## Algorithms/Divide and Conquer

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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 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. 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 binary search program in c using divide and conquer method 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.

Any of the examples can be converted into an equivalent example using "exclusive" upper bound binary search program in c using divide and conquer method.

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. 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 orderingor 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. 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. Note that these algorithms are almost exactly the same as the leftmost-insertion-point algorithms, except for how the inequality treats equal values.

This can occur if the array size is greater than half the maximum integer value. 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. 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. Both solutions are generic. Note that the completion condition is different from one given in the pseudocode example above. The example assumes that the binary search program in c using divide and conquer method index type does not overflow when mid is incremented or decremented beyond the corresponding array bound.

This is a wrong assumption for Ada, where array bounds can start or end at the very first or last value of the index type. To deal with this, the exit condition is rather directly expressed as crossing the corresponding array bound by the coming interval middle. Note that the difference between the bounds is the number of elements equal to the element you want. It does not give you any information as to where it is. The following solution is based on the one described in: Analysis, Classification, and Examples.

Also available at http: It includes detailed loop invariants and pre- and postconditions, which make the running time linear instead of logarithmic when full contract checking is enabled.

User-defined implementations of the same would be considerably slower. Nonetheless, here they are in order to comply with the task requirements. Factor already includes a binary search in its standard library.

The following code offers an interface compatible with the requirement of this task, and returns either the index of the element if it has been found or f otherwise. This binary search program in c using divide and conquer method is designed for maintaining a sorted array.

If the item is not found, then then location returned is the proper insertion point for the item. This could be used in an optimized Insertion sortfor example. This has the array indexed from binary search program in c using divide and conquer method to N, and the "not found" return code is zero or negative. Changing the search to be for A first: For the more advanced fortrans, declaring the parameters to be INTENT IN may help, as despite passing arrays "by reference" being the norm, the newer compilers may generate copy-in, copy-out code, vitiating the whole point of using a fast binary search instead of a slow linear search.

In such a situation however, preparing in-line code may be the better move: Later compilers offer features allowing the development of "generic" functions so that the same function name may be used yet the actual routine invoked will be selected according to how the parameters are integers or floating-point, and of different precisions.

There would still need to be a version of the function for each type combination, each with its own name. Unfortunately, there is no three-way comparison test for character data. The use of "exclusive" bounds simplifies the adjustment of the bounds: The "inclusive" bounds version by contrast requires two manipulations of L and R at every step - once to see if the span is empty, and a second time to locate the index to test.

Imagine a test array containing the even numbers: A count could be kept of the number of probes required to find binary search program in c using divide and conquer method of those four values, and likewise with a search for the odd numbers 1,3,5,7,9 that would probe all the places where a value might be not found.

Plot the average number of probes for the two cases, plus the maximum number of probes for any case, and then repeat for another number of elements to search. With only one element in the array to be searched, all values are the same: The point of this is that the IF-test is going to initiate some jumps, so why not arrange that one of the bound adjustments needs no subsequent jump to the start of the next iteration - in the first version, both bound adjustments needed such a jump, the GO TO 1 statements.

This was done by shifting the code for label 2 up to precede the code for label 1 binary search program in c using divide and conquer method and removing its now pointless GO TO 1 executed each timebut adding an initial GO TO 1, executed once only. This sort of change is routine when manipulating spaghetti code It is because the method involves such a small amount of effort per iteration that minor changes offer a significant benefit. A lot depends on the implementation of the three-way test: These branch tests may in turn be made in an order that notes which option if any involves "falling through" to the next statement, thus it may be better to swap the order of labels 3 and 4.

Further, the compiler may itself choose to re-order the various code pieces. An assembler version of this routine attended to all these details. Some compilers do not produce machine code directly, but instead translate the source code into another language which is then compiled, and a common choice for that is C. This is all very well, but C is one of the many languages that do not have a three-way test option and so cannot represent Fortran's three-way IF statement directly.

Before emitting asservations of faith that pseudocode such as. That is, does the compiler make any remark, and does the resulting machine code contain a redundant test?

However, despite all the binary search program in c using divide and conquer method, the three-way IF statement has been declared deprecated in later versions of Fortran, with no alternative **binary search program in c using divide and conquer method** repeated testing offered. Incidentally, the exclusive-bounds version leads to a good version of the interpolation search whereby the probe position is interpolated, not just in the middle of the spanunlike the version based on inclusive-bounds.

Further, the unsourced offering in Wikipedia contains a bug - try searching an array of two equal elements for that value. There are also functions sort. SearchStringsand a very general sort. Search function that allows you to binary search a range of numbers based on any condition not necessarily just search for an index of an element in an array.

The algorithm itself, parametrized by an "interrogation" predicate p in the spirit of the explanation above:. The algorithm uses tail recursion, so the iterative and the recursive approach are identical in Haskell the compiler will convert recursive calls into jumps. J already includes a binary search primitive I. The following code offers an interface compatible with the requirement of this task, and returns either the index of the element if it has been found or 'Not Found' otherwise:.

If the input array is sorted, then binarySearch value as defined here will return an index i. Can be tested in http: The calculation of "mid" cannot overflow, since Maple uses arbitrary precision integer arithmetic, and the largest list or array is far, far smaller than the effective range of integers.

The following is another implementation that takes a more manual approach. Instead of using an intrinsic function, a general binary search algorithm is implemented using the language alone. Returns a positive index if found, otherwise the negative index where it would go if inserted now.

Incidentally, REXX doesn't care if the values are integers or even numbersas long as they're in binary search program in c using divide and conquer method. The calls to helper are in tail position, so since Scheme implementations support proper tail-recursion the computation proces is iterative. SPARK does not allow recursion, so only the iterative solution is provided.

This example shows the use of a loop assertion. Search is a procedure, rather than a function, so that it can return a Found flag and a Position for Item, if found. This is better design than having a Position value that means 'item not found'.