Why is processing a sorted array faster than processing an unsorted array

Question

Here is a piece of C   code that shows some very peculiar behavior  For some strange reason  sorting the data miraculously makes the code almost six times faster   include  lt algorithm gt   include  lt ctime gt   include  lt iostream gt   int main            Generate data     const unsigned arraySize   32768      int data arraySize        for  unsigned c   0  c  lt  arraySize    c          data c    std  rand     256              With this  the next loop runs faster      std  sort data  data   arraySize           Test     clock t start   clock        long long sum   0       for  unsigned i   0  i  lt  100000    i                   Primary loop         for  unsigned c   0  c  lt  arraySize    c                        if  data c   gt   128                  sum    data c                        double elapsedTime   static cast lt double gt  clock   - start    CLOCKS PER SEC       std  cout  lt  lt  elapsedTime  lt  lt  std  endl      std  cout  lt  lt   quot sum    quot   lt  lt  sum  lt  lt  std  endl      Without std  sort data  data   arraySize    the code runs in 11 54 seconds  With the sorted data  the code runs in 1 93 seconds    Initially  I thought this might be just a language or compiler anomaly  so I tried Java  import java util Arrays  import java util Random   public class Main       public static void main String   args                   Generate data         int arraySize   32768          int data     new int arraySize            Random rnd   new Random 0           for  int c   0  c  lt  arraySize    c          data c    rnd nextInt     256                  With this  the next loop runs faster         Arrays sort data               Test         long start   System nanoTime            long sum   0           for  int i   0  i  lt  100000    i                           Primary loop             for  int c   0  c  lt  arraySize    c                                if  data c   gt   128                      sum    data c                                    System out println  System nanoTime   - start    1000000000 0           System out println  quot sum    quot    sum            With a similar but less extreme result   My first thought was that sorting brings the data into the cache  but then I thought how silly that was because the array was just generated   What is going on  Why is processing a sorted array faster than processing an unsorted array   The code is summing up some independent terms  so the order should not matter

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Frequently used Boolean operations in C   produce many branches in the compiled program  If these branches are inside loops and are hard to predict they can slow down execution significantly  Boolean variables are stored as 8-bit integers with the value 0 for false and 1 for true   Boolean variables are overdetermined in the sense that all operators that have Boolean variables as input check if the inputs have any other value than 0 or 1  but operators that have Booleans as output can produce no other value than 0 or 1  This makes operations with Boolean variables as input less efficient than necessary  Consider example   bool a  b  c  d  c   a  amp  amp  b  d   a    b    This is typically implemented by the compiler in the following way   bool a  b  c  d  if  a    0        if  b    0            c   1            else           goto CFALSE          else       CFALSE      c   0    if  a    0        if  b    0            d   0            else           goto DTRUE          else       DTRUE      d   1      This code is far from optimal  The branches may take a long time in case of mispredictions  The Boolean operations can be made much more efficient if it is known with certainty that the operands have no other values than 0 and 1  The reason why the compiler does not make such an assumption is that the variables might have other values if they are uninitialized or come from unknown sources  The above code can be optimized if a and b has been initialized to valid values or if they come from operators that produce Boolean output  The optimized code looks like this   char a   0  b   1  c  d  c   a  amp  b  d   a   b    char is used instead of bool in order to make it possible to use the bitwise operators   amp  and    instead of the Boolean operators   amp  amp  and      The bitwise operators are single instructions that take only one clock cycle  The OR operator     works even if a and b have other values than 0 or 1  The AND operator   amp   and the EXCLUSIVE OR operator     may give inconsistent results if the operands have other values than 0 and 1     can not be used for NOT  Instead  you can make a Boolean NOT on a variable which is known to be 0 or 1 by XOR ing it with 1   bool a  b  b    a    can be optimized to   char a   0  b  b   a   1    a  amp  amp  b cannot be replaced with a  amp  b if b is an expression that should not be evaluated if a is false    amp  amp  will not evaluate b   amp  will   Likewise  a    b can not be replaced with a   b if b is an expression that should not be evaluated if a is true   Using bitwise operators is more advantageous if the operands are variables than if the operands are comparisons   bool a  double x  y  z  a   x  gt  y  amp  amp  z  lt  5 0    is optimal in most cases  unless you expect the  amp  amp  expression to generate many branch mispredictions

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As data is distributed between 0 and 255 when the array is sorted  around the first half of the iterations will not enter the if-statement  the if statement is shared below    if  data c   gt   128      sum    data c     The question is  What makes the above statement not execute in certain cases as in case of sorted data  Here comes the  branch predictor   A branch predictor is a digital circuit that tries to guess which way a branch  e g  an if-then-else structure  will go before this is known for sure  The purpose of the branch predictor is to improve the flow in the instruction pipeline  Branch predictors play a critical role in achieving high effective performance   Let s do some bench marking to understand it better  The performance of an if-statement depends on whether its condition has a predictable pattern  If the condition is always true or always false  the branch prediction logic in the processor will pick up the pattern  On the other hand  if the pattern is unpredictable  the if-statement will be much more expensive   Let   s measure the performance of this loop with different conditions   for  int i   0  i  lt  max  i        if  condition          sum      Here are the timings of the loop with different true-false patterns   Condition                Pattern             Time  ms  -------------------------------------------------------  i  amp  0  80000000     0    T repeated          322   i  amp  0xffffffff     0    F repeated          276   i  amp  1     0             TF alternating      760   i  amp  3     0             TFFFTFFF              513   i  amp  2     0             TTFFTTFF              1675   i  amp  4     0             TTTTFFFFTTTTFFFF      1275   i  amp  8     0             8T 8F 8T 8F           752   i  amp  16     0            16T 16F 16T 16F       490   A    bad    true-false pattern can make an if-statement up to six times slower than a    good    pattern  Of course  which pattern is good and which is bad depends on the exact instructions generated by the compiler and on the specific processor   So there is no doubt about the impact of branch prediction on performance

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Branch prediction  With a sorted array  the condition data c   gt   128 is first false for a streak of values  then becomes true for all later values  That s easy to predict  With an unsorted array  you pay for the branching cost

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This question is rooted in branch prediction models on CPUs  I d recommend reading this paper  Increasing the Instruction Fetch Rate via Multiple Branch Prediction and a Branch Address Cache When you have sorted elements  the IR can not be bothered to fetch all CPU instructions  again and again  It fetches them from the cache

