , and  2 - 3  is a diagram of a b-tree structure according to an alternate embodiment of the invention. This embodiment duplicates count values so that each node includes, not only count values of referenced substrees, but the total count for all subtrees. Thus, with reference to FIG. 2 a , first level node  110   a  includes counts  121 - 125  for respective subtrees and includes a total count value 120. 
     FIGS. 3-1,  3 - 2 , and  3 - 3  is a diagram of another b-tree structure according to still a further embodiment of the invention in which a count of entries referenced by a node is included only at that node, i.e., there is no subdivision count indicating the individual counts contained in each subtree referenced. Thus, as shown in FIG. 3 a , first level node  110   b  includes a count  120  indicating the total number of entries contained in subtrees referenced by pointers  131 - 135 . This embodiment would therefore require an additional indication to determine the count values for each subtree. 
     Maintaining a count of the number of entries stored in each subtree referenced by a node provides for both the determination of the ordinality of a key and, conversely, access of the key based on its ordinality, without complete traversal of all leading (or trailing) keys. Instead, count values are accumulated during tree searching to obtain the ordinality of the target key or to access a key based on its ordinality. 
     FIG. 4 is a flow diagram of a recursive routine for determining the ordinality of a key, i.e., the ordered position of the key in the tree. Upon initial entry of the Keycount routine at terminal  401 , at step  403  the node being visited is examined to determine if it is a terminal leaf node. This test provides for recursion termination when there are no more nodes to be visited. In this case, the routine traverses the leaf node at step  409  to determine the ordinal position of the key (if found) or the number of keys less than the key if the key is not found. This value is then returned as the Keycount to the calling routine. Since the routine is recursive, the calling routine is typically an upper level of Keycount which will add the results to its findings, and execute its own return. 
     Thus, if Keycount is visiting an interior or root node, processing continues at step  405  where a subtotal is computed of the counts for all subtrees referenced by the node that have only keys less than the targeted key, i.e., have “expanses” that do not include and are less than the key. At step  407  Keycount “calls itself,” adding the subtree subtotal to the result from traversal of the subtree including the key in its expanse, i.e., range of keys stored. That is, upon return from the called verion(s) of Keycount, the present Keycount has returned to it the count of entries less than or equal to the targeted key in that subtree. Keycount then adds to that value the count of any additional entries in the next level subtree that are less than the key, and returns that value to the calling routine at step  411 . 
     A flow chart for a “Keyfind” routine is given in FIG.  5 . Keyfind is also recursive and provides for access to an entry based on its ordinality rather than its key value. Keyfind is entered by a calling routine at terminal  501  and recursive test  503  is used to determine if the node being visited is a terminal leaf node. If the current node is a leaf, the routine uses the count supplied by the calling routine at step  509  to index through the entries and return the key found at the count position at step  511 . Conversely, if the node being visited is an interior or root node, at step  505  Keyfind computes the sum of all subtrees having counts which, when summed, are less than the requested count, i.e., all subtrees “to the left” of the subtree having an entry corresponding to the count supplied by the calling routine. To traverse this latter subtree (i.e., the subtree including the target ordinal entry), Keyfind calls itself, supplying the remainder of the count not accounted for by the subtrees to the left. The key value returned by the called version is then returned at step  511  to each calling version of Keyfind until all versions are completed and exited. 
     In addition to the Keycount and Keyfind routines, pseudo-code presented in Appendix A provides a nonrecursive function named Ordinal returning an ordinal value of a key and a recursive implementation of Keyfind named Findkey. 
     In addition to b-trees, the invention is applicable to, and may be used in combination with other hierarchical data structures. For example, FIGS. 6-1 and  6 - 2  is a diagram of a preferred embodiment of the invention applied to a hybrid “Judy” tree structure, the latter fully described in and by the previously referenced and incorporated pending U.S. patent application Ser. No. 09/457,164 entitled “A Fast, Efficient, Adaptive, Hybrid Tree”. The algorithm and data structure contained in that patent application is also known as Judy. The Judy tree structure is augmented by the addition of count values for each subordinate structure referenced by a node. Thus, for example, root node  600  includes a pointer  601  to Judy node  610  and a count value  603  associated with the number of entries addressed by the pointer. Similarly, Judy nodes  613 ,  640 ,  650 ,  660  and  670  each associate a count with each pointer originating at those nodes. For example, pointer array  611  includes pointers  631 ,  632 ,  633  and  634  to respective lower level Judy nodes  640 ,  650 ,  660  and  670 . Associated with each of these pointers are respective count values stored in count array  613 , indicating the total number of entries referenced by each of the respective pointers. 
