Highly dynamic Radix fast search algorithm with easy sort/deletion - Embedded.com

Highly dynamic Radix fast search algorithm with easy sort/deletion

Note: This article was originally printed in Embedded Systems Programming magazine, March 1997. A follow-up to this article will appear in the June 2007 issue of Embedded Systems Design magazine, and June 1st here on Embedded.com.

Good search algorithms are hard to find. Embedded systems in particular have some peculiar requirements. This article presents a search algorithm suitable for embedded applications.

Every program, even the simplest, needs to store and retrieve data according to some key. Simple arrays are the favorite and most common choice. Storage and access are simple and predictable. Every high-level language has built in facilities for this kind of search. As keys get lengthier and the key space gets sparsely occupied, this solution becomes impractical. Large, mostly empty, arrays result. Search algorithms that have more desirable properties are sought. Always desirable are the predictable allocation of storage, linearly related to the number of keys, and a fast average search with tolerable worst-case search times. Sometimes, other algorithm characteristics are important. For dynamic environments, key deletion is necessary. Some applications require the return of a sorted list of key values. Over decades, numerous algorithms have been developed, each with more or less desirable characteristics. Their usage depends on matching their most desirable properties with the needs of the application. An often recurring theme in search algorithm evaluation is the difficulties encountered in deletion. Very often comparatively complex storage structure repairs are required to restore the links and structure consistency. As I will show later, ease of deletion was very important in the selection and modification of the algorithm that I ultimately used.

Embedded applications, in particular, have some peculiar requirements. Usually, their real-time nature limits both space and time resources that can be spent on searching. Regular periodic deadlines and highly dynamic search/insertion/deletion environments make undesirable those algorithms that have periodic complex restructuring requirements, such as rebalancing of trees or elimination of lazy deletion nodes. Algorithms that produce unpredictable characteristics in the storage structure, based on key length, key insertion order, or key value, introduce too much uncertainty into the dynamic allocation requirements and can unpredictably break deadlines. For example, in binary search trees, one of the most fundamental search algorithms in computer science, if the keys inserted are not in random order, the tree can be very unbalanced and result in worst-case searches requiring N key comparisons for N nodes.

I do not intend here to survey the vast field of search algorithms. Nor do I intend to justify/quantify even the most typical parameters characteristic of the algorithm that I selected, such as average search/worst-case search/insertion/deletion times or sort complexity. Many fine texts cover, in great depth, the mathematically precise calculations of these characteristic parameters. The text I relied on most heavily was Robert Sedgewick's classic Algorithms in C .1 I intend merely to present an implementation of an existing algorithm, modified for deletion, that I have found to be extremely useful in highly dynamic situations.

Application environment
I started out with a specific application in mind, and the particular requirements of that application drove my investigation of search algorithms. I work in the Air Traffic Surveillance group at MIT Lincoln Laboratory on the GPS-Squitter project. We do aircraft surveillance research for the FAA with a combination of the GPS satellite system, Mode S transponders on the aircraft, and Mode S receivers on the ground, all linked together in a Sun SPARCstation central control computer. The basic surveillance concept is as follows: GPS signals, available globally, are received by the target aircraft, decoded into latitude, longitude, and altitude, and encoded (with additional information such as Mode S address, time, squitter type, and turn indicator) into a 112-bit long squitter that is broadcast by the Mode S transponder every half-second. The ground Mode S receivers—there are usually several of these at an airport or in the general surveillance area—pick up these broadcasts and link them to the central control computer. The key here, literally, is the Mode S address. These addresses are 24 bits long and uniquely identify each aircraft. They translate into the tail identification numbers seen on aircraft. The central control computer picks up receptions from all the Mode S receivers and decodes the information for storage, based on the target Mode S address. Currently, there are relatively few long squitter broadcasting aircraft. Most aircraft broadcast short squitters that have only the Mode S address for identification purposes. My initial central control computer implementation used a simple linked list to store long squitter targets only. This arrangement is fine with only several test aircraft targets. Later versions of the software required monitoring short squitter activity also, and because most commercial aircraft broadcast these, the world of aircraft to be monitored in a high-traffic terminal control area went immediately from several to several hundred. Thus my search for a new storage algorithm.

