Another Look at my old Benchmark

This is a follow-up of my previous post. Here I change the table to several charts. Hope it seems more friendly to readers. You can find the links to these libraries in that table. Their source codes, including my testing code, are available here. You may also want to see my previous posts in the last few days for my interpretation to the results.

On C string (char*) keys, I fail to use JE_rb_old and JE_rb_new to get the correct result on Mac and so they are not showed in the charts. I would really appreciate if someone may give me the correct implementation using these libraries. In addition, tr1_unordered_map uses a lot of memory according to my program. The memory for string keys are faked.

For conveniece, here are some brief descriptions of these libraries (with no order):

• google_dense and google_sparse: google’s sparsehash library. Google_dense is fast but memory hungery while google_sparse is the opposite.
• sgi_hash_map and sgi_map: SGI’s STL that comes with g++-4. The backend of sgi_map is a three-pointer red-black tree.
• tr1::unordered_map: GCC’s TR1 library that comes with g++-4. It implements a hash table.
• rdestl::hash_map: from RDESTL, another implementation of STL.
• uthash: a hash library in C
• JG_btree: John-Mark Gurney’s btree library.
• JE_rb_new, JE_rb_old, JE_trp_hash and JE_trp_prng: Jason Evans’ binary search tree libraries. JE_rb_new implements a left-leaning red-black tree; JE_rb_old a three-pointer red-black tree; both JE_trp_hash and JE_trp_prng implement treaps but with different strategies on randomness.
• libavl_rb, libavl_prb, libavl_avl and libavl_bst: from GNU libavl. They implment a two-pointer red-black tree, a three-pointer red-black tree, an AVL tree and a unbalanced binary search tree, respectively.
• NP_rbtree and NP_splaytree: Niels Provos’ tree library for FreeBSD. A three-pointer red-black tree and a splay tree.
• TN_rbtree: Thomas Niemann’s red-black tree. I ported it to C++.
• sglib_rbtree: from SGLIB. It implements a two-pointer recursive red-black tree (all the other binary search trees are implemented without recursion).
• libavl_avl_cpp, libavl_rb_cpp and libavl_rb_cpp2: incomplete C++ version of libavl (no iterator), ported by me. Libavl_rb_cpp2 further uses the same technique in JE_rb_new to save the color bit. Source codes available in the package.
• khash and kbtree: my hash table and B-tree implementation. kbtree is based on JG_rbtree.

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Futher Discussion on Search Trees

Over the weekend, I have done a more comprehensive benchmark of various libraries on search trees. Two AVL, seven red-black tree, one Splay tree, two treap implementations are involved, together with seven hash table libraries. As I need to present a big table, I have to write it in a free-style HTML page. You can find the complete benchmark here and all the source codes here. I only copy the “concluding remarks” in the benchmark page as follows:

• Hash table is preferred over search trees if we do not require order.
• In applications similar to my example, B-tree is better than most of binary search trees in terms of both speed and memory.
• AVL tree and red-black tree are the best general-purposed BSTs. They are very close in efficiency.
• For pure C libraries, using macros is usually more efficient than using void* to achieve generic programming.

You can find the result and much more discussions in that page. If you think the source codes or the design of benchmark can be improved, please leave comments here or send me E-mail. In addition, I failed to use several libraries and so you can see some blank in the table. I would also appreciate if someone could show me how to use those libraries correctly.

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Thoughts on Generic Programming in C

I came across two interviews (here and here) of Alexander Stepanov, the father of STL. There are quite a lot of interesting bits. For example, he thinks C++ is the best programming language to realize his goal, but he is also strongly against OOP at the same time. In addition, he has paid a lot of efforts on efficiency, which we can see from STL. He said: “It is silly to abstract an algorithm in such a way that when you instantiate it back it becomes inefficient”. I like these two interviews because I think in the same way. The only exception is I do not use STL, although I think it is the best generic library and I like it a lot. But why?