     As with the b-tree implementation, the Judy structure implementation supports counting to determine ordinality of keys and to determine the number of keys between ordinal values. For example, Appendix B contains an example C program code of a preferred implementation of Judy  1  Count( ). This routine returns the “count of Indexes” between (inclusively) the two limits by traversing the a “Digital tree” data structure. Thus, the routine determines an ordinal value for the specified keys to compute a population count present within the specified range. 
     FIGS. 7 and 8 are diagrams of binary trees according to the invention, in which each node includes a count of the number of keys addressed by the respective left and right pointers (FIG. 7) and wherein each node includes a total count of keys addressable by that node, but without allocation between subtrees (FIG.  8 ). 
     When implemented in software, the elements of the present invention are essentially the code segments to perform the necessary tasks. The program or code segments can be stored in a processor readable medium or transmitted by a computer data signal embodied in a carrier wave, or a signal modulated by a carrier, over a transmission medium. The “processor readable medium” may include any medium that can store or transfer information. Examples of the processor readable medium include an electronic circuit, a semiconductor memory device, a ROM, a flash memory, an erasable ROM (EROM), a floppy diskette, a compact disk CD-ROM, an optical disk, a hard disk, a fiber optic medium, a radio frequency (RF) link, etc. The computer data signal may include any signal that can propagate over a transmission medium such as electronic network channels, optical fibers, air, electromagnetic, RF links, etc. The code segments may be downloaded via computer networks such as the Internet, Intranet, etc. 
     FIG. 9 illustrates computer system  900  adapted to use the present invention. Central processing unit (CPU)  901  is coupled to system bus  902 . The CPU  901  may be any general purpose CPU, such as an HP PA-8500 or Intel Pentium processor. However, the present invention is not restricted by the architecture of CPU  901  as long as CPU  901  supports the inventive operations as described herein. Bus  902  is coupled to random access memory (RAM)  903 , which may be SRAM, DRAM, or SDRAM. ROM  904  is also coupled to bus  902 , which may be PROM, EPROM, or EEPROM. RAM  903  and ROM  904  hold user and system data and programs as is well known in the art. 
     Bus  902  is also coupled to input/output (I/O) controller card  905 , communications adapter card  911 , user interface card  908 , and display card  909 . The I/O card  905  connects to storage devices  906 , such as one or more of a hard drive, a CD drive, a floppy disk drive, a tape drive, to the computer system. Communications card  911  is adapted to couple the computer system  900  to a network  912 , which may be one or more of a telephone network, a local (LAN) and/or a wide-area (WAN) network, an Ethernet network, and/or the Internet network. User interface card  908  couples user input devices, such as keyboard  913  and pointing device  907 , to the computer system  900 . The display card  909  is driven by CPU  901  to control the display on display device  910 . 
     While illustrative and presently preferred embodiments of the invention have been described in detail herein, it is to be understood that the inventive concepts may be otherwise variously embodied and employed and that the appended claims are intended to be construed to include such variations except insofar as limited by the prior art. For example, the invention is applicable to a wide range of data structures and is not limited to either digital, b-trees or binary trees. Further, while the invention includes traversal of a data structure using count information stored throughout the structure to obtain ordinality of keys and entries, derive population counts over specified key ranges, and access and retrieve data based on key ordinality, the invention is not limited to these specific methods or uses of such count information or data structure. Additionally, while the term “root” node has been used to refer to an initial node of a tree, it is understood by those skilled in the art that a tree structure may be recursively defined and thereby decompose into a plurality of subtrees, each subtree having a “head” or “root” node that is an interior node of the larger tree. 
     
