e.r diagram on hspital management

1. Capital expenditures (CAPEX or capex) are expenditures creating future benefits.
A capital expenditure is incurred when a business spends money either to buy fixed assets
or to add to the value of an existing fixed asset with a useful life that extends beyond the
taxable year. Included in capital expenditures are amounts spent on:
1. acquiring fixed assets
2. fixing problems with an asset that existed prior to acquisition
3. preparing an asset to be used in business
4. legal costs of establishing or maintaining one's right of ownership in a piece of
property
5. restoring property or adapting it to a new or different use
6. starting a new business
For tax purposes, capital expenditures are costs that cannot be deducted in the year in which they are paid or
incurred, and must be capitalized.
An operating expense, operating expenditure, operational expense, operational
expenditure or OPEX is an on-going cost for running a product, business, or systemFor
example, the purchase of a photocopier is the CAPEX, and the annual paper and toner cost
is the OPEXOn an income statement, quot;operating expensesquot; is the sum of a business's
operating expenses for a period of time, such as a month or year.

2. Dijkstra's algorithm, conceived by Dutch computer
scientist Edsger Dijkstra in 1959, [1] is a graph search algorithm that solves the single-
source shortest path problem for a graph with nonnegative edge path costs, producing a
shortest path tree. This algorithm is often used in routing.
For a given source vertex (node) in the graph, the algorithm finds the path with lowest cost
(i.e. the shortest path) between that vertex and every other vertex. It can also be used for
finding costs of shortest paths from a single vertex to a single destination vertex by
stopping the algorithm once the shortest path to the destination vertex has been determined.
For example, if the vertices of the graph represent cities and edge path costs represent
driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm
can be used to find the shortest route between one city and all other cities. As a result, the
shortest path first is widely used in network routing protocols, most notably IS-IS and
OSPF (Open Shortest Path First).
Contents
[hide]
1 Algorithm
•
2 Description of the algorithm
•
3 Pseudocode
•
4 Running time
•
5 Python implementation
•
6 Related problems and algorithms
•
7 See also
•
8 Notes
•
9 References
•
10 External links
•
[edit] Algorithm

3. Let's call the node we are starting with an initial node. Let a distance of a node X be the
distance from the initial node to it. Dijkstra's algorithm will assign some initial distance
values and will try to improve them step-by-step.
1. Assign to every node a distance value. Set it to zero for our initial node and to
infinity for all other nodes.
2. Mark all nodes as unvisited. Set initial node as current.
3. For current node, consider all its unvisited neighbours and calculate their distance
(from the initial node). For example, if current node (A) has distance of 6, and an
edge connecting it with another node (B) is 2, the distance to B through A will be
6+2=8. If this distance is less than the previously recorded distance (infinity in the
beginning, zero for the initial node), overwrite the distance.
4. When we are done considering all neighbours of the current node, mark it as
visited. A visited node will not be checked ever again; its distance recorded now is
final and minimal.
5. Set the unvisited node with the smallest distance (from the initial node) as the next
quot;current nodequot; and continue from step 3
[edit] Description of the algorithm
Suppose you create a knotted web of strings, with each knot corresponding to a node, and
the strings corresponding to the edges of the web: the length of each string is proportional
to the weight of each edge. Now you compress the web into a small pile without making
any knots or tangles in it. You then grab your starting knot and pull straight up. As new
knots start to come up with the original, you can measure the straight up-down distance to
these knots: this must be the shortest distance from the starting node to the destination
node. The acts of quot;pulling upquot; and quot;measuringquot; must be abstracted for the computer, but
the general idea of the algorithm is the same: you have two sets, one of knots that are on
the table, and another of knots that are in the air. Every step of the algorithm, you take the
closest knot from the table and pull it into the air, and mark it with its length. If any knots
are left on the table when you're done, you mark them with the distance infinity.
Or, using a street map, suppose you're marking over the streets (tracing the street with a
marker) in a certain order, until you have a route marked in from the starting point to the
destination. The order is conceptually simple: from all the street intersections of the already
marked routes, find the closest unmarked intersection - closest to the starting point (the
quot;greedyquot; part). It's the whole marked route to the intersection, plus the street to the new,
unmarked intersection. Mark that street to that intersection, draw an arrow with the
direction, then repeat. Never mark to any intersection twice. When you get to the
destination, follow the arrows backwards. There will be only one path back against the
arrows, the shortest one.
[edit] Pseudocode