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Bjarne Stroustrup s Answer to this question  That sounds like an interview question  Is it true  How would you know  It is a bad idea to answer questions about efficiency without first doing some measurements  so it is important to know how to measure  So  I tried with a vector of a million integers and got  Already sorted    32995 milliseconds Shuffled          125944 milliseconds  Already sorted    18610 milliseconds Shuffled          133304 milliseconds  Already sorted    17942 milliseconds Shuffled          107858 milliseconds  I ran that a few times to be sure  Yes  the phenomenon is real  My key code was  void run vector lt int gt  amp  v  const string amp  label        auto t0   system clock  now        sort v begin    v end         auto t1   system clock  now        cout  lt  lt  label           lt  lt  duration cast lt microseconds gt  t1     t0  count             lt  lt   quot  milliseconds n quot      void tst         vector lt int gt  v 1 000 000       iota v begin    v end    0       run v   quot already sorted  quot        std  shuffle v begin    v end    std  mt19937  std  random device             run v   quot shuffled     quot       At least the phenomenon is real with this compiler  standard library  and optimizer settings  Different implementations can and do give different answers  In fact  someone did do a more systematic study  a quick web search will find it  and most implementations show that effect  One reason is branch prediction  the key operation in the sort algorithm is    if v i   lt  pivot          or equivalent  For a sorted sequence that test is always true whereas  for a random sequence  the branch chosen varies randomly  Another reason is that when the vector is already sorted  we never need to move elements to their correct position  The effect of these little details is the factor of five or six that we saw  Quicksort  and sorting in general  is a complex study that has attracted some of the greatest minds of computer science  A good sort function is a result of both choosing a good algorithm and paying attention to hardware performance in its implementation  If you want to write efficient code  you need to know a bit about machine architecture

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Sorted arrays are processed faster than an unsorted array  due to a phenomena called branch prediction  The branch predictor is a digital circuit  in computer architecture  trying to predict which way a branch will go  improving the flow in the instruction pipeline  The circuit computer predicts the next step and executes it  Making a wrong prediction leads to going back to the previous step  and executing with another prediction  Assuming the prediction is correct  the code will continue to the next step  A wrong prediction results in repeating the same step  until a correct prediction occurs  The answer to your question is very simple  In an unsorted array  the computer makes multiple predictions  leading to an increased chance of errors  Whereas  in a sorted array  the computer makes fewer predictions  reducing the chance of errors  Making more predictions requires more time  Sorted Array  Straight Road                                                                                      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT  Unsorted Array  Curved Road                               Branch prediction  Guessing predicting which road is straight and following it without checking                                             Straight road                                             Longer road  Although both the roads reach the same destination  the straight road is shorter  and the other is longer  If then you choose the other by mistake  there is no turning back  and so you will waste some extra time if you choose the longer road  This is similar to what happens in the computer  and I hope this helped you understand better   Also I want to cite  Simon Weaver from the comments   It doesn   t make fewer predictions - it makes fewer incorrect predictions  It still has to predict for each time through the loop

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One way to avoid branch prediction errors is to build a lookup table  and index it using the data   Stefan de Bruijn discussed that in his answer  But in this case  we know values are in the range  0  255  and we only care about values  gt   128   That means we can easily extract a single bit that will tell us whether we want a value or not  by shifting the data to the right 7 bits  we are left with a 0 bit or a 1 bit  and we only want to add the value when we have a 1 bit   Let s call this bit the  quot decision bit quot   By using the 0 1 value of the decision bit as an index into an array  we can make code that will be equally fast whether the data is sorted or not sorted   Our code will always add a value  but when the decision bit is 0  we will add the value somewhere we don t care about   Here s the code     Test clock t start   clock    long long a      0  0   long long sum   for  unsigned i   0  i  lt  100000    i           Primary loop     for  unsigned c   0  c  lt  arraySize    c                int j    data c   gt  gt  7           a j     data c            double elapsedTime   static cast lt double gt  clock   - start    CLOCKS PER SEC  sum   a 1    This code wastes half of the adds but never has a branch prediction failure   It s tremendously faster on random data than the version with an actual if statement  But in my testing  an explicit lookup table was slightly faster than this  probably because indexing into a lookup table was slightly faster than bit shifting   This shows how my code sets up and uses the lookup table  unimaginatively called lut for  quot LookUp Table quot  in the code    Here s the C   code     Declare and then fill in the lookup table int lut 256   for  unsigned c   0  c  lt  256    c      lut c     c  gt   128    c   0      Use the lookup table after it is built for  unsigned i   0  i  lt  100000    i           Primary loop     for  unsigned c   0  c  lt  arraySize    c                sum    lut data c             In this case  the lookup table was only 256 bytes  so it fits nicely in a cache and all was fast   This technique wouldn t work well if the data was 24-bit values and we only wanted half of them    the lookup table would be far too big to be practical   On the other hand  we can combine the two techniques shown above  first shift the bits over  then index a lookup table   For a 24-bit value that we only want the top half value  we could potentially shift the data right by 12 bits  and be left with a 12-bit value for a table index   A 12-bit table index implies a table of 4096 values  which might be practical  The technique of indexing into an array  instead of using an if statement  can be used for deciding which pointer to use   I saw a library that implemented binary trees  and instead of having two named pointers  pLeft and pRight or whatever  had a length-2 array of pointers and used the  quot decision bit quot  technique to decide which one to follow   For example  instead of  if  x  lt  node- gt value      node   node- gt pLeft  else     node   node- gt pRight   this library would do something like  i    x  lt  node- gt value   node   node- gt link i    Here s a link to this code  Red Black Trees  Eternally Confuzzled

