User:Subh83/CommonsContrib

Animated GIFs illustrating Dijkstra's, A* and weighted A* search algorithms

Dijkstra's algorithm uses a heuristic identically equal to 0. Nodes in all the different directions are explored uniformly.

A* search is informed by a heuristic, hence more efficient than Dijkstra's, yet optimal when a consistent heuristic is used.

A* search that uses a heuristic that is 5.0(=ε) times a consistent heuristic, and obtains a suboptimal path (weighted A*).

The above figures illustrate a typical example from motion planning in robotics using 3 different search algorithms (all variants of A*). The graph is created by uniform square discretization of a 2-dimensional planar region, placing a node in each discretized cell, and connecting each node with its 8 neighbors using bidirectional edges. Cost of edges are same as their Euclidean lengths. The gray shape represents an obstacle. The filled circles in red & green represent expanded nodes (nodes in closed set). The color indicate the g-score (red: lower g-score, green: higher g-score). The empty nodes with blue boundary are the ones in open set. The nodes of the graph are generated on the fly, and nodes falling inside the obstacle are discarded as inaccessible. The objective is to find the least cost path (restricted to the graph) from a given start to a given goal node. A consistent heuristic in this problem is the Euclidean distance of a node to goal. That is, h_a(n) = |g - n|₂, where n represents the Euclidean coordinate of the node n, g is the goal node, and | . |₂ is the 2-norm.

These animations were created using YGSBPL graph-search library and OpenCV.

Animated GIF illustrating simple construction of a quotient space

Illustration of quotient space S2 obtained by gluing the boundary (in blue) of the disk D2 to a single point.

The adjacent figure illustrates construction of a simple quotient space. The boundary (in blue) of a 2-disk when identified (glued together) to a single point, gives the topological 2-sphere. In general, $S^{n}=D^{n}/\partial D^{n}$. More precisely, it is the adjunction space $x\cup _{f}D^{n}$, where $x$ is a single point, and with $f:\partial D^{n}\rightarrow x$ such that $f(p)=x,\forall p\in \partial D^{n}$. The animation was created using MATLAB and ImageMagick.

Functors and Category theory related Illustrations

Below are a couple of simple illustrations for the Hom functor, hoping it will help people to grasp the concept more easily, as well as help in remembering the definition of the functor.

Illustration of Hom functor (covariant).

Fix $A\in {\mathcal {C}}$. Then there are sets of morphisms $Hom(A,X)=\{\alpha _{1},\alpha _{2},\cdots \}$, $Hom(A,Y)=\{\beta _{1},\beta _{2},\cdots \}$, etc. Also, for $f:X\rightarrow Y$, we define a map $Hom(A,f):Hom(A,X)\rightarrow Hom(A,Y)$, i.e. $Hom(A,f):\{\alpha _{1},\alpha _{2},\cdots \}\rightarrow \{\beta _{1},\beta _{2},\cdots \}$ by $\alpha \mapsto f\circ \alpha$ (illustrated by the dashed arrow). Thus $Hom(A,-)$ is a covariant functor.