Tree Vertex Splitting In Data Structure

[Yvone Samiento]

Tree vertex splitting is often used in tree-related algorithms, such as tree traversal algorithms, for example, bfs and dfs and tree decomposition algorithms, for instance, finding tree decompositions for graph problems and dynamic programming on trees.

In the Design and Analysis of Algorithms (DAA) context, "tree vertex splitting" generally refers to a technique used in algorithms involving trees. One example of an operation that could be considered a tree split is found in a data structure with dynamic operations. This operation divides the tree containing vertex v into two parts by deleting the edge from v to its parent. This operation assumes that v is not a tree root.

In tree vertex splitting, a single vertex is divided into multiple vertices. Each new vertex retains one of the edges initially connected to the original vertex. This process results in a forest of trees, where each tree corresponds to one of the new vertices.

Mathematically, if we have a tree T and a vertex v in T, the vertex split operation on v results in a new graph T'. If v were of degree d in T, it would result in d new vertices in T', each of degree 1.

In Tree Vertex Splitting, you choose a node in the tree. This selected vertex is then "split" or separated from the tree, resulting in two separate trees. The first one contains the chosen vertex and all vertices in the subtree rooted at this node.

The second tree includes all the other vertices. For example, if we consider a tree with vertices 1, 2, 3, 4, and 5, 1 is the root with two children: 2 and 3. 2 has a child: 4. 3 has a child: 5. If we perform a Tree Vertex Splitting at node 3, we end up with two trees. The first tree contains vertices 3 and 5. The second tree has vertices 1, 2, and 4.

Tree vertex splitting has various applications, such as placing flip-flops in partial scan designs, latches in pipelined circuits, signal boosters in courses and networks, and more. This technique divides networks into subnetworks for better efficiency and security. It simplifies complex problems in algorithms and is used in the greedy method for process scheduling in operating systems.

Tree vertex splitting is a fundamental operation used in many graph algorithms. It is beneficial in algorithms that deal with connectivity and network flow problems. By splitting vertices, these algorithms can simplify the problem and make it easier to find optimal solutions.

The complexity of tree vertex splitting can vary depending on the specific algorithm or data structure used. For example, the time complexity of the vCover function is O(n), where n is the number of nodes in the binary hierarchical data structure. This is because each node is visited only once, and its vertex cover size is calculated in constant time. The space complexity of the program is O(n), where n is the number of nodes in the binary tree.

Tree vertex splitting can be compared with other techniques like the greedy and divide and conquer algorithms. The greedy algorithm builds up a solution piece by piece, always choosing the following article that offers the most prominent and immediate benefit. In the divide and conquer algorithm, the problem is broken down into small parts, called sub-problems, and then solved individually and combined to get the final solution.

While tree vertex splitting is a powerful technique, it presents several challenges. The operation can significantly increase the graph's size, leading to increased computational complexity. Care must also be taken to ensure the algorithm correctly handles the split vertices.

Tree vertex splitting is a powerful technique in graph theory that allows for the simplification of complex problems. By breaking down a single vertex into multiple vertices, we can transform a complex tree structure into a simpler forest of trees. This operation is particularly useful in algorithms dealing with connectivity and network flow problems.

Hello, I would like to present some Questions i got about space Partitioning in OpenGL.
First and foremost i want to ask about the data structure types that are used to divide the space in slices. I am interested in those:

BSP trees are the most general. Each node partitions the space with arbitrary plane. Kd trees have the restriction that the planes are axis-aligned, and successive levels cycle through the axes. Octrees are further constrained in that each plane passes through the centroid of parent region, splitting into two equal halves. A consequence of that is that groups of 3 consecutive levels split the space into 8 equal cubes, so you normally consider groups of three planes (one for each axis) simultaneously.

Kd trees and octrees work best when the geometry tends to be axis-aligned. Octrees work best when the geometry is aligned to a regular grid (e.g. geometry generated from voxel data by the Marching Cubes algorithm).