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One way to avoid branch prediction errors is to build a lookup table  and index it using the data  Stefan de Bruijn discussed that in his answer  But in this case  we know values are in the range  0  255  and we only care about values  gt   128  That means we can easily extract a single bit that will tell us whether we want a value or not  by shifting the data to the right 7 bits  we are left with a 0 bit or a 1 bit  and we only want to add the value when we have a 1 bit  Let s call this bit the  quot decision bit quot   By using the 0 1 value of the decision bit as an index into an array  we can make code that will be equally fast whether the data is sorted or not sorted  Our code will always add a value  but when the decision bit is 0  we will add the value somewhere we don t care about  Here s the code     Test clock t start   clock    long long a      0  0   long long sum   for  unsigned i   0  i  lt  100000    i           Primary loop     for  unsigned c   0  c  lt  arraySize    c                int j    data c   gt  gt  7           a j     data c            double elapsedTime   static cast lt double gt  clock   - start    CLOCKS PER SEC  sum   a 1    This code wastes half of the adds but never has a branch prediction failure  It s tremendously faster on random data than the version with an actual if statement  But in my testing  an explicit lookup table was slightly faster than this  probably because indexing into a lookup table was slightly faster than bit shifting  This shows how my code sets up and uses the lookup table  unimaginatively called lut for  quot LookUp Table quot  in the code   Here s the C   code     Declare and then fill in the lookup table int lut 256   for  unsigned c   0  c  lt  256    c      lut c     c  gt   128    c   0      Use the lookup table after it is built for  unsigned i   0  i  lt  100000    i           Primary loop     for  unsigned c   0  c  lt  arraySize    c                sum    lut data c             In this case  the lookup table was only 256 bytes  so it fits nicely in a cache and all was fast  This technique wouldn t work well if the data was 24-bit values and we only wanted half of them    the lookup table would be far too big to be practical  On the other hand  we can combine the two techniques shown above  first shift the bits over  then index a lookup table  For a 24-bit value that we only want the top half value  we could potentially shift the data right by 12 bits  and be left with a 12-bit value for a table index  A 12-bit table index implies a table of 4096 values  which might be practical  The technique of indexing into an array  instead of using an if statement  can be used for deciding which pointer to use  I saw a library that implemented binary trees  and instead of having two named pointers  pLeft and pRight or whatever  had a length-2 array of pointers and used the  quot decision bit quot  technique to decide which one to follow  For example  instead of  if  x  lt  node- gt value      node   node- gt pLeft  else     node   node- gt pRight  this library would do something like   i    x  lt  node- gt value   node   node- gt link i    It s a nice solution and maybe it will work

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That s for sure     Branch prediction makes the logic run slower  because of the switching which happens in your code  It s like you are going a straight street or a street with a lot of turnings  for sure the straight one is going to be done quicker     If the array is sorted  your condition is false at the first step  data c   gt   128  then becomes a true value for the whole way to the end of the street  That s how you get to the end of the logic faster  On the other hand  using an unsorted array  you need a lot of turning and processing which make your code run slower for sure    Look at the image I created for you below  Which street is going to be finished faster   So programmatically  branch prediction causes the process to be slower    Also at the end  it s good to know we have two kinds of branch predictions that each is going to affect your code differently  1  Static 2  Dynamic   Static branch prediction is used by the microprocessor the first time a conditional branch is encountered  and dynamic branch prediction is used for succeeding executions of the conditional branch code  In order to effectively write your code to take advantage of these rules  when writing if-else or switch statements  check the most common cases first and work progressively down to the least common  Loops do not necessarily require any special ordering of code for static branch prediction  as only the condition of the loop iterator is normally used

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As what has already been mentioned by others  what behind the mystery is Branch Predictor    I m not trying to add something but explaining the concept in another way   There is a concise introduction on the wiki which contains text and diagram  I do like the explanation below which uses a diagram to elaborate the Branch Predictor intuitively      In computer architecture  a branch predictor is a   digital circuit that tries to guess which way a branch  e g  an   if-then-else structure  will go before this is known for sure  The   purpose of the branch predictor is to improve the flow in the   instruction pipeline  Branch predictors play a critical role in   achieving high effective performance in many modern pipelined   microprocessor architectures such as x86       Two-way branching is usually implemented with a conditional jump   instruction  A conditional jump can either be  not taken  and continue   execution with the first branch of code which follows immediately   after the conditional jump  or it can be  taken  and jump to a   different place in program memory where the second branch of code is   stored  It is not known for certain whether a conditional jump will be   taken or not taken until the condition has been calculated and the   conditional jump has passed the execution stage in the instruction   pipeline  see fig  1       Based on the described scenario  I have written an animation demo to show how instructions are executed in a pipeline in different situations    Without the Branch Predictor        Without branch prediction  the processor would have to wait until the   conditional jump instruction has passed the execute stage before the   next instruction can enter the fetch stage in the pipeline    The example contains three instructions and the first one is a conditional jump instruction  The latter two instructions can go into the pipeline until the conditional jump instruction is executed      It will take 9 clock cycles for 3 instructions to be completed    Use Branch Predictor and don t take a conditional jump  Let s assume that the predict is not taking the conditional jump      It will take 7 clock cycles for 3 instructions to be completed    Use Branch Predictor and take a conditional jump  Let s assume that the predict is not taking the conditional jump      It will take 9 clock cycles for 3 instructions to be completed      The time that is wasted in case of a branch misprediction is equal to   the number of stages in the pipeline from the fetch stage to the   execute stage  Modern microprocessors tend to have quite long   pipelines so that the misprediction delay is between 10 and 20 clock   cycles  As a result  making a pipeline longer increases the need for a   more advanced branch predictor    As you can see  it seems we don t have a reason not to use Branch Predictor   It s quite a simple demo that clarifies the very basic part of Branch Predictor  If those gifs are annoying  please feel free to remove them from the answer and visitors can also get the live demo source code from BranchPredictorDemo

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Besides the fact that the branch prediction may slow you down  a sorted array has another advantage      You can have a stop condition instead of just checking the value  this way you only loop over the relevant data  and ignore the rest  The branch prediction will miss only once       sort backwards  higher values first   may be in some other part of the code  std  sort data  data   arraySize  std  greater lt int gt        for  unsigned c   0  c  lt  arraySize    c           if  data c   lt  128                  break                  sum    data c