Two reasons. Firstly, STL is written in C++, which makes it unavailable to all C projects. It is possible to only use STL and forget all the other features in C++, but people rarely do so. At least I have not seen such a project where STL is combined with procedural programming. In addition, C++ projects are usually less portable than C projects and STL makes it worse. It puts a lot of stress on C++ compilers. Even Stepanov agreeed, by the time of the interview, that “The unfortunate reality is that a lot of code in the present implementation of STL is suboptimal because of the compiler limitations and bugs of the compilers I had to use when I was developing STL”. Secondly, using STL also means much longer compiling time. I remembered I used to compile a customized Linux kernel for my old laptop in an hour. Probably I would spend more than a day to compile if it was written using C++/STL.

A generic container library would benefit a lot of C programmers, but so far I am not aware of any efficient implementation. Glib tries to achieve so, but it uses void* and this inevitably will incur overhead and complicate interfaces. And finally, I decide to write my own one. Ideally (but probably impractically) I want to achieve four goals: a) efficiency in speed and space; b) elegance in interface; c) independency between functinality and d) simplicity in codes. However, currently I am not competent enough to achieve all these goals and I am not a professional programmer at all (and so cannot invest enough time). As I said in my About page, I mainly do this to please myself.

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C Array vs. C++ Vector

Here is a piece of source codes that compare C arrays and C++ vectors. It tests six scenarios: a) preallocated C array; b) dynamically growing C array; c) dynamical C vector calling kv_a macro (in my kvec.h); d) dynamical C vector calling kv_push macro (in my kvec.h); e) preallocated C++ vector and f) dynamically growing C++ vector. You can find my kvec.h on my blog.

#include
#include
#include
#include
#include “kvec.h”

int main()
{
int M = 10, N = 20000000, i, j;
clock_t t;
t = clock();
for (i = 0; i < M; ++i) { int *array = (int*)malloc(N * sizeof(int)); for (j = 0; j < N; ++j) array[j] = j; free(array); } printf("C array, preallocated: %.3f sec\n", (float)(clock() - t) / CLOCKS_PER_SEC); t = clock(); for (i = 0; i < M; ++i) { int *array = 0, max = 0; for (j = 0; j < N; ++j) { if (j == max) { max = !max? 1 : max << 1; array = (int*)realloc(array, sizeof(int)*max); } array[j] = j; } free(array); } printf("C array, dynamic: %.3f sec\n", (float)(clock() - t) / CLOCKS_PER_SEC); t = clock(); for (i = 0; i < M; ++i) { kvec_t(int) array; kv_init(array); kv_resize(int, array, N); for (j = 0; j < N; ++j) kv_a(int, array, j) = j; kv_destroy(array); } printf("C vector, dynamic (kv_a): %.3f sec\n", (float)(clock() - t) / CLOCKS_PER_SEC); t = clock(); for (i = 0; i < M; ++i) { kvec_t(int) array; kv_init(array); for (j = 0; j < N; ++j) kv_push(int, array, j); kv_destroy(array); } printf("C vector, dynamic (kv_push): %.3f sec\n", (float)(clock() - t) / CLOCKS_PER_SEC); t = clock(); for (i = 0; i < M; ++i) { std::vector array;
array.reserve(N);
for (j = 0; j < N; ++j) array[j] = j; } printf("C++ vector, preallocated: %.3f sec\n", (float)(clock() - t) / CLOCKS_PER_SEC); t = clock(); for (i = 0; i < M; ++i) { std::vector array;
for (j = 0; j < N; ++j) array.push_back(j); } printf("C++ vector, dynamic: %.3f sec\n", (float)(clock() - t) / CLOCKS_PER_SEC); return 0; } [/sourcecode] Here is the result on two machines (compiled with g++ -O2 -fomit-frame-pointer -finline-functions):

type	MacIntel	LinuxIntel
C array, preallocated	1.589	1.180
C array, dynamic	2.064	1.340
C vector, dynamic (kv_a)	2.051	1.600
C vector, dynamic (kv_push)	1.932	1.250
C++ vector, preallocated	2.119	1.590
C++ vector, dynamic	5.095	3.770

Such result may vary with different machines/compilers, but not much.