       
         
               
               
             
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
             
               
               
             
               
             
               
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                 APPENDIX A 
               
               
                   
               
             
             
               
                 ORDINAL(ROOTPOINTER, FINDKEY): INTEGER 
                 Return ordinal position of key in tree 
               
             
          
           
               
                   
                 BEGIN 
               
             
          
           
               
                   
                 COUNT = 0; 
                 Initialize counter to accumulate count 
               
               
                   
                 IF NODETYPE(ROOTPOINTER) = LEAF THEN BEGIN 
                 If pointer is to leaf node, count leafs less than 
               
               
                   
                   
                 or equal to key 
               
             
          
           
               
                   
                 WHILE (LEAFKEY &lt;= FINDKEY) BEGIN 
                 Step through leaf node until key found 
               
             
          
           
               
                   
                 INCREMENT COUNT; 
                 Increment count of keys &lt;= target key 
               
               
                   
                 IF MORELEAVES THEN NEXTLEAFKEY; 
                 Advance to test next leaf if more leaves exist 
               
             
          
           
               
                   
                 END WHILE 
                 Exit while loop if key found or search 
               
               
                   
                   
                 finished 
               
               
                   
                 IF LEAFKEY = FINDKEY THEN FOUND=TRUE; 
                 If key found, set boolean FOUND to true 
               
             
          
           
               
                   
                 ELSE BEGIN 
                 If node points to subtrees, add count from 
               
               
                   
                   
                 subtrees outside expanse, recurse subtree 
               
               
                   
                   
                 having expanse including FINDKEY 
               
             
          
           
               
                   
                 WHILE (SUBTREEMAXKEY &lt;= FINDKEY) BEGIN; 
                 Top of loop to scan subtrees outside expanse 
               
             
          
           
               
                   
                 COUNT = COUNT + TREECOUNT; 
                 Add subtree ordinality to total 
               
               
                   
                 IF MORESUBTREES THEN NEXTSUBTREE; 
                 Advance to test next subtree if one exists 
               
             
          
           
               
                   
                 END WHILE; 
                 Exit While loop if all &lt;= subtrees acounted 
               
               
                   
                   
                 for 
               
               
                   
                 IF SUBTREEMAXKEY &lt; FINDKEY THEN 
                 Reurse subtree if FINDKEY not found 
               
             
          
           
               
                   
                 COUNT = COUNT + ORDINAL(SUBTREE, FINDKEY) 
               
             
          
           
               
                   
                 END ELSE; 
               
               
                   
                 ORDINAL = COUNT; 
               
               
                   
                 RETURN 
               
               
                   
                 END ORDINAL 
               
             
          
           
               
                 FINDKEY(ROOTPOINTER, N):REAL 
               
             
          
           
               
                   
                 BEGIN; 
                 Return Nth key in tree 
               
               
                   
                 IF NODE.TYPE(ROOTPONTER) &lt;&gt; LEAF THEN BEGIN 
                 Check to see if at interior node vice leaf 
               
             
          
           
               
                 LISTPOINTER = 1; 
                 Initialize list pointer to first entry in list 
               
               
                 COUNT = 0; NEXTCOUNT = COUNT(LISTPOINTER); 
                 Initialize local counter to zero 
               
               
                 DO WHILE (COUNT + NEXTCOUNT &lt; N); BEGIN; 
                 Traverse tree until count = N 
               
             
          
           
               
                   
                 COUNT = COUNT + NEXTCOUNT; 
                 Add count of next to total 
               
               
                   
                 LISTPOINTER = LISTPOINTER + 1; 
                 Advance to next node entry in list 
               
               
                   
                 NEXTCOUNT = COUNT(LISTPOINTER); 
                 Get next count value in list 
               
             
          
           
               
                 END DO 
                 Exit loop when next entry would exceed cnt 
               
               
                 IF NODE.TYPE.NODEPOINTER = LEAF THEN N = 
               
               
                 COUNT = COUNT + FINDKEY(NODEPOINTER(LISTPOINTER), 
                 Call FINDKEY to traverse subtree 
               
             
          
           
               
                   
                 (N-COUNT); 
               
             
          
           
               
                 ELSE 
                 If not interior node, parse count leaves 
               
               
                 FINDKEY = KEY(COUNT); 
               
               
                 RETURN; 
                 Return 
               
               
                 END FINDKEY; 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
               
               
               
             
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
               
             
               
               
             
               
               
             
               
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
               
               
             
               
               
               
             
               
               
               
             
               
               
             
               
               
               
               
             
               
               
             
               
               
               
             
               
               
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
               
               
             
               
               
             
               
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
               
             
               
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
               
             
           
               
                   
                 APPENDIX B 
               
               
                   
                   
               
             
             
               
                   
                 // 
                 INCLUDE FILES 
               
             
          
           
               
                   
                 #include 
                 “Judy1.h” 
               
               
                   
                 /* 
               
             
          
           
               
                   
                 Judy1Count () returns the “count of Indexes” between 
               
               
                   
                 (inclusively) the two limits. It traverses the 
               
               
                   
                 a “Digital tree” data structure. This Count code is 
               
               
                   
                 the GENERIC untuned version (minimum code size). 
               