This environment implies several requirements on the storage algorithm. First, because we are dealing with high-speed moving targets that must be monitored continuously, fast searches without any bad worst-case searches due to key value are necessary. Also, because the environment is highly dynamic with targets moving into and out of the surveillance area continuously, the algorithm had to have key deletion that was very efficient. Over long periods of time, the storage structure could not accumulate characteristics that required periodic restructuring of the entire structure. Lazy deletion was intolerable. Hash tables were rejected because of deletion problems and the need to maintain a relatively sparse structure to avoid multiple hash hits from the same hash value. I wanted the algorithm to behave just as nicely on allocation of the last available storage node as on the first. Additionally, when a target moved out of contact, the central control computer would stop receiving squitters from all of the Mode S receivers. Eventually, the target would have to be dropped or aged out. To do this, the entire storage structure would have to be searched for targets that had been lost from contact for greater than the age out seconds time, and all of these targets would have to be deleted. In some cases, I also wanted to be able to sort the targets by some other field than target node key. For example, displaying the targets in order of strength of squitter contact was desirable in some circumstances. Traversing each target node, calculating the number of squitters received by all the Mode S receivers in the last n seconds, and sorting by this number was very useful—on occasion.

After considering various trees, tries, and hash tables, the algorithm I selected as my starting point was among the class of algorithms referred to as radix search methods, characterized by examining the keys one bit at a time rather than full key comparisons at each stage in node traversal. This algorithm, Practical Algorithm To Retrieve Information Coded In Alphanumeric (PATRICIA) (due to Donald R. Morrison), was outlined in some detail by Sedgewick with some rudimentary example code for searching and insertion.2 I was attracted to this algorithm because it solved some of the characteristic problems of the more elementary radix search methods.

Radix searching
Digital search trees, for example, have some desirable properties. The longest node path is the length of the longest leading bit key string match. They use about lg N (the number of bits needed to represent N ) key comparisons, on average, and have a worst case of b comparisons for b -bit keys. Digital search trees are usually well balanced. However, they require a full key comparison at each node traversal and they have a structure dependent on the order of insertion.

An alternative radix search algorithm that presents some additional advantages is the radix search trie. This method branches according to each successive key bit, but without storing keys in each node. Keys are only stored in leaf nodes. This introduces some additional complexity. The two types of nodes are branch and leaf nodes. One traverses this structure by examining only successive key bits and doing a final key comparison when a leaf node is reached. The trie structure is also independent of insertion order. Whereas the digital search tree required about lg N key comparisons, the digital search trie requires about lg Nbit comparisons. For longer keys, this difference can be significant. The same holds for the worst case. Two disadvantages come with tries: a multinode implementation and creation of trees with long paths of branch nodes for keys with leading bits in common. The resulting tree has unequal numbers of branch and leaf nodes. Both of these disadvantages complicate the code. Sedgewick mentions multiway radix algorithms, but these algorithms did not really solve the problems, except for situations in which the algorithm was tailored to the key length or value and the size of N .

PATRICIA
PATRICIA examines key bits also, with two way branches based on the bit value. However, PATRICIA eliminates the long path branch nodes problem by storing, in each node, the bit number within the key that is to be compared. When a link that would connect to a leaf node in digital search tries is encountered, an upward link to an exiting node is established and the key is stored there. This upward link is to the only node in the tree that could possibly store that key. Recognition of an upward link is easy, because the node it points to will have a bit comparison number larger than the node from which the link originates. That is, as the search proceeds down the tree, the bit indices get smaller as from left to right in the key. This process also has the effect of equalizing the number of nodes and keys allocated. Every node will thus have a key, but they will only be used for full key comparison at the end of the search on the upward link. Frequently, the terminating link in a node points to itself, in which case the key would be stored in that node, but not always. Sometimes upward terminating links point to nodes much further up the tree. For arbitrarily long keys, PATRICIA requires at most n bit comparisons, where n is the number of bits in the key, and one full key comparison at the upward link. An example using five bit keys can be seen in Figure 17.8 on page 254 of Sedgewick.