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The above behavior is happening because of Branch prediction   To understand branch prediction one must first understand Instruction Pipeline   Any instruction is broken into a sequence of steps so that different steps can be executed concurrently in parallel  This technique is known as instruction pipeline and this is used to increase throughput in modern processors  To understand this better please see this example on Wikipedia   Generally  modern processors have quite long pipelines  but for ease let s consider these 4 steps only        IF -- Fetch the instruction from memory    ID -- Decode the instruction   EX -- Execute the instruction    WB -- Write back to CPU register   4-stage pipeline in general for 2 instructions    Moving back to the above question let s consider the following instructions                           A  if  data c   gt   128                                                                                                                                      true          false                                                                                                                                                                                 B  sum    data c            C  for loop or print      Without branch prediction  the following would occur   To execute instruction B or instruction C the processor will have to wait till the instruction A doesn t reach till EX stage in the pipeline  as the decision to go to instruction B or instruction C depends on the result of instruction A  So the pipeline will look like this   when if condition returns true    When if condition returns false    As a result of waiting for the result of instruction A  the total CPU cycles spent in the above case  without branch prediction  for both true and false  is 7   So what is branch prediction   Branch predictor will try to guess which way a branch  an if-then-else structure  will go before this is known for sure  It will not wait for the instruction A to reach the EX stage of the pipeline  but it will guess the decision and go to that instruction  B or C in case of our example    In case of a correct guess  the pipeline looks something like this    If it is later detected that the guess was wrong then the partially executed instructions are discarded and the pipeline starts over with the correct branch  incurring a delay   The time that is wasted in case of a branch misprediction is equal to the number of stages in the pipeline from the fetch stage to the execute stage  Modern microprocessors tend to have quite long pipelines so that the misprediction delay is between 10 and 20 clock cycles  The longer the pipeline the greater the need for a good branch predictor   In the OP s code  the first time when the conditional  the branch predictor does not have any information to base up prediction  so the first time it will randomly choose the next instruction  Later in the for loop  it can base the prediction on the history   For an array sorted in ascending order  there are three possibilities     All the elements are less than 128  All the elements are greater than 128  Some starting new elements are less than 128 and later it become greater than 128   Let us assume that the predictor will always assume the true branch on the first run   So in the first case  it will always take the true branch since historically all its predictions are correct  In the 2nd case  initially it will predict wrong  but after a few iterations  it will predict correctly  In the 3rd case  it will initially predict correctly till the elements are less than 128  After which it will fail for some time and the correct itself when it sees branch prediction failure in history    In all these cases the failure will be too less in number and as a result  only a few times it will need to discard the partially executed instructions and start over with the correct branch  resulting in fewer CPU cycles    But in case of a random unsorted array  the prediction will need to discard the partially executed instructions and start over with the correct branch most of the time and result in more CPU cycles compared to the sorted array

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I just read up on this question and its answers  and I feel an answer is missing   A common way to eliminate branch prediction that I ve found to work particularly good in managed languages is a table lookup instead of using a branch  although I haven t tested it in this case    This approach works in general if    it s a small table and is likely to be cached in the processor  and you are running things in a quite tight loop and or the processor can preload the data    Background and why  From a processor perspective  your memory is slow  To compensate for the difference in speed  a couple of caches are built into your processor  L1 L2 cache   So imagine that you re doing your nice calculations and figure out that you need a piece of memory  The processor will get its  load  operation and loads the piece of memory into cache -- and then uses the cache to do the rest of the calculations  Because memory is relatively slow  this  load  will slow down your program   Like branch prediction  this was optimized in the Pentium processors  the processor predicts that it needs to load a piece of data and attempts to load that into the cache before the operation actually hits the cache  As we ve already seen  branch prediction sometimes goes horribly wrong -- in the worst case scenario you need to go back and actually wait for a memory load  which will take forever  in other words  failing branch prediction is bad  a memory load after a branch prediction fail is just horrible     Fortunately for us  if the memory access pattern is predictable  the processor will load it in its fast cache and all is well   The first thing we need to know is what is small  While smaller is generally better  a rule of thumb is to stick to lookup tables that are  lt   4096 bytes in size  As an upper limit  if your lookup table is larger than 64K it s probably worth reconsidering   Constructing a table  So we ve figured out that we can create a small table  Next thing to do is get a lookup function in place  Lookup functions are usually small functions that use a couple of basic integer operations  and  or  xor  shift  add  remove and perhaps multiply   You want to have your input translated by the lookup function to some kind of  unique key  in your table  which then simply gives you the answer of all the work you wanted it to do   In this case     128 means we can keep the value   lt  128 means we get rid of it  The easiest way to do that is by using an  AND   if we keep it  we AND it with 7FFFFFFF  if we want to get rid of it  we AND it with 0  Notice also that 128 is a power of 2 -- so we can go ahead and make a table of 32768 128 integers and fill it with one zero and a lot of 7FFFFFFFF s   Managed languages  You might wonder why this works well in managed languages  After all  managed languages check the boundaries of the arrays with a branch to ensure you don t mess up     Well  not exactly     -   There has been quite some work on eliminating this branch for managed languages  For example     for  int i   0  i  lt  array Length    i          Use array i      In this case  it s obvious to the compiler that the boundary condition will never be hit  At least the Microsoft JIT compiler  but I expect Java does similar things  will notice this and remove the check altogether  WOW  that means no branch  Similarly  it will deal with other obvious cases   If you run into trouble with lookups in managed languages -- the key is to add a  amp  0x something FFF to your lookup function to make the boundary check predictable -- and watch it going faster   The result of this case     Generate data int arraySize   32768  int   data   new int arraySize    Random random   new Random 0   for  int c   0  c  lt  arraySize    c        data c    random Next 256        To keep the spirit of the code intact  I ll make a separate lookup table  I assume we cannot modify  data  or the number of loops     int   lookup   new int 256    for  int c   0  c  lt  256    c        lookup c     c  gt   128    c   0        Test DateTime startTime   System DateTime Now  long sum   0   for  int i   0  i  lt  100000    i           Primary loop     for  int j   0  j  lt  arraySize    j                   Here you basically want to use simple operations - so no         random branches  but things like  amp         -     etc  are fine             sum    lookup data j             DateTime endTime   System DateTime Now  Console WriteLine endTime - startTime   Console WriteLine  sum       sum   Console ReadLine