             
          
           
               
                   
                 */ 
               
             
          
           
               
                   
                 // 
                 Proto for internal routine 
               
             
          
           
               
                   
                 static ulong_t 
               
               
                   
                 _JudyAi ( Pvoid_t, ulong_t, ulong_t); 
               
               
                   
                 ulong_t 
               
               
                   
                 Judy1Count 
               
             
          
           
               
                   
                 ( 
                   
                   
               
               
                   
                 Pvoid_t 
                 PJudy, 
                 // Root Pointer to 1st Branch in tree 
               
               
                   
                 ulong_t 
                 I1, 
                 // Starting Index 
               
               
                   
                 ulong_t 
                 I2 
                 // Ending Index 
               
               
                   
                 ) 
               
             
          
           
               
                   
                 { 
               
             
          
           
               
                   
                 ulong_t Asize; 
               
               
                   
                 Slot_t Slot; 
               
               
                   
                 PSlot_t PSlot; 
               
             
          
           
               
                   
                 Slot 
                 = (Slot_t) PJudy; 
                 // For clarity - no code generated 
               
             
          
           
               
                   
                 // 
                 Check if we have anything to do (Negative space?) 
               
               
                   
                   
                 if (I1 &gt; I2) return(0L); 
               
               
                   
                 // 
                 Check if the tree has been allocated 
               
               
                   
                   
                 if (Slot == 0L || Slot == cJU_NULL) { return (0L); } 
               
               
                   
                 // 
                 Since there is not enough room for a “count” in the root pointer derive 
               
               
                   
                 // 
                 the tree total population by counting all entrys in the 1st node 
               
               
                   
                 // 
                 Is the root pointer point to a Node? 
               
               
                   
                   
                 if (JU_IS_BRANCH_PTR1 (Slot)) 
               
               
                   
                   
                 { 
               
             
          
           
               
                   
                 // 
                 Calculate the Judy Array Size 
               
               
                   
                   
                 PSlot = JU_SLOT_TO_COUNTER_POINTER1 (Slot); 
               
             
          
           
               
                   
                 Asize 
                   = 
                 *PSlot++; 
               
               
                   
                 Asize 
                 += 
                 *PSlot++; 
               
               
                   
                 Asize 
                 += 
                 *PSlot++; 
               
               
                   
                 Asize 
                 += 
                 *PSlot; 
               
             
          
           
               
                   
                 } 
               
             
          
           
               
                   
                 // 
                 Root pointer points to a linear sorted tree of Indexex 
               
             
          
           
               
                   
                 else 
                 // Must be a Leaf 
               
             
          
           
               
                   
                 { 
               
             
          
           
               
                   
                 Asize 
                 = 
                 JU_SLOT_TO_FLWR_COLOR1 (Slot); 
               
             
          
           
               
                   
                 } 
               
             
          
           
               
                   
                 // 
                 Got the total population of the Array 
               
               
                   
                 // 
                 Leave out the special case of I1 == I2 for testing 
               
               
                   
                 // 
                 if (I1 == I2) return ((Judy1Get (PJudy, I1) : 1L ? 0L); 
               
               
                   
                 // 
                 Check for other special cases (algorithm by AJS) 
               
               
                   
                   
                 return ( 
               
             
          
           
               
                   
                 ((I1 ==  0L) ? Asize 
                 : _JudyAi (PJudy, I1, 
                 Asize)) - 
               
               
                   
                 ((I2 == ˜0L) ? 0L 
                 : _JudyAi (PJudy, I2+1L, 
                 Asize)) 
               
             
          
           
               
                   
                 ); 
               
             
          
           
               
                   
                 } 
               
               
                   
                 /* 
               
             
          
           
               
                   
                 This internal routine is called to determine the number of 
               
               
                   
                 Indexes from the passed Index to the end (inclusive). No error 
               
               
                   
                 checking because it is called after error checking is done in 
               
               
                   
                 Judy1First (). 
               
