The first part represents the stiffness of the games and the second part represents the sampling of the graph. Reversi is proved to be PSPACE-Complete using Bounded 2CL, Peg Solitaire is proved to be NP-Complete using Bounded NCL. The objective of the study of the pin diamond of the special graphic classes is to find the maximum number of marbles that can be removed from a completely filled board, if the player is given the privilege of removing a marble from each cell first, then following the post-move rules initial.
Game
As computer scientists, we find that games and puzzles act as powerful models of computation, quite different from the usual models of automata and circuits, and offer a new way of thinking about computation.
Bounded Games
Unbounded Games
One Player Games
Two Player Games
Team Games
Constraint Logic Formalism
Constraint Graph
Basic vertices for One player games
Bounded Games
It looks like the 2nd weight from the blue end is blowing through the two red ends. It looks like we came from a lead and chose one of the 2 remaining ends to move forward.
Basic vertices for Two Player Games
Bounded Games
If the white player chooses this variable first, then he flips the white edge and now the black player cannot choose this variable and flip this edge. Similarly, if the black player chooses first, only the black player can turn the black edge.
Introduction
Basic Gadgets
Then the black player can tap the G cell and turn the white facing cell H. So if the white player can tap the B cell, he will finally touch the D cell. If the white player taps the R cell, then it is not a valid move to tap the B cell.
Peg Solitaire
Introduction
Basic Gadgets
If a player can place a marble on cell marked A, he can jump over the peg and reach cell D. Or if the player can place a marble on cell marked B, they can jump over the peg and reach cell D. the player can reach cell A and then reach cell B by making a series of moves.
Peg solitaire on special Graph Classes
Graphs Studied
If the initial hole is on a hanging vertex, then only one peg can be removed from the graph. To solve this path, you should be able to fill the holes of the empty bridge. If the initial hole is at point 0, then the bridge cannot be filled and therefore P2 is not freely solvable.
In a single move, two pegs are removed from one group and one peg is added to another group. Every time two pegs are removed from the odd side, an empty bridge is formed, and the size of nodes on the odd side decreases by 2. After K such moves, there will be an empty bridge that has only one peg on one side, which is not soluble.
Since the number of vertices is odd, the last empty bridge cannot be filled to solve the path. The problem reduced to Pn-2 which is solvable when the hole is placed at (n-1)th vertex. The problem reduced to Pn−2 which is solvable when the hole is placed at (n-1)th vertex.
As we know Pn where (n-2) is odd is 2 distance solvable, then the cycle Cn where n is odd is 2 distance solvable Figure 2.28 follows the steps described above.
Badland
- Introduction
- Version of the Game
- Decision Problem
- Theoretical Problem definition corresponding to a single level
Given a complete level in a single screen, the player can reach the end of this level facing the obstacles on the way. Given a complete level of Badland on a single screen, it is NP-complete to decide whether the player can reach the end of this level to face the obstacles on the way. Badland is clearly in NP: The player can guess a path and verify it by playing the level in polynomial time.
The player spawns through the portal at the top and exits this gadget through one of the portals at the bottom. If this variable gadget matches variable The player spawns from one of the portals at the top and taps the corresponding switch that turns off the high-speed fan and exits this gadget through the portal at the bottom.
After playing all the variable gadgets and clause gadgets, the player has to go through a tunnel with death balls and fast fans. If any of the high-speed fans are turned on, the fan will force the player into deathballs and the game will be over. Once all the fans are turned off, the player can go through the tunnel and reach the green portal and win the level.
A tunnel is created with high-speed fans and death balls, which correspond to clauses in F (x, y, z).
Introduction
Problem definition
Sampling Methods
- Random Node Sampling (RN)
- Random Edge Sampling (RE)
- Tightly Induced Edge Sampling (TIES)
- Forest Fire Sampling (FFS)
- Karger Sampling (KS)
- Breadth First Search Sampling
- Tightly Induced Breadth First Search Sampling
The random node sampling produces a sample by randomly selecting a subset of vertices and taking the induced subgraph. Random Edge Sampling produces a sample by randomly selecting a subset of edges from the original graph. The dense induced edge sampling produces a sample by randomly selecting a subset of edges from the original graph and sampling the induced subgraph.