4. In the following algorithm, the code u := node in Q with smallest dist[], searches
for the vertex u in the vertex set Q that has the least dist[u] value. That vertex is removed
from the set Q and returned to the user. dist_between(u, v) calculates the length
between the two neighbor-nodes u and v. alt on line 11 is the length of the path from the
root node to the neighbor node v if it were to go through u. If this path is shorter than the
current shortest path recorded for v, that current path is replaced with this alt path. The
previous array is populated with a pointer to the quot;next-hopquot; node on the source graph to get
the shortest route to the source.
1 function Dijkstra(Graph, source):
2 for each vertex v in Graph: // Initializations
3 dist[v] := infinity // Unknown distance
function from source to v
4 previous[v] := undefined // Previous node in
optimal path from source
5 dist[source] := 0 // Distance from source
to source
6 Q := the set of all nodes in Graph // All nodes in the graph
are unoptimized - thus are in Q
7 while Q is not empty: // The main loop
8 u := vertex in Q with smallest dist[]
9 remove u from Q
10 for each neighbor v of u: // where v has not yet
been removed from Q.
11 alt := dist[u] + dist_between(u, v) // be careful
in 1st step - dist[u] is infinity yet
12 if alt < dist[v] // Relax (u,v,a)
13 dist[v] := alt
14 previous[v] := u
15 return previous[]
If we are only interested in a shortest path between vertices source and target, we can
terminate the search at line 10 if u = target. Now we can read the shortest path from source
to target by iteration:
1 S := empty sequence
2 u := target
3 while defined previous[u]
4 insert u at the beginning of S
5 u := previous[u]
Now sequence S is the list of vertices constituting one of the shortest paths from target to
source, or the empty sequence if no path exists.
A more general problem would be to find all the shortest paths between source and target
(there might be several different ones of the same length). Then instead of storing only a
single node in each entry of previous[] we would store all nodes satisfying the relaxation
condition. For example, if both r and source connect to target and both of them lie on
different shortest paths through target (because the edge cost is the same in both cases),
then we would add both r and source to previous[target]. When the algorithm completes,
previous[] data structure will actually describe a graph that is a subset of the original graph

5. with some edges removed. Its key property will be that if the algorithm was run with some
starting node, then every path from that node to any other node in the new graph will be the
shortest path between those nodes in the original graph, and all paths of that length from
the original graph will be present in the new graph. Then to actually find all these short
paths between two given nodes we would use a path finding algorithm on the new graph,
such as depth-first search.
[edit] Running time
An upper bound of the running time of Dijkstra's algorithm on a graph with edges E and
vertices V can be expressed as a function of |E| and |V| using the Big-O notation.
For any implementation of set Q the running time is O(|E|*decrease_key_in_Q + |V|
*extract_minimum_in_Q), where decrease_key_in_Q and extract_minimum_in_Q are
times needed to perform that operation in set Q.
The simplest implementation of the Dijkstra's algorithm stores vertices of set Q in an
ordinary linked list or array, and operation Extract-Min(Q) is simply a linear search
through all vertices in Q. In this case, the running time is O(|V|2+|E|)=O(|V|2).
For sparse graphs, that is, graphs with fewer than |V|2 edges, Dijkstra's algorithm can be
implemented more efficiently by storing the graph in the form of adjacency lists and using
a binary heap, pairing heap, or Fibonacci heap as a priority queue to implement the Extract-
Min function efficiently. With a binary heap, the algorithm requires O((|E|+|V|) log |V|)
time (which is dominated by O(|E| log |V|) assuming every vertex is connected, that is, |E| ≥
|V| - 1), and the Fibonacci heap improves this to O( | E | + | V | log | V | ).
[edit] Python implementation
import heapq
from collections import defaultdict
class Edge(object):
def __init__(self, start, end, weight):
self.start, self.end, self.weight = start, end, weight
# For heapq.
def __cmp__(self, other): return cmp(self.weight, other.weight)
class Graph(object):
def __init__(self):
# The adjacency list.
self.adj = defaultdict(list)
def add_e(self, start, end, weight = 0):
self.adj[start].append(Edge(start, end, weight))
def s_path(self, src):
quot;quot;quot;