Update: My example passed valgrind check on a Linux (Debian etch, g++-4.1), but Tom_ pointed out that it did not pass VC++’s debugger as vector::operator[] writes outside vector::size(). Anyway, it is good not to use operator[] in my way. You can replace reserve()+[] with resize()+[] or reserve()+push_back(). On my machine, the replacement gives a little bit slower speed, but it is safer/more portable in that way. Thanks for all the comments.

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Comparison of Hash Table Libraries

As a Perl programmer, I enjoy a lot using hash tables. I keep this habit in C/C++ programming. Then what C/C++ hash libraries are available? How are they compared to each other? In this post, I will give a brief review of hash libraries and present a small benchmark showing their practical performance.

Hash table libraries

In C++, the most widely used hash table implementation is hash_map/set in SGI STL, which is part of the GCC compiler. Note that hash_map/set is SGI’s extention to STL, but is not part of STL. TR1 (technical report 1) tries to standardize hash tables. It provides unordered_map/set with similar API to hash_map/set. Most of TR1 routines are available since gcc-4.0. Google sparse hash is another C++ hash table template library with similar API to hash_map/set. It provides two implementations, one is efficient in speed and the other is in memory.

In contrast, there are few good C libraries around. I have tried SunriseDD, uthash, glibc hash table, hashit, Christopher Clark’s hashtable, glib hash table and ghthash. SunriseDD sounds a great library that implements a lock-free hash table. However, I am not sure how to install it or use it, although the code itself is well documented. Uthash is a single header file. It is quite complex to use and incompatiable with C++. It also lacks basic APIs such as counting how many elements in the hash table. Glibc hash and hashit seem to only implement static hash tables. Glibc hash even does not have deletion operation. Only glib hash, CC’s hashtable and ghthash implement most of common operations. And they still have their weakness in comparison to C++ implementations (see below).

Design of the benchmark

The benchmark is comprised of two experiments. In the first experiment, a random integer array of 5 million elements is generated with about 1.25 million distinct keys. Each element is then tested whether it is present in the hash. If the element is in the hash, it will be removed; otherwise, it will be inserted. 625,792 distinct keys will be in the hash after this process. To test performance on string input, I convert integers to strings with sprintf().

The second experiment is designed by Craig Silverstein, the author of sparsehash. I am using his source codes. This experiment tests the performance of insertion from zero sized hash, insertion from preallocated hash, replacement, query, query of empty hash, and removal.

Results

The following table gives the results in the first experiment:

Library	Mac-intCPU (sec)	Mac-strCPU (sec)	Mac PeakMem (MB)	Linux-intCPU (sec)	Linux-strCPU (sec)	Linux PeakMem (MB)
glib	1.904	2.436	11.192	3.490	4.720	24.968
ghthash	2.593	2.869	29.0/39.0	3.260	3.460	61.232
CC’s hashtable	2.740	3.424	59.756	3.040	4.050	129.020
TR1	1.371	2.571	16.140	1.750	3.300	28.648
STL hash_set	1.631	2.698	14.592	2.070	3.430	25.764
google-sparse	2.957	6.098	4.800	2.560	6.930	5.42/8.54
google-dense	0.700	2.833	24.616	0.550	2.820	24.7/49.3
khash (C++)	1.089	2.372	6.772	1.100	2.900	6.88/13.1
khash (C)	0.987	2.294	6.780	1.140	2.940	6.91/13.1
STL set (RB)	5.898	12.978	19.868	7.840	18.620	29.388
kbtree (C)	3.080	13.413	3.268	4.260	17.620	4.86/9.59
NP’s splaytree	8.455	23.369	8.936	11.180	27.610	19.024

Notes:

• Please be aware that changing the size of input data may change the ranking of speed and memory. The speed of a library may vary up to 10% in two different runs.
• CPU time is measured in seconds. Memory denotes the peak memory, measured in MB.
• For string hash, only the pointer to a string is inserted. Memory in the table does not count the space used by strings.
• If two numbers are given for memory, the first is for integer keys and the second for string keys.
• For all C++ libraries and khash.h, one operation is needed to achieve “insert if absent; delete otherwise”. Glib and ghthash require two operations, which does not favour these two libraries.
• The speed may also be influenced by the efficiency of hash funtions. Khash and Glib use the same hash function. TR1/SGI-STL/google-hash use another hash function. Fortunately, to my experiment, the two string hash functions have quite similar performance and so the benchmark reflects the performance of the overall hash libraries instead of just hash functions.
• For glib and ghthash, what is inserted is the pointer to the integer instead of the integer itself.
• Ghthash supports dynamic hash table. However, the results do not seem correct when this is switched on. I am using fixed-size hash table. This favours ghthash.
• CC’s hashtable will force to free a key, which is not implemented in all the other libraries. This behaviour will add overhead on both speed and memory in my benchmark (but probably not in other applications). The memory is measured for integer keys.
• This simple benchmark does not test the strength and weakness of splay tree.

And here is the result of the second experiment:

Library	grow	pred/grow	replace	fetch	fetchnull	remove	Memory
TR1	194.2	183.9	30.7	15.6	15.2	83.4	224.6
STL hash_map	149.0	110.5	35.6	11.5	14.0	87.2	204.2
STL map	289.9	289.9	141.3	134.3	7.0	288.6	236.8
google-sparse	417.2	237.6	89.5	84.0	12.1	100.4	85.4
google-dense	108.4	39.4	17.8	8.3	2.8	18.0	256.0
khash (C++)	111.2	99.2	26.1	11.5	3.0	17.4	198.0

Notes:

• CPU time is measured in nanosecond for each operation. Memory is measured by TCmalloc. It is the memory difference before and after the allocation of the hash table, instead of the peak memory.
• In this experiment, integers are inserted in order and there are no collisions in the hash table.
• All these libraries provide similar API.

Discussions

• Speed and memory. The larger the hash table, the fewer collisions may occur and the faster the speed. For the same hash library, increasing memory always increases speed. When we compare two libraries, both speed and memory should be considered.
• C vs. C++. All C++ implementations have similar API. It is also very easy to use for any type of keys. Both C libraries, ghthash and glib, can only keep pointers to the keys, which complicates API and increases memory especially for 64-bit systems where a pointer takes 8 bytes. In general, C++ libraries is perferred over C ones. Surprisingly, on 32-bit Mac OS X, glib outperforms TR1 and STL for string input. This might indicate that the glib implementation itself is very efficient, but just the lack of functionality in C affects the performance.
• Generic programming in C. Except my khash.h, all the other C hash libraries use (void*) to achieve generic typing. Using void* is okey for strings, but will cause overhead for integers. This is why all C libraries, except khash.h, is slower than C++ libraries on integer keys, but close to on string keys.
• Open addressing vs. chaining hash. Khash and google hash implement open addressing hash while the remaining implement chaining hash. In open addressing hash, the size of each bucket equals the size of a key plus 0.25 byte. Google sparsehash further compresses unused bucket to 1 bit, achieving high memory efficiency. In chaining hash, the memory overhead of each bucket is at least 4 bytes on 32bit machines, or 8 bytes on 64bit machines. However, chaining hash is less affected when the hash table is nearly full. In practice, both open addressing and chaining hash occupy similar memory under similar speed. Khash takes less peak memory mainly due to its advanced technique in rehashing which reduces memory usage. So far as speed is concerned, chaining hash may have fewer comparison between keys. We can see this from the fact that the speed of chaining hash approaches that of open addressing hash on string keys but much slower on integer keys.
• Memory usage of search trees. B-tree is the winner here. Each element in the B-tree only needs one additional pointer. When there are enough elements, a B-tree is at least halfly full; on average it should be around 75% full. And so on 64-bit systems, for a B-tree with N elements, we need additional N*8/0.75=10N bytes memory. Splay tree will need N*8*2=16N extra space. RB tree is the worst.
• Other issues. a) Google hash becomes unbearably slow when I try to put a lot of strings in the hash table. All the other libraries do not have this problem. b) Google hash performs more comparisons than khash. This is obvious because google-dense is clearly faster on integer keys but comparable to khash on string keys.

