A binary search divides a range of values into halves, and continues to narrow down the field of search until the unknown value is found. It is the classic example of a "divide and conquer" algorithm.

As an analogy, consider the children's game "guess a number." The scorer has a secret number, and will only tell the player if their guessed number is higher than, lower than, or equal to the secret number. The player then uses this information to guess a new number.

As the player, an optimal strategy for the general case is to start by choosing the range's midpoint as the guess, and then asking whether the guess was higher, lower, or equal to the secret number. If the guess was too high, one would select the point exactly between the range midpoint and the beginning of the range. If the original guess was too low, one would ask about the point exactly between the range midpoint and the end of the range. This process repeats until one has reached the secret number.



Task

Given the starting point of a range, the ending point of a range, and the "secret value", implement a binary search through a sorted integer array for a certain number. Implementations can be recursive or iterative (both if you can). Print out whether or not the number was in the array afterwards. If it was, print the index also.

There are several binary search algorithms commonly seen. They differ by how they treat multiple values equal to the given value, and whether they indicate whether the element was found or not. For completeness we will present pseudocode for all of them.

All of the following code examples use an "inclusive" upper bound (i.e. high = N-1 initially). Any of the examples can be converted into an equivalent example using "exclusive" upper bound (i.e. high = N initially) by making the following simple changes (which simply increase high by 1):

 change high = N-1 to high = N
 change high = mid-1 to high = mid
 (for recursive algorithm) change if (high < low) to if (high <= low)
 (for iterative algorithm) change while (low <= high) to while (low < high)
Traditional algorithm
The algorithms are as follows (from Wikipedia). The algorithms return the index of some element that equals the given value (if there are multiple such elements, it returns some arbitrary one). It is also possible, when the element is not found, to return the "insertion point" for it (the index that the value would have if it were inserted into the array).

Recursive Pseudocode:

  // initially called with low = 0, high = N-1
  BinarySearch(A[0..N-1], value, low, high) {
      // invariants: value > A[i] for all i < low
                     value < A[i] for all i > high
      if (high < low)
          return not_found // value would be inserted at index "low"
      mid = (low + high) / 2
      if (A[mid] > value)
          return BinarySearch(A, value, low, mid-1)
      else if (A[mid] < value)
          return BinarySearch(A, value, mid+1, high)
      else
          return mid
  }

Iterative Pseudocode:

  BinarySearch(A[0..N-1], value) {
      low = 0
      high = N - 1
      while (low <= high) {
          // invariants: value > A[i] for all i < low
                         value < A[i] for all i > high
          mid = (low + high) / 2
          if (A[mid] > value)
              high = mid - 1
          else if (A[mid] < value)
              low = mid + 1
          else
              return mid
      }
      return not_found // value would be inserted at index "low"
  }

Leftmost insertion point
The following algorithms return the leftmost place where the given element can be correctly inserted (and still maintain the sorted order). This is the lower (inclusive) bound of the range of elements that are equal to the given value (if any). Equivalently, this is the lowest index where the element is greater than or equal to the given value (since if it were any lower, it would violate the ordering), or 1 past the last index if such an element does not exist. This algorithm does not determine if the element is actually found. This algorithm only requires one comparison per level.

Recursive Pseudocode:

  // initially called with low = 0, high = N - 1
  BinarySearch_Left(A[0..N-1], value, low, high) {
      // invariants: value > A[i] for all i < low
                     value <= A[i] for all i > high
      if (high < low)
          return low
      mid = (low + high) / 2
      if (A[mid] >= value)
          return BinarySearch_Left(A, value, low, mid-1)
      else
          return BinarySearch_Left(A, value, mid+1, high)
  }

Iterative Pseudocode:

  BinarySearch_Left(A[0..N-1], value) {
      low = 0
      high = N - 1
      while (low <= high) {
          // invariants: value > A[i] for all i < low
                         value <= A[i] for all i > high
          mid = (low + high) / 2
          if (A[mid] >= value)
              high = mid - 1
          else
              low = mid + 1
      }
      return low
  }

Rightmost insertion point
The following algorithms return the rightmost place where the given element can be correctly inserted (and still maintain the sorted order). This is the upper (exclusive) bound of the range of elements that are equal to the given value (if any). Equivalently, this is the lowest index where the element is greater than the given value, or 1 past the last index if such an element does not exist. This algorithm does not determine if the element is actually found. This algorithm only requires one comparison per level. Note that these algorithms are almost exactly the same as the leftmost-insertion-point algorithms, except for how the inequality treats equal values.