This process above is applied recursively to each burnt neighbor until our budget (ie sample size) is reached. This is a contraction based algorithm where an edge is contracted to produce a new vertex labeled with the vertices of the edge, all the edges from the two vertices are added to this newly formed vertex. So this contraction of edges continues until a vertex meets the budget or there is no edge left to contract.
In the first case we take the induced subgraph formed by a vertex, in the second case we take a fraction of the vertices from each vertex to achieve our budget. This is the same traditional BFS algorithm, in which the initial node is chosen randomly, and then nodes are discovered one by one, adding nodes to the sample until the budget is reached. delete points from sampleNodes if exceeds sampleNodeCount */. 37) if size(sampleNodes)≥sampleNodeCount then. Then, in the induction step, all the edges present in the graph G are added between each sampled node.
BFS has been used by researchers in the past, but BFS together with the careful induction step is not studied before.
Experimental Evaluation
Datasets
Properties
The clustering coefficient of a vertex is a measure of how close its neighbors are to a clique. A connected component is a subgraph in which every vertex is connected to every other vertex and is not connected to any other vertex in the supergraph. In our work we looked at the component with the largest size among all components in the given graph.
If the LCC of the sample graph is very less compared to the expected size, it is hardly possible that A can be reached from B or vice versa. On the other hand, if the LCC is the same as the expected LCC of the sample, it is highly likely that both nodes are connected. If we prioritize node availability more, we should consider an algorithm that creates a sample whose LCC is the same as the expected LCC of the sample.
Examples include the average number of hops to transport data packets from source to destination, the average number of clicks that will take you from one website to another. If the LCC of two samples is almost the same, the algorithm that exhibits lower ASPL is better than the other algorithm. Consider an unweighted directed graph G = (V, E), where V is the set of vertices and E is the set of edges.
Evaluation Measures
Kolmogorov-Smirnov (KS)
Kullback-Leibler (KL)
To solve this problem, we use Skew-Divergence which smoothes the PDFS before calculating the KL divergence. After smoothing the binomial distribution, we can use the KL formula to evaluate the KL divergence.
Results
Largest Connected Component Size Figures 3.12 a, 3.12 b and 3.12 c represent the largest connected component (LCC) size of graphs. The probability of selecting a seed node from LCC is high compared to nodes in other components. So each of them samples all the nodes from that component, thus preserving the LCC size completely.
TIES performs better in case of denser graphs and KARGER, TIBFS performs better in case of sparse graph. ES, TIES, NS, BFS divides the tree into several small components as it also appears from the Fig.3.13 a, b, c. The LCC of trees sampled by BFS, TIBFS, KARGER is the same but KARGER has the largest ASPL, so BFS and TIBFS are preferable to KARGER in the case of trees.
In the case of general graphs, TIES works well, but TIBFS and KARGER are a bit far from the actual curve. The traversal-based algorithm (TIBFS, BFS) overestimates the size of the LCC by sampling all nodes from the LCC of the original graph. For trees, KARGER and TIBFS did not perform well because it does not split the tree into components unlike other algorithms.
In case of trees, again traversal based algorithm like TIBFS, BFS, KARGER have less bias than other algorithms.
Conclusion
-statisticFor each factor, we have already done 10 samples. we find the average KS distance for these 10 samples and we do the same for all datasets. we have considered degree distribution and cluster coefficient distribution for evaluation. Overall, the sampling algorithms, which include induction steps as part of its algorithm, perform well for both properties. Coming to trees TIES lose their advantage at induction stages and do not work well.
Skewed divergence KS statistic evaluates the maximum distance between two given distributions, but the skewed divergence shown in Fig. We can observe that for graphs TIES, KARGER, TIBFS exhibit less skewness than any other algorithm. In the future, more analysis of the algorithms can be done on special graphs such as a complete graph, cycle graph, bipartite graph and so on.