Concluding remarks

• C++ hash library is much easier to use than C libraries. This is definitely where C++ is preferred over C.
• TR1 hash implementation is no faster than STL implementation. They may outperform one another under certain input or settings.
• SGI hash_map is faster and takes less memory than STL map. Unless ordering is important, hash_map is a better container than map.
• Google hash is a worthy choice when we understand why it is slow for many string keys.
• My khash library, which is a single-file C++ template header, achieves good balance between speed and memory. All my source codes are available at the Programs page.

Update

1. C interface can be elegant, too, if we implement it cleverly. See this post.
2. I realize that we just need one lookup to achieve “insert if absent; delete otherwise”. This further improves the speed for all C++ libraries.
3. I have analyzed google dense hash table in this post which explains why it is faster than khash on integer keys but close to or slower than on string keys.
4. This thread directed me to gcc hashtable, and cocom hashtable. They are more or less independent of other source codes, but it would still take time to separate the source codes. So, I have not benchmarked them. Just keep a record.
5. Python dictionary is in fact a hash table. The dictnotes.txt in that directory gives some quite interesting discussion about how to implement hash efficiently.
6. hashlib library. A bit hard to use and I cannot get it running correctly. Possibly I have not provided a proper second hash function for rehashing.
7. Added results for STL set (based on red-black tree) and John-Mark Gurney’s B-tree implementation (JG’s btree). Both libraries are considerably slower than hash tables. Of course search trees provide more functionality than hash tables, and every nice thing comes with a price. I have also tried Jason Evans’s and Niels Provos’ red-black tree implementations. On integer keys, JE’s takes 6.110 seconds on Mac-Intel using 18.884 MB memory and NP’s taks 6.611 seconds using the same amount of memory. This performance is close to that of STL set. They appear to be slower mainly due to the additional malloc/free calls I have to made under their APIs. Unlike hash table which have a variety of ways to implement it, red-black tree usually has one way (well, can be more. See also Jason’s blog.). And so I only show the performance of STL set as a representitive.
8. Replaced JG’s B-tree with a modified version. The new version is both faster and more light-weighted.

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Comparison of Internal Sorting Algorithms

Sorting algorithm

Given an array of size N, sorting can be done in O(N log(N)) in average. The most frequently used sorting algorithms that can achieve this time complexity are quicksort, heapsort and mergesort. They usually require O(log(N)), O(1) and O(N) working space, respectively (the space complexity of mergesort can be improved at the cost of speed). Most people believe quicksort is the fastest sorting algorithm. However, the fact is quicksort is only fast in terms of the number of swaps. When comparison is expensive, mergesort is faster than quicksort because mergesort uses less comparisons. GNU sort uses mergesort. Replacing it with a quicksort reduces the speed on typical text input. In addition, of the three algorithms, only mergesort is a stable sort. Stability is sometimes useful for a general tool like GNU sort.

The worst-case time complexity of quicksort is O(N^2). In practice, we combine quicksort and heapsort to avoid worst-case performance while retaining the fast average speed. The resulting algorithm is called introsort (introspective sort).

Implementation

The two most widely used implementations are glibc qsort and STL (unstable) introsort. Libc qsort calls a function for comparison. For simple comparison, a function call is expensive, which may greatly hurt the efficiency of qsort. STL sort does not have this problem. It is one of the fastest implementations I am aware of. My own implementation of introsort is similar but not as fast as STL introsort.

GNU sort implements a top-down recursive sort. On integer sorting, it is twice slower than introsort (see below). Iterative top-down mergesort is hard to implement. Iterative bottom-up mergesort is much easier. My implementation is a bottom-up one.