6. Returns the distance to every vertex from the source and the
array representing, at index i, the node visited before
visiting node i. This is in the form (dist, previous).
quot;quot;quot;
dist, visited, previous, queue = {src: 0}, {}, {}, []
heapq.heappush(queue, (dist[src],src))
while len(queue) > 0:
distance, current = heapq.heappop(queue)
if current in visited:
continue
visited[current] = True
for edge in self.adj[current]:
relaxed = dist[current] + edge.weight
end = edge.end
if end not in dist or relaxed < dist[end]:
previous[end], dist[end] = current, relaxed
heapq.heappush(queue, (dist[end],end))
return dist, previous
For the example graph in the Applet by Carla Laffra of Pace University we do:
g = Graph()
g.add_e(1,2,4)
g.add_e(1,4,1)
g.add_e(2,1,74)
g.add_e(2,3,2)
g.add_e(2,5,12)
g.add_e(3,2,12)
g.add_e(3,10,12)
g.add_e(3,6,74)
g.add_e(4,7,22)
g.add_e(4,5,32)
g.add_e(5,8,33)
g.add_e(5,4,66)
g.add_e(5,6,76)
g.add_e(6,10,21)
g.add_e(6,9,11)
g.add_e(7,3,12)
g.add_e(7,8,10)
g.add_e(8,7,2)
g.add_e(8,9,72)
g.add_e(9,10,7)
g.add_e(9,6,31)
g.add_e(9,8,18)
g.add_e(10,6,8)
# Find a shortest path from vertex 'a' (1) to 'j' (10).
dist, prev = g.s_path(1)
# Trace the path back using the prev array.
path, current, end = [], 10, 10
while current in prev:
path.insert(0, prev[current])
current = prev[current]

7. print path
print dist[end]
Output:
[1, 2, 3] (namely a, b, c)
18
[edit] Related problems and algorithms
The functionality of Dijkstra's original algorithm can be extended with a variety of
modifications. For example, sometimes it is desirable to present solutions which are less
than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the
optimal solution is first calculated. A single edge appearing in the optimal solution is
removed from the graph, and the optimum solution to this new graph is calculated. Each
edge of the original solution is suppressed in turn and a new shortest-path calculated. The
secondary solutions are then ranked and presented after the first optimal solution.
Dijkstra's algorithm is usually the working principle behind link-state routing protocols,
OSPF and IS-IS being the most common ones.
Unlike Dijkstra's algorithm, the Bellman-Ford algorithm can be used on graphs with
negative edge weights, as long as the graph contains no negative cycle reachable from the
source vertex s. (The presence of such cycles means there is no shortest path, since the total
weight becomes lower each time the cycle is traversed.)
The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of
the subgraph that must be explored, if additional information is available that provides a
lower-bound on the quot;distancequot; to the target.
The process that underlies Dijkstra's algorithm is similar to the greedy process used in
Prim's algorithm. Prim's purpose is to find a minimum spanning tree for a graph.
For the solution of nonconvex cost trees (typical for real-world costs exhibiting economies
of scale) one solution allowing application of this algorithm is to successively divide the
problem into convex subtrees (using bounding linear costs) and to pursue subsequent
divisions using branch and bound methods. Such methods have been superseded by more
efficient direct methods[citation needed].

8. Graph search algorithms
Search
A*
•
B*
•
Bellman-Ford
•
algorithm
Best-first search
•
quot;Floyd's algorithmquot; redirects here. For Bidirectional search
•
cycle detection, see Floyd's cycle-finding algorithm. Breadth-first search
•
D*
•
In computer science, the Floyd–Warshall algorithm Depth-first search
•
Depth-limited
(sometimes known as the WFI Algorithm or Roy–Floyd •
search
algorithm, since Bernard Roy described this algorithm in
Dijkstra's algorithm
•
1959) is a graph analysis algorithm for finding shortest paths
Floyd–Warshall
•
in a weighted, directed graph. A single execution of the
algorithm
algorithm will find the shortest paths between all pairs of
Hill climbing
•
vertices. The Floyd–Warshall algorithm is an example of
Iterative deepening
•
dynamic programming.
depth-first search
Johnson's algorithm
•
Contents
Uniform-cost
•
[hide] search
1 Algorithm
•
2 Pseudocode
•
3 Behaviour with negative cycles
•
4 Analysis
•
5 Applications and generalizations
•
6 Implementations
•
7 References
•
8 See also
•
9 External links
•
[edit] Algorithm
The Floyd-Warshall algorithm compares all possible paths through the graph between each
pair of vertices. It is able to do this with only V3 comparisons. This is remarkable
considering that there may be up to V2 edges in the graph, and every combination of edges

