Sort a large set of floating point numbers which are in range from 0.0 to 1.0 and are uniformly distributed across the range.

Note: Bucket sort can be seen as a generalization of counting sort; in fact, if each bucket has size 1 then bucket sort degenerates to counting sort.

The variable bucket size of bucket sort allows it to use O ( n ) memory instead of O ( M ) memory, where M is the number of distinct values; in exchange, it gives up counting sort s O ( n M ) worst - case behavior.

The lower bound for Comparison based sorting algorithm (Merge Sort, Heap Sort, Quick-Sort.

Can we sort the array in linear time Counting sort can not be applied here as we use keys as index in counting sort.

I mentioned above that the worst case for bucket sort is counting sort.

Thats because in the worst case, bucket sort may end up with one bucket per key.

Even after allowing for some probabilistic variance, to sort each bucket would still take 2-1n, which is still a constant.

Additional informationassumption: Integer numbers in the range 0..

The idea: determine for each input element x its rank: the number of elements less than x.

Once we know the rank r of x, we can place it in position r1 Example: if there are 6 elements smaller than 17, then we can place 17 in the 7 th position.

Repetitions: when there are several elements with the same value, locate them one after the other in the order in which they appear in the input this is called stable sorting, Data Structures, Spring 2004 L.

Joskowicz 4 5 For each Ai, count the number of elements to it.

This rank of Ai is the index indicating where it goes When there are no repetitions and n k, RankAi Ai and BRankAi Ai 4.

Joskowicz 3 3 4 When n k or there are repetitions, place them one after the other in the order in which they appear in the input and adjust the index by one this is called stable sorting 6.

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Start from the least significant digit, working up to the most significant one.

Since there are only 10 different digits 0..

For example, we can use Counting-Sort for each call, with k 9.

Joskowicz 720 355 436 457 657 329 839 720 329 436 839 355 457 657 329 355 436 457 657 720 839 13.

Define xl as the number formed by the last l digits of x, for l d.

For example, x 2345 then x 1 5, x 2 45, x 3 345 Base: for d 1, Radix-Sort uses a stable sorting algorithm to sort n numbers in the range 0..

So if x 1.

By induction, x and y remain in the same order they appeared before the ith iteration, and snde the ith iteration is stable, they will remain so after the additional iteration.

Each pass of Counting-Sort takes (nk) so we get (n2 r) and there are d passes, so the total running time is (d(n2 r)), or ((br)(n2 r)).

For given values of n and b, we can choose r b to be optimum minimize ((br)(n2 r)).

Since the numbers are uniformly distributed, not too many elements will be placed in each bucket.

If we insert them in order (using Insertion-Sort), the buckets and the elements in them will always be in sorted order.

Joskowicz 0 1 2 3 4 5 6 7 8 9.

Concatenate lists B0, B1, Bn 1 in order Data Structures, Spring 2004 L.

In the worst case, O(n) numbers will end up in the same bucket, so in the worst case, it will take O(n 2) time.

However, in the average case, only a constant number of elements will fall in each bucket, so it will take O(n) (see proof in book).

Extensions: use a different indexing scheme to distribute the numbers (hashing later in the course) Data Structures, Spring 2004 L.

Bucket Sort Example