Paul Hsieh has also implemented quicksort, heapsort and mergesort. His implementation should be very efficient from what I can tell. To see whether my implementation is good enough, I copied and pasted his codes in my program, and applied “inline” where necessary.

Comparison

I designed a small benchmark on sorting 50 million random integers. As comparison is cheap in this case, the number of swaps dominate the performance. I compiled and run the program on three machines: MacIntel (Core2-2G/Mac/g++-4.2.1), LinuxIntel (Xeon-1.86G/Linux/g++-4.1.2) and LinuxAMD (Opteron-2G/Linux/g++-3.4.4). On all the three platforms, the program was compiled with “-O2 -fomit-frame-pointer”. The time (in seconds) spent on sorting is showed in the following table:

Algorithm	MacIntel	LinuxIntel	LinuxAMD	Linux_icc
STL sort	7.749	8.260	7.170	8.400
STL stable_sort	9.684	11.990	10.270	10.770
libc qsort	16.579	81.190	30.490	81.120
introsort	7.887	8.880	7.670	9.320
iterative mergesort	10.371	12.480	10.110	10.130
binary heapsort	36.651	45.710	42.460	40.820
combsort11	18.131	19.290	19.370	19.490
isort (func call)	16.760	17.380	13.390	16.740
isort (template func)	7.902	8.800	7.690	9.010
Paul’s heapsort	34.790	43.680	40.740	39.060
Paul’s quicksort	8.410	8.940	7.810	9.450
Paul’s mergesort	11.103	13.390	10.680	13.030

As for the algorithm itself, we can see that introsort is the fastest and heapsort is the slowest. Mergesort is also very fast. Combsort11 is claimed to approach quicksort, but I do not see this in sorting large integer arrays. As for the implementation of quicksort/introsort, STL is the best, with my implementation following very closely. Paul’s implmentation is also very efficient. Libc qsort is clearly slower, which cannot simply attribute to the use of function calls. My implementation with function calls, although slower than without function calls, outperforms libc qsort on both Linux machines. As for the implementation of mergesort, my version has similar performance to STL stable_sort. Note that stable_sort uses buffered recursive mergesort when a temporary array can be allocated. When memory is insufficient, it will use in-place mergesort which is not evaluated here.

Availability and alternative benchmarks

My implementation is available here as a single C++ template header file. The program for benchmark is also available. Programs in plain text can be acquired by chopping .html in the two links.

Paul Hsieh’s benchmark is here, including the original source codes. He also discussed how algorithms perform when the initial array is not completely random (I am one of “naive people” in his standard). Please note that in his benchmark, he was sorting an array of size 60,000 for 10000 times, while in my benchmark I more focus on very large arrays. Notably, heapsort approaches introsort on small arrays, but far slower on large arrays. Presumably this is because the bad cache performance of heapsort. Both quicksort and mergesort are very cache efficient.

In addition to Paul’s benchmark, you can also find alternative ones here and here. They seem to be more interested in the theoretical issues rather than efficient practical implementations.

If you search “benchmark sorting algorithms” in google, the first result is this page, which was implemented in D by Stewart Gordon. This benchmark aims to evaluate the performance on small arrays. It also tests the speed when the array is sorted or reverse sorted. However, the implementation is not optimized enough at least for quicksort. Using insertion sort when the array is nearly sorted is always preferred. You can also find this report from google search, but the implementation of quicksort is too naive to be efficient.

Concluding remarks

Although in the table introsort performs the best, we may want to use mergesort if we want to perform stable sorting, or the comparison is very expensive. Mergesort is also faster than introsort if the array is nearly sorted. STL sort seems to take particular care in this case, which makes it still fast when the array is sorted.

In common cases when comparison is cheap, introsort is the best choice. Of the various implementations, STL is the fastest. If you do not use STL or you just want to use C, you can use/adapt my implmentation which is very close to STL sort in speed. Do not use libc qsort, especially on Linux. It is not well implemented.