Insertion has two cases—internal and external. These cases are shown in Figures 17.9 and 17.10 on page 255 of Sedgewick. External insertion is the most simple. These cases end at self-pointing nodes. When a search ends, it is, by the defining property of the tree, the only node at which a search for the new key could end. The new key is inserted in a node connected to the terminating node, with a search bit index of the leftmost bit in the two keys which differ. The upward links must now be adjusted—one pointing to the old terminating node and the other pointing to the new terminating node. In the example, X =11000 is the terminating node and Z =11010 is to be inserted. The three leading bits (four, three, and two) are the same. The first differing bit is bit one. The new node is thus inserted with bit search index one. Internal insertion starts out similarly, but ends when the current key bit is at a bit index that is between the search bit indexes already in the tree, and no branch is already in the tree for that bit index. This situation is shown in Figure 17.10 for the value T =10100. Let's step down the tree. At the head of the tree the bit search index is four. Bit four of T is one, so go right. The next node in the tree has bit search index three. Bit three of T is zero, so go left. The next node in the tree has search bit index one, but this is to the right of bit two of T , which is one. At this point in the tree there is no place to go right at bit index two. Thus a more leftward bit index, which was skipped previously, must be inserted between X and P . This insertion requires a more complex repair than external insertion. The XP link must be broken and a new node inserted, and the other link must be set to point to the just-inserted node. If this verbiage seems complicated, just examine Figures 17.9 and 17.10. These figures are labeled with the bit search indexes and all the links.

To quote Sedgewick, “PATRICIA is the quintessential radix search method: it manages to identify the bits which distinguish the search keys and build them into a data structure (with no surplus nodes) that quickly leads from any search key to the only key in the data structure that could be equal.” But this method has one serious flaw from the point of view of my application: deletion. Horowitz suggests the construction of the delete function as an exercise, but does not give an example, and indicates that the deletion complexity is a function of the free depth.3 Although the structure of the nodes is independent of the order of insertion, the location of the keys is not. All is not lost however. If one changes the algorithm by eliminating the upward links and inserting links pointing to leaf nodes instead, the structure repairs become doable. Once the key node is found, the deletion cost is predictable. The double node implementation resurfaces however. Examining the structure reveals that every insertion/deletion results in the addition/deletion of two nodes, one branch node, and one data node. The branch nodes will be small, with only pointers and indexes. The data nodes will not have upward links and will contain the key and the application data. I was willing to tolerate additional code complexity for ease of deletion. Knuth has a detailed calculation of successful and unsuccessful searching times, which should be unchanged even in this modification.4

Implementation and usage
The code for my implementation is shown in rdx_sch.c , which contains all of the access routines. Header files rdx_sch.h and rdx_dta.h contain the typedefs of the data nodes and the application data structure. An extensive comment at the beginning of rdx_sch.c has prototypes, explanatory/cautionary notes and examples of usage. The first thing you will want to do is open rdx_sch.h and set MAX_NUM_RDX_NODES and NUM_KEY_BYTES to the number of key nodes and number of key bytes you desire. For my application, 1,024 seemed more than enough for the number of aircraft targets in a terminal surveillance area, and three bytes was the Mode S address size. The data node typedef follows and includes the APP_DATA data structure in rdx_dta.h for the node data. I won't show my actual data, as this is extensive and not relevant to a discussion of the algorithm. The first three members of both data and branch nodes are identical: a boolean indicating branch(0) or data(1) node id, a left(0)/right(1) parent branch indicator, and a pointer to the parent node. The data nodes have the full key plus the data structure. The branch nodes have the bit index used for search comparison against the search key and the left and right node pointers. These pointers may point to other branch or data nodes.