9. is tested. It does so by incrementally improving an estimate on the shortest path between
two vertices, until the estimate is known to be optimal.
Consider a graph G with vertices V, each numbered 1 through N. Further consider a
function shortestPath(i,j,k) that returns the shortest possible path from i to j using only
vertices 1 through k as intermediate points along the way. Now, given this function, our
goal is to find the shortest path from each i to each j using only nodes 1 through k + 1.
There are two candidates for this path: either the true shortest path only uses nodes in the
set (1...k); or there exists some path that goes from i to k + 1, then from k + 1 to j that is
better. We know that the best path from i to j that only uses nodes 1 through k is defined by
shortestPath(i,j,k), and it is clear that if there were a better path from i to k + 1 to j, then the
length of this path would be the concatenation of the shortest path from i to k + 1 (using
vertices in (1...k)) and the shortest path from k + 1 to j (also using vertices in (1...k)).
Therefore, we can define shortestPath(i,j,k) in terms of the following recursive formula:
This formula is the heart of Floyd Warshall. The algorithm works by first computing
shortestPath(i,j,1) for all (i,j) pairs, then using that to find shortestPath(i,j,2) for all (i,j)
pairs, etc. This process continues until k=n, and we have found the shortest path for all (i,j)
pairs using any intermediate vertices.
[edit] Pseudocode
Conveniently, when calculating the kth case, one can overwrite the information saved from
the computation of k − 1. This means the algorithm uses quadratic memory. Be careful to
note the initialization conditions:
1 /* Assume a function edgeCost(i,j) which returns the cost of the
edge from i to j
2 (infinity if there is none).
3 Also assume that n is the number of vertices and edgeCost(i,i)=0
4 */
5
6 int path[][];
7 /* A 2-dimensional matrix. At each step in the algorithm, path[i][j]
is the shortest path
8 from i to j using intermediate vertices (1..k-1). Each path[i]
[j] is initialized to
9 edgeCost(i,j) or infinity if there is no edge between i and j.
10 */
11
12 procedure FloydWarshall ()
13 for k: = 1 to n
14 for each (i,j) in {1,..,n}2
15 path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );
[edit] Behaviour with negative cycles

10. For numerically meaningful output, Floyd-Warshall assumes that there are no negative
cycles (in fact, between any pair of vertices which form part of a negative cycle, the
shortest path is not well-defined because the path can be arbitrarily negative). Nevertheless,
if there are negative cycles, Floyd–Warshall can be used to detect them. A negative cycle
can be detected if the path matrix contains a negative number along the diagonal. If path[i]
[i] is negative for some vertex i, then this vertex belongs to at least one negative cycle.
Please help improve this section by expanding it. Further information might be found
on the talk page. (June 2008)
[edit] Analysis
To find all n2 of from those of requires 2n2 bit operations. Since we begin with and
compute the sequence of n zero-one matrices , , ..., , the total number of bit operations used
is . Therefore, the complexity of the algorithm is Θ(n3) and can be solved by a deterministic
machine in polynomial time.
[edit] Applications and generalizations
The Floyd–Warshall algorithm can be used to solve the following problems, among others:
Shortest paths in directed graphs (Floyd's algorithm).
•
Transitive closure of directed graphs (Warshall's algorithm). In Warshall's original
•
formulation of the algorithm, the graph is unweighted and represented by a Boolean
adjacency matrix. Then the addition operation is replaced by logical conjunction
(AND) and the minimum operation by logical disjunction (OR).
Finding a regular expression denoting the regular language accepted by a finite
•
automaton (Kleene's algorithm)
Inversion of real matrices (Gauss-Jordan algorithm).
•
Optimal routing. In this application one is interested in finding the path with the
•
maximum flow between two vertices. This means that, rather than taking minima as
in the pseudocode above, one instead takes maxima. The edge weights represent
fixed constraints on flow. Path weights represent bottlenecks; so the addition
operation above is replaced by the minimum operation.
Testing whether an undirected graph is bipartite.
•
Fast computation of Pathfinder Networks.
•
Maximum Bandwidth Paths in Flow Networks
•
[edit] Implementations
A Perl implementation is available in the Graph module
•
A Javascript implementation is available at Alex Le's Blog
•
A Python implementation is available in the NetworkX package
•
A C implementation is available at ece.rice.edu
•