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You are a victim of branch prediction fail   What is Branch Prediction  Consider a railroad junction   Image by Mecanismo  via Wikimedia Commons  Used under the CC-By-SA 3 0 license  Now for the sake of argument  suppose this is back in the 1800s - before long distance or radio communication  You are the operator of a junction and you hear a train coming  You have no idea which way it is supposed to go  You stop the train to ask the driver which direction they want  And then you set the switch appropriately  Trains are heavy and have a lot of inertia  So they take forever to start up and slow down  Is there a better way  You guess which direction the train will go   If you guessed right  it continues on  If you guessed wrong  the captain will stop  back up  and yell at you to flip the switch  Then it can restart down the other path   If you guess right every time  the train will never have to stop  If you guess wrong too often  the train will spend a lot of time stopping  backing up  and restarting   Consider an if-statement  At the processor level  it is a branch instruction   You are a processor and you see a branch  You have no idea which way it will go  What do you do  You halt execution and wait until the previous instructions are complete  Then you continue down the correct path  Modern processors are complicated and have long pipelines  So they take forever to  quot warm up quot  and  quot slow down quot   Is there a better way  You guess which direction the branch will go   If you guessed right  you continue executing  If you guessed wrong  you need to flush the pipeline and roll back to the branch  Then you can restart down the other path   If you guess right every time  the execution will never have to stop  If you guess wrong too often  you spend a lot of time stalling  rolling back  and restarting   This is branch prediction  I admit it s not the best analogy since the train could just signal the direction with a flag  But in computers  the processor doesn t know which direction a branch will go until the last moment  So how would you strategically guess to minimize the number of times that the train must back up and go down the other path  You look at the past history  If the train goes left 99  of the time  then you guess left  If it alternates  then you alternate your guesses  If it goes one way every three times  you guess the same    In other words  you try to identify a pattern and follow it  This is more or less how branch predictors work  Most applications have well-behaved branches  So modern branch predictors will typically achieve  gt 90  hit rates  But when faced with unpredictable branches with no recognizable patterns  branch predictors are virtually useless  Further reading   quot Branch predictor quot  article on Wikipedia   As hinted from above  the culprit is this if-statement  if  data c   gt   128      sum    data c    Notice that the data is evenly distributed between 0 and 255  When the data is sorted  roughly the first half of the iterations will not enter the if-statement  After that  they will all enter the if-statement  This is very friendly to the branch predictor since the branch consecutively goes the same direction many times  Even a simple saturating counter will correctly predict the branch except for the few iterations after it switches direction  Quick visualization  T   branch taken N   branch not taken  data     0  1  2  3  4      126  127  128  129  130      250  251  252      branch   N  N  N  N  N        N    N    T    T    T        T    T    T                NNNNNNNNNNNN     NNNNNNNTTTTTTTTT     TTTTTTTTTT   easy to predict   However  when the data is completely random  the branch predictor is rendered useless  because it can t predict random data  Thus there will probably be around 50  misprediction  no better than random guessing   data     226  185  125  158  198  144  217  79  202  118   14  150  177  182      branch     T    T    N    T    T    T    T   N    T    N    N    T    T    T                TTNTTTTNTNNTTT        completely random - impossible to predict    So what can be done  If the compiler isn t able to optimize the branch into a conditional move  you can try some hacks if you are willing to sacrifice readability for performance  Replace  if  data c   gt   128      sum    data c    with  int t    data c  - 128   gt  gt  31  sum     t  amp  data c    This eliminates the branch and replaces it with some bitwise operations   Note that this hack is not strictly equivalent to the original if-statement  But in this case  it s valid for all the input values of data     Benchmarks  Core i7 920   3 5 GHz C   - Visual Studio 2010 - x64 Release     Scenario Time  seconds      Branching - Random data 11 777   Branching - Sorted data 2 352   Branchless - Random data 2 564   Branchless - Sorted data 2 587     Java - NetBeans 7 1 1 JDK 7 - x64     Scenario Time  seconds      Branching - Random data 10 93293813   Branching - Sorted data 5 643797077   Branchless - Random data 3 113581453   Branchless - Sorted data 3 186068823     Observations   With the Branch  There is a huge difference between the sorted and unsorted data  With the Hack  There is no difference between sorted and unsorted data  In the C   case  the hack is actually a tad slower than with the branch when the data is sorted   A general rule of thumb is to avoid data-dependent branching in critical loops  such as in this example    Update   GCC 4 6 1 with -O3 or -ftree-vectorize on x64 is able to generate a conditional move  So there is no difference between the sorted and unsorted data - both are fast   Or somewhat fast  for the already-sorted case  cmov can be slower especially if GCC puts it on the critical path instead of just add  especially on Intel before Broadwell where cmov has 2 cycle latency  gcc optimization flag -O3 makes code slower than -O2   VC   2010 is unable to generate conditional moves for this branch even under  Ox   Intel C   Compiler  ICC  11 does something miraculous  It interchanges the two loops  thereby hoisting the unpredictable branch to the outer loop  So not only is it immune to the mispredictions  it is also twice as fast as whatever VC   and GCC can generate  In other words  ICC took advantage of the test-loop to defeat the benchmark     If you give the Intel compiler the branchless code  it just out-right vectorizes it    and is just as fast as with the branch  with the loop interchange     This goes to show that even mature modern compilers can vary wildly in their ability to optimize code

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In the same line  I think this was not highlighted by any answer  it s good to mention that sometimes  specially in software where the performance matters   like in the Linux kernel  you can find some if statements like the following   if  likely  everything is ok             Do something        or similarly   if  unlikely very improbable condition            Do something            Both likely   and unlikely   are in fact macros that are defined by using something like the GCC s   builtin expect to help the compiler insert prediction code to favour the condition taking into account the information provided by the user  GCC supports other builtins that could change the behavior of the running program or emit low level instructions like clearing the cache  etc  See this documentation that goes through the available GCC s builtins   Normally this kind of optimizations are mainly found in hard-real time applications or embedded systems where execution time matters and it s critical  For example  if you are checking for some error condition that only happens 1 10000000 times  then why not inform the compiler about this  This way  by default  the branch prediction would assume that the condition is false