Open rdx_sch.c and examine the code. Note that MAX_KEY_BITS is set to MAX_KEY_BYTES*8 , but that the definition of key [MAX_KEY_BYTES+1] contained one extra byte. The algorithm starts with a data structure with one branch and one data node always allocated with an “impossible” key of at least one more bit length than the normal search keys. I set this key to all 0xff. Next, follow the branch node typedef and the static storage of the branch and data nodes. Static storage was chosen for its obvious efficiency in a highly dynamic environment. Everything should be static, as only the access routines should provide access to the data structure. The selection of 1,024 nodes, although a power of two, is not important—any number will do. Note also that the allocated storage for both node types has the extra node allocated for the initial root impossible key. Thus, a full 1,024 data nodes are available. The head pointers to the search tree and the free lists of both types of nodes are declared as pointers to a specific node type. However, on occasion, these pointers are cast to point to the other type of node. This implementation problem is encountered with any type of multi-node type tree. It is important to note that the first three members of each type of node may not be moved or altered, because casts to the opposite type refer to these members.

Initialization is with rdx_ini() . The initial branch and data node with the impossible key are allocated and initialized. The remaining branch and data nodes are linked on free lists. Searching is with DNODE *rdx_sch(key) . The key passed is without any extra byte added at the beginning. For my application, this key is just a pointer to the 3-byte Mode S addresses. The return points to a data node with the found key or is NULL if the key was not found. All routines that take a key argument copy the key to storage with an extra zero byte for subsequent comparison. Searching always starts at the left branch of the always-allocated initial branch node. The simple while() loop does the job. The macro GBIT() gets the key index bit needed for that branch node (in b ) and branches. The search ends when a data node is encountered. A single key comparison determines search success or failure.

Insertion is by far the most complex operation. There are three cases:

  • The key is not already in the tree: insert it, pass back a pointer to the data node in the argument, and return one
  • The key already exists: pass back a pointer to the data node and return zero
  • The nodes are exhausted: pass back NULL argument and return -1

Start by searching for the key. If the key is found, this amounts to doing rdx_sch() —set the node pointer, return value, and return. If the key is not found, we begin actual insertion. The availability of free nodes is checked next, and the error case returned if none are free. Insertion proceeds by finding the leftmost bit that the search key is different from the terminating key actually found in the tree. A while() loop with GBIT() decrementing key_bit performs this. When this bit index is determined, the new branch and data nodes are allocated, with t1 pointing to the new branch node and t2 pointing to the new data node. We now begin traversing down the tree, looking for the internal or external location to insert, all the while saving the parent (p pointer) and child (c pointer) pointers to the nodes to insert between. If the search terminates with c-&#62d = 1, then the insertion is external. If the search terminates with c-&#62b&#60 = key_bit , the insertion is internal. The variable lr saves the link type—node is a right link of its parent (lr =1) or node is a left link (lr =0) of its parent, that is then used to set the brvariable in all nodes and is needed for deletion. With the new nodes allocated (pointers t1 and t2) and the parent and child nodes to be inserted between (pointers p and c ), the only remaining task is to change all the pointers among four nodes correctly. There are two cases (surprise!). The new branch node is between a parent that points right to its child or left to its child. The new data node will then be pointed to by the new branch nodes' other opposite direction pointer. There are six forward and backward pointers to set, the br variables in the two new nodes, the b -bit index in the new branch node, and the key in the new data node. I will not even begin to describe setting each of these. Just examine the code and draw a four node tree with arrows for links. This solution will probably be much clearer than a verbal description. If all this does not read exactly like a novel, please don't be discouraged. It took me some time to get all the pointers for both cases straight as well.

Deletion proceeds similarly, but in opposite fashion to insertion. There are two cases:

  • If the key is found, delete it and return one
  • If the key is not found, return zero

Start by searching. When the terminal data node is reached, its parent is the paired branch node to delete also. Thus, the other child of this branch node will have to point to the parent of the to-be-deleted branch node. Repairing the pointers of this other child and the parent of the to-be-deleted branch completes the deletion. The free branch and data nodes are now returned to the free lists. Sorting in this kind of structure comes simply by recursively traversing the tree and storing pointers to the nodes in order as they are reached by the recursions. A static array of node pointers, DNODE *node_ptrs [MAX_NUM_RDX_NODES+1], is allocated. This array will contain pointers to the key-sorted nodes. One extra node pointer allocated in this array will point to the impossible key node. Examining the key in the node pointed to by node_ptrs [MAX_NUM_RDX_NODES+1] should yield all 0xff bytes. As the recursive invocations reach data nodes the node_ptrs_cnt is incremented. When rdx_srt() returns, it sets its argument to a pointer to the head of the sorted pointers array and the function returns the number of elements. This scheme allows the use of qsort() to resort the nodes by resorting the array of pointers according to node data other than the key. An example of this is given in the test program.