I was looking more for an answer to my previous post , what i dont seem to get is if the nodes arent supposed to contain data(read that on another thread) , but only the leaves how is the testing perfromed ? what do you test with ?
I cant seem to grasp how they work on algorithmic level not geometrical .
For example i want to take a model and push it into an ocree , partition its vertices (triangles?) and perfrom more efficient ray-triangle intersection.

I was looking more for an answer to my previous post , what i dont seem to get is if the nodes arent supposed to contain data(read that on another thread) , but only the leaves how is the testing perfromed ? what do you test with ?

[QUOTE=Asmodeus;1279075]- what happens if the model is translated in space (or is translatING in space during the run time)
[/QUOTE]
The octree would be in object space. To test for intersection of a ray with a transformed object, you transform the ray into the space of the object and perform calculations in object space. Consequently, the geometry remains fixed relative to the octree.

IIRC, the idTech engines (which were based upon BSP trees) used BSP trees for static geometry and also for objects which moved along a constrained path (doors, elevator platforms, etc). They treated the swept path as a single object for the purpose of generating the tree, so that only the corresponding subtree(s) needed to be modified at run time as the object moved.

Also the other obvious solution to that is maybe to have a VBO/VAO for each leaf , then only rebind the leaves which intersect the frustum. So here arises the Question is rebinding bettter than rebuilding ? Guess yes ?

I will post this as a final note at what i want to acomplish, using multiple levels of tests.
The idea is that i have some kind of a BVH(maybe octree of AABBs) which will contain the Bounding Volumes of each object in a given scene. Then i will perfrom frustum tests on that structure to determine the nodes (and respectivly the models in that node) that are visible or not. Those that are not visible are simply not drawn, Those that are visible have 2 options:

ALSO: How would i fit multiple objects bounding boxes in an octree (leaf/node?!) for the first test pass, where i should test against the objects as a whole. I cant actually grasp how i can devide the world space and put the bounding boxes(spheres) of the objects inside that octree

I worked out the above Questions after some time. But i was wondering, since my geometry is split between all leaves, so each leaf has some chunk of the terrain vertices (for now just vertices,no indices involved to avoid complications). So i am using flat out buffer where all vertices are ordered according to the indices, ready to be drawn. But since my geometry is split , how sould i specifiy offsets to the glVertexAttribPointer (or rather how would one obtain those offsets per leaf)
I want to draw with just specifying offset to the buffer without having to rebind or recreate a vbo for each leaf every time.
Thanks in Advance !

Yea thats what i figured , but how would one find start. I have the count available, but how about the start. Should i get the first vertex from the node to be the start vertex ?
Also cant i use glDrawElements/Arrays to specify offset such as:
glDrawArrays(GL_TRIANGLES,first,count) ?

In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time[1] and has several applications in dynamic graph algorithms and graph drawing.

The basic structures underlying the SPQR tree, the triconnected components of a graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by Saunders Mac Lane (1937); these structures were used in efficient algorithms by several other researchers[2] prior to their formalization as the SPQR tree by Di Battista and Tamassia (1989, 1990, 1996).

An SPQR tree takes the form of an unrooted tree in which for each node x there is associated an undirected graph or multigraph Gx. The node, and the graph associated with it, may have one of four types, given the initials SPQR:

Typically, it is not allowed within an SPQR tree for two S nodes to be adjacent, nor for two P nodes to be adjacent, because if such an adjacency occurred the two nodes could be merged into a single larger node. With this assumption, the SPQR tree is uniquely determined from its graph. When a graph G is represented by an SPQR tree with no adjacent P nodes and no adjacent S nodes, then the graphs Gx associated with the nodes of the SPQR tree are known as the triconnected components of G.

The problem of constructing the triconnected components of a graph was first solved in linear time by Hopcroft & Tarjan (1973). Based on this algorithm, Di Battista & Tamassia (1996) suggested that the full SPQR tree structure, and not just the list of components, should be constructible in linear time. After an implementation of a slower algorithm for SPQR trees was provided as part of the GDToolkit library, Gutwenger & Mutzel (2001) provided the first linear-time implementation. As part of this process of implementing this algorithm, they also corrected some errors in the earlier work of Hopcroft & Tarjan (1973).