             
          
           
               
                   
                 */ 
               
               
                   
                 static ulong_t 
               
               
                   
                 _JudyAi 
               
             
          
           
               
                   
                 (  
               
             
          
           
               
                   
                 Pvoid_t 
                 PJudy, 
                 // Root Pointer to 1st Branch in tree 
               
               
                   
                 ulong_t 
                 Index, 
                 // Index or Key to lookup 
               
               
                   
                 ulong_t 
                 TotalIndexes 
                 // Array Size 
               
               
                   
                 ) 
               
             
          
           
               
                   
                 { 
               
             
          
           
               
                   
                 Slot_t 
                 Slot; 
                 // The Slot in current Branch 
               
               
                   
                 PSlot_t 
                 PSlot; 
                 // The Slot in current Branch 
               
               
                   
                 ulong_t 
                 Digit; 
                 // Offset into current Branch pointing to Slot 
               
               
                   
                 ulong_t 
                 SftIdx; 
                 // Shifted Index to just relevant bits 
               
               
                   
                 ulong_t 
                 PosIdx; 
                 // Current Leaf index 
               
               
                   
                 ulong_t 
                 ii; 
                 // Total Indexes in tree 
               
             
          
           
               
                   
                 SftIdx = Index; 
                 // For clarity 
               
             
          
           
               
                   
                 // 
                 Search and find the closest actual Index in the tree 
               
               
                   
                   
                 if (Judy1First (PJudy, &amp;SftIdx) == cJU_FALSE) return (0L); 
               
             
          
           
               
                   
                 Slot 
                 = (Slot_t) PJudy; 
                 // For clarity - no code generated 
               
             
          
           
               
                   
                 // 
                 Traverse the tree until pointer is to a leaf 
               
               
                   
                   
                 while (JU_IS_BRANCH_PTR1 (Slot)) 
               
               
                   
                   
                 { 
               
             
          
           
               
                   
                 // 
                 Get next decoded Digit 
               
             
          
           
               
                   
                   
                 Digit 
                 = JU_SFTIDX_TO_DIGIT1 (SftIdx); 
               
             
          
           
               
                   
                 // 
                 Get the pointer to the Counter area 
               
               
                   
                   
                 PSlot = JU_SLOT_TO_COUNTER_POINTER1 (Slot); 
               
               
                   
                 // 
                 Move down to next Level 
               
             
          
           
               
                   
                 Slot 
                 = *((PSlot_t) Slot + Digit); 
               
             
          
           
               
                   
                 // 
                 Subtract Indexes below the requested one 
               
               
                   
                   
                 for (ii = 0L; ii &lt; Digit; ii++) 
               
               
                   
                   
                 { 
               
             
          
           
               
                   
                 TotalIndexes −= *PSlot++; 
               
             
          
           
               
                   
                 } 
               
             
          
           
               
                   
                 // 
                 Shift out decoded Index bits 
               
               
                   
                   
                 JU_SFTIDX_DIGITS1 (SftIdx); 
               
             
          
           
               
                   
                 } 
               
             
          
           
               
                   
                 // 
                 Slot 
                 : Is the Leaf pointer 
               
               
                   
                 // 
                 SftIdx 
                 : Is the Index to decode at this level 
               
             
          
           
               
                   
                 // 
                 Change Leaf pointer to a normal pointer. 
               
               
                   
                   
                 PSlot = JU_SLOT_TO_FLWR_POINTER1 (Slot); 
               
               
                   
                 // 
                 Search Leaf for the required matching Index 
               
               
                   
                   
                 PosIdx = 0L; 
               
               
                   
                   
                 do 
               
               
                   
                   
                 { 
               
             
          
           
               
                   
                 if (SftIdx == *(PSlot + PosIdx)) 
               
               
                   
                 { 
               
             
          
           
               
                   
                 return(TotalIndexes - PosIdx); 
               
             
          
           
               
                   
                 } 
               
             
          
           
               
                   
                 } while (++PosIdx); 
               
             
          
           
               
                   
                 // 
                 Not a possible return, since Index was found by Judy1First () 
               
               
                   
                 }