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**1 - 7**of**7**### 6.851: Advanced Data Structures Spring 2007

, 2007

"... The topic of this lecture is the fusion tree, another data structure to solve the predecessor/successor problem. Given a static set S ⊆ {0, 1, 2,..., 2 w − 1}, fusion trees can answer predecessor/successor queries in O(log w n). Essentially, fusion trees are B-trees with a branching factor of k = Θ( ..."

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The topic of this lecture is the fusion tree, another data structure to solve the predecessor/successor problem. Given a static set S ⊆ {0, 1, 2,..., 2 w − 1}, fusion trees can answer predecessor/successor queries in O(log w n). Essentially, fusion trees are B-trees with a branching factor of k = Θ(w1/5). Tree height is

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"... Last lecture we covered dynamic trees, also known as link-cut trees. Link-cut trees are able to represent a dynamic forest of rooted trees in O(log n) amortized time per operation. In this lecture we will see how to maintain connectivity information for general graphs. We will start by examining a s ..."

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Last lecture we covered dynamic trees, also known as link-cut trees. Link-cut trees are able to represent a dynamic forest of rooted trees in O(log n) amortized time per operation. In this lecture we will see how to maintain connectivity information for general graphs. We will start by examining a simpler, although not strictly better, alternative to link-cut trees known as Euler-tour trees. Then, for the special case of Decremental Connectivity, in which edges may only be deleted, we will achieve constant runtime. We will then use Euler-tour trees to achieve dynamic connectivity in general graphs in O(log 2 n) time. Finally we will survey some of what is and is not known for dynamic graphs. The next class will be about lower bounds. 2 Dynamic Connectivity Our goal is to maintain an undirected graph subject to: • insert/delete of edges or vertices (with no edges). • connected(u, v): whether u and v are connected, on general undirected graphs that are subject to vertex and edge insertion and deletion. • Another possible query is to determine whether the entire graph is connected; though this may seem easier, the current lower and upper bounds are the same as for pairwise connectivity.

### 6.851: Advanced Data Structures Spring 2011

, 2012

"... In the next two lectures we study the question of dynamic optimality, or whether there exists a binary search tree algorithm that performs ”as well ” as all other algorithms on any input string. In this lecture we will define a binary search tree as a formal model of computation, show some analytic ..."

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In the next two lectures we study the question of dynamic optimality, or whether there exists a binary search tree algorithm that performs ”as well ” as all other algorithms on any input string. In this lecture we will define a binary search tree as a formal model of computation, show some analytic bounds that a dynamically optimal binary search tree needs to satisfy, and show two search trees that are conjectured to be dynamically optimal. The first is the splay tree, which we will cover only briefly. The second will require us to build up a geometric view of a sequence of binary search tree queries. 2 Binary Search Trees The question we will explore in this lecture is whether or not there is a ”best ” binary search tree. We know that there are self-balancing binary search trees that take O(logn) per query. Can we do better? In order to explore this question, we need a more rigorous definition of binary search tree. In this lecture we will treat the binary search tree (BST) as a model of computation that is a subset of the pointer machine model of computation. 2.1 Model of Computation A BST can be viewed is a model of computation where data must be stored as keys in a binary search tree. Each key has a pointer to its parent (unless it is the root) and a pointer to its left and right children, or a null pointer if they do not exist. The value of the key stored in the left child of a node must be less than or equal to the value of the key stored at the node, which is in turn less than or equal to the value of the key stored at the right child. The model supports the following unit-cost operations: • Walk to left child. • Walk to right child. • Walk to parent. • Rotate node x. 1 p x x

### The Vulcan game of Kal-toh: Finding

"... or making triconnected planar subgraphs by ..."

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