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The reason why performance improves drastically when the data is sorted is that the branch prediction penalty is removed  as explained beautifully in Mysticial s answer  Now  if we look at the code if  data c   gt   128      sum    data c    we can find that the meaning of this particular if    else    branch is to add something when a condition is satisfied  This type of branch can be easily transformed into a conditional move statement  which would be compiled into a conditional move instruction  cmovl  in an x86 system  The branch and thus the potential branch prediction penalty is removed  In C  thus C    the statement  which would compile directly  without any optimization  into the conditional move instruction in x86  is the ternary operator                  So we rewrite the above statement into an equivalent one  sum    data c   gt  128   data c    0   While maintaining readability  we can check the speedup factor  On an Intel Core i7-2600K   3 4  GHz and Visual Studio 2010 Release Mode  the benchmark is  x86     Scenario Time  seconds      Branching - Random data 8 885   Branching - Sorted data 1 528   Branchless - Random data 3 716   Branchless - Sorted data 3 71     x64     Scenario Time  seconds      Branching - Random data 11 302   Branching - Sorted data 1 830   Branchless - Random data 2 736   Branchless - Sorted data 2 737     The result is robust in multiple tests  We get a great speedup when the branch result is unpredictable  but we suffer a little bit when it is predictable  In fact  when using a conditional move  the performance is the same regardless of the data pattern  Now let s look more closely by investigating the x86 assembly they generate  For simplicity  we use two functions max1 and max2  max1 uses the conditional branch if    else      int max1 int a  int b        if  a  gt  b          return a      else         return b     max2 uses the ternary operator                  int max2 int a  int b        return a  gt  b   a   b     On a x86-64 machine  GCC -S generates the assembly below   max1     movl     edi  -4  rbp      movl     esi  -8  rbp      movl    -4  rbp    eax     cmpl    -8  rbp    eax     jle      L2     movl    -4  rbp    eax     movl     eax  -12  rbp      jmp      L4  L2      movl    -8  rbp    eax     movl     eax  -12  rbp   L4      movl    -12  rbp    eax     leave     ret   max2     movl     edi  -4  rbp      movl     esi  -8  rbp      movl    -4  rbp    eax     cmpl     eax  -8  rbp      cmovge  -8  rbp    eax     leave     ret  max2 uses much less code due to the usage of instruction cmovge  But the real gain is that max2 does not involve branch jumps  jmp  which would have a significant performance penalty if the predicted result is not right  So why does a conditional move perform better  In a typical x86 processor  the execution of an instruction is divided into several stages  Roughly  we have different hardware to deal with different stages  So we do not have to wait for one instruction to finish to start a new one  This is called pipelining  In a branch case  the following instruction is determined by the preceding one  so we cannot do pipelining  We have to either wait or predict  In a conditional move case  the execution conditional move instruction is divided into several stages  but the earlier stages like Fetch and Decode do not depend on the result of the previous instruction  only latter stages need the result  Thus  we wait a fraction of one instruction s execution time  This is why the conditional move version is slower than the branch when the prediction is easy  The book Computer Systems  A Programmer s Perspective  second edition explains this in detail  You can check Section 3 6 6 for Conditional Move Instructions  entire Chapter 4 for Processor Architecture  and Section 5 11 2 for special treatment for Branch Prediction and Misprediction Penalties  Sometimes  some modern compilers can optimize our code to assembly with better performance  sometimes some compilers can t  the code in question is using Visual Studio s native compiler   Knowing the performance difference between a branch and a conditional move when unpredictable can help us write code with better performance when the scenario gets so complex that the compiler can not optimize them automatically

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I tried the same code with MATLAB 2011b with my MacBook Pro  Intel i7  64 bit  2 4 GHz  for the following MATLAB code     Processing time with Sorted data vs unsorted data                                                                               Generate data arraySize   32768 sum   0    Generate random integer data from range 0 to 255 data   randi 256  arraySize  1      Sort the data data1  sort data     data1  data  when no sorting done    Start a stopwatch timer to measure the execution time tic   for i 1 100000      for j 1 arraySize          if data1 j  gt  128             sum sum   data1 j           end     end end  toc   ExeTimeWithSorting   toc - tic    The results for the above MATLAB code are as follows     a  Elapsed time  without sorting    3479 880861 seconds    b  Elapsed time  with sorting     2377 873098 seconds    The results of the C code as in  GManNickG I get     a  Elapsed time  without sorting    19 8761 sec    b  Elapsed time  with sorting     7 37778 sec    Based on this  it looks MATLAB is almost 175 times slower than the C implementation without sorting and 350 times slower with sorting  In other words  the effect  of branch prediction  is 1 46x for MATLAB implementation and 2 7x for the C implementation

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It s about branch prediction  What is it   A branch predictor is one of the ancient performance improving techniques which still finds relevance into modern architectures  While the simple prediction techniques provide fast lookup and power efficiency they suffer from a high misprediction rate   On the other hand  complex branch predictions    either neural based or variants of two-level branch prediction    provide better prediction accuracy  but they consume more power and complexity increases exponentially   In addition to this  in complex prediction techniques the time taken to predict the branches is itself very high    ranging from 2 to 5 cycles    which is comparable to the execution time of actual branches   Branch prediction is essentially an optimization  minimization  problem where the emphasis is on to achieve lowest possible miss rate  low power consumption  and low complexity with minimum resources    There really are three different kinds of branches  Forward conditional branches - based on a run-time condition  the PC  program counter  is changed to point to an address forward in the instruction stream  Backward conditional branches - the PC is changed to point backward in the instruction stream  The branch is based on some condition  such as branching backwards to the beginning of a program loop when a test at the end of the loop states the loop should be executed again  Unconditional branches - this includes jumps  procedure calls and returns that have no specific condition  For example  an unconditional jump instruction might be coded in assembly language as simply  quot jmp quot   and the instruction stream must immediately be directed to the target location pointed to by the jump instruction  whereas a conditional jump that might be coded as  quot jmpne quot  would redirect the instruction stream only if the result of a comparison of two values in a previous  quot compare quot  instructions shows the values to not be equal   The segmented addressing scheme used by the x86 architecture adds extra complexity  since jumps can be either  quot near quot   within a segment  or  quot far quot   outside the segment   Each type has different effects on branch prediction algorithms   Static dynamic Branch Prediction  Static branch prediction is used by the microprocessor the first time a conditional branch is encountered  and dynamic branch prediction is used for succeeding executions of the conditional branch code  References   Branch predictor  A Demonstration of Self-Profiling  Branch Prediction Review  Branch Prediction  Using wayback machine

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An official answer would be from   Intel - Avoiding the Cost of Branch Misprediction Intel - Branch and Loop Reorganization to Prevent Mispredicts Scientific papers - branch prediction computer architecture Books  J L  Hennessy  D A  Patterson  Computer architecture  a quantitative approach Articles in scientific publications  T Y  Yeh  Y N  Patt made a lot of these on branch predictions    You can also see from this lovely diagram why the branch predictor gets confused     Each element in the original code is a random value  data c    std  rand     256    so the predictor will change sides as the std  rand   blow   On the other hand  once it s sorted  the predictor will first move into a state of strongly not taken and when the values change to the high value the predictor will in three runs through change all the way from strongly not taken to strongly taken

