Data Structures and Algorithms with Object-Oriented Design Patterns (2 -- under construction)

 

Heap and Priority Queue

       Priority Tree are often used to the implementation of algorithms. It can improve the performance of these algorithms including Backtracking, Sorting and Graph algorithms.

       Binary Heap

              Complete Binary Tree, Perfect Binary Tree

              Put: put the item to the next position in the complete binary tree and make it heap ordered by moving it from bottom to the root. O (log n)

              Remove: remove the root element and move the last element in complete binary tree to the root and make it heap ordered by moving it form root to bottom. O (log n)

              Build Heap from Array: move every internal node to the bottom to make the heap ordered. O (n)

       Leftist Heap

              Leftist Tree: path length of the right subtree is no more than left one for every subtree

              Merge: using recursion, the tree with the bigger root merge with the right subtree of another tree. And then, if the path length of right subtree is more than left one, exchange them. O (log n1+log n2)

              Put: Merge the original tree with a new leftist heap containing only the new node O (log n)

              Remove: remove the root and merge the two children of the root O (log n)

       Binomial Queue

              Binomial Tree: B0={R}, Bk={R, Bk-1, Bk-2….B0}

                     The number of nodes at level l in the binomial tree of order k is given by the binomial coefficient (k, l)

              Binomial Queue: A forest of binomial tree with heap order

                     Remove tree: best O (1), worst O (log n)

                     Add tree: O (1)

                     Find minimum: O (log n)

                     Merge: O (log (m + n))

                     Put: Merge the original binomial queue with a new one containing the only node O (log n)

                     Remove: remove the root of the minimum binomial tree and then add all its children to the binomial queue

 

Set

       Array Set

       Bit-Vector Set

       Multiset

       Partition

              Implement a partition using a forest (using array)

              The nodes in the partition tree have three fields – items, parent and rank

              Find: find the root of the partition tree

              Join: join two partition tree

              Collapsing find: 1 find the root 2 assign all the nodes on the finding path to the root

              Union by size: attach the small tree to the root node of the larger one.

                     The height of partition tree is guaranteed to be O (log n)

              Union by rank (height): attach the tree with lower height to the one with higher height

                     The height of partition tree is guaranteed to be O (log n)

              Application: processing the Equivalence Relation (reflexive, symmetric, transitive)