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- In the sorting problem, we are:
- given a collection C of n elements that can be compared according to a total order relation
- the task is to rearrange the elements in C in increasing (or at least non-decreasing if there are ties) order.

Complexity of Algorithms

- Priority queue is a container of elements, each having an associated key
- keys determine the ’’priority’’ used to pick elements to be removed
- PQ fundamental methods
- insertItem(k,e): insert e to PQ
- removeMin(k): remove min. element
- minElement(): return min. element
- minKey(): return key of min. el.

Complexity of Algorithms

- In the first phase we put the elements of C into an initially empty priority queue P by means of a series of ninsertItem operations
- In the second phase, we extract the elements from P in non-decreasing order by means of series of nremoveMin operations

Complexity of Algorithms

Complexity of Algorithms

- A heap is a realisation of PQ that is efficient for both insertions and removals
- heap allows to perform both insertions and removals in logarithmic time
- In heap the elements and their keys are stored in (almost complete) binary tree

Complexity of Algorithms

- In a heapT, for every nodev other than the root, the key stored at v is greater than (or equal) to the key stored at its parent

Complexity of Algorithms

- heap: complete binary treeT containing elements with keys satisfying heap-order property; implemented using a vector representation
- last: reference to the last used node of T
- comp: comparator that defines the total order relation on keys and maintains the minimum element at the root of T

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

- Thm: The heap-sort algorithm sorts a sequence of S of n comparable elements in O(n log n) time, where
- Bottom-up construction of heap with n items takes O(n) time, and
- Extraction of n elements (in increasing order) from the heap takes O(n log n) time

Complexity of Algorithms

- Divide: if the input size is small then solve the problem directly; otherwise divide the input data into two or more disjoint subsets
- Recur:recursively solve the sub-problems associated with the subsets
- Conquer: take the solutions to the sub-problems and merge them into a solution to the original problem

Complexity of Algorithms

- Divide: if input sequence S has 0 or 1 element then return S; otherwise split S into two sequences S1 and S2, each containing about ½ elements of S
- Recur:recursively sort sequences S1 and S2
- Conquer: Put the elements back into S by merging the sorted sequences S1 and S2 into a single sorted sequence

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

- Thm:merging two sorted sequencesS1 and S2 takes O(n1+n2) time, where n1 is the size of S1 and n2 is the size of S2 (see comp108 notes)
- Thm:Merge-sort runs in O(n log n) time in the worst (and average) case

Complexity of Algorithms

Complexity of Algorithms

- Worst-case running time of merge-sort t(n) can be expressed by recurrence equation:

- Assuming that n is a power of 2 we get:

- t(n) = 2(2t(n/22) + (cn/2)) + cn = 22t(n/22) + 2cn = … = 2it(n/2i) + icn = O(n log n), for i=log n (closed form)

Complexity of Algorithms

- Divide: if ¦S¦>1, select a pivotx in S and create three sequences: L, E and G, s.t.,
- L stores elements in S < x
- E stores elements in S = x
- G stores elements in S > x

- Recur: recursively sort sequences L & G
- Conquer: put sorted elements from L, E and finally from G back to S.

Complexity of Algorithms

Complexity of Algorithms

Complexity of Algorithms

- Let si be the sum of the input sizes of the nodes at depth i in a quick sort tree T
- si n-i (and si = n-i when use of pivots lead always to only one nonempty sequence: either L or G)
- The worst-case complexity is bounded by:

- which is O(n2).

Complexity of Algorithms

- Thm: the expected running time of randomised (pivot is chosen in random) quick-sort is O(n log n)
- Proof:
- Fact: the expected number of times that a fair coin must be flipped until it shows heads k times is 2k.
- We say that a random chosen pivot is right if neither of the groups L nor G is > ¾ ¦S¦
- The probability of a success in choosing a right pivot is ½
- Any path in the quick-sort tree can contain at most log4/3 n nodes with right pivots
- Hence, the expected length of each path is 2log4/3 n

Complexity of Algorithms

Complexity of Algorithms

- In comparison-based model the input elements can be compared only with themselves and the result of each comparison xi xj is always yes or no
- Thm: the running time of any comparison-based sorting algorithm is (n log n) in the worst case
- Proof:
- Sorting of n elements can be identified with recognising a particular permutation of n elements
- There is n!=n·(n-1) ·…·2·1 permutations of n elements
- Each comparison splits a group of permutations into two groups (one that satisfies the inequality and one that doesn’t)
- In order to ensure that the size of each group of permutations is brought down to one we need log2(n!) > log (n/2)n/2=n/2·log n/2 = (n log n) comparisons

Complexity of Algorithms

Complexity of Algorithms

- Bucket-sort is not based on comparisons but rather on using keys as indices of a bucket array B that has entries within an integer range [0,…,N-1]
- Initially all items from input sequence S are moved to appropriate buckets, i.e., an item with key k is moved to bucket B[k]
- Then we move all items back into S according to their order of appearance in consecutive buckets B[0], B[1], …, B[N]

Complexity of Algorithms

Complexity of Algorithms

- In selection problem we are interested in identifying a single element in terms of its rank relative to an ordering of the entire set
- Examples include identifying the minimum and the maximum elements, but we may be also interested in identifying the median or general kth element
- The selection problem can be solved with a help of efficient sorting algorithm in time O(n log n)
- However, the selection problem can be solved in time O(n) using more accurate prune-and search (decrease-and-conquer) method

Complexity of Algorithms

- In prune-and-search method we solve a given problem by pruning away a fraction of input objects and recursively solving a smaller problem
- When the problem is reduced to constant size it is solved by some brute-force method
- The solution to the original problem is completed by returning back from all the recursive calls

Complexity of Algorithms

- Prune: pick an element x from S at random and use it as a pivot to subdivide S into three groups L, E and G, where
- L stores elements in S < x
- E stores elements in S = x
- G stores elements in S > x

- Search: based on the value of k, we determine on which of these sets to recur

Complexity of Algorithms

Complexity of Algorithms

- Thm: the expected running time of randomised quick-select on a sequence of size n is O(n)
- Thm: there exists a deterministic algorithm for a selection problem that works (in the worst-case) in time O(n)

Complexity of Algorithms