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The assumption by other answers that one needs to sort the data is not correct   The following code does not sort the entire array  but only 200-element segments of it  and thereby runs the fastest   Sorting only k-element sections completes the pre-processing in linear time  O n   rather than the O n log n   time needed to sort the entire array    include  lt algorithm gt   include  lt ctime gt   include  lt iostream gt   int main         int data 32768   const int l   sizeof data   sizeof data 0        for  unsigned c   0  c  lt  l    c          data c    std  rand     256          sort 200-element segments  not the whole array     for  unsigned c   0  c   200  lt   l  c    200          std  sort  amp data c    amp data c   200         clock t start   clock        long long sum   0       for  unsigned i   0  i  lt  100000    i            for  unsigned c   0  c  lt  sizeof data   sizeof int     c                if  data c   gt   128                  sum    data c                        std  cout  lt  lt  static cast lt double gt  clock   - start    CLOCKS PER SEC  lt  lt  std  endl      std  cout  lt  lt   sum      lt  lt  sum  lt  lt  std  endl      This also  proves  that it has nothing to do with any algorithmic issue such as sort order  and it is indeed branch prediction

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No doubt some of us would be interested in ways of identifying code that is problematic for the CPU s branch-predictor  The Valgrind tool cachegrind has a branch-predictor simulator  enabled by using the --branch-sim yes flag  Running it over the examples in this question  with the number of outer loops reduced to 10000 and compiled with g    gives these results   Sorted     32551   Branches         656 645 130     656 609 208 cond      35 922 ind    32551   Mispredicts          169 556         169 095 cond         461 ind    32551   Mispred rate             0 0             0 0              1 2        Unsorted     32555   Branches         655 996 082     655 960 160 cond    35 922 ind    32555   Mispredicts      164 073 152     164 072 692 cond       460 ind    32555   Mispred rate            25 0            25 0            1 2        Drilling down into the line-by-line output produced by cg annotate we see for the loop in question   Sorted             Bc    Bcm Bi Bim       10 001      4  0   0      for  unsigned i   0  i  lt  10000    i                                                                           primary loop  327 690 000 10 016  0   0          for  unsigned c   0  c  lt  arraySize    c                                         327 680 000 10 006  0   0              if  data c   gt   128             0      0  0   0                  sum    data c                                                                             Unsorted             Bc         Bcm Bi Bim       10 001           4  0   0      for  unsigned i   0  i  lt  10000    i                                                                                     primary loop  327 690 000      10 038  0   0          for  unsigned c   0  c  lt  arraySize    c                                              327 680 000 164 050 007  0   0              if  data c   gt   128             0           0  0   0                  sum    data c                                                                                       This lets you easily identify the problematic line - in the unsorted version the if  data c   gt   128  line is causing 164 050 007 mispredicted conditional branches  Bcm  under cachegrind s branch-predictor model  whereas it s only causing 10 006 in the sorted version     Alternatively  on Linux you can use the performance counters subsystem to accomplish the same task  but with native performance using CPU counters   perf stat   sumtest sorted   Sorted    Performance counter stats for    sumtest sorted      11808 095776 task-clock                     0 998 CPUs utilized                    1 062 context-switches               0 090 K sec                               14 CPU-migrations                 0 001 K sec                              337 page-faults                    0 029 K sec                   26 487 882 764 cycles                         2 243 GHz                     41 025 654 322 instructions                   1 55  insns per cycle          6 558 871 379 branches                     555 455 M sec                          567 204 branch-misses                  0 01  of all branches            11 827228330 seconds time elapsed   Unsorted    Performance counter stats for    sumtest unsorted      28877 954344 task-clock                     0 998 CPUs utilized                    2 584 context-switches               0 089 K sec                               18 CPU-migrations                 0 001 K sec                              335 page-faults                    0 012 K sec                   65 076 127 595 cycles                         2 253 GHz                     41 032 528 741 instructions                   0 63  insns per cycle          6 560 579 013 branches                     227 183 M sec                    1 646 394 749 branch-misses                 25 10  of all branches            28 935500947 seconds time elapsed   It can also do source code annotation with dissassembly   perf record -e branch-misses   sumtest unsorted perf annotate -d sumtest unsorted      Percent        Source code  amp  Disassembly of sumtest unsorted ------------------------------------------------                                     sum    data c       0 00          400a1a        mov    -0x14  rbp   eax    39 97          400a1d        mov     eax  eax     5 31          400a1f        mov    -0x20040  rbp  rax 4   eax     4 60          400a26        cltq        0 00          400a28        add     rax -0x30  rbp        See the performance tutorial for more details

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In the sorted case  you can do better than relying on successful branch prediction or any branchless comparison trick  completely remove the branch   Indeed  the array is partitioned in a contiguous zone with data  lt  128 and another with data  gt   128  So you should find the partition point with a dichotomic search  using Lg arraySize    15 comparisons   then do a straight accumulation from that point   Something like  unchecked   int i  0  j  k  arraySize  while  i  lt  k      j   i   k   gt  gt  1    if  data j   gt   128      k  j    else     i  j    sum  0  for    i  lt  arraySize  i      sum   data i     or  slightly more obfuscated  int i  k  j   i   k   gt  gt  1  for  i  0  k  arraySize  i  lt  k   data j   gt   128   k   i   j    j   i   k   gt  gt  1  for  sum  0  i  lt  arraySize  i      sum   data i     A yet faster approach  that gives an approximate solution for both sorted or unsorted is  sum  3137536   assuming a truly uniform distribution  16384 samples with expected value 191 5   -

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This question has already been answered excellently many times over  Still I d like to draw the group s attention to yet another interesting analysis  Recently this example  modified very slightly  was also used as a way to demonstrate how a piece of code can be profiled within the program itself on Windows  Along the way  the author also shows how to use the results to determine where the code is spending most of its time in both the sorted  amp  unsorted case  Finally the piece also shows how to use a little known feature of the HAL  Hardware Abstraction Layer  to determine just how much branch misprediction is happening in the unsorted case  The link is here  A Demonstration of Self-Profiling

