Sp1ayTreeh SplayTreecpp TestSplaylkeecpp

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Contains the interface for the SplayTree class Contains the implementation for the SplayTree class Contains a test program for the SplayTree class

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Exercises

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In Short 221 Show the result of inserting 3, I, 4 , 5 , 2 , 9 , 6 , and 8 into a a bottom-up splay tree b top-down splay tree

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222 Show the result of deleting 3 from the splay tree shown in Exercise 221

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for both the bottom-up and top-down versions

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In Theory

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223 Prove that the amortized cost of a top-down splay is O(1og N) 224 Prove that if all nodes in a splay tree are accessed in sequential

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order, the resulting tree consists of a chain of left children

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225 Suppose that, in an attempt to save time, we splay on every second

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tree operation Does the amortized cost remain logarithmic

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226 Nodes I through N = 1024 form a splay tree of left children

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a What is the internal path length of the tree (exactly) b Calculate the internal path length after each of find ( 1), find (2) , and find ( 3 ) when a bottom-up splay is performed By changing the potential function, you can prove different bounds for splaying Let the weight function W ( i )be some function assigned to each node in the tree and S(i) be the sum of the weights of all nodes in the subtree rooted at i, including i itself The special case W(i) I for all nodes corresponds to the function used in the proof = of the splaying bound Let N be the number of nodes in the tree and M be the number of accesses Prove the following two theorems a The total access time is O(M + (M + N ) log N) b If q, is the total number of times that item i is accessed and qi > 0 N for all i, then the total access time is O(M + = q, log(Mlq,))

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xi ,

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In Practice

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228 Use the splay tree to implement a priority queue class 229 Modify the splay tree to support order statistics 2210 Implement the SplayTree operator= and copy constructor correctly If necessary, you may change the rhs parameter to be a nonconstant reference (such a change is allowed by the standard)

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Programming Projects

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2211 Compare empirically the simplified top-down splay implemented in Section 226 with the original top-down splay discussed in Section 225

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2212 Unlike balanced search trees, splay trees incur overhead during a

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find operation that can be undesirable if the access sequence is sufficiently random Experiment with a strategy that splays on a find operation only after a certain depth d is traversed in the top-down search The splay does not move the accessed item all the way to the root, but rather to the point at depth d where the splaying is started

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2213 Compare empirically a top-down splay tree priority queue implementation with a binary heap by using a random insert and deleteMin operations b insert and deleteMin operations corresponding to an eventdriven simulation C insert and deleteMin operations corresponding to Dijkstra's algorithm

References

The splay tree is described in the paper [3] The concept of amortized analysis is discussed in the survey paper [4] and also in greater detail in [5] A comparison of splay trees and AVL trees is given in [I], and [2] shows that splay trees perform well in some types of event-driven simulations

1 J Bell and G Gupta, "An Evaluation of Self-Adjusting Binary Search Tree Techniques," Sofmure-Practice and Experience 23 (1993), 369-382 2 D W Jones, "An Empirical Comparison of Priority-Queue and Event-Set Implementations," Communications of the ACM 29 (1986), 300-31 1 3 D D Sleator and R E Tarjan, "Self-adjusting Binary Search Trees," Jourrzal of the ACM 32 (1985), 652-686 4 R E Tarjan, "Amortized Computational Complexity," SlAM Journal on Algebraic and Discrete Methods 6 (I 985), 306-3 18 5 M A Weiss, Data Structures and Algorithm Analysis irz C++, 2d ed, Addison-Wesley, Reading, Mass, 1999

23

Merging Priority Queues

In this chapter we examine priority queues that support an additional operation: The merge operation, which is important in advanced algorithm design, combines two priority queues into one (and logically destroys the originals) We represent the priority queues as general trees, which simplifies somewhat the decreaseKey operation and is important in some applications In this chapter, we show: how the skew heap-a mergeable priority queue implemented with binary trees-works how the pairing heap-a mergeable priority queue based on the M-ary tree-works The pairing heap appears to be a practical alternative to the binary heap even if the merge operation is not needed