Update

1. This website gives severl good implementations of sorting algorithms. I also believe the programmer behind is very capable. Highly recommended.

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C++ Reduces Coding Time?

Just now I got an email from a mailing list, saying that C++ helps to greatly reduce coding time in comparison to C. I have heard a lot about this argument. But is that true?

C++ can possibly accelerate development in two ways: firstly, OOP (Object-Oriented Programming) helps to organize large projects, and secondly, STL (Standard Template Library) saves time on reimplementing frequently used subroutines. However, I do not find C++ OOP greatly helps me. To me, it is not right to clearly classify a programming language as a procedure-oriented or object-oriented language. It is only right to say a development methodology is procedure-oriented or object-oriented. We can effectively mimic the fundamental OOP ideas in C, a socalled procedure-oriented language, by packaging related data in a struct and transfer the a pointer to the struct to subroutines. I know C++ programmers would argue doing in this way is far from OOP, but it has captured the essence of OOP and in practice sufficient to organize large projects with this simple and natural idea. The large amount of existing C projects, such as Linux kernel, gcc and Emacs, prove this is the truth. With OOP ideas, we can use C to organize large projects without difficulty. C++ does not provide more power except introducing more complicated concepts.

I do not use STL most of time. I have implemented most of useful subroutines in C/C++ by myself. I actually spend less time in using my own library than using STL as I am very familiar with my own codes. Of course, implementing an efficient and yet generic library by myself takes a lot of time, but I really learn a lot in this invaluable process. I can hardly imagine how a programmer who does not get a firm grasp of data structures, which can only be achieved by implementing by him/herself, can ever write good programs. To this end, I agree that for elementary programmers using STL reduces coding time; but this is achieved at the cost of weakening the ability to write better programs. And for an advanced programmer, using STL may help but probably does not save much time.

Note that I am not saying C++ is a bad language as a whole. In fact, I use C++ template functions a lot and C++ template classes at times. In this post, I just want to emphasize the importantance to focusing on the art of programming instead of on the artificial concepts or on the degree of laziness a language can provide.

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Derivative-Free Optimization (DFO)

You can find good subroutines in GSL for multivariable nonlinear optimization, but the only method it provides for DFO is Nelder-Mead simplex method, which, claimed by Numerical Recipes, is “almost surely” slower than the Powell’s direct set method “in all likely applications”. You can find some DFO solvers here, but few of them are implemented in C/C++.

I have reimplemented a Hooke-Jeeve’s solver, adapted Powell’s direct set method from Numerical Recipes in C and ported NEWUOA solver from Fortran to C++ with f2c. Of the three solvers, Hooke-Jeeve’s method is the simplest. It simply searches the nearby regions around a given point and reduces radius in each iteration. It is purely heuristic and is hardly related to any “algorithm”. The Powell’s direct set method minimizes the objective function by minimizing along a direction using Brent’s method. It is believed to outperform Nelder-Mead method. NEWUOA is a much more advanced method. I do not know how it works, frankly. A benchmark shows that NEWUOA outperforms NMSMAX, an implementation of Nelder-Mead method, and APPSPACK as well. All the three solvers I implement work well on the problem I want to solve.

The three solvers are implemented as C++ template headers, one header file for each solver. The Hooke-Jeeve’s method is here and NEWUOA is here. I cannot release the source codes for Powell’s direct set as Numerical Recipes disallows this.

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C++ iostream can be Slooow…

A colleague of mine just told me that C++ iostream is typically an order of magnitude slower than printf. His example shows that printing out a string like “%s\t%d\tabc\t%s\t%s\n” with C++ iostream is 3 times slower than printf in Perl! This observation agrees my experience, although I have never done any benchmark. I abandoned iostream after I tried it the first time in my program.

Update: In another test, C++ iostream is ~30% slower, which is not too bad. Anyway, be aware that C++ iostream can be very slow in some cases. This thread also provides helpful information.

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