Recursive Pseudocode:

  // initially called with low = 0, high = N - 1
  BinarySearch_Right(A[0..N-1], value, low, high) {
      // invariants: value >= A[i] for all i < low
                     value < A[i] for all i > high
      if (high < low)
          return low
      mid = (low + high) / 2
      if (A[mid] > value)
          return BinarySearch_Right(A, value, low, mid-1)
      else
          return BinarySearch_Right(A, value, mid+1, high)
  }

Iterative Pseudocode:

  BinarySearch_Right(A[0..N-1], value) {
      low = 0
      high = N - 1
      while (low <= high) {
          // invariants: value >= A[i] for all i < low
                         value < A[i] for all i > high
          mid = (low + high) / 2
          if (A[mid] > value)
              high = mid - 1
          else
              low = mid + 1
      }
      return low
  }

Extra credit
Make sure it does not have overflow bugs.

The line in the pseudo-code above to calculate the mean of two integers:

mid = (low + high) / 2
could produce the wrong result in some programming languages when used with a bounded integer type, if the addition causes an overflow. (This can occur if the array size is greater than half the maximum integer value.) If signed integers are used, and low + high overflows, it becomes a negative number, and dividing by 2 will still result in a negative number. Indexing an array with a negative number could produce an out-of-bounds exception, or other undefined behavior. If unsigned integers are used, an overflow will result in losing the largest bit, which will produce the wrong result.

One way to fix it is to manually add half the range to the low number:

mid = low + (high - low) / 2
Even though this is mathematically equivalent to the above, it is not susceptible to overflow.

Another way for signed integers, possibly faster, is the following:

mid = (low + high) >>> 1
where  >>>  is the logical right shift operator. The reason why this works is that, for signed integers, even though it overflows, when viewed as an unsigned number, the value is still the correct sum. To divide an unsigned number by 2, simply do a logical right shift.



Related task

 Guess the number/With Feedback (Player)


See also

 wp:Binary search algorithm
 Extra, Extra - Read All About It: Nearly All Binary Searches and Mergesorts are Broken.


 
/* ARM assembly AARCH64 Raspberry PI 3B */
/*  program binSearch64.s   */
 
/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
 
/*********************************/
/* Initialized data              */
/*********************************/
.data
sMessResult:        .asciz "Value find at index : @ \n"
szCarriageReturn:   .asciz "\n"
sMessRecursif:      .asciz "Recursive search : \n"
sMessNotFound:      .asciz "Value not found. \n"
 
TableNumber:        .quad   4,6,7,10,11,15,22,30,35
                    .equ NBELEMENTS,  (. - TableNumber) / 8
/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:          .skip 24
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main 
main:                                           // entry of program 
    mov x0,4                                    // search first value
    ldr x1,qAdrTableNumber                      // address number table
    mov x2,NBELEMENTS                           // number of élements 
    bl bSearch
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
    mov x0,#11                                  // search median value
    ldr x1,qAdrTableNumber
    mov x2,#NBELEMENTS
    bl bSearch
    ldr x1,qAdrsZoneConv
    bl conversion10                             // decimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
    mov x0,#12                                  //value not found
    ldr x1,qAdrTableNumber
    mov x2,#NBELEMENTS
    bl bSearch
    cmp x0,#-1
    bne 2f
    ldr x0,qAdrsMessNotFound
    bl affichageMess 
    b 3f
2:
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
3:
    mov x0,#35                                  // search last value
    ldr x1,qAdrTableNumber
    mov x2,#NBELEMENTS
    bl bSearch
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
/****************************************/
/*       recursive                      */
/****************************************/
    ldr x0,qAdrsMessRecursif
    bl affichageMess                            // display message
 
