3/7/13. Binomial Tree. Binomial Tree. Binomial Tree. Binomial Tree. Number of nodes with respect to k? N(B o ) = 1 N(B k ) = 2 N(B k-1 ) = 2 k

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1 //1 Adapted from: Kevin Wayne B k B k B k : a binomial tree with the addition of a left child with another binomial tree Number of nodes with respect to k? N(B o ) = 1 N(B k ) = 2 N( ) = 2 k B 1 B 2 B 1 B 2 B B 4 B 1 B 2 B B 4 B k B k Height? Degree of root node? B 1 B 2 B 1 B 2 H(B o ) = 1 H(B k ) = 1 H( ) = k k, each time we add another binomial tree B 1 B 2 B B 4 B 1 B 2 B B 4 1

2 //1 What are the children of the root? B k B 1 B 2 Why is it called a binomial tree? k binomial trees:, B k-2,, depth 0 depth 1 depth 2 depth B 1 B 2 B B 4 depth 4 B 4 Binomial Heap k i B k has nodes at depth i. 4 = 2 depth 0 Binomial heap Vuillemin, 198. Sequence of binomial trees that satisfy binomial heap property: each tree is min-heap ordered top level: full or empty binomial tree of order k which are empty or full is based on the number of elements depth 1 depth depth 45 2 depth 4 B 4 2

3 //1 Binomial Heap Like our set data structure from last time, except binomial tree heaps instead of arrays A 0 : [] A 1 : [, ] A 2 : empty A : empty A : [, 8, 29, 10,,, 2, 22, 48, 1, 1, 45, 2,, ] How many heaps? Binomial Heap: Properties O(log n) binary number representation N = 19 # trees = height = 4 binary = N = 19 # trees = height = 4 binary = Binomial Heap: Properties Binomial Heap: Properties Where is the max/min? Must be one of the roots of the heaps Runtime of max/min? O(log n) N = 19 # trees = height = 4 binary = N = 19 # trees = height = 4 binary = 10011

4 //1 Height? Binomial Heap: Properties floor(log 2 n) - largest tree = B log n - height of that tree is log n How can we merge two binomial tree heaps of the same size (2 k )? connect roots of H' and H'' choose smaller key to be root of H Runtime? O(1) N = 19 # trees = height = 4 binary = H' H'' Go through each tree size starting at 0 and merge as we go How can we combine/merge binomial heaps (i.e. a combination of binomial tree heaps)? 19 =

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6 // Analogous to binary addition Running time? Proportional to number of trees in root lists 2 O(log 2 N) O(log N) Binomial Heap: Delete Min/Max We can find the min/max in O(log n). How can we extract it? Hint: B k consists of binomial trees:, B k-2,, 19 = H

7 //1 Binomial Heap: Delete Min Delete node with minimum key in binomial heap H. Find root x with min key in root list of H, and delete H' broken binomial trees H Union(H', H) Binomial Heap: Delete Min Delete node with minimum key in binomial heap H. Find root x with min key in root list of H, and delete H' broken binomial trees H Union(H', H) Running time? O(log N) H H Binomial Heap: Decrease Key Just call Decrease-Key/Increase-Key of Heap Suppose x is in binomial tree B k Bubble node x up the tree if x is too small Running time: O(log N) Proportional to depth of node x Binomial Heap: Delete Delete node x in binomial heap H Decrease key of x to - Delete min Running time: O(log N) depth = H x 2

8 //1 Binomial Heap: Insert Insert a new node x into binomial heap H H' MakeHeap(x) H Union(H', H) Running time. O(log N) Build-Heap Call insert n times Runtime? O(n log n) Can we get a tighter bound? x H H' 45 2 Build-Heap Build-Heap Call insert n times Consider inserting n numbers times how many times will be empty? how many times will we need to merge with? how many times will we need to merge with B 1? how many times will we need to merge with B 2? how many times will we need to merge with B log n? cost Call insert n times Consider inserting n numbers times cost how many times will be empty? n/2 O(1) how many times will we need to merge with? n/2 O(1) how many times will we need to merge with B 1? n/4 O(1) how many times will we need to merge with B 2? n/8 O(1) how many times will we need to merge with B log n? 1 O(1) Runtime? Θ(n) 8

9 //1 Heaps Fibonacci Heaps Similar to binomial heap A Fibonacci heap consists of a sequence of heaps More flexible Heaps do not have to be binomial trees More complicated Min [H] Heaps Heaps Should you always use a Fibonacci heap? Extract-Max and Delete are O(n) worst case Constants can be large on some of the operations Complicated to implement 9

10 //1 Heaps Can we do better? 10