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d-heaps d-heap— 5-heap children of node in position i are in positions parent of node in position i is in advantages disadvantages Leftist Heaps operations A leftist heap The null path length (x) denoted npl(x) Nodes in the tree below are labeled with their null path lengths. Leftist Heap operations Basic operation: merge H1 and H2 are leftist heaps 1. 2. 3. Example: Other leftist heap operations Insert What will the tree look like if the keys are inserted in decreasing order: 9, 7, 5, 3, 1? inserting keys in increasing order: 3, 5, 7, 9, 11, 13 The Delete Min Operation Leftist heap merge merge1(H1, H2: PRIORITY_Q): PRIORITY_Q // Recursive function to merge two leftist heaps. The root of H1 is smaller than the root of H2. When merge is called it determines which heap has the smaller root value and then calls merge1. p. 232 Weiss Data Structures and Algorithm Analysis in C++ 2nd Ed. Begin if (H1^.left = null) then // H1 is a single node H1^.left H2 else begin H1^.right merge(H1^.right, H2) if (H1^.left^.npl < H1^.right^.npl) then swap(H1^.left, H1^.right); H1^.npl H1^.right^.npl + 1 end return(H1) end Skew Heaps Binomial Heaps A binomial heap is a data structure for which A binomial heap is a forest of heap-ordered trees satisfying: 1. 2. 3. Height 0 Height 1 Height 2 Observations about the trees a. b. c. Height 3 Binomial heap operations Merging two binomial heaps H 1: H 2: Step 1: Merge trees of height 0 to get Step 2: Choose one of the trees of height 1 to go into the result and merge the other two. (The choice is arbitrary.) Step 3: We now have two trees of height 2 so merge them. Insertions and Deletions H 1: H 2: 1st step 2nd step Complexity of building a binomial heap of size n TW(n) = 3rd step Delete min Complexity Implementation Considerations in deciding the representation: 1. 2. 3. One implementation solution