Merging Priority Queues

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A long right path is still possible However, it rarely occurs and must be preceded by many merges involving short right paths

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We expect the result of the child swapping to be that the length of the right path will not be unduly large all the time For instance, if we merge a pair of long right-path trees, the nodes involved in the path do not reappear on a right path for quite some time in the future Obtaining trees that have the property that every node appears on a right path is still possible, but that can be done only as a result of a large number of relatively inexpensive merges In Section 2314, we prove this assertion rigorously by establishing that the amortized cost of a merge operation is only logarithmic

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2314 Analysis of the Skew Heap

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The actual cost of a merge is the number of nodes on the right paths of the two trees that are merged

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Suppose that we have two heaps, H, and H, and that there are r l and r2 nodes on their respective right paths Then the time required to perform the merge is proportional to r , + r, When we charge 1 unit for each node on the right paths, the cost of the merge is proportional to the number of charges Because the trees have no structure, all the nodes in both trees may lie on the right path This condition would give a O(N) worst-case bound for merging the trees (in Exercise 234 you are asked to construct such a tree) As we demonstrate shortly, the amortized time needed to merge two skew heaps is O(l0g N) As with the splay tree, we introduce a potential function that cancels the varying costs of skew heap operations We want the potential function to increase by a total of O(1og N) - ( r , + r,) so that the total of the merge cost and potential change is only O(1og N) If the potential is minimal prior to the first operation, applying the telescoping sum guarantees that the total spent for any M operations is O(M log N),as with the splay tree What we need is some potential function that captures the effect of skew heap operations Finding such a function is quite challenging Once we have found one, however, the proof is relatively short

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DEFINITION: A node is a heavy node if the size of its right subtree is larger than the size of its left subtree Otherwise, it is a light node; a node is light if its subtrees are of equal size

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The potential function is the number of heavy nodes Only nodes on the merged path have their heavy or light status changedThe number of light nodes on a right path is logarithmic

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In Figure 233, prior to the merge, nodes 3 and 4 are heavy After the merge, only node 3 is heavy Three facts are easily shown First, as a result of a merge, only nodes on the right path can have their heavy or light status changed because no other nodes have their subtrees altered Second, a leaf is light Third the number of light nodes on the right path of an N node tree is at most h o g N ] + 1 The reason is that the right child of a light node is less than half the size of the light node itself, and the halving principle applies The additional +I is a result of the leaf's being light With these preliminaries, we can now state and prove Theorems 231 and 232

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The Skew ~

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Change in the heavy or light status of nodes after a merge

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Let HI and H , be two skew heaps with Nl and N2 nodes, respectively, and let N be their combined size (that is, N1 + N2)Suppose that the right path of H I has 1, light nodes and h, heavs nodes,for a total of I , + h , , whereas the right path of H , has 1, light nodes and h2 heavy nodes, for a total of I - + h2 If the potential is dejined as the total number of heavy nodes in the , collection of skew heaps, then the merge costs at most 2 log N + ( h , + h2), but the change in potential is at most 2 log N - ( h , + h2) The cost of the merge is merely the total number of nodes on the right paths, I , + 1, + h, + h2 The number of light nodes is logarithmic, so 1,1LlogN,]+ 1 and I , 1 ~ l o g N , ~ + 1 T h u s l , + 1 ~ 5 l o g N ~ + l o g N ~ + 2 5 2 log N, where the last inequali~ follows from Theorem 224 The merge cost is thus at most 2 log N + ( h l + h2)The bound on the potential change follows from thefact that only the nodes involved in the merge can have their heavy/light status changed and from the fact that any heavy node on the path must become light because its children are swapped Even if all the light nodes became heavy, the potential change would still be limited to 1, + l2 - (11, + h2)Based on the same argument as before, that is at most 2 log N - ( h , + h2)

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