11. A C++ implementation is available in the boost::graph library
•
A Java implementation is in the Apache commons graph library.
•
A Java implementation is on Algowiki
•
A C# implementation is in QuickGraph
•
A PHP implementation and PL/pgSQL implementation are available at Microshell.
•
[edit] References
Cormen, Thomas H.; Leiserson, Charles E., Rivest, Ronald L. (1990). Introduction
•
to Algorithms (1st ed.). MIT Press and McGraw-Hill. ISBN 0-262-03141-8.
o Section 26.2, quot;The Floyd–Warshall algorithmquot;, pp. 558–565;
o Section 26.4, quot;A general framework for solving path problems in directed
graphsquot;, pp. 570–576.
Floyd, Robert W. (June 1962). quot;Algorithm 97: Shortest Pathquot;. Communications of
•
the ACM 5 (6): 345. doi:10.1145/367766.368168.
Kleene, S. C. (1956). quot;Representation of events in nerve nets and finite automataquot;.
•
in C. E. Shannon and J. McCarthy. Automata Studies. Princeton University Press.
pp. 3–42.
Warshall, Stephen (January 1962). quot;A theorem on Boolean matricesquot;. Journal of
•
the ACM 9 (1): 11–12. doi:10.1145/321105.321107.
Kenneth H. Rosen (2003). Discrete Mathematics and Its Applications, 5th Edition.
•
Addison Wesley. ISBN 0-07-119881-4 (ISE).
[edit] See also
Robert Floyd
•
Stephen Warshall
•
Dijkstra's algorithm
•
Johnson's algorithm (in a sparse graph)
•
[edit] External links
Analyze Floyd's algorithm in an online Javascript IDE
•
Interactive animation of Floyd–Warshall algorithm
•
Retrieved from quot;http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmquot;

12. Dijkstra'
s algorithm
We now study path problems for graphs with edge weights, which we call costs. For
convenience we also call a (slightly modied) weighted adjacency matrix of a graph (or
digraph) simply a cost matrix. Although the shortest path problem seems much harder with
edge-weights than without, Dijkstra has invented an algorithm that computes the shortest
distances from a vertex s to all other vertices in time . This is only slightly slower
than the time which is needed for our method maxDistances.
We assume below that the `cost' between two vertices without an edge (arc) is some
suciently large number (representing 1 in the cost matrix).
algorithm Dijkstra
(digraph , cost matrix , vertex )
1 W = {s} and d[s] = 0
2 for each &#;`
{s} do
d[ ] =
3 while do
find d[x] = min(d[y] | y W)

13. W=W
4 for all &#;`
W do
d[y] = min(d[y], d[x] + )
endwhile
5 return shortest distance vector d[ ]
end
The idea behind Dijkstra's algorithm is to start with a set of vertices W (initially )
reachable by some total cost C and all other vertices will require higher cost than C. We
gradually increase the value of C until all vertices are reachable from the start vertex . The
increment to C is calculated by taking the smallest cost edge between vertices of and
vertices of V &#;`
W. Line 4 in the algorithm updates these distances after a new vertex is added to W. The
while loop on line 3 is repeated times. Thus, one can check that the total time of the
algorithm is . A priority queue can be used to efficiently pick the next closest vertex
x.
Example 29. An application of Dijkstra's algorithm on the second digraph of Example 26
is given below for every starting vertex.
This example illustrates that the distance vector is updated at most times (only before
a new vertex is selected and added to Thus we could have omitted the lines with
above.