Two additional simple functions finish our tour. An access function, rdx_kys() , is provided to find out how many nodes are allocated, and a debugging function, rdx_prt() , is provided to print out the contents of all the allocated tree nodes in two columns, with the associated branch and data nodes side by side.

Testing and examples
Nothing in software is as useful as an example. I provide rdx_test.c to both test and illustrate the usage of these routines. I generate 128 random three byte keys. The data node information is random() numbers also. The code is self-explanatory. Note the qsort() resort by node data. Timing this on a SPARCstation 10 results in a run time of about 100ms. This time is reduced by more than half by eliminating all the print out stuff. A more accurate test would do only algorithm manipulations of a random nature for more lengthy periods. This testing, however, was more than adequate for my application. For applications of less than 50 to 100 nodes, linked lists would probably be faster, but as the number of nodes increases, radix search has the advantage, especially for long keys.

One caveat: if you are retrieving keys from a data node and want to subsequently delete this key, then you must be careful to skip the first key byte— rdx_del( &(((DNODE *)dn[i])->key[1]) ) , because the data node keys are all one byte longer than keys passed to the routines. This byte will always be zero except for the impossible key head node.

Enhancements
Imagining several useful variations on this theme is entirely possible. My implementation assumes uniform and identical data nodes. It would be relatively easy to modify this implementation to malloc() data storage at key insertion. Rather than containing data, the terminal nodes would contain a pointer returned by malloc() and the size of the allocation. Deletion would free() this storage. Both branch and what would now be data pointer nodes would be allocated as before. A second possibility would be to extend, dynamically, the free list and allocate a new block of nodes upon exhaustion. I happen to like multithreaded code. I have rewritten applications from multiprocess shared memory architectures to multithreaded architectures with considerable savings in code and complexity, plus an improved responsiveness. These routines are not thread safe. There are two ways to proceed. The first is simply to enclose each access from any thread about a mutex_(un)lock pair. This brute force method works, but limits access of the entire structure to one thread at a time. A more fine grain approach would be to lock only the data or branch nodes involved and any child nodes of these. If many threads are to read the structure and only one thread is to modify the structure, then a readers/writer lock, rw_[wrlrd]lock() , could be used.

The structure as it stands is designed for records of unique keys. For applications with duplicate keys the data nodes could be modified to point to linked lists. I prefer to simply extend the key to include enough record data to make to keys unique. However, not all applications will want to do this. This algorithm, and my implementation of it, should be highly portable to environments other than the Sun I developed on. There are no system or library calls to worry about. If you want to use multiple trees in your application, I suggest duplicating the code and adding a descriptive prefix or postfix to the routine names. Everything is pretty much standard C.

www.embedded.com/code.htmhouses the code associated with this article.

Richard Hogaboom works for Comsys Information Technology Services at MIT Lincoln Laboratory. Reach him at rah@atc.ll.mit.edu for a tar file or with comments/bugs.

Endnotes:
1. Sedgewick, Robert. Algorithms in C. Reading, MA: Addison-Wesley, 1990.
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2. Morrison, Donald R. “PATRICIA—Practical Algorithm To Retrieve Information Coded in Alphanumeric,” Journal of the Association for Computing Machinery, Oct. 1968, pp. 514-534.
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3. Horowitz, Ellis, Sartaj Sahni, and Susan Anderson-Freed. Fundamentals of Data Structures in C. Salt Lake City, UT: Computer Science Press, 1993.
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4. Knuth, Donald E. The Art of Computer Programming: Sorting and Searching. Reading, MA: Addison-Wesley, 1973.
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