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Branch-prediction gain  It is important to understand that branch misprediction doesn t slow down programs  The cost of a missed prediction is just as if branch prediction didn t exist and you waited for the evaluation of the expression to decide what code to run  further explanation in the next paragraph   if  expression           Run 1   else          Run 2    Whenever there s an if-else   switch statement  the expression has to be evaluated to determine which block should be executed  In the assembly code generated by the compiler  conditional branch instructions are inserted  A branch instruction can cause a computer to begin executing a different instruction sequence and thus deviate from its default behavior of executing instructions in order  i e  if the expression is false  the program skips the code of the if block  depending on some condition  which is the expression evaluation in our case  That being said  the compiler tries to predict the outcome prior to it being actually evaluated  It will fetch instructions from the if block  and if the expression turns out to be true  then wonderful  We gained the time it took to evaluate it and made progress in the code  if not then we are running the wrong code  the pipeline is flushed  and the correct block is run  Visualization  Let s say you need to pick route 1 or route 2  Waiting for your partner to check the map  you have stopped at    and waited  or you could just pick route1 and if you were lucky  route 1 is the correct route   then great you didn t have to wait for your partner to check the map  you saved the time it would have taken him to check the map   otherwise you will just turn back  While flushing pipelines is super fast  nowadays taking this gamble is worth it  Predicting sorted data or a data that changes slowly is always easier and better than predicting fast changes   O      Route 1   -------------------------------                       ---------                                               Route 2   --------------------------------

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On ARM  there is no branch needed  because every instruction has a 4-bit condition field  which tests  at zero cost  any of 16 different different conditions that may arise in the Processor Status Register  and if the condition on an instruction is false  the instruction is skipped  This eliminates the need for short branches  and there would be no branch prediction hit for this algorithm  Therefore  the sorted version of this algorithm would run slower than the unsorted version on ARM  because of the extra overhead of sorting  The inner loop for this algorithm would look something like the following in ARM assembly language  MOV R0   0      R0   sum   0 MOV R1   0      R1   c   0 ADR R2  data    R2   addr of data array  put this instruction outside outer loop   inner loop     Inner loop branch label     LDRB R3   R2  R1       R3   data c      CMP R3   128           compare R3 to 128     ADDGE R0  R0  R3       if R3  gt   128  then sum    data c  -- no branch needed      ADD R1  R1   1         c       CMP R1   arraySize     compare c to arraySize     BLT inner loop         Branch to inner loop if c  lt  arraySize  But this is actually part of a bigger picture  CMP opcodes always update the status bits in the Processor Status Register  PSR   because that is their purpose  but most other instructions do not touch the PSR unless you add an optional S suffix to the instruction  specifying that the PSR should be updated based on the result of the instruction  Just like the 4-bit condition suffix  being able to execute instructions without affecting the PSR is a mechanism that reduces the need for branches on ARM  and also facilitates out of order dispatch at the hardware level  because after performing some operation X that updates the status bits  subsequently  or in parallel  you can do a bunch of other work that explicitly should not affect  or be affected by  the status bits  then you can test the state of the status bits set earlier by X  The condition testing field and the optional  quot set status bit quot  field can be combined  for example   ADD R1  R2  R3 performs R1   R2   R3 without updating any status bits  ADDGE R1  R2  R3 performs the same operation only if a previous instruction that affected the status bits resulted in a Greater than or Equal condition  ADDS R1  R2  R3 performs the addition and then updates the N  Z  C and V flags in the Processor Status Register based on whether the result was Negative  Zero  Carried  for unsigned addition   or oVerflowed  for signed addition   ADDSGE R1  R2  R3 performs the addition only if the GE test is true  and then subsequently updates the status bits based on the result of the addition   Most processor architectures do not have this ability to specify whether or not the status bits should be updated for a given operation  which can necessitate writing additional code to save and later restore status bits  or may require additional branches  or may limit the processor s out of order execution efficiency  one of the side effects of most CPU instruction set architectures forcibly updating status bits after most instructions is that it is much harder to tease apart which instructions can be run in parallel without interfering with each other  Updating status bits has side effects  therefore has a linearizing effect on code  ARM s ability to mix and match branch-free condition testing on any instruction with the option to either update or not update the status bits after any instruction is extremely powerful  for both assembly language programmers and compilers  and produces very efficient code  When you don t have to branch  you can avoid the time cost of flushing the pipeline for what would otherwise be short branches  and you can avoid the design complexity of many forms of speculative evalution  The performance impact of the initial naive imlementations of the mitigations for many recently discovered processor vulnerabilities  Spectre etc   shows you just how much the performance of modern processors depends upon complex speculative evaluation logic  With a short pipeline and the dramatically reduced need for branching  ARM just doesn t need to rely on speculative evaluation as much as CISC processors   Of course high-end ARM implementations do include speculative evaluation  but it s a smaller part of the performance story   If you have ever wondered why ARM has been so phenomenally successful  the brilliant effectiveness and interplay of these two mechanisms  combined with another mechanism that lets you  quot barrel shift quot  left or right one of the two arguments of any arithmetic operator or offset memory access operator at zero additional cost  are a big part of the story  because they are some of the greatest sources of the ARM architecture s efficiency  The brilliance of the original designers of the ARM ISA back in 1983  Steve Furber and Roger  now Sophie  Wilson  cannot be overstated

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If you are curious about even more optimizations that can be done to this code  consider this   Starting with the original loop   for  unsigned i   0  i  lt  100000    i        for  unsigned j   0  j  lt  arraySize    j                if  data j   gt   128              sum    data j             With loop interchange  we can safely change this loop to   for  unsigned j   0  j  lt  arraySize    j        for  unsigned i   0  i  lt  100000    i                if  data j   gt   128              sum    data j             Then  you can see that the if conditional is constant throughout the execution of the i loop  so you can hoist the if out   for  unsigned j   0  j  lt  arraySize    j        if  data j   gt   128                for  unsigned i   0  i  lt  100000    i                        sum    data j                       Then  you see that the inner loop can be collapsed into one single expression  assuming the floating point model allows it   fp fast is thrown  for example   for  unsigned j   0  j  lt  arraySize    j        if  data j   gt   128                sum    data j    100000            That one is 100 000 times faster than before

[java] Why is processing a sorted array faster than processing an unsorted array?

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