    mov x0,#4                                   // search first value
    ldr x1,qAdrTableNumber
    mov x2,#0                                   // low index of elements
    mov x3,#NBELEMENTS - 1                      // high index of elements
    bl bSearchR
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
    mov x0,#11
    ldr x1,qAdrTableNumber
    mov x2,#0
    mov x3,#NBELEMENTS - 1
    bl bSearchR
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
    mov x0,#12
    ldr x1,qAdrTableNumber
    mov x2,#0
    mov x3,#NBELEMENTS - 1
    bl bSearchR
    cmp x0,#-1
    bne 4f
    ldr x0,qAdrsMessNotFound
    bl affichageMess 
    b 5f
4:
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
5:
    mov x0,#35
    ldr x1,qAdrTableNumber
    mov x2,#0
    mov x3,#NBELEMENTS - 1
    bl bSearchR
    ldr x1,qAdrsZoneConv
    bl conversion10                             // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message
 
 
100:                                            // standard end of the program 
    mov x0, #0                                  // return code
    mov x8, #EXIT                               // request to exit program
    svc #0                                      // perform the system call
 
//qAdrsMessValeur:          .quad sMessValeur
qAdrsZoneConv:            .quad sZoneConv
qAdrszCarriageReturn:     .quad szCarriageReturn
qAdrsMessResult:          .quad sMessResult
qAdrsMessRecursif:        .quad sMessRecursif
qAdrsMessNotFound:        .quad sMessNotFound
qAdrTableNumber:          .quad TableNumber
 
/******************************************************************/
/*     binary search   iterative                                  */ 
/******************************************************************/
/* x0 contains the value to search */
/* x1 contains the adress of table */
/* x2 contains the number of elements */
/* x0 return index or -1 if not find */
bSearch:
    stp x1,lr,[sp,-16]!              // save  registers
    stp x2,x3,[sp,-16]!              // save  registers
    stp x4,x5,[sp,-16]!              // save  registers
    mov x3,#0                        // low index
    sub x4,x2,#1                     // high index = number of elements - 1
1:
    cmp x3,x4
    bgt 99f
    add x2,x3,x4                     // compute (low + high) /2
    lsr x2,x2,#1
    ldr x5,[x1,x2,lsl #3]            // load value of table at index x2
    cmp x5,x0
    beq 98f
    bgt 2f
    add x3,x2,#1                     // lower -> index low = index + 1
    b 1b                             // and loop
2:
    sub x4,x2,#1                     // bigger -> index high = index - 1
    b 1b                             // and loop
98:
    mov x0,x2                        // find !!!
    b 100f
99:
    mov x0,#-1                       //not found
100:
    ldp x4,x5,[sp],16                // restaur  2 registers
    ldp x2,x3,[sp],16                // restaur  2 registers
    ldp x1,lr,[sp],16                // restaur  2 registers
    ret                              // return to address lr x30
/******************************************************************/
/*     binary search   recursif                                  */ 
/******************************************************************/
/* x0 contains the value to search */
/* x1 contains the adress of table */
/* x2 contains the low index of elements */
/* x3 contains the high index of elements */
/* x0 return index or -1 if not find */
bSearchR:
    stp x2,lr,[sp,-16]!              // save  registers
    stp x3,x4,[sp,-16]!              // save  registers
    stp x5,x6,[sp,-16]!              // save  registers
    cmp x3,x2                        // index high < low ?
    bge 1f
    mov x0,#-1                       // yes -> not found
    b 100f
1:
    add x4,x2,x3                                     // compute (low + high) /2
    lsr x4,x4,#1
    ldr x5,[x1,x4,lsl #3]                            // load value of table at index x4
    cmp x5,x0
    beq 99f 
    bgt 2f                                           // bigger ?
    add x2,x4,#1                                     // no new search with low = index + 1
    bl bSearchR
    b 100f
2:                                                   // bigger
    sub x3,x4,#1                                     // new search with high = index - 1
    bl bSearchR
    b 100f
99:
    mov x0,x4                                      // find !!!
    b 100f 
100:
    ldp x5,x6,[sp],16                // restaur  2 registers
    ldp x3,x4,[sp],16                // restaur  2 registers
    ldp x2,lr,[sp],16                // restaur  2 registers
    ret                              // return to address lr x30
/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"
 