Sequential Searching

Search Algorithm

Lecture 6

Sequential Searching



If the elements are randomly distributed and

sequentially stored in a data structure, we use the sequential search.



We assume inputs are stored in an array or in a linked list.



We search data one by one comparing the search key.

•

worst case time complexity= O(logn)

For each search (while loop), the size of records gets the half

Binary Search



Binary search is fast but if we want to insert or delete a key, we have to move n/2 records on

average to keep the sorted structure, thus it’s not appropriate for applications, which need insertion or deletion.

Binary Search

Binary Search Tree



If the data records are dynamically inserted/deleted, we need a way to make insertion and deletion easy.



Binary search tree: Binary tree that each node as left and right link.



left side of node x is smaller than x and right side of node x is bigger than x

Binary Search Tree



average time complexity= O(logn)

proportion to the number of comparisons. Thus, proportion to the length of the paths.

average number of comparisons

= average length of tree paths/n

Binary Search Tree



average time complexity= O(logn)

proportion to the number of comparisons. Thus, proportion to the length of the paths



insertion and deletion time depend on the search time.

 worst time complexity= O(n)

 average time complexity= O(logn)

Balanced Tree



The worst cases using binary search tree

 three cases

inserted in increasing order A, B, C, D, E, F, G, H

Balanced Tree



The worst cases using binary search tree

 three cases

inserted in decreasing order A, B, C, D, E, F, G, H

Z, Y, X, W, V, U, T, S

…

Balanced Tree

 The worst cases using binary search tree

 three cases

big and small key insert in turn A, B, C, D, E, F, G, H

Z, Y, X, W, V, U, T, S A, Z, B, Y, C, X, D, W

Balanced Tree



avoid skewed tree



balance the height of left and the right



^{2-3-4 tree}



red black tree

2-3-4 tree

 insertion – insert G

 find the insertion location for G, fail at HIJ, the node already contains 3 elements (4-node), thus, we cannot insert into the node. Create a new node to insert the

element by increasing the height of the tree. Better solution?

 splitting nodes only occur through the search

In other words, splitting occurs only on the search path

 data structure is complicated

 Implementation is not easy

 For some cases, it is slower then the binary search tree

2-3-4 tree